Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 492421 20 pageshttpdxdoiorg1011552013492421
Review ArticleA Review of Ranking Models in Data Envelopment Analysis
F Hosseinzadeh Lotfi1 G R Jahanshahloo1 M Khodabakhshi2
M Rostamy-Malkhlifeh1 Z Moghaddas1 and M Vaez-Ghasemi1
1 Department of Mathematics Science and Research Branch Islamic Azad University Tehran Iran2Department of Mathematics Faculty of Science Lorestan University Khorramabad Iran
Correspondence should be addressed to F Hosseinzadeh Lotfi farhadhosseinzadehir
Received 26 January 2013 Accepted 27 May 2013
Academic Editor Hadi Nasseri
Copyright copy 2013 F Hosseinzadeh Lotfi et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
In the course of improving various abilities of data envelopment analysis (DEA) models many investigations have been carried outfor ranking decision-making units (DMUs) This is an important issue both in theory and practice There exist a variety of paperswhich apply different ranking methods to a real data set Here the ranking methods are divided into seven groups As each of theexistingmethods can be viewed fromdifferent aspects it is possible that somewhat these groups have an overlappingwith the othersThe first group conducts the evaluation by a cross-efficiency matrix where the units are self- and peer-evaluated In the second onethe ranking units are based on the optimal weights obtained from multiplier model of DEA technique In the third group super-efficiency methods are dealt with which are based on the idea of excluding the unit under evaluation and analyzing the changes offrontierThe fourth group involvesmethods based on benchmarking which adopts the idea of being a useful target for the inefficientunitsThe fourth group uses themultivariate statistical techniques usually applied after conducting the DEA classificationThe fifthresearch area ranks inefficient units through proportional measures of inefficiency The sixth approach involves multiple-criteriadecision methodologies with the DEA technique In the last group some different methods of ranking units are mentioned
1 Introduction
Data envelopment analysis as a mathematical tool was ini-tiated by Farrell [1] and Charnes et al [2] They formulateda linear programming problem with which it is possible toevaluate decision-making units (DMUs) withmultiple inputsand outputs Note that in this technique it is not necessaryto know the production function In this technique an LPproblem is solved for each DMU and the relative efficiency ofeach unit obtained as a linear combination of correspondingoptimal weights In this problem weights are free to gettheir value to show the under evaluation unit in optimisticviewpointThose units with optimal objective function equalto one are called ldquobest practicerdquo These units are locatedonto the efficient frontier and those far away from thisfrontier are called inefficient As proved in DEA literatureat least one of the units is located onto this frontier Note
that units located unto this frontier can be considered asbenchmarks for inefficient units Based onwhat Charnes et al[2] provided many extensions to DEA Models are presentedin the literature As the example of the most important oneBanker et al [3] can be mentioned Also multiplicative andadditive models developed in the literature by Charnes et al[4ndash6] In that timeThrall [7] provided a complete comparisonof all classic DEA models
One of the important issues discussed in DEA literatureis ranking efficient units since the efficient units obtained inthe efficiency score of one cannot be compared with eachother on the basis of this criterion any more Therefore itseems necessary to providemodels for further discriminationamong these units Many papers are presented in the litera-ture review for ranking the efficient units Note that Adler andGolany [8] used principle component analysis for improvingthe discrimination ofDEA But this attemptwas not sufficient
2 Journal of Applied Mathematics
and just made a reduction in number of efficient units notrank units completelyMany papers presented in the literaturefor ranking efficient units one of the first papers is Young andHamer [9] One important field in ranking is cross-efficiencyto name a few consider Sexton et al [10] Rodder andReucher[11] Orkcu and Bal [12] Wu et al [13] Jahanshahloo et al[14] Wang et al [15] Ramon et al [16] Guo and Wu [17]Contreras [18] Wu et al [19] Zerafat Angiz et al [20] andWashio and Yamada [21]
In the literature there exist other methods based onfinding optimal weights in DEA analysis as Jahanshahloo etal [22] Wang et al [23] Alirezaee and Afsharian [24] LiuandHsuan Peng [25]Wang et al [26] Hatefi and Torabi [27]Hosseinzadeh Lotfi et al [28] Wang et al [29] and Ramon etal [30]
One of the important fields in ranking is super efficiencypresented by Andersen and Petersen [31] Mehrabian et al[32] Tone [33] Jahanshahloo et al [34] Jahanshahloo etal [35] Chen and Sherman [36] Amirteimoori et al [37]Jahanshahloo et al [38] Li et al [39] Sadjadi et al [40]Gholam Abri et al [41] Jahanshahloo et al [42] Noura et al[43] Ashrafi et al [44] Chen et al [45] Rezai Balf et al [46]and Chen et al [47]
Another important field in ranking is benchmarkingmethods such as Torgersen et al [48] Sueyoshi [49] Jahan-shahloo et al [50] Lu and Lo [51] and Chen and Deng [52]
One important field is using statistical tools for rankingunits first suggested by Friedman and Sinuany-Stern [53] andMecit and Alp [54]
One of the significant fields in ranking is unseeingmulticriteria decision-making (MCDM) methodologies andDEA analysis To mention a few consider Joro et al [55] Liand Reeves [56] Belton and Stewart [57] Sinuany-Stern etal [58] Strassert and Prato [59] Chen [60] Jablonsky [61]Wang and Jiang [62] and Hosseinzadeh Lotfi et al [63]
Also there exist some other ranking methods not muchdeveloped and extended in the literature Seiford and Zhu[64] Jahanshahloo [65] Jahanshahloo et al [66] Jahan-shahloo et al [67] Jahanshahloo and Afzalinejad [68]Amirteimoori [69] Kao [70] Khodabakhshi and Aryavash[71] and Zerafat Angiz et al [72]
In addition to the theoretical papers presented in rankingliterature there exist a variety of papers which used these newmodels in applications such as Charnes et al [73] Cook andKress [74] Cook et al [75] Martic and Savic [76] De Leeneerand Pastijn [77] Lins et al [78] Paralikas and Lygeros [79]Ali and Nakosteen [80] Martin and Roman [81] Raab andFeroz [82] Wang et al [26] Williams and Van Dyke [83]Jurges and Schneider [84] Giokas and Pentzaropoulos [85]Darvish et al [86] Lu and Lo [51] Feroz et al [87] Sadjadi etal [40] Ramon et al [30] and Sitarz [88]
There exist some papers which reviewed ranking meth-ods as Adler et al [89] In this paper most of the rankingmethods specially the new ones described in the literatureare reviewed Here the different ranking methods are classi-fied into seven groups after reviewing the basic DEAmethodin Section 2 In Section 3 the cross-efficiency technique will
be discussed In this method first suggested by Sexton etal [10] the DMUs are self- and peer-assessed In Section 4some of the ranking methods based on optimal weightsobtained from DEA models common set of weights arebriefly reviewed Super-efficiency methods first introducedby Andersen and Petersen [31] will be reviewed in Section 5The basic idea is based on the idea of leaving out one unitand assessing by the remaining units Section 6 discusses theevaluation of DMUs through benchmarking an approachoriginating in Torgersen et al [48] Section 7 will reviewthe papers which use the statistical tools for ranking unitsfirst suggested by Friedman and Sinuany-Stern [53] suchas canonical correlation analysis and discriminant analysisSection 8 discusses the ranking of units based on multi-criteria decision-making (MCDM)methodologies and DEASection 9 discusses some different ranking methods existingin the DEA literature Section 10 presents the results of thevarious methodologies applied to an example
2 Data Envelopment Analysis
Data envelopment analysis is a mathematical programmingtechnique for performance evaluation of a set of decision-making units
Let a set consists of 119899 homogeneous decision-makingunits to be evaluated Assume that each of these units uses119898 inputs 119909
119894119895(119894 = 1 119898) to produce 119904 outputs 119910
119903119895(119903 =
1 119904) Moreover 119883119895isin 119877
119898 and 119884119895isin 119877
119904 are consideredto be nonnegative vectors We define the set of productionpossibility as 119879 = (119883 119884) | 119883 can produce 119884
When variable returns to scale form of technology isassumed we have 119879 = 119879BCC and
119879BCC =
(119909 119910) | 119909 ge
119899
sum
119895=1
120582119895119909119895
119910 le
119899
sum
119895=1
120582119895119910119895
119899
sum
119895=1
120582119895= 1 120582
119895ge 0 119895 = 1 119899
(1)
and when constant returns to scale form of technology isassumed we have 119879 = 119879CCR and
119879CCR =
(119909 119910) | 119909 ge
119899
sum
119895=1
120582119895119909119895 119910 le
119899
sum
119895=1
120582119895119910119895
120582119895ge 0 119895 = 1 119899
(2)
Journal of Applied Mathematics 3
The two-phase enveloping problem with constant returns toscale form of technology first provided by Charnes et al [2]is as follows
min 120579 minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1
120582119895119909119894119895+ 119904
minus
119894= 120579119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895minus 119904
+
119903= 119910
119903119900 119903 = 1 119904
120582119895ge 119895 = 1 119899
(3)
The two-phase enveloping problem with variable returns toscale form of technology first provided by Banker et al [3] isas follows
min 120579 minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1
120582119895119909119894119895+ 119904
minus
119894= 120579119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895minus 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119899
(4)
As regards the above-mentioned problems and due to corre-sponding feasible region it is evident that 120579lowastCCR le 120579
lowastBCCAccording to the definition of 119879CCR and 119879BCC an envelop
constructed through units called best practice or efficientConsidering mentioned problems if a DMU
119900is not CCR
(BCC) efficient it is possible to project this DMU ontothe CCR (BCC) efficiency frontier considering the followingformulas
119909119894119900= 120579
lowast119909119894119900minus 119904
minuslowast
119894=
119899
sum
119895=1
120582lowast
119895119909119894119895 119894 = 1 119898
119910119903119900= 119910
119903119900+ 119904
+lowast
119903=
119899
sum
119895=1
120582lowast
119895119910119903119895 119903 = 1 119904
(5)
The dual model corresponding to the following model isas follows
max119904
sum
119903=1
119906119903119910119903119900
st119898
sum
119894=1
V119894119909119894119900= 1
119904
sum
119903=1
119906119903119910119903119895minus
119898
sum
119894=1
V119894119909119894119895le 0 119895 = 1 119899
119880 ge 0 119881 ge 0
(6)
For overcoming the problem of zero weights and variabilityof weights use of assurance region is suggested byThompsonet al [90ndash92]
As mentioned in the literature usually there exist morethan one efficient units and these units cannot be furthercompared to each other on basis of efficiency scores Thusit felt necessary to provide new models for ranking theseunits There exist a variety of ranking models in context ofdata envelopment analysis In the remaining of this paper wereview some of these models
3 Cross-Efficiency Ranking Techniques
Sexton et al [10] provided a method for ranking units basedon this idea that units are self- and peer-evaluated Forderiving the cross-efficiency of any DMU
119895using weights
chosen by DMU119900 they proposed the following equation
120579119900119895=
119880lowast
119900119884119895
119881lowast
119900119883
119895
(7)
where 119880lowast 119881
lowast are optimal weights obtained from the follow-ing model for DMU
119900under assessment
min 119881119905119883
119900
st 119880119905119884119900= 1
119880119905119884119895minus 119881
119905119883
119895le 0 119895 = 1 119899
119880 ge 0 119881 ge 0
(8)
NowDMU119900received the average cross-efficiency score as
120579119900= sum
119899
119895=1120579119900119895119899 for further details about this averaging see
also Green et al [93] Doyle and Green [94] also used cross-efficiency matrix for ranking units According to this methodfor ranking DMUs many investigations have been done asreviewed in Adler et al [89]
Rodder and Reucher [11] presented a consensual peer-based DEA model for ranking units As the authors saidthis method is generalized twofold The first is an optimalefficiency improving input allocation the second aim isthe choice of a peer DMU whose corresponding price isacceptable for the other units Consider
max 119881lowast119879
119896119882
119897
st 119880lowast119879
119896119884119897minus 119881
lowast119879
119896119882
119897le 0
119882119897minussum
119895
120583119897119895119883
119895ge 0
sum
119895
120583119897119895119884119895ge 119884
119897
120583119897119895ge 0 forall119895
(9)
4 Journal of Applied Mathematics
The higher the degree of input variation is the better thechance to be efficient will be
Orkcu and Bal [12] provided a goal programming tech-nique to be used in the second stage of the cross-evaluationTheir modified model is as follows
As the authors noted there exist alternative optimal solutionsWu et al [13] described the main suffering of cross-
efficiency when the ultimate average cross-efficiency utilizedfor ranking units For removing this shortcoming they elim-inated the assumption of average and utilized the Shannonentropy in order to obtain the weights for ultimate cross-efficiency scores Jahanshahloo et al [14] provided a methodfor selecting symmetric weights to be used in DEA cross-efficacy
Step 1 Efficiency of DMUs needs to be computed
Step 2 Choose the solutions in accordance with the sec-ondary goal for each DMU as follows
min 119890119879119885119900119890
st 119906119900119910119900= 1
V119900119883
119900= 120579
119900
119906119900119884 minus V
119900119883 le 0
119906119900119894119910119900119894minus 119906
119900119895119910119900119895le 119911
119900119894119895 forall119894 119895
119906119900119895119910119900119895minus 119906
119900119894119910119900119894le 119911
119900119894119895 forall119894 119895
119906119900 V
119900ge 119889
(11)
Step 3 The cross-efficiency for any DMU119895 using the weights
that DMU119900has chosen in the previous model is then used as
follows
120579119900119895=
119906lowast
119900119884119895
Vlowast119900119883
119895
(12)
Wang et al [15] provided a cross-efficiency evaluation basedon ideal and anti-ideal units for ranking As the authorsmentioned a DMU could choose a unique set of input andoutput weights to make its distance from ideal DMU assmall as possible or the distance from anti-ideal DMU aslarge as possible or the both Thus according to this ideathey proposed the following procedure for cross-efficiencyevaluation
Model 1 Minimization of the distance from ideal DMU
Model 2 Maximization of the distance from anti-ideal DMU
Model 3 Maximization of the distance between ideal DMUand anti-ideal DMU
Model 4 Maximization of the relative closeness
The authors mentioned that the bigger the relative close-ness of a DMU is the better performance it will have
In a paper Ramon et al [16] selected the profiles ofweights used in cross-efficiency assessment As the authorssaid they tried to prevent unrealistic weighting They havediscussed the zero weights as they excluded variables fromthe evaluation In the calculation of cross-efficiency scoresthey proposed to ignore the profiles of those weights of theunit under evaluation that among their alternate optimalsolutions cannot choose nonzeroweightsThey also proposedthe ldquopeer-restrictedrdquo cross-efficiency evaluation where theunits assessed in a peer evaluation which means profiles ofweights of some inefficient units are not considered Finallythe presented approach extended to derive a common set ofweights Guo and Wu [17] provided a complete ranking ofDMUs with undesirable outputs using restriction in DEAAs the author mentioned this model is presented to realizea unique ranking of units by ldquomaximal balanced indexrdquoaccording to the obtained optimal shadow prices
119896+1ge 119889 (119896 120576) 119896 = 1 119899 minus 1
120583119899ge 119889 (119899 120576)
(19)
where 120579119895119896obtained from Step 5 and 120573lowast
119895is the optimal solution
of the following model Finally the DMUs are ranked basedon their 119911lowast
119895= sum
119899
119896=1120583lowast
119896120579119895119896values
Washio and Yamada [21] discussed that in real casesfinding the best ranking is more significant than acquiringthe most advantage weight and maximizing the efficiencyThus they presented a model called rank-based measure(RBM) for evaluating units from different viewpoint Thusthey suggested a method for acquiring those weight resultedfrom the best ranking as long as calculating those weightthat maximizes the efficiency score Finally they applied thepresented model to the cross-efficiency assessment
4 Ranking Techniques Based on FindingOptimal Weights in DEA Analysis
Jahanshahloo et al [22] gave a note on some of the DEAmodels for complete ranking using common set of weightsThey proved that by solving only one problem it is possibleto determine the common set of weights
where 119860 is the set of efficient units of the following model
max
sum119904
119903=11199061199031199101199031+ 119906
0
sum119898
119894=1V1198941199091198941
sdot sdot sdot
sum119904
119903=1119906119903119910119903119899+ 119906
0
sum119898
119894=1V119894119909119894119899
stsum
119904
119903=1119906119903119910119903119895+ 119906
0
sum119898
119894=1V119894119909119894119895
le 1 119895 = 1 119899
119880 119881 120578 ge 120576 1199060free
(21)
Note that DMUs can be ranked based on the evaluation oftheir efficiencies
Wang et al [23] provided an aggregating preference rank-ing In this paper use of ordered weighted averaging (OWA)operator is proposed for aggregating preference rankings Let
119908119895be the relative importance weight given to the jth ranking
place and V119894119895the vote candidate i receives in the jth ranking
place The total score of each candidate is defined as
119911119894=
119898
sum
119894=1
V119894119895119882
119895 119894 = 1 119898 (22)
Alirezaee and Afsharian [24] discussed multiplier model inwhich the variables are considered as shadow prices note thatsum
119894=1V119894119909119894119895are total revenue and cost of DMU
119895
which are considered in optimization problem The authorsclaimed that sum119904
119903=1119906119903119910119903119895minus sum
119898
119894=1V119894119909119894119895le 0 119895 = 1 119899 is the
profit restriction for DMU119895 If 119865(119909 119910) = 0 is considered to be
the efficient production function then
sum
119894
120597119865
120597119909119894
119909119894+sum
119895
120597119865
120597119910119895
119910119895= 0 (23)
They mentioned that the connected profit of the DMU iszero when shadow prices are derived from the technologyand called this situation as balance situation As the authorsmentioned therefore in the case that DMU
1is efficient but
DMU2is inefficient or the efficiency score of both DMUs is
the same and it obtains more negative quantity in balanceindex it can be concluded that DMU
1has a better rank than
DMU2
Liu and Hsuan Peng [25] in their paper proposed amethod for determining the common set of weights forranking units In common weights analysis methodologythey provided the following model
Δlowast= min sum
119895isin119864
Δ119900
119895+ Δ
119894
119895
stsum
119904
119903=1119910119903119895119880119903+ Δ
119900
119895
sum119898
119894=1119909119894119895119881119894minus Δ
119894
119895
= 1 119895 isin 119864
Δ119900
119895 Δ
119894
119895ge 0 forall119895 isin 119864
119880119903ge 120576 119903 = 1 119904
119881119894ge 120576 119894 = 1 119898
(24)
The mentioned ratio form the linear equationsWang et al [26] proposed a paper for ranking decision-
making units by imposing a minimum weight restrictionin DEA The authors noted that using data envelopmentanalysis it is not possible to distinguish between DEA effi-cient units Thus they presented a method for ranking unitsusing imposing minimum weight restriction for the input-output data As they mentioned these weights restrictionsare decided by a decision maker (DM) or an assessor asregards the solutions to a series of LP models considered fordetermining a maximin weight for each efficient DMU
Hatefi and Torabi [27] proposed a common weightmulti criteria decision analysis (MCDA)-data envelopmentanalysis (DEA) for constructing composite indicators (CIs)As the authors proved the presented model can discrimi-nate between efficient units The obtained common weightshave discriminating power more than those obtained from
Journal of Applied Mathematics 7
previous models Finally they studied the robustness anddiscriminating power of the proposed method by Spearmanrsquosrank correlation coefficient
Hosseinzadeh Lotfi et al [28] proposed one DEA rankingmodel based on applying aggregate units In doing so artificialunits called aggregate units are defined as follows Theaggregate unit is shown by DMU
119886
119909119901
119894119886= sum
119896isin119877119901
119909119894119896 119910
119901
119903119886= sum
119896isin119877119901
119910119903119896
119894 = 1 119898 119903 = 1 119904
(25)
where 119877119901= 119895 | DMU
119895isin 119864
119901 119864
119901= 119864DMU
119901 Note that 119864
is the set of efficient unitsFirst it is tried to maximize the efficiency score of the
DMU119886and then to maximize the efficiency score of the
DMU119901
119886 For resolving the existence of alternative solutions
the authors presented an approach comprising (119898+119904) simplelinear problems to achieve the most appropriate optimalsolutions among all alternative optimal solutions Finallythey proposed theRI index for ranking all efficientDMUs Let119880119886 119881
119875119886 and119880119901
119886 119881
119901
119886be the optimal solutions of the multiplier
model for assentDMU119886andDMU119901
119886 Consider 120578
119886= sum
119904
119903=1119906119903119886minus
sum119898
119894=1V119894119886 120578
119901
119886= sum
119904
119903=1119906119901
119903119886minus sum
119898
119894=1V119901119894119886 thus RI
119901= 120578
119900
119886minus 120578
119886
Wang et al [29] presented two nonlinear regressionmodels for deriving common set of weights for fully rankunits
min 119911 =
119899
sum
119895=1
(120579lowast
119895minus
sum119904
119903=1119906119903119910119903119895
sum119898
119894=1V119894119909119894119895
)
2
st 119880 119881 ge 0
min119899
sum
119895=1
(
119904
sum
119903=1
119906119903119910119903119895minus 120579
lowast
119895
119898
sum
119894=1
V119894119909119894119895)
2
st119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895) +
119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) = 119899
119880 119881 ge 0
(26)
Ramon et al [30] aimed at deriving a common set of weightsfor ranking units As the authors mentioned the idea is basedupon minimization of the deviations of the common weightsfrom the nonzero weights obtained fromDEA Furthermoreseveral norms are used for measuring such differences
5 Super-Efficiency Ranking Techniques
Super efficiency models introduced in DEA technique arebased upon the idea of leave one out and assessing this unittrough the remanding units
Andersen and Petersen [31] introduced a model for rank-ing efficient units The proposed model is as follows
Although this idea is useful for further discriminatingefficient units it has been shown in the literature that itmay be infeasible and nonstable Thrall [7] mentioned theinfeasibility of super-efficiency CCRmodel Also a conditionunder which infeasibility occurred in super-efficiency DEAmodels is mentioned by Zhu [95] Seiford and Zhu [96] andDula and Hickman [97]
Hashimoto [98] provided a model based on the idea ofone leave out and assurance region for ranking units
Mehrabian et al [32] presented a complete ranking forefficiency units in DEA context As the authors mentionedthis model does not have difficulties of AP model
min 119908119901+ 1
st119899
sum
119895=1 119895 = 119901
120582119895119909119894119895le 119909
119894119901+ 119908
1199011 119894 = 1 119898
119899
sum
119895=1 119895 = 119901
120582119895119910119903119895ge 119910
119903119901 119903 = 1 119904
120582119895ge 0 119895 = 119901
(28)
With this method it is not possible to rank nonextremeefficient units
Tone [33] presented super efficiency of SBM model Thismodel has the advantages of nonradial models and it isalways feasible and stable
minsum
119898
119894=1119909119894119909
119894119900
sum119904
119903=1119910119903119910
119903119900
st119899
sum
119895=1119895 = 119900
120582119895119909119894119895le 119909
119894 119894 = 1 119898
119899
sum
119895=1119895 = 119900
120582119895119910119903119895ge 119910
119903 119903 = 1 119904
119909119894ge 119909
119894119900 0 le 119910
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119900
(29)
8 Journal of Applied Mathematics
Jahanshahloo et al [34] added some ratio constraints to themultiplier form of AP model and introduced a new methodfor ranking DMUs
In their paper Chen and Sherman [36] presented a non-radial super-efficiency method and discussed the advantageof it They verified that this model is invariant to units ofinputoutput measurement Let 119869119900 = 119869DMU
119900
Step 1 Solve the followingmodel to find the extreme efficientunits in 119869119900
This method cannot rank nonextreme efficient units
Jahanshahloo et al [38] presented modified MAJ modelfor ranking efficiency units in DEA technique
min 119908119901+ 1
st119899
sum
119895=1119895 = 119901
120582119895
119909119894119895
119872119894
le
119909119894119901
119872119894
+ 1199081199011 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895ge 119910
119903119901 119903 = 1 119904
120582119895ge 0 119895 = 119901
(39)
where 119872119894= Max 119909
119894119895| DMU
119895is efficient It cannot rank
nonextreme efficient unitsLi et al [39] presented a new method for ranking which
does not have difficulties of earlier methods The presentedmodel is always feasible and stable
min 1 +
1
119898
119898
sum
119894=1
119904+
1198942
119877minus
119894
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895+ 119904
minus
1198941minus 119904
+
1198942= 119909
119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895minus 119904
+
119903= 119910
119903119901 119903 = 1 119904
119904minus
1198942ge 0 119904
minus
1198941ge 0 119904
+
119903ge 0 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 119901
(40)
In this case extreme efficient units cannot be ranked
In a paper Khodabakhshi [99] addresses super efficiencyon improved outputs He mentioned that as AP modelmay be infeasible under variable returns to scale technol-ogy using the presented model gives a complete rankingwhen getting an input combination for improving outputs issuitable
Sadjadi et al [40] presented a robust super-efficiencyDEA for ranking efficient units They noted that as inmost of the times exact data do not exist and the sto-chastic super-efficiency model presented in their paperincorporates the robust counterpart of super-efficiencyDEA
where 119883119884 are input-output data This model does notrank nonextreme efficient units It may be unstable andinfeasible
Gholam Abri et al [41] proposed a model for rankingefficient units They used representation theory and rep-resented the DMU under assessment as a convex combi-nation of extreme efficient units As the authors noted itis expected that the performance of DMU
119900is the same
as the performance of convex combination of extremeefficient units Thus it is possible to represent DMU
119900as
follows
(119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119899
sum
119895=1
120582119895= 1 119895 = 1 119904 (42)
As regards representation theorem this system has119898 + 119904 minus 1
constraints and 119904 variables (1205821 120582
119904) If this system has a
10 Journal of Applied Mathematics
unique solution we will have 120579lowast119900= sum
119899
119895=1= 1 120582
lowast
119895120579119895 otherwise
two models should be considered
min119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
max119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
(43)
Consider 1205791and 120579
2as the optimal solution of the above-
mentioned models respectively If 1205791= 120579
2 this is the same
as what has been mentioned previously If 1205791lt 120579
2 then
mentionedmodels provide an interval which helps rank unitsfrom the worst to the best Sometimes it will obtain theranking score with a bounded interval [120579
1 120579
2]
Jahanshahloo et al [42] presented models for rankingefficient units The presented models are somehow a modifi-cation of cross-efficiency model that overcomes the difficultyof alternative optimal weightsThe authors in their paperwithregard to the changes and also utilizing TOPSIS techniquepresented a new super-efficientmethod for ranking unitsThepresented model is as follows
120579119894119895=
119880119905119884119894
119881119905119883
119894
|
119880119905119884119897
119881119905119883
119897
le 1
119880119905119884119895
119881119905119883
119895
= 120579119895119895
119897 = 1 119899 119897 = 119894 119880 ge 0 119881 ge 0
(44)
where 120579119895119895is the efficiency score of DMU
119869using correspond-
ing weights Also 120579119894119895is the efficiency of DMU
119894using optimal
weights of DMU119895 Noura et al [43] provided a method for
ranking efficient units based on this idea that more effectiveand useful units in society should have better rank
Step 1 For each efficient unit choose the lower and upperlimit for each inputs and outputs Let 119864 be the set of efficientunits
where 120585 is representative of a small and nonzero number usedfor not dividing by zero Now consider119863minus119895 as follows whichshows that more successful DMU
119895will be if the larger value
of the119863119895is
119863119895= sum
119894isin119868
119889119894119895+ sum
119903isin119877
119889119903119895 (48)
where 119868 = 119863+
119894cup 119863
minus
119894 119877 = 119863
+
119903cup 119863
minus
119903
Ashrafi et al [44] introduced an enhanced Russell mea-sure of super efficiency for ranking efficient units inDEAThelinear counterpart of the proposed model is as follows
119900 le 120573 le V119903ge 120573 119903 = 1 119904
(49)
It cannot rank nonextreme efficient units
Journal of Applied Mathematics 11
Chen et al [45] proposed a modified super-efficiencymethod for ranking units based on simultaneous input-output projectionThe presentedmodel overcomes the infea-sibility problem
1199011= min
120579119904119903
119900
120601119904119903
119900
st sum
119895=1
119895 = 119900
120582119895119909119894119895le 120579
119904119903
119900119909119894119900 119894 = 1 119898
sum
119895=1
119895 = 119900
120582119895119910119903119895ge 120601
119904119903
119900119910119903119900 119903 = 1 119904
sum
119895=1
119895 = 119900
120582119895= 1
0 lt 120579119904119903
119900le 1 120601
119904119903
119900ge 1 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(50)
Rezai Balf et al [46] provide a model for ranking units basedon Tchebycheff norm As proved that this model is alwaysfeasible and stable it seems to have superiority over othermodels
max 119881119901
st 119881119901ge
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901 119894 = 1 119898
119881119901ge 119910
119903119901minus
119899
sum
119895=1
119895 = 119900
120582119895119910119903119895 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(51)
where
119881119901= Max
(
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901)119894 = 1 119898
(119910119903119901minus
119899
sum
119895=1
119895 = 119901
120582119895119910119903119895)119903 = 1 119904
(52)
Chen et al [47] for overcoming the infeasibility problem thatoccurred in variable returns to scale super-efficiency DEAmodel according to a directional distance function developedNerlove-Luenberger (N-L) measure of super-efficiency
6 Benchmarking Ranking Techniques
Sueyoshi et al [49] proposes a ldquobenchmark approachrdquo forbaseball evaluation This method is the combination of DEA
and (Offensive earned-run average) OERA As the authorsnoted using this method it is possible to select best unitsand also their ranking orders They mentioned that usingonly DEAmay result in a shortcoming in assessment as manyefficient units can be identifiedThus the authors used slack-adjusted DEA model and OERA to overcome this difficulty
Jahanshahloo et al [50] presented a new model forranking DMUs based on alteration in reference set The ideais based on this fact that efficient units can be the target unitfor inefficient units
min 120597119886119887
= 120579 minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st sum
119895isin119869minus119887
120582119895119909119894119895+ 119904
minus
119894minus 120579119909
119894119886 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
120579 free 119904minus119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 isin 119869 minus 119887
(53)
Finally the ranking order for each efficient unit b can becomputing by
119887= sum
119886isin119869119899
120597119886119887
119899
(54)
Note that this method cannot rank nonextreme efficientunits
Lu and Lo [51] provided an interactive benchmark modelfor ranking units The idea is based upon considering a fixedunit as a benchmark and calculating the efficiency of otherunits pair by pair to this unitThis procedure continueswhenall units are accounted for as a benchmark unit ConsiderDMU
119900as a unit under evaluation andDMU
119887as a benchmark
Assume
119909119894= 119909
119894119900(1 + 120601
119894) 119910
119903= 119910
119903119900(1 minus 120593
119903)
min 120579lowast119887
119900=
1 + (1119898)sum119898
119894=1120601119894
1 minus (1119904)sum119904
119903=1120593119903
st 120582119887119909119894119887minus 119909
119894119900120601119894le 119909
119894119900 119894 = 1 119898
120582119887119910119903119887minus 119910
119903119900120593119903ge 119910
119903119900 119903 = 1 119904
120601119894ge 119900 120593
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(55)
Then using (sum119899
119887=1120579119887lowast
119900)119899 119900 = 1 119899 the efficiency of
benchmark DMU119900can be obtained Using the following
index which indicates the increment in efficiency of a unitby moving from peer appraisal to self-appraisal it is nowpossible to rank units Note that the less the magnitude of
12 Journal of Applied Mathematics
this index is the better rank for corresponding unit will beobtained
FPIIBM119870
=
(TEBCC119896
minus STDIBM119896
)
(STDIBM119896
)
(56)
where TEBCC119896
and STDIBM119896
are respectively efficiency in BCCmodel and normalization of TEIBM of DMU
119896 Chen [52]
provided a paper for ranking efficient and inefficient unitsin DEA They noted that the evaluation of efficient unitsis based upon the alterations in efficiency of all inefficientunits by omitting in reference set At first solve the followingmodel which measures the efficiency of DMU
119886when DMU
119887
is eliminated from the reference set
min 120588119886119887
= 120579119904
119886minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1119895 = 119887
120582119895119909119894119895+ 119904
minus
119894= 120579
119904
119886119909119894119886 119894 = 1 119898
119899
sum
119895=1119895 = 119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
119899
sum
119895=1119895 = 119887
= 1
120579119904
119886ge 0 119904
minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119887
(57)
Then again solve the previous model this time for calculat-ing the efficiency score of each inefficient unit when each ofefficient units in turn is eliminated from the reference set 120578lowast
119886
and calculate the efficiency change
120591119886119887
= 120588lowast
119886119887minus 120578
lowast
119886 (58)
Then calculate the following index called as ldquoMCDErdquo indexfor each efficient and inefficient unit
119864119886= sum
119887isin119881119864
119882lowast
119887120588lowast
119886119887 119864
119887= sum
119887isin119881119868
119882lowast
119887120588lowast
119886119887 (59)
Now in accordance with themagnitude of the acquired indexit is possible to rank unitsThose units with higher score havebetter ranking order
7 Ranking Techniques by MultivariateStatistics in the DEA
As described in DEA literature in DEA technique frontier istaken into consideration rather than central tendency consid-ered in regression analysis DEA technique considers that anenvelope encompasses through all the observations as tight aspossible and does not try to fit regression planes in center ofdata In DEA methodology each unit is considered initiallyand compared to the efficient frontier but in regressionanalysis a procedure is considered in which a single functionfits to the data DEA uses different weights for differentunits but does not let the units use weights of other units
Canonical correlation analysis as Friedman and Sinuany-Stern [53] noted can be used for ranking units This methodis somehow the extension of regression analysis Adler et al[89] The aim in canonical correlation analysis is to find asingle vector commonweight for the inputs and outputs of allunits Consider119885
119895119882
119895as the composite input and output and
as corresponding weights respectively The presentedmodel by Tatsuoka and Lohnes [100] is as follows
max 119903119911119908
=
119878119909119910119880
(119878119909119909119881) (119878
119910119910119880)
(12)
st 119909119909119881 = 1
119910119910119880 = 1
(60)
where 119878119909119909 119878
119910119910 and 119878
119909119910are respectively defined as the matri-
ces of the sums of squares and sums of products of thevariables
Friedman and Sinuany-Stern [53] while defining the ratioof linear combinations of the inputs and outputs 119879
119895=
119882119895119885
119895used canonical correlation analysis As they noted that
scaling ratio 119879119895of the canonical correlation analysisDEA is
unboundedSinuany-Stern et al [101] used linear discriminant analy-
sis for ranking units They defined
119863119895=
119904
sum
119903=1
119906119903119910119903119895+
119898
sum
119894=1
V119903(minus119909
119894119895) (61)
DMU119895is said to be efficient if 119863
119895gt 119863
119888where 119863
119888is a critical
value based on themidpoint of themeans of the discriminantfunction value of the two groups Morrison [102] The largerthe amount of119863
119895is the better rank DMU
119895will have
Friedman and Sinuany-Stern [53] noted that as cross-efficiencyDEA canonical correlation analysisDEA and dis-criminant analysisDEA ranking orders may vary from eachother thus it seems necessary to introduce the combinedranking (CODEA) Combined ranking for each unit con-sidered all the ranks obtained from the above-mentionedrankings Moreover statistical tests are one for goodness offit between DEA and a specific ranking and the other fortesting correlation between variety of ranking orders Siegeland Castellan [103]
8 Ranking with MulticriteriaDecision-Making (MCDM) Methodologiesand DEA
Li and Reeves [56] for increasing discrimination power ofDEA presented amultiple-objective linear program (MOLP)As the authors mentioned using minimax and minsumefficiency in addition to the standard DEA objective functionhelp to increase discrimination power of DEA
Strassert and Prato [59] presented the balancing andranking method which uses a three-step procedure forderiving an overall complete or partial final order of optionsIn the first step derive an outranking matrix for all options
Journal of Applied Mathematics 13
from the criteria values Considering this matrix it is possibleto show the frequencywithwhich one option is ranked higherthan the other options In the second step by triangularizingthe outranking matrix establish an implicit preordering orprovisional ordering of optionsTheoutrankingmatrix showsthe degree to which there is a complete overall order ofoptions In the third step based on information given in anadvantages-disadvantages table the provisional ordering issubjected to different screening and balancing operations
Chen [60] utilized a nonparametric approach DEA toestimate and rank the efficiency of association rules withmultiple criteria in following steps Proposed postprocessingapproach is as follows
Step 1 Input data for association rule mining
Step 2 Mine association rules by using the a priori algorithmwith minimum support and minimum confidence
Step 3 Determine subjective interestingness measures byfurther considering the domain related knowledge
Step 4 Calculate the preference scores of association rulesdiscovered in Step 2 by using Cook and Kressrsquos DEA model
Step 5 Discriminate the efficient association rules found inStep 3 by using Obata and Ishiirsquos [104] discriminate model
Step 6 Select rules for implementation by considering thereference scores generated in Step 5 and domain relatedknowledge
Jablonsky [61] presented two original models super SBMand AHP for ranking of efficient units in DEA As theauthormentioned thesemodels are based onmultiple criteriadecision-making techniques-goal programming and analytichierarchy process Super SBM model for ranking units is asfollows
min 120579119866
119902= 1 + 119905120574 + (1 minus 119905) (
sum119898
119894=1119904+
1119894
119909119894119902
+
sum119898
119894=1119904minus
2119896
119910119896119902
)
st119899
sum
119895=1119895 = 119902
120582119895119909119894119895+ 119904
minus
1119894minus 119904
+
1119894= 119909
119894119902 i = 1 119898
119899
sum
119895=1119895 = 119902
120582119895119910119896119895+ 119904
minus
2119896minus 119904
+
2119896= 119910
119896119902 119896 = 1 119903
119904+
1119894le 120574 119894 = 1 119898
119904minus
2119896le 120574 119896 = 1 119903
119905 isin 0 1 120582119895ge 0 119878
+
1 119878
+
2ge 0 119878
minus
1 119878
minus
2ge 0
119895 = 1 119899
(62)
Wang and Jiang [62] presented an alternative mixedinteger linear programming models in order to identifythe most efficient units in DEA technique As the authors
mentioned presented models can make full use of input-output information with no need to specify any assuranceregions for input and output weights to avoid zero weights
min119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) minus
119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895)
st119904
sum
119903=1
119906119903119910119903119895minus
119898
sum
119894=1
V119894119909119894119895le 119868
119895 119895 = 1 119899
119899
sum
119895=1
119868119895= 1
119868119895isin 0 1 119895 = 1 119899
119906119903ge
1
(119898 + 119904)max119895119910
119903119895
119903 = 1 119904
V119894ge
1
(119898 + 119904)max119895119909
119894119895
119894 = 1 119898
(63)
Hosseinzadeh Lotfi et al [63] provided an improved three-stage method for ranking alternatives in multiple criteriadecision analysis In the first stage based on the bestand worst weights in the optimistic and pessimistic casesobtain the rank position of each alternative respectively Theobtained weight in the first stage is not unique thus it seemsnecessary to introduce a secondary goal that is used in thesecond stage Finally in the third stage the ranks of thealternatives compute in the optimistic or pessimistic case Itis mentionable that the model proposed in the third stageis a multicriteria decision-making (MCDM) model and itis solved by mixed integer programming As the authorsmentioned and provided in their paper this model can beconverted into an LP problem
The previous model with 119897 = 1 is the original CCR modelNote that DMUs in 119864
1 show the first level efficient frontier119897 = 2 indicates the second level efficient frontier when thefirst level efficient frontier is omittedThe following algorithmfinds these efficient frontiers
Step 1 Set 119897 = 1 and evaluate the entire set of DMUs 1198691 Inthis way the first level efficient DMU 1198641 is identified
Step 2 Exclude the efficient DMUs from future DEA runsand set 119869119897+1 = 119869
119897minus 119864
119897 (If 119869119897+1 = oslash stop)
Step 3 Evaluate the new subset of ldquoinefficientrdquo DMUs 119869119897+1to obtain a new set of efficient DMUs 119864119897+1
Step 4 Let l = l + 1 Go to Step 2
When 119869119897+1 = oslash the algorithm stopsAs the authors proved in this way it is possible to rank
the DMUs in the first efficient frontier based upon theirattractiveness scores and identify the best one
Jahanshahloo et al [65] provided a paper for rankingunits using gradient line As the authors mentioned theadvantage of this model is stability and robustness
supports on 119879119888 the obtained PPS by omitting theDMUunder
assessment in 119879119888 As the authors proved a unit is efficient
iff the optimal objective function of the following model isgreater than zero Consider
1198750= (119883 119884) 119883 = 120572119883
119900 119884 = 120573119884
119900
1198780= (119883 119884) isin 1198750
119883 = 120572119883119900 119884 = 120573119884
119900 120572 ge 0 120573 ge 0
(67)
Intersection of 1198780and efficient surface of
119879119888119888 is a half line
where its equation is (minus119881lowast119879119883
119900)120572 + (119880
lowast119879119884119900)120573 = 0 To rank
DMU119900the length of connecting arc DMU
119900with intersection
point of line and previous ellipse is calculated in (120572 120573) spaceThis intersection is as follows
120572lowast= (
1198702
120572119870
2
120573
1198702
120573+ (119881
lowast119879119883
119900119880
lowast119879119884119900)2119870
2
120572
)
12
120573lowast= (
119881lowast119879119883
119900
119880lowast119879119884119900
)120572lowast
(68)
Now the length of connecting arc DMU119900to the point
corresponding to 120572lowast 120573lowast in (120572 120573) plan is calculated as follows119868 = int
120572lowast
1(1 + 120573)
12119889120572 where
119870120572= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119899
119895=11199092
119894119900
)
12
119870120573= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119904
119903=11199102
119903119900
)
12
(69)
Jahanshahloo et al [66] also provided a paper with theconcept of advantage in data envelopment analysis
In their paper Jahanshahloo et al [67] consideringMonteCarlo method presented a new method of ranking
Step 1 Generate a uniformly distributed sequence of 1198801198952119899
119895=1
on(0 1)
Step 2 Random numbers should be classified into 119873 pairslike (119880
1 119880
1015840
1) (119880
119873 119880
1015840
119873) in a way that each number is used
just one time
Step 3 Compute119883119894= 119886 + 119880
119894(119887 minus 119886) and 119891(119883
119894) gt 119888119880
1015840
119894
Step 4 Estimate the integral 119868 by 120579119868= 119888(119887 minus 119886)(119873
119867119873)
Now consider DMU119900as an efficient unit measure those
units that are dominated DMU119900 As for dome DMUs this
would be unbounded thus the authors for each unitbounded the region Then for DMU
119900if (minus119883
119900 119884
119900) ge (minus119883 119884)
then (minus119883 119884) is in the RED of DMU119900 Now by using 119881
119901=
119881lowast(119873
119867119873) all the hits (the above condition) can be counted
where 119881lowast is the measure of the whole region JahanshahlooandAfzalinejad [68] presented amethod based on distance of
Journal of Applied Mathematics 15
the unit under evaluation to the full inefficient frontier Theypresented two models radian and nonradial
min 120601
st119899
sum
119895=1
120582119895119909119894119895= 120601119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
120582119895ge 0 119895 = 1 119898
max119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903
st sum
119895isin119869minus119887
120582119895119909119894119895minus 119904
minus
119894= 119909
119894119900 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895+ 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
119904minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(70)
Amirteimoori [69] based on the same idea considered bothefficient and antiefficient frontiers for efficiency analysis andranking units
Kao [70] mentioned that determining the weights ofindividual criteria in multiple criteria decision analysis in away that all alternatives can be compared according to theaggregate performance of all criteria is of great importanceAs Kao noted this problem relates to search for alternativeswith a shorter and longer distance respectively to the idealand anti-ideal units He proposed a measure that consideredthe calculation of the relative position of an alternativebetween the ideal and anti-ideal for finding an appropriaterankings
Khodabakhshi and Aryavash [71] presented a method forranking all units using DEA concept
First Compute the minimum and maximum efficiencyvalues of eachDMU in regard to this assumption that the sumof efficiency values of all DMUs equals to 1
Second determine the rank of each DMU in relation to acombination of itsminimumandmaximumefficiency values
Zerafat Angiz et al [72] proposed a technique in orderto aggregate the opinions of experts in voting system Asthe authors mentioned the presented method uses fuzzyconcept and it is computationally efficient and can fullyrank alternatives At first number of votes given to a rankposition was grouped to construct fuzzy numbers and thenthe artificial ideal alternative introduced Furthermore byperforming DEA the efficiency measure of alternatives was
obtained considering artificial ideal alternative comparedby each of the alternatives pair by pair Thus alternativesare ranked in accordance with their efficiency scores Ifthis method cannot completely rank alternatives weightrestrictions based on fuzzy concept are imposed into theanalysis
10 Different Applications in Ranking Units
As discussed formerly there exist a variety of ranking meth-ods and applications in theliterature Nowadays DEAmodelsare widely used in different areas for efficiency evaluationbenchmarking and target setting ranking entities and soforth Ranking units as one of the important issues in DEAhas been performed in different areas Jahanshahloo et al [42]ranked cities in Iran to find the best place for creating a datafactory Hosseinzadeh Lotfi et al [28] used a new methodfor ranking in order to find the best place for power plantlocation Ali andNakosteen [80] Amirteimoori et al [37] andAlirezaee and Afsharian [24] Soltanifar and HosseinzadehLotfi [105] Zerafat Angiz et al [72] Hosseinzadeh Lotfiet al [28] Jahanshahloo [50] Chen and Deng [52] andJablonsky [61] used different ranking methods in bankingsystem Sadjadi [40] ranked provincial gas companies in IranMehrabian et al [32] Li et al [39] Orkcu and Bal [12] andWu et al [19] ranked different departments of universitiesJahanshahloo et al [35] Jahanshahloo and Afzalinejad [68]utilized the presented models in ranking 28 Chinese citiesJahanshahloo at al [35] provided an application to burdensharing amongst NATOmember nations Jahanshahloo et al[14] utilized the presented model for ranking nursery homesContreras [18] andWang et al [23] used the provided rankingtechniques in ranking candidates Lu and Lo [51] applied theirmethod on an application to financial holding companiesHosseinzadeh Lotfi et al [63] consider an empirical exampleinwhich voters are asked to rank twoout of seven alternativesWang and Jiang [62] utilized the provided method in facilitylayout design in manufacturing systems and performanceevaluation of 30OECD countries Chen [60] used an exampleof market basket data in order to illustrate the providedapproach
11 Application
In this section some of the reviewed models are applied onthe example used in Soltanifar and Hosseinzadeh Lotfi [105]Consider twenty commercial banks of Iran with input-outputdata tabulated in Table 1 and summarized as follows Also theresults of CCRmodel are listed in this table As it can be seenseven units are efficient DMUS 14 7 12 15 17 and 20 Inputsare staff computer terminal and spaceOutputs are depositsloans granted and charge
Consider some of the important ranking methods inliterature as follows
RM1 AP model [31] is based upon the idea of leaveunit evaluation out and measuring the distance of theunit under evaluation from the production possibilityset constructed by the remaining DMUs
RM2MAJmodel [32] presented for ranking efficientwhich is always stable but might be infeasible in somecasesRM3 Modified MAJ model [38] overcomes theproblem which might occurr in MAJ modelRM4 A new model based on the idea of alterationsin the reference set of the inefficient units [50]RM5 A model presented by Li et al [39] whichis a super-efficiency method that does not have thesuffering in previous methodsRM6 Slack-basedmodel [33] is based upon the inputand output variables at the same timeRM7 SA DEAmodel [49] overcomes the problem ofinfeasibility which existed in AP modelRM8 Cross-efficiency [10] is provided based onusing weights of each unit under evaluation in opti-mality for other unitsRM9 A model based on finding common set ofweights [25] which determine the common set ofweights for DMUs and ranked DMUs based this ideaRM10 A model based on finding common set ofweights [22 67] for ranking efficient unitsRM11 L1-norm model [34 35 65 66] the idea isbased upon the leave-one-out efficient unit and l1-norm which is always feasible and stableRM12 119871
infin-norm model Rezai Balf et al [46] pro-
vided a method with more ability over other existingmethods based on Tchebycheff normRM13 An enhanced Russel measure of super-efficiency model for ranking units [44]RM14 A rankingmodel which considers the distanceof unit from the full inefficient frontier [38]RM15 Amodified super-efficiencymodel [45] whichovercomes the infeasibility that may happen in prob-lem This model is based on simultaneous projectionof input output
In accordance with properties of different ranking models inorder to rank efficient units consider Table 2
1199011 Feasibility
1199012 Ranking extreme efficient units
1199013 Complexity in computation
1199014 Instability
1199015 Absence of multiple optimal solution
1199016 Dependency to 120579 and slacks
1199017 Dependency to the number of efficient and ineffi-
cient units
In Tables 3 and 4 the rank of efficient units consideringthe above mentioned methods is listed Note that ED showsefficient DMUs (ED) and RM
119895 (119895 = 1 15) are those
explained previously
12 Conclusion
In this paper the DEA ranking was reviewed and classifiedinto seven general groups In the first group those papersbased on a cross-efficiencymatrix were reviewed In this fieldDMUs have been evaluated by self- and peer pressure Thesecond group of papers is based on those papers lookingfor optimal weights in DEA analysis The third one is thesuper-efficiency method By omitting the under evaluationunit and constructing a new frontier by the remaining unitsthe unit under evaluation can get an equal or greater scoreThe fourth group is based on benchmarking idea In thisclass the effect of an efficient unit considered as a targetfor inefficient units is investigated This idea is very usefulfor the managers in decision making Another class thefifth one involves the application of multivariate statisticaltools The sixth section discusses the ranking methods basedon multicriteria decision-making (MCDM) methodologiesand DEA The final section includes some various rankingmethods presented in the literature Finally in an applicationthe result of some of the above-mentioned ranking methodsis presented
References
[1] M J Farrell ldquoThe measurement of productive efficiencyrdquoJournal of the Royal Statistical Society A vol 120 pp 253ndash2811957
[2] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
18 Journal of Applied Mathematics
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
and just made a reduction in number of efficient units notrank units completelyMany papers presented in the literaturefor ranking efficient units one of the first papers is Young andHamer [9] One important field in ranking is cross-efficiencyto name a few consider Sexton et al [10] Rodder andReucher[11] Orkcu and Bal [12] Wu et al [13] Jahanshahloo et al[14] Wang et al [15] Ramon et al [16] Guo and Wu [17]Contreras [18] Wu et al [19] Zerafat Angiz et al [20] andWashio and Yamada [21]
In the literature there exist other methods based onfinding optimal weights in DEA analysis as Jahanshahloo etal [22] Wang et al [23] Alirezaee and Afsharian [24] LiuandHsuan Peng [25]Wang et al [26] Hatefi and Torabi [27]Hosseinzadeh Lotfi et al [28] Wang et al [29] and Ramon etal [30]
One of the important fields in ranking is super efficiencypresented by Andersen and Petersen [31] Mehrabian et al[32] Tone [33] Jahanshahloo et al [34] Jahanshahloo etal [35] Chen and Sherman [36] Amirteimoori et al [37]Jahanshahloo et al [38] Li et al [39] Sadjadi et al [40]Gholam Abri et al [41] Jahanshahloo et al [42] Noura et al[43] Ashrafi et al [44] Chen et al [45] Rezai Balf et al [46]and Chen et al [47]
Another important field in ranking is benchmarkingmethods such as Torgersen et al [48] Sueyoshi [49] Jahan-shahloo et al [50] Lu and Lo [51] and Chen and Deng [52]
One important field is using statistical tools for rankingunits first suggested by Friedman and Sinuany-Stern [53] andMecit and Alp [54]
One of the significant fields in ranking is unseeingmulticriteria decision-making (MCDM) methodologies andDEA analysis To mention a few consider Joro et al [55] Liand Reeves [56] Belton and Stewart [57] Sinuany-Stern etal [58] Strassert and Prato [59] Chen [60] Jablonsky [61]Wang and Jiang [62] and Hosseinzadeh Lotfi et al [63]
Also there exist some other ranking methods not muchdeveloped and extended in the literature Seiford and Zhu[64] Jahanshahloo [65] Jahanshahloo et al [66] Jahan-shahloo et al [67] Jahanshahloo and Afzalinejad [68]Amirteimoori [69] Kao [70] Khodabakhshi and Aryavash[71] and Zerafat Angiz et al [72]
In addition to the theoretical papers presented in rankingliterature there exist a variety of papers which used these newmodels in applications such as Charnes et al [73] Cook andKress [74] Cook et al [75] Martic and Savic [76] De Leeneerand Pastijn [77] Lins et al [78] Paralikas and Lygeros [79]Ali and Nakosteen [80] Martin and Roman [81] Raab andFeroz [82] Wang et al [26] Williams and Van Dyke [83]Jurges and Schneider [84] Giokas and Pentzaropoulos [85]Darvish et al [86] Lu and Lo [51] Feroz et al [87] Sadjadi etal [40] Ramon et al [30] and Sitarz [88]
There exist some papers which reviewed ranking meth-ods as Adler et al [89] In this paper most of the rankingmethods specially the new ones described in the literatureare reviewed Here the different ranking methods are classi-fied into seven groups after reviewing the basic DEAmethodin Section 2 In Section 3 the cross-efficiency technique will
be discussed In this method first suggested by Sexton etal [10] the DMUs are self- and peer-assessed In Section 4some of the ranking methods based on optimal weightsobtained from DEA models common set of weights arebriefly reviewed Super-efficiency methods first introducedby Andersen and Petersen [31] will be reviewed in Section 5The basic idea is based on the idea of leaving out one unitand assessing by the remaining units Section 6 discusses theevaluation of DMUs through benchmarking an approachoriginating in Torgersen et al [48] Section 7 will reviewthe papers which use the statistical tools for ranking unitsfirst suggested by Friedman and Sinuany-Stern [53] suchas canonical correlation analysis and discriminant analysisSection 8 discusses the ranking of units based on multi-criteria decision-making (MCDM)methodologies and DEASection 9 discusses some different ranking methods existingin the DEA literature Section 10 presents the results of thevarious methodologies applied to an example
2 Data Envelopment Analysis
Data envelopment analysis is a mathematical programmingtechnique for performance evaluation of a set of decision-making units
Let a set consists of 119899 homogeneous decision-makingunits to be evaluated Assume that each of these units uses119898 inputs 119909
119894119895(119894 = 1 119898) to produce 119904 outputs 119910
119903119895(119903 =
1 119904) Moreover 119883119895isin 119877
119898 and 119884119895isin 119877
119904 are consideredto be nonnegative vectors We define the set of productionpossibility as 119879 = (119883 119884) | 119883 can produce 119884
When variable returns to scale form of technology isassumed we have 119879 = 119879BCC and
119879BCC =
(119909 119910) | 119909 ge
119899
sum
119895=1
120582119895119909119895
119910 le
119899
sum
119895=1
120582119895119910119895
119899
sum
119895=1
120582119895= 1 120582
119895ge 0 119895 = 1 119899
(1)
and when constant returns to scale form of technology isassumed we have 119879 = 119879CCR and
119879CCR =
(119909 119910) | 119909 ge
119899
sum
119895=1
120582119895119909119895 119910 le
119899
sum
119895=1
120582119895119910119895
120582119895ge 0 119895 = 1 119899
(2)
Journal of Applied Mathematics 3
The two-phase enveloping problem with constant returns toscale form of technology first provided by Charnes et al [2]is as follows
min 120579 minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1
120582119895119909119894119895+ 119904
minus
119894= 120579119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895minus 119904
+
119903= 119910
119903119900 119903 = 1 119904
120582119895ge 119895 = 1 119899
(3)
The two-phase enveloping problem with variable returns toscale form of technology first provided by Banker et al [3] isas follows
min 120579 minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1
120582119895119909119894119895+ 119904
minus
119894= 120579119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895minus 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119899
(4)
As regards the above-mentioned problems and due to corre-sponding feasible region it is evident that 120579lowastCCR le 120579
lowastBCCAccording to the definition of 119879CCR and 119879BCC an envelop
constructed through units called best practice or efficientConsidering mentioned problems if a DMU
119900is not CCR
(BCC) efficient it is possible to project this DMU ontothe CCR (BCC) efficiency frontier considering the followingformulas
119909119894119900= 120579
lowast119909119894119900minus 119904
minuslowast
119894=
119899
sum
119895=1
120582lowast
119895119909119894119895 119894 = 1 119898
119910119903119900= 119910
119903119900+ 119904
+lowast
119903=
119899
sum
119895=1
120582lowast
119895119910119903119895 119903 = 1 119904
(5)
The dual model corresponding to the following model isas follows
max119904
sum
119903=1
119906119903119910119903119900
st119898
sum
119894=1
V119894119909119894119900= 1
119904
sum
119903=1
119906119903119910119903119895minus
119898
sum
119894=1
V119894119909119894119895le 0 119895 = 1 119899
119880 ge 0 119881 ge 0
(6)
For overcoming the problem of zero weights and variabilityof weights use of assurance region is suggested byThompsonet al [90ndash92]
As mentioned in the literature usually there exist morethan one efficient units and these units cannot be furthercompared to each other on basis of efficiency scores Thusit felt necessary to provide new models for ranking theseunits There exist a variety of ranking models in context ofdata envelopment analysis In the remaining of this paper wereview some of these models
3 Cross-Efficiency Ranking Techniques
Sexton et al [10] provided a method for ranking units basedon this idea that units are self- and peer-evaluated Forderiving the cross-efficiency of any DMU
119895using weights
chosen by DMU119900 they proposed the following equation
120579119900119895=
119880lowast
119900119884119895
119881lowast
119900119883
119895
(7)
where 119880lowast 119881
lowast are optimal weights obtained from the follow-ing model for DMU
119900under assessment
min 119881119905119883
119900
st 119880119905119884119900= 1
119880119905119884119895minus 119881
119905119883
119895le 0 119895 = 1 119899
119880 ge 0 119881 ge 0
(8)
NowDMU119900received the average cross-efficiency score as
120579119900= sum
119899
119895=1120579119900119895119899 for further details about this averaging see
also Green et al [93] Doyle and Green [94] also used cross-efficiency matrix for ranking units According to this methodfor ranking DMUs many investigations have been done asreviewed in Adler et al [89]
Rodder and Reucher [11] presented a consensual peer-based DEA model for ranking units As the authors saidthis method is generalized twofold The first is an optimalefficiency improving input allocation the second aim isthe choice of a peer DMU whose corresponding price isacceptable for the other units Consider
max 119881lowast119879
119896119882
119897
st 119880lowast119879
119896119884119897minus 119881
lowast119879
119896119882
119897le 0
119882119897minussum
119895
120583119897119895119883
119895ge 0
sum
119895
120583119897119895119884119895ge 119884
119897
120583119897119895ge 0 forall119895
(9)
4 Journal of Applied Mathematics
The higher the degree of input variation is the better thechance to be efficient will be
Orkcu and Bal [12] provided a goal programming tech-nique to be used in the second stage of the cross-evaluationTheir modified model is as follows
As the authors noted there exist alternative optimal solutionsWu et al [13] described the main suffering of cross-
efficiency when the ultimate average cross-efficiency utilizedfor ranking units For removing this shortcoming they elim-inated the assumption of average and utilized the Shannonentropy in order to obtain the weights for ultimate cross-efficiency scores Jahanshahloo et al [14] provided a methodfor selecting symmetric weights to be used in DEA cross-efficacy
Step 1 Efficiency of DMUs needs to be computed
Step 2 Choose the solutions in accordance with the sec-ondary goal for each DMU as follows
min 119890119879119885119900119890
st 119906119900119910119900= 1
V119900119883
119900= 120579
119900
119906119900119884 minus V
119900119883 le 0
119906119900119894119910119900119894minus 119906
119900119895119910119900119895le 119911
119900119894119895 forall119894 119895
119906119900119895119910119900119895minus 119906
119900119894119910119900119894le 119911
119900119894119895 forall119894 119895
119906119900 V
119900ge 119889
(11)
Step 3 The cross-efficiency for any DMU119895 using the weights
that DMU119900has chosen in the previous model is then used as
follows
120579119900119895=
119906lowast
119900119884119895
Vlowast119900119883
119895
(12)
Wang et al [15] provided a cross-efficiency evaluation basedon ideal and anti-ideal units for ranking As the authorsmentioned a DMU could choose a unique set of input andoutput weights to make its distance from ideal DMU assmall as possible or the distance from anti-ideal DMU aslarge as possible or the both Thus according to this ideathey proposed the following procedure for cross-efficiencyevaluation
Model 1 Minimization of the distance from ideal DMU
Model 2 Maximization of the distance from anti-ideal DMU
Model 3 Maximization of the distance between ideal DMUand anti-ideal DMU
Model 4 Maximization of the relative closeness
The authors mentioned that the bigger the relative close-ness of a DMU is the better performance it will have
In a paper Ramon et al [16] selected the profiles ofweights used in cross-efficiency assessment As the authorssaid they tried to prevent unrealistic weighting They havediscussed the zero weights as they excluded variables fromthe evaluation In the calculation of cross-efficiency scoresthey proposed to ignore the profiles of those weights of theunit under evaluation that among their alternate optimalsolutions cannot choose nonzeroweightsThey also proposedthe ldquopeer-restrictedrdquo cross-efficiency evaluation where theunits assessed in a peer evaluation which means profiles ofweights of some inefficient units are not considered Finallythe presented approach extended to derive a common set ofweights Guo and Wu [17] provided a complete ranking ofDMUs with undesirable outputs using restriction in DEAAs the author mentioned this model is presented to realizea unique ranking of units by ldquomaximal balanced indexrdquoaccording to the obtained optimal shadow prices
119896+1ge 119889 (119896 120576) 119896 = 1 119899 minus 1
120583119899ge 119889 (119899 120576)
(19)
where 120579119895119896obtained from Step 5 and 120573lowast
119895is the optimal solution
of the following model Finally the DMUs are ranked basedon their 119911lowast
119895= sum
119899
119896=1120583lowast
119896120579119895119896values
Washio and Yamada [21] discussed that in real casesfinding the best ranking is more significant than acquiringthe most advantage weight and maximizing the efficiencyThus they presented a model called rank-based measure(RBM) for evaluating units from different viewpoint Thusthey suggested a method for acquiring those weight resultedfrom the best ranking as long as calculating those weightthat maximizes the efficiency score Finally they applied thepresented model to the cross-efficiency assessment
4 Ranking Techniques Based on FindingOptimal Weights in DEA Analysis
Jahanshahloo et al [22] gave a note on some of the DEAmodels for complete ranking using common set of weightsThey proved that by solving only one problem it is possibleto determine the common set of weights
where 119860 is the set of efficient units of the following model
max
sum119904
119903=11199061199031199101199031+ 119906
0
sum119898
119894=1V1198941199091198941
sdot sdot sdot
sum119904
119903=1119906119903119910119903119899+ 119906
0
sum119898
119894=1V119894119909119894119899
stsum
119904
119903=1119906119903119910119903119895+ 119906
0
sum119898
119894=1V119894119909119894119895
le 1 119895 = 1 119899
119880 119881 120578 ge 120576 1199060free
(21)
Note that DMUs can be ranked based on the evaluation oftheir efficiencies
Wang et al [23] provided an aggregating preference rank-ing In this paper use of ordered weighted averaging (OWA)operator is proposed for aggregating preference rankings Let
119908119895be the relative importance weight given to the jth ranking
place and V119894119895the vote candidate i receives in the jth ranking
place The total score of each candidate is defined as
119911119894=
119898
sum
119894=1
V119894119895119882
119895 119894 = 1 119898 (22)
Alirezaee and Afsharian [24] discussed multiplier model inwhich the variables are considered as shadow prices note thatsum
119894=1V119894119909119894119895are total revenue and cost of DMU
119895
which are considered in optimization problem The authorsclaimed that sum119904
119903=1119906119903119910119903119895minus sum
119898
119894=1V119894119909119894119895le 0 119895 = 1 119899 is the
profit restriction for DMU119895 If 119865(119909 119910) = 0 is considered to be
the efficient production function then
sum
119894
120597119865
120597119909119894
119909119894+sum
119895
120597119865
120597119910119895
119910119895= 0 (23)
They mentioned that the connected profit of the DMU iszero when shadow prices are derived from the technologyand called this situation as balance situation As the authorsmentioned therefore in the case that DMU
1is efficient but
DMU2is inefficient or the efficiency score of both DMUs is
the same and it obtains more negative quantity in balanceindex it can be concluded that DMU
1has a better rank than
DMU2
Liu and Hsuan Peng [25] in their paper proposed amethod for determining the common set of weights forranking units In common weights analysis methodologythey provided the following model
Δlowast= min sum
119895isin119864
Δ119900
119895+ Δ
119894
119895
stsum
119904
119903=1119910119903119895119880119903+ Δ
119900
119895
sum119898
119894=1119909119894119895119881119894minus Δ
119894
119895
= 1 119895 isin 119864
Δ119900
119895 Δ
119894
119895ge 0 forall119895 isin 119864
119880119903ge 120576 119903 = 1 119904
119881119894ge 120576 119894 = 1 119898
(24)
The mentioned ratio form the linear equationsWang et al [26] proposed a paper for ranking decision-
making units by imposing a minimum weight restrictionin DEA The authors noted that using data envelopmentanalysis it is not possible to distinguish between DEA effi-cient units Thus they presented a method for ranking unitsusing imposing minimum weight restriction for the input-output data As they mentioned these weights restrictionsare decided by a decision maker (DM) or an assessor asregards the solutions to a series of LP models considered fordetermining a maximin weight for each efficient DMU
Hatefi and Torabi [27] proposed a common weightmulti criteria decision analysis (MCDA)-data envelopmentanalysis (DEA) for constructing composite indicators (CIs)As the authors proved the presented model can discrimi-nate between efficient units The obtained common weightshave discriminating power more than those obtained from
Journal of Applied Mathematics 7
previous models Finally they studied the robustness anddiscriminating power of the proposed method by Spearmanrsquosrank correlation coefficient
Hosseinzadeh Lotfi et al [28] proposed one DEA rankingmodel based on applying aggregate units In doing so artificialunits called aggregate units are defined as follows Theaggregate unit is shown by DMU
119886
119909119901
119894119886= sum
119896isin119877119901
119909119894119896 119910
119901
119903119886= sum
119896isin119877119901
119910119903119896
119894 = 1 119898 119903 = 1 119904
(25)
where 119877119901= 119895 | DMU
119895isin 119864
119901 119864
119901= 119864DMU
119901 Note that 119864
is the set of efficient unitsFirst it is tried to maximize the efficiency score of the
DMU119886and then to maximize the efficiency score of the
DMU119901
119886 For resolving the existence of alternative solutions
the authors presented an approach comprising (119898+119904) simplelinear problems to achieve the most appropriate optimalsolutions among all alternative optimal solutions Finallythey proposed theRI index for ranking all efficientDMUs Let119880119886 119881
119875119886 and119880119901
119886 119881
119901
119886be the optimal solutions of the multiplier
model for assentDMU119886andDMU119901
119886 Consider 120578
119886= sum
119904
119903=1119906119903119886minus
sum119898
119894=1V119894119886 120578
119901
119886= sum
119904
119903=1119906119901
119903119886minus sum
119898
119894=1V119901119894119886 thus RI
119901= 120578
119900
119886minus 120578
119886
Wang et al [29] presented two nonlinear regressionmodels for deriving common set of weights for fully rankunits
min 119911 =
119899
sum
119895=1
(120579lowast
119895minus
sum119904
119903=1119906119903119910119903119895
sum119898
119894=1V119894119909119894119895
)
2
st 119880 119881 ge 0
min119899
sum
119895=1
(
119904
sum
119903=1
119906119903119910119903119895minus 120579
lowast
119895
119898
sum
119894=1
V119894119909119894119895)
2
st119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895) +
119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) = 119899
119880 119881 ge 0
(26)
Ramon et al [30] aimed at deriving a common set of weightsfor ranking units As the authors mentioned the idea is basedupon minimization of the deviations of the common weightsfrom the nonzero weights obtained fromDEA Furthermoreseveral norms are used for measuring such differences
5 Super-Efficiency Ranking Techniques
Super efficiency models introduced in DEA technique arebased upon the idea of leave one out and assessing this unittrough the remanding units
Andersen and Petersen [31] introduced a model for rank-ing efficient units The proposed model is as follows
Although this idea is useful for further discriminatingefficient units it has been shown in the literature that itmay be infeasible and nonstable Thrall [7] mentioned theinfeasibility of super-efficiency CCRmodel Also a conditionunder which infeasibility occurred in super-efficiency DEAmodels is mentioned by Zhu [95] Seiford and Zhu [96] andDula and Hickman [97]
Hashimoto [98] provided a model based on the idea ofone leave out and assurance region for ranking units
Mehrabian et al [32] presented a complete ranking forefficiency units in DEA context As the authors mentionedthis model does not have difficulties of AP model
min 119908119901+ 1
st119899
sum
119895=1 119895 = 119901
120582119895119909119894119895le 119909
119894119901+ 119908
1199011 119894 = 1 119898
119899
sum
119895=1 119895 = 119901
120582119895119910119903119895ge 119910
119903119901 119903 = 1 119904
120582119895ge 0 119895 = 119901
(28)
With this method it is not possible to rank nonextremeefficient units
Tone [33] presented super efficiency of SBM model Thismodel has the advantages of nonradial models and it isalways feasible and stable
minsum
119898
119894=1119909119894119909
119894119900
sum119904
119903=1119910119903119910
119903119900
st119899
sum
119895=1119895 = 119900
120582119895119909119894119895le 119909
119894 119894 = 1 119898
119899
sum
119895=1119895 = 119900
120582119895119910119903119895ge 119910
119903 119903 = 1 119904
119909119894ge 119909
119894119900 0 le 119910
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119900
(29)
8 Journal of Applied Mathematics
Jahanshahloo et al [34] added some ratio constraints to themultiplier form of AP model and introduced a new methodfor ranking DMUs
In their paper Chen and Sherman [36] presented a non-radial super-efficiency method and discussed the advantageof it They verified that this model is invariant to units ofinputoutput measurement Let 119869119900 = 119869DMU
119900
Step 1 Solve the followingmodel to find the extreme efficientunits in 119869119900
This method cannot rank nonextreme efficient units
Jahanshahloo et al [38] presented modified MAJ modelfor ranking efficiency units in DEA technique
min 119908119901+ 1
st119899
sum
119895=1119895 = 119901
120582119895
119909119894119895
119872119894
le
119909119894119901
119872119894
+ 1199081199011 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895ge 119910
119903119901 119903 = 1 119904
120582119895ge 0 119895 = 119901
(39)
where 119872119894= Max 119909
119894119895| DMU
119895is efficient It cannot rank
nonextreme efficient unitsLi et al [39] presented a new method for ranking which
does not have difficulties of earlier methods The presentedmodel is always feasible and stable
min 1 +
1
119898
119898
sum
119894=1
119904+
1198942
119877minus
119894
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895+ 119904
minus
1198941minus 119904
+
1198942= 119909
119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895minus 119904
+
119903= 119910
119903119901 119903 = 1 119904
119904minus
1198942ge 0 119904
minus
1198941ge 0 119904
+
119903ge 0 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 119901
(40)
In this case extreme efficient units cannot be ranked
In a paper Khodabakhshi [99] addresses super efficiencyon improved outputs He mentioned that as AP modelmay be infeasible under variable returns to scale technol-ogy using the presented model gives a complete rankingwhen getting an input combination for improving outputs issuitable
Sadjadi et al [40] presented a robust super-efficiencyDEA for ranking efficient units They noted that as inmost of the times exact data do not exist and the sto-chastic super-efficiency model presented in their paperincorporates the robust counterpart of super-efficiencyDEA
where 119883119884 are input-output data This model does notrank nonextreme efficient units It may be unstable andinfeasible
Gholam Abri et al [41] proposed a model for rankingefficient units They used representation theory and rep-resented the DMU under assessment as a convex combi-nation of extreme efficient units As the authors noted itis expected that the performance of DMU
119900is the same
as the performance of convex combination of extremeefficient units Thus it is possible to represent DMU
119900as
follows
(119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119899
sum
119895=1
120582119895= 1 119895 = 1 119904 (42)
As regards representation theorem this system has119898 + 119904 minus 1
constraints and 119904 variables (1205821 120582
119904) If this system has a
10 Journal of Applied Mathematics
unique solution we will have 120579lowast119900= sum
119899
119895=1= 1 120582
lowast
119895120579119895 otherwise
two models should be considered
min119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
max119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
(43)
Consider 1205791and 120579
2as the optimal solution of the above-
mentioned models respectively If 1205791= 120579
2 this is the same
as what has been mentioned previously If 1205791lt 120579
2 then
mentionedmodels provide an interval which helps rank unitsfrom the worst to the best Sometimes it will obtain theranking score with a bounded interval [120579
1 120579
2]
Jahanshahloo et al [42] presented models for rankingefficient units The presented models are somehow a modifi-cation of cross-efficiency model that overcomes the difficultyof alternative optimal weightsThe authors in their paperwithregard to the changes and also utilizing TOPSIS techniquepresented a new super-efficientmethod for ranking unitsThepresented model is as follows
120579119894119895=
119880119905119884119894
119881119905119883
119894
|
119880119905119884119897
119881119905119883
119897
le 1
119880119905119884119895
119881119905119883
119895
= 120579119895119895
119897 = 1 119899 119897 = 119894 119880 ge 0 119881 ge 0
(44)
where 120579119895119895is the efficiency score of DMU
119869using correspond-
ing weights Also 120579119894119895is the efficiency of DMU
119894using optimal
weights of DMU119895 Noura et al [43] provided a method for
ranking efficient units based on this idea that more effectiveand useful units in society should have better rank
Step 1 For each efficient unit choose the lower and upperlimit for each inputs and outputs Let 119864 be the set of efficientunits
where 120585 is representative of a small and nonzero number usedfor not dividing by zero Now consider119863minus119895 as follows whichshows that more successful DMU
119895will be if the larger value
of the119863119895is
119863119895= sum
119894isin119868
119889119894119895+ sum
119903isin119877
119889119903119895 (48)
where 119868 = 119863+
119894cup 119863
minus
119894 119877 = 119863
+
119903cup 119863
minus
119903
Ashrafi et al [44] introduced an enhanced Russell mea-sure of super efficiency for ranking efficient units inDEAThelinear counterpart of the proposed model is as follows
119900 le 120573 le V119903ge 120573 119903 = 1 119904
(49)
It cannot rank nonextreme efficient units
Journal of Applied Mathematics 11
Chen et al [45] proposed a modified super-efficiencymethod for ranking units based on simultaneous input-output projectionThe presentedmodel overcomes the infea-sibility problem
1199011= min
120579119904119903
119900
120601119904119903
119900
st sum
119895=1
119895 = 119900
120582119895119909119894119895le 120579
119904119903
119900119909119894119900 119894 = 1 119898
sum
119895=1
119895 = 119900
120582119895119910119903119895ge 120601
119904119903
119900119910119903119900 119903 = 1 119904
sum
119895=1
119895 = 119900
120582119895= 1
0 lt 120579119904119903
119900le 1 120601
119904119903
119900ge 1 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(50)
Rezai Balf et al [46] provide a model for ranking units basedon Tchebycheff norm As proved that this model is alwaysfeasible and stable it seems to have superiority over othermodels
max 119881119901
st 119881119901ge
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901 119894 = 1 119898
119881119901ge 119910
119903119901minus
119899
sum
119895=1
119895 = 119900
120582119895119910119903119895 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(51)
where
119881119901= Max
(
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901)119894 = 1 119898
(119910119903119901minus
119899
sum
119895=1
119895 = 119901
120582119895119910119903119895)119903 = 1 119904
(52)
Chen et al [47] for overcoming the infeasibility problem thatoccurred in variable returns to scale super-efficiency DEAmodel according to a directional distance function developedNerlove-Luenberger (N-L) measure of super-efficiency
6 Benchmarking Ranking Techniques
Sueyoshi et al [49] proposes a ldquobenchmark approachrdquo forbaseball evaluation This method is the combination of DEA
and (Offensive earned-run average) OERA As the authorsnoted using this method it is possible to select best unitsand also their ranking orders They mentioned that usingonly DEAmay result in a shortcoming in assessment as manyefficient units can be identifiedThus the authors used slack-adjusted DEA model and OERA to overcome this difficulty
Jahanshahloo et al [50] presented a new model forranking DMUs based on alteration in reference set The ideais based on this fact that efficient units can be the target unitfor inefficient units
min 120597119886119887
= 120579 minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st sum
119895isin119869minus119887
120582119895119909119894119895+ 119904
minus
119894minus 120579119909
119894119886 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
120579 free 119904minus119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 isin 119869 minus 119887
(53)
Finally the ranking order for each efficient unit b can becomputing by
119887= sum
119886isin119869119899
120597119886119887
119899
(54)
Note that this method cannot rank nonextreme efficientunits
Lu and Lo [51] provided an interactive benchmark modelfor ranking units The idea is based upon considering a fixedunit as a benchmark and calculating the efficiency of otherunits pair by pair to this unitThis procedure continueswhenall units are accounted for as a benchmark unit ConsiderDMU
119900as a unit under evaluation andDMU
119887as a benchmark
Assume
119909119894= 119909
119894119900(1 + 120601
119894) 119910
119903= 119910
119903119900(1 minus 120593
119903)
min 120579lowast119887
119900=
1 + (1119898)sum119898
119894=1120601119894
1 minus (1119904)sum119904
119903=1120593119903
st 120582119887119909119894119887minus 119909
119894119900120601119894le 119909
119894119900 119894 = 1 119898
120582119887119910119903119887minus 119910
119903119900120593119903ge 119910
119903119900 119903 = 1 119904
120601119894ge 119900 120593
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(55)
Then using (sum119899
119887=1120579119887lowast
119900)119899 119900 = 1 119899 the efficiency of
benchmark DMU119900can be obtained Using the following
index which indicates the increment in efficiency of a unitby moving from peer appraisal to self-appraisal it is nowpossible to rank units Note that the less the magnitude of
12 Journal of Applied Mathematics
this index is the better rank for corresponding unit will beobtained
FPIIBM119870
=
(TEBCC119896
minus STDIBM119896
)
(STDIBM119896
)
(56)
where TEBCC119896
and STDIBM119896
are respectively efficiency in BCCmodel and normalization of TEIBM of DMU
119896 Chen [52]
provided a paper for ranking efficient and inefficient unitsin DEA They noted that the evaluation of efficient unitsis based upon the alterations in efficiency of all inefficientunits by omitting in reference set At first solve the followingmodel which measures the efficiency of DMU
119886when DMU
119887
is eliminated from the reference set
min 120588119886119887
= 120579119904
119886minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1119895 = 119887
120582119895119909119894119895+ 119904
minus
119894= 120579
119904
119886119909119894119886 119894 = 1 119898
119899
sum
119895=1119895 = 119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
119899
sum
119895=1119895 = 119887
= 1
120579119904
119886ge 0 119904
minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119887
(57)
Then again solve the previous model this time for calculat-ing the efficiency score of each inefficient unit when each ofefficient units in turn is eliminated from the reference set 120578lowast
119886
and calculate the efficiency change
120591119886119887
= 120588lowast
119886119887minus 120578
lowast
119886 (58)
Then calculate the following index called as ldquoMCDErdquo indexfor each efficient and inefficient unit
119864119886= sum
119887isin119881119864
119882lowast
119887120588lowast
119886119887 119864
119887= sum
119887isin119881119868
119882lowast
119887120588lowast
119886119887 (59)
Now in accordance with themagnitude of the acquired indexit is possible to rank unitsThose units with higher score havebetter ranking order
7 Ranking Techniques by MultivariateStatistics in the DEA
As described in DEA literature in DEA technique frontier istaken into consideration rather than central tendency consid-ered in regression analysis DEA technique considers that anenvelope encompasses through all the observations as tight aspossible and does not try to fit regression planes in center ofdata In DEA methodology each unit is considered initiallyand compared to the efficient frontier but in regressionanalysis a procedure is considered in which a single functionfits to the data DEA uses different weights for differentunits but does not let the units use weights of other units
Canonical correlation analysis as Friedman and Sinuany-Stern [53] noted can be used for ranking units This methodis somehow the extension of regression analysis Adler et al[89] The aim in canonical correlation analysis is to find asingle vector commonweight for the inputs and outputs of allunits Consider119885
119895119882
119895as the composite input and output and
as corresponding weights respectively The presentedmodel by Tatsuoka and Lohnes [100] is as follows
max 119903119911119908
=
119878119909119910119880
(119878119909119909119881) (119878
119910119910119880)
(12)
st 119909119909119881 = 1
119910119910119880 = 1
(60)
where 119878119909119909 119878
119910119910 and 119878
119909119910are respectively defined as the matri-
ces of the sums of squares and sums of products of thevariables
Friedman and Sinuany-Stern [53] while defining the ratioof linear combinations of the inputs and outputs 119879
119895=
119882119895119885
119895used canonical correlation analysis As they noted that
scaling ratio 119879119895of the canonical correlation analysisDEA is
unboundedSinuany-Stern et al [101] used linear discriminant analy-
sis for ranking units They defined
119863119895=
119904
sum
119903=1
119906119903119910119903119895+
119898
sum
119894=1
V119903(minus119909
119894119895) (61)
DMU119895is said to be efficient if 119863
119895gt 119863
119888where 119863
119888is a critical
value based on themidpoint of themeans of the discriminantfunction value of the two groups Morrison [102] The largerthe amount of119863
119895is the better rank DMU
119895will have
Friedman and Sinuany-Stern [53] noted that as cross-efficiencyDEA canonical correlation analysisDEA and dis-criminant analysisDEA ranking orders may vary from eachother thus it seems necessary to introduce the combinedranking (CODEA) Combined ranking for each unit con-sidered all the ranks obtained from the above-mentionedrankings Moreover statistical tests are one for goodness offit between DEA and a specific ranking and the other fortesting correlation between variety of ranking orders Siegeland Castellan [103]
8 Ranking with MulticriteriaDecision-Making (MCDM) Methodologiesand DEA
Li and Reeves [56] for increasing discrimination power ofDEA presented amultiple-objective linear program (MOLP)As the authors mentioned using minimax and minsumefficiency in addition to the standard DEA objective functionhelp to increase discrimination power of DEA
Strassert and Prato [59] presented the balancing andranking method which uses a three-step procedure forderiving an overall complete or partial final order of optionsIn the first step derive an outranking matrix for all options
Journal of Applied Mathematics 13
from the criteria values Considering this matrix it is possibleto show the frequencywithwhich one option is ranked higherthan the other options In the second step by triangularizingthe outranking matrix establish an implicit preordering orprovisional ordering of optionsTheoutrankingmatrix showsthe degree to which there is a complete overall order ofoptions In the third step based on information given in anadvantages-disadvantages table the provisional ordering issubjected to different screening and balancing operations
Chen [60] utilized a nonparametric approach DEA toestimate and rank the efficiency of association rules withmultiple criteria in following steps Proposed postprocessingapproach is as follows
Step 1 Input data for association rule mining
Step 2 Mine association rules by using the a priori algorithmwith minimum support and minimum confidence
Step 3 Determine subjective interestingness measures byfurther considering the domain related knowledge
Step 4 Calculate the preference scores of association rulesdiscovered in Step 2 by using Cook and Kressrsquos DEA model
Step 5 Discriminate the efficient association rules found inStep 3 by using Obata and Ishiirsquos [104] discriminate model
Step 6 Select rules for implementation by considering thereference scores generated in Step 5 and domain relatedknowledge
Jablonsky [61] presented two original models super SBMand AHP for ranking of efficient units in DEA As theauthormentioned thesemodels are based onmultiple criteriadecision-making techniques-goal programming and analytichierarchy process Super SBM model for ranking units is asfollows
min 120579119866
119902= 1 + 119905120574 + (1 minus 119905) (
sum119898
119894=1119904+
1119894
119909119894119902
+
sum119898
119894=1119904minus
2119896
119910119896119902
)
st119899
sum
119895=1119895 = 119902
120582119895119909119894119895+ 119904
minus
1119894minus 119904
+
1119894= 119909
119894119902 i = 1 119898
119899
sum
119895=1119895 = 119902
120582119895119910119896119895+ 119904
minus
2119896minus 119904
+
2119896= 119910
119896119902 119896 = 1 119903
119904+
1119894le 120574 119894 = 1 119898
119904minus
2119896le 120574 119896 = 1 119903
119905 isin 0 1 120582119895ge 0 119878
+
1 119878
+
2ge 0 119878
minus
1 119878
minus
2ge 0
119895 = 1 119899
(62)
Wang and Jiang [62] presented an alternative mixedinteger linear programming models in order to identifythe most efficient units in DEA technique As the authors
mentioned presented models can make full use of input-output information with no need to specify any assuranceregions for input and output weights to avoid zero weights
min119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) minus
119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895)
st119904
sum
119903=1
119906119903119910119903119895minus
119898
sum
119894=1
V119894119909119894119895le 119868
119895 119895 = 1 119899
119899
sum
119895=1
119868119895= 1
119868119895isin 0 1 119895 = 1 119899
119906119903ge
1
(119898 + 119904)max119895119910
119903119895
119903 = 1 119904
V119894ge
1
(119898 + 119904)max119895119909
119894119895
119894 = 1 119898
(63)
Hosseinzadeh Lotfi et al [63] provided an improved three-stage method for ranking alternatives in multiple criteriadecision analysis In the first stage based on the bestand worst weights in the optimistic and pessimistic casesobtain the rank position of each alternative respectively Theobtained weight in the first stage is not unique thus it seemsnecessary to introduce a secondary goal that is used in thesecond stage Finally in the third stage the ranks of thealternatives compute in the optimistic or pessimistic case Itis mentionable that the model proposed in the third stageis a multicriteria decision-making (MCDM) model and itis solved by mixed integer programming As the authorsmentioned and provided in their paper this model can beconverted into an LP problem
The previous model with 119897 = 1 is the original CCR modelNote that DMUs in 119864
1 show the first level efficient frontier119897 = 2 indicates the second level efficient frontier when thefirst level efficient frontier is omittedThe following algorithmfinds these efficient frontiers
Step 1 Set 119897 = 1 and evaluate the entire set of DMUs 1198691 Inthis way the first level efficient DMU 1198641 is identified
Step 2 Exclude the efficient DMUs from future DEA runsand set 119869119897+1 = 119869
119897minus 119864
119897 (If 119869119897+1 = oslash stop)
Step 3 Evaluate the new subset of ldquoinefficientrdquo DMUs 119869119897+1to obtain a new set of efficient DMUs 119864119897+1
Step 4 Let l = l + 1 Go to Step 2
When 119869119897+1 = oslash the algorithm stopsAs the authors proved in this way it is possible to rank
the DMUs in the first efficient frontier based upon theirattractiveness scores and identify the best one
Jahanshahloo et al [65] provided a paper for rankingunits using gradient line As the authors mentioned theadvantage of this model is stability and robustness
supports on 119879119888 the obtained PPS by omitting theDMUunder
assessment in 119879119888 As the authors proved a unit is efficient
iff the optimal objective function of the following model isgreater than zero Consider
1198750= (119883 119884) 119883 = 120572119883
119900 119884 = 120573119884
119900
1198780= (119883 119884) isin 1198750
119883 = 120572119883119900 119884 = 120573119884
119900 120572 ge 0 120573 ge 0
(67)
Intersection of 1198780and efficient surface of
119879119888119888 is a half line
where its equation is (minus119881lowast119879119883
119900)120572 + (119880
lowast119879119884119900)120573 = 0 To rank
DMU119900the length of connecting arc DMU
119900with intersection
point of line and previous ellipse is calculated in (120572 120573) spaceThis intersection is as follows
120572lowast= (
1198702
120572119870
2
120573
1198702
120573+ (119881
lowast119879119883
119900119880
lowast119879119884119900)2119870
2
120572
)
12
120573lowast= (
119881lowast119879119883
119900
119880lowast119879119884119900
)120572lowast
(68)
Now the length of connecting arc DMU119900to the point
corresponding to 120572lowast 120573lowast in (120572 120573) plan is calculated as follows119868 = int
120572lowast
1(1 + 120573)
12119889120572 where
119870120572= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119899
119895=11199092
119894119900
)
12
119870120573= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119904
119903=11199102
119903119900
)
12
(69)
Jahanshahloo et al [66] also provided a paper with theconcept of advantage in data envelopment analysis
In their paper Jahanshahloo et al [67] consideringMonteCarlo method presented a new method of ranking
Step 1 Generate a uniformly distributed sequence of 1198801198952119899
119895=1
on(0 1)
Step 2 Random numbers should be classified into 119873 pairslike (119880
1 119880
1015840
1) (119880
119873 119880
1015840
119873) in a way that each number is used
just one time
Step 3 Compute119883119894= 119886 + 119880
119894(119887 minus 119886) and 119891(119883
119894) gt 119888119880
1015840
119894
Step 4 Estimate the integral 119868 by 120579119868= 119888(119887 minus 119886)(119873
119867119873)
Now consider DMU119900as an efficient unit measure those
units that are dominated DMU119900 As for dome DMUs this
would be unbounded thus the authors for each unitbounded the region Then for DMU
119900if (minus119883
119900 119884
119900) ge (minus119883 119884)
then (minus119883 119884) is in the RED of DMU119900 Now by using 119881
119901=
119881lowast(119873
119867119873) all the hits (the above condition) can be counted
where 119881lowast is the measure of the whole region JahanshahlooandAfzalinejad [68] presented amethod based on distance of
Journal of Applied Mathematics 15
the unit under evaluation to the full inefficient frontier Theypresented two models radian and nonradial
min 120601
st119899
sum
119895=1
120582119895119909119894119895= 120601119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
120582119895ge 0 119895 = 1 119898
max119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903
st sum
119895isin119869minus119887
120582119895119909119894119895minus 119904
minus
119894= 119909
119894119900 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895+ 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
119904minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(70)
Amirteimoori [69] based on the same idea considered bothefficient and antiefficient frontiers for efficiency analysis andranking units
Kao [70] mentioned that determining the weights ofindividual criteria in multiple criteria decision analysis in away that all alternatives can be compared according to theaggregate performance of all criteria is of great importanceAs Kao noted this problem relates to search for alternativeswith a shorter and longer distance respectively to the idealand anti-ideal units He proposed a measure that consideredthe calculation of the relative position of an alternativebetween the ideal and anti-ideal for finding an appropriaterankings
Khodabakhshi and Aryavash [71] presented a method forranking all units using DEA concept
First Compute the minimum and maximum efficiencyvalues of eachDMU in regard to this assumption that the sumof efficiency values of all DMUs equals to 1
Second determine the rank of each DMU in relation to acombination of itsminimumandmaximumefficiency values
Zerafat Angiz et al [72] proposed a technique in orderto aggregate the opinions of experts in voting system Asthe authors mentioned the presented method uses fuzzyconcept and it is computationally efficient and can fullyrank alternatives At first number of votes given to a rankposition was grouped to construct fuzzy numbers and thenthe artificial ideal alternative introduced Furthermore byperforming DEA the efficiency measure of alternatives was
obtained considering artificial ideal alternative comparedby each of the alternatives pair by pair Thus alternativesare ranked in accordance with their efficiency scores Ifthis method cannot completely rank alternatives weightrestrictions based on fuzzy concept are imposed into theanalysis
10 Different Applications in Ranking Units
As discussed formerly there exist a variety of ranking meth-ods and applications in theliterature Nowadays DEAmodelsare widely used in different areas for efficiency evaluationbenchmarking and target setting ranking entities and soforth Ranking units as one of the important issues in DEAhas been performed in different areas Jahanshahloo et al [42]ranked cities in Iran to find the best place for creating a datafactory Hosseinzadeh Lotfi et al [28] used a new methodfor ranking in order to find the best place for power plantlocation Ali andNakosteen [80] Amirteimoori et al [37] andAlirezaee and Afsharian [24] Soltanifar and HosseinzadehLotfi [105] Zerafat Angiz et al [72] Hosseinzadeh Lotfiet al [28] Jahanshahloo [50] Chen and Deng [52] andJablonsky [61] used different ranking methods in bankingsystem Sadjadi [40] ranked provincial gas companies in IranMehrabian et al [32] Li et al [39] Orkcu and Bal [12] andWu et al [19] ranked different departments of universitiesJahanshahloo et al [35] Jahanshahloo and Afzalinejad [68]utilized the presented models in ranking 28 Chinese citiesJahanshahloo at al [35] provided an application to burdensharing amongst NATOmember nations Jahanshahloo et al[14] utilized the presented model for ranking nursery homesContreras [18] andWang et al [23] used the provided rankingtechniques in ranking candidates Lu and Lo [51] applied theirmethod on an application to financial holding companiesHosseinzadeh Lotfi et al [63] consider an empirical exampleinwhich voters are asked to rank twoout of seven alternativesWang and Jiang [62] utilized the provided method in facilitylayout design in manufacturing systems and performanceevaluation of 30OECD countries Chen [60] used an exampleof market basket data in order to illustrate the providedapproach
11 Application
In this section some of the reviewed models are applied onthe example used in Soltanifar and Hosseinzadeh Lotfi [105]Consider twenty commercial banks of Iran with input-outputdata tabulated in Table 1 and summarized as follows Also theresults of CCRmodel are listed in this table As it can be seenseven units are efficient DMUS 14 7 12 15 17 and 20 Inputsare staff computer terminal and spaceOutputs are depositsloans granted and charge
Consider some of the important ranking methods inliterature as follows
RM1 AP model [31] is based upon the idea of leaveunit evaluation out and measuring the distance of theunit under evaluation from the production possibilityset constructed by the remaining DMUs
RM2MAJmodel [32] presented for ranking efficientwhich is always stable but might be infeasible in somecasesRM3 Modified MAJ model [38] overcomes theproblem which might occurr in MAJ modelRM4 A new model based on the idea of alterationsin the reference set of the inefficient units [50]RM5 A model presented by Li et al [39] whichis a super-efficiency method that does not have thesuffering in previous methodsRM6 Slack-basedmodel [33] is based upon the inputand output variables at the same timeRM7 SA DEAmodel [49] overcomes the problem ofinfeasibility which existed in AP modelRM8 Cross-efficiency [10] is provided based onusing weights of each unit under evaluation in opti-mality for other unitsRM9 A model based on finding common set ofweights [25] which determine the common set ofweights for DMUs and ranked DMUs based this ideaRM10 A model based on finding common set ofweights [22 67] for ranking efficient unitsRM11 L1-norm model [34 35 65 66] the idea isbased upon the leave-one-out efficient unit and l1-norm which is always feasible and stableRM12 119871
infin-norm model Rezai Balf et al [46] pro-
vided a method with more ability over other existingmethods based on Tchebycheff normRM13 An enhanced Russel measure of super-efficiency model for ranking units [44]RM14 A rankingmodel which considers the distanceof unit from the full inefficient frontier [38]RM15 Amodified super-efficiencymodel [45] whichovercomes the infeasibility that may happen in prob-lem This model is based on simultaneous projectionof input output
In accordance with properties of different ranking models inorder to rank efficient units consider Table 2
1199011 Feasibility
1199012 Ranking extreme efficient units
1199013 Complexity in computation
1199014 Instability
1199015 Absence of multiple optimal solution
1199016 Dependency to 120579 and slacks
1199017 Dependency to the number of efficient and ineffi-
cient units
In Tables 3 and 4 the rank of efficient units consideringthe above mentioned methods is listed Note that ED showsefficient DMUs (ED) and RM
119895 (119895 = 1 15) are those
explained previously
12 Conclusion
In this paper the DEA ranking was reviewed and classifiedinto seven general groups In the first group those papersbased on a cross-efficiencymatrix were reviewed In this fieldDMUs have been evaluated by self- and peer pressure Thesecond group of papers is based on those papers lookingfor optimal weights in DEA analysis The third one is thesuper-efficiency method By omitting the under evaluationunit and constructing a new frontier by the remaining unitsthe unit under evaluation can get an equal or greater scoreThe fourth group is based on benchmarking idea In thisclass the effect of an efficient unit considered as a targetfor inefficient units is investigated This idea is very usefulfor the managers in decision making Another class thefifth one involves the application of multivariate statisticaltools The sixth section discusses the ranking methods basedon multicriteria decision-making (MCDM) methodologiesand DEA The final section includes some various rankingmethods presented in the literature Finally in an applicationthe result of some of the above-mentioned ranking methodsis presented
References
[1] M J Farrell ldquoThe measurement of productive efficiencyrdquoJournal of the Royal Statistical Society A vol 120 pp 253ndash2811957
[2] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
18 Journal of Applied Mathematics
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
The two-phase enveloping problem with constant returns toscale form of technology first provided by Charnes et al [2]is as follows
min 120579 minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1
120582119895119909119894119895+ 119904
minus
119894= 120579119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895minus 119904
+
119903= 119910
119903119900 119903 = 1 119904
120582119895ge 119895 = 1 119899
(3)
The two-phase enveloping problem with variable returns toscale form of technology first provided by Banker et al [3] isas follows
min 120579 minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1
120582119895119909119894119895+ 119904
minus
119894= 120579119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895minus 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119899
(4)
As regards the above-mentioned problems and due to corre-sponding feasible region it is evident that 120579lowastCCR le 120579
lowastBCCAccording to the definition of 119879CCR and 119879BCC an envelop
constructed through units called best practice or efficientConsidering mentioned problems if a DMU
119900is not CCR
(BCC) efficient it is possible to project this DMU ontothe CCR (BCC) efficiency frontier considering the followingformulas
119909119894119900= 120579
lowast119909119894119900minus 119904
minuslowast
119894=
119899
sum
119895=1
120582lowast
119895119909119894119895 119894 = 1 119898
119910119903119900= 119910
119903119900+ 119904
+lowast
119903=
119899
sum
119895=1
120582lowast
119895119910119903119895 119903 = 1 119904
(5)
The dual model corresponding to the following model isas follows
max119904
sum
119903=1
119906119903119910119903119900
st119898
sum
119894=1
V119894119909119894119900= 1
119904
sum
119903=1
119906119903119910119903119895minus
119898
sum
119894=1
V119894119909119894119895le 0 119895 = 1 119899
119880 ge 0 119881 ge 0
(6)
For overcoming the problem of zero weights and variabilityof weights use of assurance region is suggested byThompsonet al [90ndash92]
As mentioned in the literature usually there exist morethan one efficient units and these units cannot be furthercompared to each other on basis of efficiency scores Thusit felt necessary to provide new models for ranking theseunits There exist a variety of ranking models in context ofdata envelopment analysis In the remaining of this paper wereview some of these models
3 Cross-Efficiency Ranking Techniques
Sexton et al [10] provided a method for ranking units basedon this idea that units are self- and peer-evaluated Forderiving the cross-efficiency of any DMU
119895using weights
chosen by DMU119900 they proposed the following equation
120579119900119895=
119880lowast
119900119884119895
119881lowast
119900119883
119895
(7)
where 119880lowast 119881
lowast are optimal weights obtained from the follow-ing model for DMU
119900under assessment
min 119881119905119883
119900
st 119880119905119884119900= 1
119880119905119884119895minus 119881
119905119883
119895le 0 119895 = 1 119899
119880 ge 0 119881 ge 0
(8)
NowDMU119900received the average cross-efficiency score as
120579119900= sum
119899
119895=1120579119900119895119899 for further details about this averaging see
also Green et al [93] Doyle and Green [94] also used cross-efficiency matrix for ranking units According to this methodfor ranking DMUs many investigations have been done asreviewed in Adler et al [89]
Rodder and Reucher [11] presented a consensual peer-based DEA model for ranking units As the authors saidthis method is generalized twofold The first is an optimalefficiency improving input allocation the second aim isthe choice of a peer DMU whose corresponding price isacceptable for the other units Consider
max 119881lowast119879
119896119882
119897
st 119880lowast119879
119896119884119897minus 119881
lowast119879
119896119882
119897le 0
119882119897minussum
119895
120583119897119895119883
119895ge 0
sum
119895
120583119897119895119884119895ge 119884
119897
120583119897119895ge 0 forall119895
(9)
4 Journal of Applied Mathematics
The higher the degree of input variation is the better thechance to be efficient will be
Orkcu and Bal [12] provided a goal programming tech-nique to be used in the second stage of the cross-evaluationTheir modified model is as follows
As the authors noted there exist alternative optimal solutionsWu et al [13] described the main suffering of cross-
efficiency when the ultimate average cross-efficiency utilizedfor ranking units For removing this shortcoming they elim-inated the assumption of average and utilized the Shannonentropy in order to obtain the weights for ultimate cross-efficiency scores Jahanshahloo et al [14] provided a methodfor selecting symmetric weights to be used in DEA cross-efficacy
Step 1 Efficiency of DMUs needs to be computed
Step 2 Choose the solutions in accordance with the sec-ondary goal for each DMU as follows
min 119890119879119885119900119890
st 119906119900119910119900= 1
V119900119883
119900= 120579
119900
119906119900119884 minus V
119900119883 le 0
119906119900119894119910119900119894minus 119906
119900119895119910119900119895le 119911
119900119894119895 forall119894 119895
119906119900119895119910119900119895minus 119906
119900119894119910119900119894le 119911
119900119894119895 forall119894 119895
119906119900 V
119900ge 119889
(11)
Step 3 The cross-efficiency for any DMU119895 using the weights
that DMU119900has chosen in the previous model is then used as
follows
120579119900119895=
119906lowast
119900119884119895
Vlowast119900119883
119895
(12)
Wang et al [15] provided a cross-efficiency evaluation basedon ideal and anti-ideal units for ranking As the authorsmentioned a DMU could choose a unique set of input andoutput weights to make its distance from ideal DMU assmall as possible or the distance from anti-ideal DMU aslarge as possible or the both Thus according to this ideathey proposed the following procedure for cross-efficiencyevaluation
Model 1 Minimization of the distance from ideal DMU
Model 2 Maximization of the distance from anti-ideal DMU
Model 3 Maximization of the distance between ideal DMUand anti-ideal DMU
Model 4 Maximization of the relative closeness
The authors mentioned that the bigger the relative close-ness of a DMU is the better performance it will have
In a paper Ramon et al [16] selected the profiles ofweights used in cross-efficiency assessment As the authorssaid they tried to prevent unrealistic weighting They havediscussed the zero weights as they excluded variables fromthe evaluation In the calculation of cross-efficiency scoresthey proposed to ignore the profiles of those weights of theunit under evaluation that among their alternate optimalsolutions cannot choose nonzeroweightsThey also proposedthe ldquopeer-restrictedrdquo cross-efficiency evaluation where theunits assessed in a peer evaluation which means profiles ofweights of some inefficient units are not considered Finallythe presented approach extended to derive a common set ofweights Guo and Wu [17] provided a complete ranking ofDMUs with undesirable outputs using restriction in DEAAs the author mentioned this model is presented to realizea unique ranking of units by ldquomaximal balanced indexrdquoaccording to the obtained optimal shadow prices
119896+1ge 119889 (119896 120576) 119896 = 1 119899 minus 1
120583119899ge 119889 (119899 120576)
(19)
where 120579119895119896obtained from Step 5 and 120573lowast
119895is the optimal solution
of the following model Finally the DMUs are ranked basedon their 119911lowast
119895= sum
119899
119896=1120583lowast
119896120579119895119896values
Washio and Yamada [21] discussed that in real casesfinding the best ranking is more significant than acquiringthe most advantage weight and maximizing the efficiencyThus they presented a model called rank-based measure(RBM) for evaluating units from different viewpoint Thusthey suggested a method for acquiring those weight resultedfrom the best ranking as long as calculating those weightthat maximizes the efficiency score Finally they applied thepresented model to the cross-efficiency assessment
4 Ranking Techniques Based on FindingOptimal Weights in DEA Analysis
Jahanshahloo et al [22] gave a note on some of the DEAmodels for complete ranking using common set of weightsThey proved that by solving only one problem it is possibleto determine the common set of weights
where 119860 is the set of efficient units of the following model
max
sum119904
119903=11199061199031199101199031+ 119906
0
sum119898
119894=1V1198941199091198941
sdot sdot sdot
sum119904
119903=1119906119903119910119903119899+ 119906
0
sum119898
119894=1V119894119909119894119899
stsum
119904
119903=1119906119903119910119903119895+ 119906
0
sum119898
119894=1V119894119909119894119895
le 1 119895 = 1 119899
119880 119881 120578 ge 120576 1199060free
(21)
Note that DMUs can be ranked based on the evaluation oftheir efficiencies
Wang et al [23] provided an aggregating preference rank-ing In this paper use of ordered weighted averaging (OWA)operator is proposed for aggregating preference rankings Let
119908119895be the relative importance weight given to the jth ranking
place and V119894119895the vote candidate i receives in the jth ranking
place The total score of each candidate is defined as
119911119894=
119898
sum
119894=1
V119894119895119882
119895 119894 = 1 119898 (22)
Alirezaee and Afsharian [24] discussed multiplier model inwhich the variables are considered as shadow prices note thatsum
119894=1V119894119909119894119895are total revenue and cost of DMU
119895
which are considered in optimization problem The authorsclaimed that sum119904
119903=1119906119903119910119903119895minus sum
119898
119894=1V119894119909119894119895le 0 119895 = 1 119899 is the
profit restriction for DMU119895 If 119865(119909 119910) = 0 is considered to be
the efficient production function then
sum
119894
120597119865
120597119909119894
119909119894+sum
119895
120597119865
120597119910119895
119910119895= 0 (23)
They mentioned that the connected profit of the DMU iszero when shadow prices are derived from the technologyand called this situation as balance situation As the authorsmentioned therefore in the case that DMU
1is efficient but
DMU2is inefficient or the efficiency score of both DMUs is
the same and it obtains more negative quantity in balanceindex it can be concluded that DMU
1has a better rank than
DMU2
Liu and Hsuan Peng [25] in their paper proposed amethod for determining the common set of weights forranking units In common weights analysis methodologythey provided the following model
Δlowast= min sum
119895isin119864
Δ119900
119895+ Δ
119894
119895
stsum
119904
119903=1119910119903119895119880119903+ Δ
119900
119895
sum119898
119894=1119909119894119895119881119894minus Δ
119894
119895
= 1 119895 isin 119864
Δ119900
119895 Δ
119894
119895ge 0 forall119895 isin 119864
119880119903ge 120576 119903 = 1 119904
119881119894ge 120576 119894 = 1 119898
(24)
The mentioned ratio form the linear equationsWang et al [26] proposed a paper for ranking decision-
making units by imposing a minimum weight restrictionin DEA The authors noted that using data envelopmentanalysis it is not possible to distinguish between DEA effi-cient units Thus they presented a method for ranking unitsusing imposing minimum weight restriction for the input-output data As they mentioned these weights restrictionsare decided by a decision maker (DM) or an assessor asregards the solutions to a series of LP models considered fordetermining a maximin weight for each efficient DMU
Hatefi and Torabi [27] proposed a common weightmulti criteria decision analysis (MCDA)-data envelopmentanalysis (DEA) for constructing composite indicators (CIs)As the authors proved the presented model can discrimi-nate between efficient units The obtained common weightshave discriminating power more than those obtained from
Journal of Applied Mathematics 7
previous models Finally they studied the robustness anddiscriminating power of the proposed method by Spearmanrsquosrank correlation coefficient
Hosseinzadeh Lotfi et al [28] proposed one DEA rankingmodel based on applying aggregate units In doing so artificialunits called aggregate units are defined as follows Theaggregate unit is shown by DMU
119886
119909119901
119894119886= sum
119896isin119877119901
119909119894119896 119910
119901
119903119886= sum
119896isin119877119901
119910119903119896
119894 = 1 119898 119903 = 1 119904
(25)
where 119877119901= 119895 | DMU
119895isin 119864
119901 119864
119901= 119864DMU
119901 Note that 119864
is the set of efficient unitsFirst it is tried to maximize the efficiency score of the
DMU119886and then to maximize the efficiency score of the
DMU119901
119886 For resolving the existence of alternative solutions
the authors presented an approach comprising (119898+119904) simplelinear problems to achieve the most appropriate optimalsolutions among all alternative optimal solutions Finallythey proposed theRI index for ranking all efficientDMUs Let119880119886 119881
119875119886 and119880119901
119886 119881
119901
119886be the optimal solutions of the multiplier
model for assentDMU119886andDMU119901
119886 Consider 120578
119886= sum
119904
119903=1119906119903119886minus
sum119898
119894=1V119894119886 120578
119901
119886= sum
119904
119903=1119906119901
119903119886minus sum
119898
119894=1V119901119894119886 thus RI
119901= 120578
119900
119886minus 120578
119886
Wang et al [29] presented two nonlinear regressionmodels for deriving common set of weights for fully rankunits
min 119911 =
119899
sum
119895=1
(120579lowast
119895minus
sum119904
119903=1119906119903119910119903119895
sum119898
119894=1V119894119909119894119895
)
2
st 119880 119881 ge 0
min119899
sum
119895=1
(
119904
sum
119903=1
119906119903119910119903119895minus 120579
lowast
119895
119898
sum
119894=1
V119894119909119894119895)
2
st119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895) +
119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) = 119899
119880 119881 ge 0
(26)
Ramon et al [30] aimed at deriving a common set of weightsfor ranking units As the authors mentioned the idea is basedupon minimization of the deviations of the common weightsfrom the nonzero weights obtained fromDEA Furthermoreseveral norms are used for measuring such differences
5 Super-Efficiency Ranking Techniques
Super efficiency models introduced in DEA technique arebased upon the idea of leave one out and assessing this unittrough the remanding units
Andersen and Petersen [31] introduced a model for rank-ing efficient units The proposed model is as follows
Although this idea is useful for further discriminatingefficient units it has been shown in the literature that itmay be infeasible and nonstable Thrall [7] mentioned theinfeasibility of super-efficiency CCRmodel Also a conditionunder which infeasibility occurred in super-efficiency DEAmodels is mentioned by Zhu [95] Seiford and Zhu [96] andDula and Hickman [97]
Hashimoto [98] provided a model based on the idea ofone leave out and assurance region for ranking units
Mehrabian et al [32] presented a complete ranking forefficiency units in DEA context As the authors mentionedthis model does not have difficulties of AP model
min 119908119901+ 1
st119899
sum
119895=1 119895 = 119901
120582119895119909119894119895le 119909
119894119901+ 119908
1199011 119894 = 1 119898
119899
sum
119895=1 119895 = 119901
120582119895119910119903119895ge 119910
119903119901 119903 = 1 119904
120582119895ge 0 119895 = 119901
(28)
With this method it is not possible to rank nonextremeefficient units
Tone [33] presented super efficiency of SBM model Thismodel has the advantages of nonradial models and it isalways feasible and stable
minsum
119898
119894=1119909119894119909
119894119900
sum119904
119903=1119910119903119910
119903119900
st119899
sum
119895=1119895 = 119900
120582119895119909119894119895le 119909
119894 119894 = 1 119898
119899
sum
119895=1119895 = 119900
120582119895119910119903119895ge 119910
119903 119903 = 1 119904
119909119894ge 119909
119894119900 0 le 119910
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119900
(29)
8 Journal of Applied Mathematics
Jahanshahloo et al [34] added some ratio constraints to themultiplier form of AP model and introduced a new methodfor ranking DMUs
In their paper Chen and Sherman [36] presented a non-radial super-efficiency method and discussed the advantageof it They verified that this model is invariant to units ofinputoutput measurement Let 119869119900 = 119869DMU
119900
Step 1 Solve the followingmodel to find the extreme efficientunits in 119869119900
This method cannot rank nonextreme efficient units
Jahanshahloo et al [38] presented modified MAJ modelfor ranking efficiency units in DEA technique
min 119908119901+ 1
st119899
sum
119895=1119895 = 119901
120582119895
119909119894119895
119872119894
le
119909119894119901
119872119894
+ 1199081199011 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895ge 119910
119903119901 119903 = 1 119904
120582119895ge 0 119895 = 119901
(39)
where 119872119894= Max 119909
119894119895| DMU
119895is efficient It cannot rank
nonextreme efficient unitsLi et al [39] presented a new method for ranking which
does not have difficulties of earlier methods The presentedmodel is always feasible and stable
min 1 +
1
119898
119898
sum
119894=1
119904+
1198942
119877minus
119894
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895+ 119904
minus
1198941minus 119904
+
1198942= 119909
119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895minus 119904
+
119903= 119910
119903119901 119903 = 1 119904
119904minus
1198942ge 0 119904
minus
1198941ge 0 119904
+
119903ge 0 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 119901
(40)
In this case extreme efficient units cannot be ranked
In a paper Khodabakhshi [99] addresses super efficiencyon improved outputs He mentioned that as AP modelmay be infeasible under variable returns to scale technol-ogy using the presented model gives a complete rankingwhen getting an input combination for improving outputs issuitable
Sadjadi et al [40] presented a robust super-efficiencyDEA for ranking efficient units They noted that as inmost of the times exact data do not exist and the sto-chastic super-efficiency model presented in their paperincorporates the robust counterpart of super-efficiencyDEA
where 119883119884 are input-output data This model does notrank nonextreme efficient units It may be unstable andinfeasible
Gholam Abri et al [41] proposed a model for rankingefficient units They used representation theory and rep-resented the DMU under assessment as a convex combi-nation of extreme efficient units As the authors noted itis expected that the performance of DMU
119900is the same
as the performance of convex combination of extremeefficient units Thus it is possible to represent DMU
119900as
follows
(119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119899
sum
119895=1
120582119895= 1 119895 = 1 119904 (42)
As regards representation theorem this system has119898 + 119904 minus 1
constraints and 119904 variables (1205821 120582
119904) If this system has a
10 Journal of Applied Mathematics
unique solution we will have 120579lowast119900= sum
119899
119895=1= 1 120582
lowast
119895120579119895 otherwise
two models should be considered
min119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
max119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
(43)
Consider 1205791and 120579
2as the optimal solution of the above-
mentioned models respectively If 1205791= 120579
2 this is the same
as what has been mentioned previously If 1205791lt 120579
2 then
mentionedmodels provide an interval which helps rank unitsfrom the worst to the best Sometimes it will obtain theranking score with a bounded interval [120579
1 120579
2]
Jahanshahloo et al [42] presented models for rankingefficient units The presented models are somehow a modifi-cation of cross-efficiency model that overcomes the difficultyof alternative optimal weightsThe authors in their paperwithregard to the changes and also utilizing TOPSIS techniquepresented a new super-efficientmethod for ranking unitsThepresented model is as follows
120579119894119895=
119880119905119884119894
119881119905119883
119894
|
119880119905119884119897
119881119905119883
119897
le 1
119880119905119884119895
119881119905119883
119895
= 120579119895119895
119897 = 1 119899 119897 = 119894 119880 ge 0 119881 ge 0
(44)
where 120579119895119895is the efficiency score of DMU
119869using correspond-
ing weights Also 120579119894119895is the efficiency of DMU
119894using optimal
weights of DMU119895 Noura et al [43] provided a method for
ranking efficient units based on this idea that more effectiveand useful units in society should have better rank
Step 1 For each efficient unit choose the lower and upperlimit for each inputs and outputs Let 119864 be the set of efficientunits
where 120585 is representative of a small and nonzero number usedfor not dividing by zero Now consider119863minus119895 as follows whichshows that more successful DMU
119895will be if the larger value
of the119863119895is
119863119895= sum
119894isin119868
119889119894119895+ sum
119903isin119877
119889119903119895 (48)
where 119868 = 119863+
119894cup 119863
minus
119894 119877 = 119863
+
119903cup 119863
minus
119903
Ashrafi et al [44] introduced an enhanced Russell mea-sure of super efficiency for ranking efficient units inDEAThelinear counterpart of the proposed model is as follows
119900 le 120573 le V119903ge 120573 119903 = 1 119904
(49)
It cannot rank nonextreme efficient units
Journal of Applied Mathematics 11
Chen et al [45] proposed a modified super-efficiencymethod for ranking units based on simultaneous input-output projectionThe presentedmodel overcomes the infea-sibility problem
1199011= min
120579119904119903
119900
120601119904119903
119900
st sum
119895=1
119895 = 119900
120582119895119909119894119895le 120579
119904119903
119900119909119894119900 119894 = 1 119898
sum
119895=1
119895 = 119900
120582119895119910119903119895ge 120601
119904119903
119900119910119903119900 119903 = 1 119904
sum
119895=1
119895 = 119900
120582119895= 1
0 lt 120579119904119903
119900le 1 120601
119904119903
119900ge 1 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(50)
Rezai Balf et al [46] provide a model for ranking units basedon Tchebycheff norm As proved that this model is alwaysfeasible and stable it seems to have superiority over othermodels
max 119881119901
st 119881119901ge
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901 119894 = 1 119898
119881119901ge 119910
119903119901minus
119899
sum
119895=1
119895 = 119900
120582119895119910119903119895 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(51)
where
119881119901= Max
(
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901)119894 = 1 119898
(119910119903119901minus
119899
sum
119895=1
119895 = 119901
120582119895119910119903119895)119903 = 1 119904
(52)
Chen et al [47] for overcoming the infeasibility problem thatoccurred in variable returns to scale super-efficiency DEAmodel according to a directional distance function developedNerlove-Luenberger (N-L) measure of super-efficiency
6 Benchmarking Ranking Techniques
Sueyoshi et al [49] proposes a ldquobenchmark approachrdquo forbaseball evaluation This method is the combination of DEA
and (Offensive earned-run average) OERA As the authorsnoted using this method it is possible to select best unitsand also their ranking orders They mentioned that usingonly DEAmay result in a shortcoming in assessment as manyefficient units can be identifiedThus the authors used slack-adjusted DEA model and OERA to overcome this difficulty
Jahanshahloo et al [50] presented a new model forranking DMUs based on alteration in reference set The ideais based on this fact that efficient units can be the target unitfor inefficient units
min 120597119886119887
= 120579 minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st sum
119895isin119869minus119887
120582119895119909119894119895+ 119904
minus
119894minus 120579119909
119894119886 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
120579 free 119904minus119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 isin 119869 minus 119887
(53)
Finally the ranking order for each efficient unit b can becomputing by
119887= sum
119886isin119869119899
120597119886119887
119899
(54)
Note that this method cannot rank nonextreme efficientunits
Lu and Lo [51] provided an interactive benchmark modelfor ranking units The idea is based upon considering a fixedunit as a benchmark and calculating the efficiency of otherunits pair by pair to this unitThis procedure continueswhenall units are accounted for as a benchmark unit ConsiderDMU
119900as a unit under evaluation andDMU
119887as a benchmark
Assume
119909119894= 119909
119894119900(1 + 120601
119894) 119910
119903= 119910
119903119900(1 minus 120593
119903)
min 120579lowast119887
119900=
1 + (1119898)sum119898
119894=1120601119894
1 minus (1119904)sum119904
119903=1120593119903
st 120582119887119909119894119887minus 119909
119894119900120601119894le 119909
119894119900 119894 = 1 119898
120582119887119910119903119887minus 119910
119903119900120593119903ge 119910
119903119900 119903 = 1 119904
120601119894ge 119900 120593
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(55)
Then using (sum119899
119887=1120579119887lowast
119900)119899 119900 = 1 119899 the efficiency of
benchmark DMU119900can be obtained Using the following
index which indicates the increment in efficiency of a unitby moving from peer appraisal to self-appraisal it is nowpossible to rank units Note that the less the magnitude of
12 Journal of Applied Mathematics
this index is the better rank for corresponding unit will beobtained
FPIIBM119870
=
(TEBCC119896
minus STDIBM119896
)
(STDIBM119896
)
(56)
where TEBCC119896
and STDIBM119896
are respectively efficiency in BCCmodel and normalization of TEIBM of DMU
119896 Chen [52]
provided a paper for ranking efficient and inefficient unitsin DEA They noted that the evaluation of efficient unitsis based upon the alterations in efficiency of all inefficientunits by omitting in reference set At first solve the followingmodel which measures the efficiency of DMU
119886when DMU
119887
is eliminated from the reference set
min 120588119886119887
= 120579119904
119886minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1119895 = 119887
120582119895119909119894119895+ 119904
minus
119894= 120579
119904
119886119909119894119886 119894 = 1 119898
119899
sum
119895=1119895 = 119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
119899
sum
119895=1119895 = 119887
= 1
120579119904
119886ge 0 119904
minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119887
(57)
Then again solve the previous model this time for calculat-ing the efficiency score of each inefficient unit when each ofefficient units in turn is eliminated from the reference set 120578lowast
119886
and calculate the efficiency change
120591119886119887
= 120588lowast
119886119887minus 120578
lowast
119886 (58)
Then calculate the following index called as ldquoMCDErdquo indexfor each efficient and inefficient unit
119864119886= sum
119887isin119881119864
119882lowast
119887120588lowast
119886119887 119864
119887= sum
119887isin119881119868
119882lowast
119887120588lowast
119886119887 (59)
Now in accordance with themagnitude of the acquired indexit is possible to rank unitsThose units with higher score havebetter ranking order
7 Ranking Techniques by MultivariateStatistics in the DEA
As described in DEA literature in DEA technique frontier istaken into consideration rather than central tendency consid-ered in regression analysis DEA technique considers that anenvelope encompasses through all the observations as tight aspossible and does not try to fit regression planes in center ofdata In DEA methodology each unit is considered initiallyand compared to the efficient frontier but in regressionanalysis a procedure is considered in which a single functionfits to the data DEA uses different weights for differentunits but does not let the units use weights of other units
Canonical correlation analysis as Friedman and Sinuany-Stern [53] noted can be used for ranking units This methodis somehow the extension of regression analysis Adler et al[89] The aim in canonical correlation analysis is to find asingle vector commonweight for the inputs and outputs of allunits Consider119885
119895119882
119895as the composite input and output and
as corresponding weights respectively The presentedmodel by Tatsuoka and Lohnes [100] is as follows
max 119903119911119908
=
119878119909119910119880
(119878119909119909119881) (119878
119910119910119880)
(12)
st 119909119909119881 = 1
119910119910119880 = 1
(60)
where 119878119909119909 119878
119910119910 and 119878
119909119910are respectively defined as the matri-
ces of the sums of squares and sums of products of thevariables
Friedman and Sinuany-Stern [53] while defining the ratioof linear combinations of the inputs and outputs 119879
119895=
119882119895119885
119895used canonical correlation analysis As they noted that
scaling ratio 119879119895of the canonical correlation analysisDEA is
unboundedSinuany-Stern et al [101] used linear discriminant analy-
sis for ranking units They defined
119863119895=
119904
sum
119903=1
119906119903119910119903119895+
119898
sum
119894=1
V119903(minus119909
119894119895) (61)
DMU119895is said to be efficient if 119863
119895gt 119863
119888where 119863
119888is a critical
value based on themidpoint of themeans of the discriminantfunction value of the two groups Morrison [102] The largerthe amount of119863
119895is the better rank DMU
119895will have
Friedman and Sinuany-Stern [53] noted that as cross-efficiencyDEA canonical correlation analysisDEA and dis-criminant analysisDEA ranking orders may vary from eachother thus it seems necessary to introduce the combinedranking (CODEA) Combined ranking for each unit con-sidered all the ranks obtained from the above-mentionedrankings Moreover statistical tests are one for goodness offit between DEA and a specific ranking and the other fortesting correlation between variety of ranking orders Siegeland Castellan [103]
8 Ranking with MulticriteriaDecision-Making (MCDM) Methodologiesand DEA
Li and Reeves [56] for increasing discrimination power ofDEA presented amultiple-objective linear program (MOLP)As the authors mentioned using minimax and minsumefficiency in addition to the standard DEA objective functionhelp to increase discrimination power of DEA
Strassert and Prato [59] presented the balancing andranking method which uses a three-step procedure forderiving an overall complete or partial final order of optionsIn the first step derive an outranking matrix for all options
Journal of Applied Mathematics 13
from the criteria values Considering this matrix it is possibleto show the frequencywithwhich one option is ranked higherthan the other options In the second step by triangularizingthe outranking matrix establish an implicit preordering orprovisional ordering of optionsTheoutrankingmatrix showsthe degree to which there is a complete overall order ofoptions In the third step based on information given in anadvantages-disadvantages table the provisional ordering issubjected to different screening and balancing operations
Chen [60] utilized a nonparametric approach DEA toestimate and rank the efficiency of association rules withmultiple criteria in following steps Proposed postprocessingapproach is as follows
Step 1 Input data for association rule mining
Step 2 Mine association rules by using the a priori algorithmwith minimum support and minimum confidence
Step 3 Determine subjective interestingness measures byfurther considering the domain related knowledge
Step 4 Calculate the preference scores of association rulesdiscovered in Step 2 by using Cook and Kressrsquos DEA model
Step 5 Discriminate the efficient association rules found inStep 3 by using Obata and Ishiirsquos [104] discriminate model
Step 6 Select rules for implementation by considering thereference scores generated in Step 5 and domain relatedknowledge
Jablonsky [61] presented two original models super SBMand AHP for ranking of efficient units in DEA As theauthormentioned thesemodels are based onmultiple criteriadecision-making techniques-goal programming and analytichierarchy process Super SBM model for ranking units is asfollows
min 120579119866
119902= 1 + 119905120574 + (1 minus 119905) (
sum119898
119894=1119904+
1119894
119909119894119902
+
sum119898
119894=1119904minus
2119896
119910119896119902
)
st119899
sum
119895=1119895 = 119902
120582119895119909119894119895+ 119904
minus
1119894minus 119904
+
1119894= 119909
119894119902 i = 1 119898
119899
sum
119895=1119895 = 119902
120582119895119910119896119895+ 119904
minus
2119896minus 119904
+
2119896= 119910
119896119902 119896 = 1 119903
119904+
1119894le 120574 119894 = 1 119898
119904minus
2119896le 120574 119896 = 1 119903
119905 isin 0 1 120582119895ge 0 119878
+
1 119878
+
2ge 0 119878
minus
1 119878
minus
2ge 0
119895 = 1 119899
(62)
Wang and Jiang [62] presented an alternative mixedinteger linear programming models in order to identifythe most efficient units in DEA technique As the authors
mentioned presented models can make full use of input-output information with no need to specify any assuranceregions for input and output weights to avoid zero weights
min119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) minus
119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895)
st119904
sum
119903=1
119906119903119910119903119895minus
119898
sum
119894=1
V119894119909119894119895le 119868
119895 119895 = 1 119899
119899
sum
119895=1
119868119895= 1
119868119895isin 0 1 119895 = 1 119899
119906119903ge
1
(119898 + 119904)max119895119910
119903119895
119903 = 1 119904
V119894ge
1
(119898 + 119904)max119895119909
119894119895
119894 = 1 119898
(63)
Hosseinzadeh Lotfi et al [63] provided an improved three-stage method for ranking alternatives in multiple criteriadecision analysis In the first stage based on the bestand worst weights in the optimistic and pessimistic casesobtain the rank position of each alternative respectively Theobtained weight in the first stage is not unique thus it seemsnecessary to introduce a secondary goal that is used in thesecond stage Finally in the third stage the ranks of thealternatives compute in the optimistic or pessimistic case Itis mentionable that the model proposed in the third stageis a multicriteria decision-making (MCDM) model and itis solved by mixed integer programming As the authorsmentioned and provided in their paper this model can beconverted into an LP problem
The previous model with 119897 = 1 is the original CCR modelNote that DMUs in 119864
1 show the first level efficient frontier119897 = 2 indicates the second level efficient frontier when thefirst level efficient frontier is omittedThe following algorithmfinds these efficient frontiers
Step 1 Set 119897 = 1 and evaluate the entire set of DMUs 1198691 Inthis way the first level efficient DMU 1198641 is identified
Step 2 Exclude the efficient DMUs from future DEA runsand set 119869119897+1 = 119869
119897minus 119864
119897 (If 119869119897+1 = oslash stop)
Step 3 Evaluate the new subset of ldquoinefficientrdquo DMUs 119869119897+1to obtain a new set of efficient DMUs 119864119897+1
Step 4 Let l = l + 1 Go to Step 2
When 119869119897+1 = oslash the algorithm stopsAs the authors proved in this way it is possible to rank
the DMUs in the first efficient frontier based upon theirattractiveness scores and identify the best one
Jahanshahloo et al [65] provided a paper for rankingunits using gradient line As the authors mentioned theadvantage of this model is stability and robustness
supports on 119879119888 the obtained PPS by omitting theDMUunder
assessment in 119879119888 As the authors proved a unit is efficient
iff the optimal objective function of the following model isgreater than zero Consider
1198750= (119883 119884) 119883 = 120572119883
119900 119884 = 120573119884
119900
1198780= (119883 119884) isin 1198750
119883 = 120572119883119900 119884 = 120573119884
119900 120572 ge 0 120573 ge 0
(67)
Intersection of 1198780and efficient surface of
119879119888119888 is a half line
where its equation is (minus119881lowast119879119883
119900)120572 + (119880
lowast119879119884119900)120573 = 0 To rank
DMU119900the length of connecting arc DMU
119900with intersection
point of line and previous ellipse is calculated in (120572 120573) spaceThis intersection is as follows
120572lowast= (
1198702
120572119870
2
120573
1198702
120573+ (119881
lowast119879119883
119900119880
lowast119879119884119900)2119870
2
120572
)
12
120573lowast= (
119881lowast119879119883
119900
119880lowast119879119884119900
)120572lowast
(68)
Now the length of connecting arc DMU119900to the point
corresponding to 120572lowast 120573lowast in (120572 120573) plan is calculated as follows119868 = int
120572lowast
1(1 + 120573)
12119889120572 where
119870120572= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119899
119895=11199092
119894119900
)
12
119870120573= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119904
119903=11199102
119903119900
)
12
(69)
Jahanshahloo et al [66] also provided a paper with theconcept of advantage in data envelopment analysis
In their paper Jahanshahloo et al [67] consideringMonteCarlo method presented a new method of ranking
Step 1 Generate a uniformly distributed sequence of 1198801198952119899
119895=1
on(0 1)
Step 2 Random numbers should be classified into 119873 pairslike (119880
1 119880
1015840
1) (119880
119873 119880
1015840
119873) in a way that each number is used
just one time
Step 3 Compute119883119894= 119886 + 119880
119894(119887 minus 119886) and 119891(119883
119894) gt 119888119880
1015840
119894
Step 4 Estimate the integral 119868 by 120579119868= 119888(119887 minus 119886)(119873
119867119873)
Now consider DMU119900as an efficient unit measure those
units that are dominated DMU119900 As for dome DMUs this
would be unbounded thus the authors for each unitbounded the region Then for DMU
119900if (minus119883
119900 119884
119900) ge (minus119883 119884)
then (minus119883 119884) is in the RED of DMU119900 Now by using 119881
119901=
119881lowast(119873
119867119873) all the hits (the above condition) can be counted
where 119881lowast is the measure of the whole region JahanshahlooandAfzalinejad [68] presented amethod based on distance of
Journal of Applied Mathematics 15
the unit under evaluation to the full inefficient frontier Theypresented two models radian and nonradial
min 120601
st119899
sum
119895=1
120582119895119909119894119895= 120601119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
120582119895ge 0 119895 = 1 119898
max119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903
st sum
119895isin119869minus119887
120582119895119909119894119895minus 119904
minus
119894= 119909
119894119900 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895+ 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
119904minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(70)
Amirteimoori [69] based on the same idea considered bothefficient and antiefficient frontiers for efficiency analysis andranking units
Kao [70] mentioned that determining the weights ofindividual criteria in multiple criteria decision analysis in away that all alternatives can be compared according to theaggregate performance of all criteria is of great importanceAs Kao noted this problem relates to search for alternativeswith a shorter and longer distance respectively to the idealand anti-ideal units He proposed a measure that consideredthe calculation of the relative position of an alternativebetween the ideal and anti-ideal for finding an appropriaterankings
Khodabakhshi and Aryavash [71] presented a method forranking all units using DEA concept
First Compute the minimum and maximum efficiencyvalues of eachDMU in regard to this assumption that the sumof efficiency values of all DMUs equals to 1
Second determine the rank of each DMU in relation to acombination of itsminimumandmaximumefficiency values
Zerafat Angiz et al [72] proposed a technique in orderto aggregate the opinions of experts in voting system Asthe authors mentioned the presented method uses fuzzyconcept and it is computationally efficient and can fullyrank alternatives At first number of votes given to a rankposition was grouped to construct fuzzy numbers and thenthe artificial ideal alternative introduced Furthermore byperforming DEA the efficiency measure of alternatives was
obtained considering artificial ideal alternative comparedby each of the alternatives pair by pair Thus alternativesare ranked in accordance with their efficiency scores Ifthis method cannot completely rank alternatives weightrestrictions based on fuzzy concept are imposed into theanalysis
10 Different Applications in Ranking Units
As discussed formerly there exist a variety of ranking meth-ods and applications in theliterature Nowadays DEAmodelsare widely used in different areas for efficiency evaluationbenchmarking and target setting ranking entities and soforth Ranking units as one of the important issues in DEAhas been performed in different areas Jahanshahloo et al [42]ranked cities in Iran to find the best place for creating a datafactory Hosseinzadeh Lotfi et al [28] used a new methodfor ranking in order to find the best place for power plantlocation Ali andNakosteen [80] Amirteimoori et al [37] andAlirezaee and Afsharian [24] Soltanifar and HosseinzadehLotfi [105] Zerafat Angiz et al [72] Hosseinzadeh Lotfiet al [28] Jahanshahloo [50] Chen and Deng [52] andJablonsky [61] used different ranking methods in bankingsystem Sadjadi [40] ranked provincial gas companies in IranMehrabian et al [32] Li et al [39] Orkcu and Bal [12] andWu et al [19] ranked different departments of universitiesJahanshahloo et al [35] Jahanshahloo and Afzalinejad [68]utilized the presented models in ranking 28 Chinese citiesJahanshahloo at al [35] provided an application to burdensharing amongst NATOmember nations Jahanshahloo et al[14] utilized the presented model for ranking nursery homesContreras [18] andWang et al [23] used the provided rankingtechniques in ranking candidates Lu and Lo [51] applied theirmethod on an application to financial holding companiesHosseinzadeh Lotfi et al [63] consider an empirical exampleinwhich voters are asked to rank twoout of seven alternativesWang and Jiang [62] utilized the provided method in facilitylayout design in manufacturing systems and performanceevaluation of 30OECD countries Chen [60] used an exampleof market basket data in order to illustrate the providedapproach
11 Application
In this section some of the reviewed models are applied onthe example used in Soltanifar and Hosseinzadeh Lotfi [105]Consider twenty commercial banks of Iran with input-outputdata tabulated in Table 1 and summarized as follows Also theresults of CCRmodel are listed in this table As it can be seenseven units are efficient DMUS 14 7 12 15 17 and 20 Inputsare staff computer terminal and spaceOutputs are depositsloans granted and charge
Consider some of the important ranking methods inliterature as follows
RM1 AP model [31] is based upon the idea of leaveunit evaluation out and measuring the distance of theunit under evaluation from the production possibilityset constructed by the remaining DMUs
RM2MAJmodel [32] presented for ranking efficientwhich is always stable but might be infeasible in somecasesRM3 Modified MAJ model [38] overcomes theproblem which might occurr in MAJ modelRM4 A new model based on the idea of alterationsin the reference set of the inefficient units [50]RM5 A model presented by Li et al [39] whichis a super-efficiency method that does not have thesuffering in previous methodsRM6 Slack-basedmodel [33] is based upon the inputand output variables at the same timeRM7 SA DEAmodel [49] overcomes the problem ofinfeasibility which existed in AP modelRM8 Cross-efficiency [10] is provided based onusing weights of each unit under evaluation in opti-mality for other unitsRM9 A model based on finding common set ofweights [25] which determine the common set ofweights for DMUs and ranked DMUs based this ideaRM10 A model based on finding common set ofweights [22 67] for ranking efficient unitsRM11 L1-norm model [34 35 65 66] the idea isbased upon the leave-one-out efficient unit and l1-norm which is always feasible and stableRM12 119871
infin-norm model Rezai Balf et al [46] pro-
vided a method with more ability over other existingmethods based on Tchebycheff normRM13 An enhanced Russel measure of super-efficiency model for ranking units [44]RM14 A rankingmodel which considers the distanceof unit from the full inefficient frontier [38]RM15 Amodified super-efficiencymodel [45] whichovercomes the infeasibility that may happen in prob-lem This model is based on simultaneous projectionof input output
In accordance with properties of different ranking models inorder to rank efficient units consider Table 2
1199011 Feasibility
1199012 Ranking extreme efficient units
1199013 Complexity in computation
1199014 Instability
1199015 Absence of multiple optimal solution
1199016 Dependency to 120579 and slacks
1199017 Dependency to the number of efficient and ineffi-
cient units
In Tables 3 and 4 the rank of efficient units consideringthe above mentioned methods is listed Note that ED showsefficient DMUs (ED) and RM
119895 (119895 = 1 15) are those
explained previously
12 Conclusion
In this paper the DEA ranking was reviewed and classifiedinto seven general groups In the first group those papersbased on a cross-efficiencymatrix were reviewed In this fieldDMUs have been evaluated by self- and peer pressure Thesecond group of papers is based on those papers lookingfor optimal weights in DEA analysis The third one is thesuper-efficiency method By omitting the under evaluationunit and constructing a new frontier by the remaining unitsthe unit under evaluation can get an equal or greater scoreThe fourth group is based on benchmarking idea In thisclass the effect of an efficient unit considered as a targetfor inefficient units is investigated This idea is very usefulfor the managers in decision making Another class thefifth one involves the application of multivariate statisticaltools The sixth section discusses the ranking methods basedon multicriteria decision-making (MCDM) methodologiesand DEA The final section includes some various rankingmethods presented in the literature Finally in an applicationthe result of some of the above-mentioned ranking methodsis presented
References
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[2] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
18 Journal of Applied Mathematics
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
As the authors noted there exist alternative optimal solutionsWu et al [13] described the main suffering of cross-
efficiency when the ultimate average cross-efficiency utilizedfor ranking units For removing this shortcoming they elim-inated the assumption of average and utilized the Shannonentropy in order to obtain the weights for ultimate cross-efficiency scores Jahanshahloo et al [14] provided a methodfor selecting symmetric weights to be used in DEA cross-efficacy
Step 1 Efficiency of DMUs needs to be computed
Step 2 Choose the solutions in accordance with the sec-ondary goal for each DMU as follows
min 119890119879119885119900119890
st 119906119900119910119900= 1
V119900119883
119900= 120579
119900
119906119900119884 minus V
119900119883 le 0
119906119900119894119910119900119894minus 119906
119900119895119910119900119895le 119911
119900119894119895 forall119894 119895
119906119900119895119910119900119895minus 119906
119900119894119910119900119894le 119911
119900119894119895 forall119894 119895
119906119900 V
119900ge 119889
(11)
Step 3 The cross-efficiency for any DMU119895 using the weights
that DMU119900has chosen in the previous model is then used as
follows
120579119900119895=
119906lowast
119900119884119895
Vlowast119900119883
119895
(12)
Wang et al [15] provided a cross-efficiency evaluation basedon ideal and anti-ideal units for ranking As the authorsmentioned a DMU could choose a unique set of input andoutput weights to make its distance from ideal DMU assmall as possible or the distance from anti-ideal DMU aslarge as possible or the both Thus according to this ideathey proposed the following procedure for cross-efficiencyevaluation
Model 1 Minimization of the distance from ideal DMU
Model 2 Maximization of the distance from anti-ideal DMU
Model 3 Maximization of the distance between ideal DMUand anti-ideal DMU
Model 4 Maximization of the relative closeness
The authors mentioned that the bigger the relative close-ness of a DMU is the better performance it will have
In a paper Ramon et al [16] selected the profiles ofweights used in cross-efficiency assessment As the authorssaid they tried to prevent unrealistic weighting They havediscussed the zero weights as they excluded variables fromthe evaluation In the calculation of cross-efficiency scoresthey proposed to ignore the profiles of those weights of theunit under evaluation that among their alternate optimalsolutions cannot choose nonzeroweightsThey also proposedthe ldquopeer-restrictedrdquo cross-efficiency evaluation where theunits assessed in a peer evaluation which means profiles ofweights of some inefficient units are not considered Finallythe presented approach extended to derive a common set ofweights Guo and Wu [17] provided a complete ranking ofDMUs with undesirable outputs using restriction in DEAAs the author mentioned this model is presented to realizea unique ranking of units by ldquomaximal balanced indexrdquoaccording to the obtained optimal shadow prices
119896+1ge 119889 (119896 120576) 119896 = 1 119899 minus 1
120583119899ge 119889 (119899 120576)
(19)
where 120579119895119896obtained from Step 5 and 120573lowast
119895is the optimal solution
of the following model Finally the DMUs are ranked basedon their 119911lowast
119895= sum
119899
119896=1120583lowast
119896120579119895119896values
Washio and Yamada [21] discussed that in real casesfinding the best ranking is more significant than acquiringthe most advantage weight and maximizing the efficiencyThus they presented a model called rank-based measure(RBM) for evaluating units from different viewpoint Thusthey suggested a method for acquiring those weight resultedfrom the best ranking as long as calculating those weightthat maximizes the efficiency score Finally they applied thepresented model to the cross-efficiency assessment
4 Ranking Techniques Based on FindingOptimal Weights in DEA Analysis
Jahanshahloo et al [22] gave a note on some of the DEAmodels for complete ranking using common set of weightsThey proved that by solving only one problem it is possibleto determine the common set of weights
where 119860 is the set of efficient units of the following model
max
sum119904
119903=11199061199031199101199031+ 119906
0
sum119898
119894=1V1198941199091198941
sdot sdot sdot
sum119904
119903=1119906119903119910119903119899+ 119906
0
sum119898
119894=1V119894119909119894119899
stsum
119904
119903=1119906119903119910119903119895+ 119906
0
sum119898
119894=1V119894119909119894119895
le 1 119895 = 1 119899
119880 119881 120578 ge 120576 1199060free
(21)
Note that DMUs can be ranked based on the evaluation oftheir efficiencies
Wang et al [23] provided an aggregating preference rank-ing In this paper use of ordered weighted averaging (OWA)operator is proposed for aggregating preference rankings Let
119908119895be the relative importance weight given to the jth ranking
place and V119894119895the vote candidate i receives in the jth ranking
place The total score of each candidate is defined as
119911119894=
119898
sum
119894=1
V119894119895119882
119895 119894 = 1 119898 (22)
Alirezaee and Afsharian [24] discussed multiplier model inwhich the variables are considered as shadow prices note thatsum
119894=1V119894119909119894119895are total revenue and cost of DMU
119895
which are considered in optimization problem The authorsclaimed that sum119904
119903=1119906119903119910119903119895minus sum
119898
119894=1V119894119909119894119895le 0 119895 = 1 119899 is the
profit restriction for DMU119895 If 119865(119909 119910) = 0 is considered to be
the efficient production function then
sum
119894
120597119865
120597119909119894
119909119894+sum
119895
120597119865
120597119910119895
119910119895= 0 (23)
They mentioned that the connected profit of the DMU iszero when shadow prices are derived from the technologyand called this situation as balance situation As the authorsmentioned therefore in the case that DMU
1is efficient but
DMU2is inefficient or the efficiency score of both DMUs is
the same and it obtains more negative quantity in balanceindex it can be concluded that DMU
1has a better rank than
DMU2
Liu and Hsuan Peng [25] in their paper proposed amethod for determining the common set of weights forranking units In common weights analysis methodologythey provided the following model
Δlowast= min sum
119895isin119864
Δ119900
119895+ Δ
119894
119895
stsum
119904
119903=1119910119903119895119880119903+ Δ
119900
119895
sum119898
119894=1119909119894119895119881119894minus Δ
119894
119895
= 1 119895 isin 119864
Δ119900
119895 Δ
119894
119895ge 0 forall119895 isin 119864
119880119903ge 120576 119903 = 1 119904
119881119894ge 120576 119894 = 1 119898
(24)
The mentioned ratio form the linear equationsWang et al [26] proposed a paper for ranking decision-
making units by imposing a minimum weight restrictionin DEA The authors noted that using data envelopmentanalysis it is not possible to distinguish between DEA effi-cient units Thus they presented a method for ranking unitsusing imposing minimum weight restriction for the input-output data As they mentioned these weights restrictionsare decided by a decision maker (DM) or an assessor asregards the solutions to a series of LP models considered fordetermining a maximin weight for each efficient DMU
Hatefi and Torabi [27] proposed a common weightmulti criteria decision analysis (MCDA)-data envelopmentanalysis (DEA) for constructing composite indicators (CIs)As the authors proved the presented model can discrimi-nate between efficient units The obtained common weightshave discriminating power more than those obtained from
Journal of Applied Mathematics 7
previous models Finally they studied the robustness anddiscriminating power of the proposed method by Spearmanrsquosrank correlation coefficient
Hosseinzadeh Lotfi et al [28] proposed one DEA rankingmodel based on applying aggregate units In doing so artificialunits called aggregate units are defined as follows Theaggregate unit is shown by DMU
119886
119909119901
119894119886= sum
119896isin119877119901
119909119894119896 119910
119901
119903119886= sum
119896isin119877119901
119910119903119896
119894 = 1 119898 119903 = 1 119904
(25)
where 119877119901= 119895 | DMU
119895isin 119864
119901 119864
119901= 119864DMU
119901 Note that 119864
is the set of efficient unitsFirst it is tried to maximize the efficiency score of the
DMU119886and then to maximize the efficiency score of the
DMU119901
119886 For resolving the existence of alternative solutions
the authors presented an approach comprising (119898+119904) simplelinear problems to achieve the most appropriate optimalsolutions among all alternative optimal solutions Finallythey proposed theRI index for ranking all efficientDMUs Let119880119886 119881
119875119886 and119880119901
119886 119881
119901
119886be the optimal solutions of the multiplier
model for assentDMU119886andDMU119901
119886 Consider 120578
119886= sum
119904
119903=1119906119903119886minus
sum119898
119894=1V119894119886 120578
119901
119886= sum
119904
119903=1119906119901
119903119886minus sum
119898
119894=1V119901119894119886 thus RI
119901= 120578
119900
119886minus 120578
119886
Wang et al [29] presented two nonlinear regressionmodels for deriving common set of weights for fully rankunits
min 119911 =
119899
sum
119895=1
(120579lowast
119895minus
sum119904
119903=1119906119903119910119903119895
sum119898
119894=1V119894119909119894119895
)
2
st 119880 119881 ge 0
min119899
sum
119895=1
(
119904
sum
119903=1
119906119903119910119903119895minus 120579
lowast
119895
119898
sum
119894=1
V119894119909119894119895)
2
st119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895) +
119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) = 119899
119880 119881 ge 0
(26)
Ramon et al [30] aimed at deriving a common set of weightsfor ranking units As the authors mentioned the idea is basedupon minimization of the deviations of the common weightsfrom the nonzero weights obtained fromDEA Furthermoreseveral norms are used for measuring such differences
5 Super-Efficiency Ranking Techniques
Super efficiency models introduced in DEA technique arebased upon the idea of leave one out and assessing this unittrough the remanding units
Andersen and Petersen [31] introduced a model for rank-ing efficient units The proposed model is as follows
Although this idea is useful for further discriminatingefficient units it has been shown in the literature that itmay be infeasible and nonstable Thrall [7] mentioned theinfeasibility of super-efficiency CCRmodel Also a conditionunder which infeasibility occurred in super-efficiency DEAmodels is mentioned by Zhu [95] Seiford and Zhu [96] andDula and Hickman [97]
Hashimoto [98] provided a model based on the idea ofone leave out and assurance region for ranking units
Mehrabian et al [32] presented a complete ranking forefficiency units in DEA context As the authors mentionedthis model does not have difficulties of AP model
min 119908119901+ 1
st119899
sum
119895=1 119895 = 119901
120582119895119909119894119895le 119909
119894119901+ 119908
1199011 119894 = 1 119898
119899
sum
119895=1 119895 = 119901
120582119895119910119903119895ge 119910
119903119901 119903 = 1 119904
120582119895ge 0 119895 = 119901
(28)
With this method it is not possible to rank nonextremeefficient units
Tone [33] presented super efficiency of SBM model Thismodel has the advantages of nonradial models and it isalways feasible and stable
minsum
119898
119894=1119909119894119909
119894119900
sum119904
119903=1119910119903119910
119903119900
st119899
sum
119895=1119895 = 119900
120582119895119909119894119895le 119909
119894 119894 = 1 119898
119899
sum
119895=1119895 = 119900
120582119895119910119903119895ge 119910
119903 119903 = 1 119904
119909119894ge 119909
119894119900 0 le 119910
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119900
(29)
8 Journal of Applied Mathematics
Jahanshahloo et al [34] added some ratio constraints to themultiplier form of AP model and introduced a new methodfor ranking DMUs
In their paper Chen and Sherman [36] presented a non-radial super-efficiency method and discussed the advantageof it They verified that this model is invariant to units ofinputoutput measurement Let 119869119900 = 119869DMU
119900
Step 1 Solve the followingmodel to find the extreme efficientunits in 119869119900
This method cannot rank nonextreme efficient units
Jahanshahloo et al [38] presented modified MAJ modelfor ranking efficiency units in DEA technique
min 119908119901+ 1
st119899
sum
119895=1119895 = 119901
120582119895
119909119894119895
119872119894
le
119909119894119901
119872119894
+ 1199081199011 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895ge 119910
119903119901 119903 = 1 119904
120582119895ge 0 119895 = 119901
(39)
where 119872119894= Max 119909
119894119895| DMU
119895is efficient It cannot rank
nonextreme efficient unitsLi et al [39] presented a new method for ranking which
does not have difficulties of earlier methods The presentedmodel is always feasible and stable
min 1 +
1
119898
119898
sum
119894=1
119904+
1198942
119877minus
119894
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895+ 119904
minus
1198941minus 119904
+
1198942= 119909
119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895minus 119904
+
119903= 119910
119903119901 119903 = 1 119904
119904minus
1198942ge 0 119904
minus
1198941ge 0 119904
+
119903ge 0 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 119901
(40)
In this case extreme efficient units cannot be ranked
In a paper Khodabakhshi [99] addresses super efficiencyon improved outputs He mentioned that as AP modelmay be infeasible under variable returns to scale technol-ogy using the presented model gives a complete rankingwhen getting an input combination for improving outputs issuitable
Sadjadi et al [40] presented a robust super-efficiencyDEA for ranking efficient units They noted that as inmost of the times exact data do not exist and the sto-chastic super-efficiency model presented in their paperincorporates the robust counterpart of super-efficiencyDEA
where 119883119884 are input-output data This model does notrank nonextreme efficient units It may be unstable andinfeasible
Gholam Abri et al [41] proposed a model for rankingefficient units They used representation theory and rep-resented the DMU under assessment as a convex combi-nation of extreme efficient units As the authors noted itis expected that the performance of DMU
119900is the same
as the performance of convex combination of extremeefficient units Thus it is possible to represent DMU
119900as
follows
(119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119899
sum
119895=1
120582119895= 1 119895 = 1 119904 (42)
As regards representation theorem this system has119898 + 119904 minus 1
constraints and 119904 variables (1205821 120582
119904) If this system has a
10 Journal of Applied Mathematics
unique solution we will have 120579lowast119900= sum
119899
119895=1= 1 120582
lowast
119895120579119895 otherwise
two models should be considered
min119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
max119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
(43)
Consider 1205791and 120579
2as the optimal solution of the above-
mentioned models respectively If 1205791= 120579
2 this is the same
as what has been mentioned previously If 1205791lt 120579
2 then
mentionedmodels provide an interval which helps rank unitsfrom the worst to the best Sometimes it will obtain theranking score with a bounded interval [120579
1 120579
2]
Jahanshahloo et al [42] presented models for rankingefficient units The presented models are somehow a modifi-cation of cross-efficiency model that overcomes the difficultyof alternative optimal weightsThe authors in their paperwithregard to the changes and also utilizing TOPSIS techniquepresented a new super-efficientmethod for ranking unitsThepresented model is as follows
120579119894119895=
119880119905119884119894
119881119905119883
119894
|
119880119905119884119897
119881119905119883
119897
le 1
119880119905119884119895
119881119905119883
119895
= 120579119895119895
119897 = 1 119899 119897 = 119894 119880 ge 0 119881 ge 0
(44)
where 120579119895119895is the efficiency score of DMU
119869using correspond-
ing weights Also 120579119894119895is the efficiency of DMU
119894using optimal
weights of DMU119895 Noura et al [43] provided a method for
ranking efficient units based on this idea that more effectiveand useful units in society should have better rank
Step 1 For each efficient unit choose the lower and upperlimit for each inputs and outputs Let 119864 be the set of efficientunits
where 120585 is representative of a small and nonzero number usedfor not dividing by zero Now consider119863minus119895 as follows whichshows that more successful DMU
119895will be if the larger value
of the119863119895is
119863119895= sum
119894isin119868
119889119894119895+ sum
119903isin119877
119889119903119895 (48)
where 119868 = 119863+
119894cup 119863
minus
119894 119877 = 119863
+
119903cup 119863
minus
119903
Ashrafi et al [44] introduced an enhanced Russell mea-sure of super efficiency for ranking efficient units inDEAThelinear counterpart of the proposed model is as follows
119900 le 120573 le V119903ge 120573 119903 = 1 119904
(49)
It cannot rank nonextreme efficient units
Journal of Applied Mathematics 11
Chen et al [45] proposed a modified super-efficiencymethod for ranking units based on simultaneous input-output projectionThe presentedmodel overcomes the infea-sibility problem
1199011= min
120579119904119903
119900
120601119904119903
119900
st sum
119895=1
119895 = 119900
120582119895119909119894119895le 120579
119904119903
119900119909119894119900 119894 = 1 119898
sum
119895=1
119895 = 119900
120582119895119910119903119895ge 120601
119904119903
119900119910119903119900 119903 = 1 119904
sum
119895=1
119895 = 119900
120582119895= 1
0 lt 120579119904119903
119900le 1 120601
119904119903
119900ge 1 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(50)
Rezai Balf et al [46] provide a model for ranking units basedon Tchebycheff norm As proved that this model is alwaysfeasible and stable it seems to have superiority over othermodels
max 119881119901
st 119881119901ge
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901 119894 = 1 119898
119881119901ge 119910
119903119901minus
119899
sum
119895=1
119895 = 119900
120582119895119910119903119895 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(51)
where
119881119901= Max
(
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901)119894 = 1 119898
(119910119903119901minus
119899
sum
119895=1
119895 = 119901
120582119895119910119903119895)119903 = 1 119904
(52)
Chen et al [47] for overcoming the infeasibility problem thatoccurred in variable returns to scale super-efficiency DEAmodel according to a directional distance function developedNerlove-Luenberger (N-L) measure of super-efficiency
6 Benchmarking Ranking Techniques
Sueyoshi et al [49] proposes a ldquobenchmark approachrdquo forbaseball evaluation This method is the combination of DEA
and (Offensive earned-run average) OERA As the authorsnoted using this method it is possible to select best unitsand also their ranking orders They mentioned that usingonly DEAmay result in a shortcoming in assessment as manyefficient units can be identifiedThus the authors used slack-adjusted DEA model and OERA to overcome this difficulty
Jahanshahloo et al [50] presented a new model forranking DMUs based on alteration in reference set The ideais based on this fact that efficient units can be the target unitfor inefficient units
min 120597119886119887
= 120579 minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st sum
119895isin119869minus119887
120582119895119909119894119895+ 119904
minus
119894minus 120579119909
119894119886 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
120579 free 119904minus119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 isin 119869 minus 119887
(53)
Finally the ranking order for each efficient unit b can becomputing by
119887= sum
119886isin119869119899
120597119886119887
119899
(54)
Note that this method cannot rank nonextreme efficientunits
Lu and Lo [51] provided an interactive benchmark modelfor ranking units The idea is based upon considering a fixedunit as a benchmark and calculating the efficiency of otherunits pair by pair to this unitThis procedure continueswhenall units are accounted for as a benchmark unit ConsiderDMU
119900as a unit under evaluation andDMU
119887as a benchmark
Assume
119909119894= 119909
119894119900(1 + 120601
119894) 119910
119903= 119910
119903119900(1 minus 120593
119903)
min 120579lowast119887
119900=
1 + (1119898)sum119898
119894=1120601119894
1 minus (1119904)sum119904
119903=1120593119903
st 120582119887119909119894119887minus 119909
119894119900120601119894le 119909
119894119900 119894 = 1 119898
120582119887119910119903119887minus 119910
119903119900120593119903ge 119910
119903119900 119903 = 1 119904
120601119894ge 119900 120593
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(55)
Then using (sum119899
119887=1120579119887lowast
119900)119899 119900 = 1 119899 the efficiency of
benchmark DMU119900can be obtained Using the following
index which indicates the increment in efficiency of a unitby moving from peer appraisal to self-appraisal it is nowpossible to rank units Note that the less the magnitude of
12 Journal of Applied Mathematics
this index is the better rank for corresponding unit will beobtained
FPIIBM119870
=
(TEBCC119896
minus STDIBM119896
)
(STDIBM119896
)
(56)
where TEBCC119896
and STDIBM119896
are respectively efficiency in BCCmodel and normalization of TEIBM of DMU
119896 Chen [52]
provided a paper for ranking efficient and inefficient unitsin DEA They noted that the evaluation of efficient unitsis based upon the alterations in efficiency of all inefficientunits by omitting in reference set At first solve the followingmodel which measures the efficiency of DMU
119886when DMU
119887
is eliminated from the reference set
min 120588119886119887
= 120579119904
119886minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1119895 = 119887
120582119895119909119894119895+ 119904
minus
119894= 120579
119904
119886119909119894119886 119894 = 1 119898
119899
sum
119895=1119895 = 119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
119899
sum
119895=1119895 = 119887
= 1
120579119904
119886ge 0 119904
minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119887
(57)
Then again solve the previous model this time for calculat-ing the efficiency score of each inefficient unit when each ofefficient units in turn is eliminated from the reference set 120578lowast
119886
and calculate the efficiency change
120591119886119887
= 120588lowast
119886119887minus 120578
lowast
119886 (58)
Then calculate the following index called as ldquoMCDErdquo indexfor each efficient and inefficient unit
119864119886= sum
119887isin119881119864
119882lowast
119887120588lowast
119886119887 119864
119887= sum
119887isin119881119868
119882lowast
119887120588lowast
119886119887 (59)
Now in accordance with themagnitude of the acquired indexit is possible to rank unitsThose units with higher score havebetter ranking order
7 Ranking Techniques by MultivariateStatistics in the DEA
As described in DEA literature in DEA technique frontier istaken into consideration rather than central tendency consid-ered in regression analysis DEA technique considers that anenvelope encompasses through all the observations as tight aspossible and does not try to fit regression planes in center ofdata In DEA methodology each unit is considered initiallyand compared to the efficient frontier but in regressionanalysis a procedure is considered in which a single functionfits to the data DEA uses different weights for differentunits but does not let the units use weights of other units
Canonical correlation analysis as Friedman and Sinuany-Stern [53] noted can be used for ranking units This methodis somehow the extension of regression analysis Adler et al[89] The aim in canonical correlation analysis is to find asingle vector commonweight for the inputs and outputs of allunits Consider119885
119895119882
119895as the composite input and output and
as corresponding weights respectively The presentedmodel by Tatsuoka and Lohnes [100] is as follows
max 119903119911119908
=
119878119909119910119880
(119878119909119909119881) (119878
119910119910119880)
(12)
st 119909119909119881 = 1
119910119910119880 = 1
(60)
where 119878119909119909 119878
119910119910 and 119878
119909119910are respectively defined as the matri-
ces of the sums of squares and sums of products of thevariables
Friedman and Sinuany-Stern [53] while defining the ratioof linear combinations of the inputs and outputs 119879
119895=
119882119895119885
119895used canonical correlation analysis As they noted that
scaling ratio 119879119895of the canonical correlation analysisDEA is
unboundedSinuany-Stern et al [101] used linear discriminant analy-
sis for ranking units They defined
119863119895=
119904
sum
119903=1
119906119903119910119903119895+
119898
sum
119894=1
V119903(minus119909
119894119895) (61)
DMU119895is said to be efficient if 119863
119895gt 119863
119888where 119863
119888is a critical
value based on themidpoint of themeans of the discriminantfunction value of the two groups Morrison [102] The largerthe amount of119863
119895is the better rank DMU
119895will have
Friedman and Sinuany-Stern [53] noted that as cross-efficiencyDEA canonical correlation analysisDEA and dis-criminant analysisDEA ranking orders may vary from eachother thus it seems necessary to introduce the combinedranking (CODEA) Combined ranking for each unit con-sidered all the ranks obtained from the above-mentionedrankings Moreover statistical tests are one for goodness offit between DEA and a specific ranking and the other fortesting correlation between variety of ranking orders Siegeland Castellan [103]
8 Ranking with MulticriteriaDecision-Making (MCDM) Methodologiesand DEA
Li and Reeves [56] for increasing discrimination power ofDEA presented amultiple-objective linear program (MOLP)As the authors mentioned using minimax and minsumefficiency in addition to the standard DEA objective functionhelp to increase discrimination power of DEA
Strassert and Prato [59] presented the balancing andranking method which uses a three-step procedure forderiving an overall complete or partial final order of optionsIn the first step derive an outranking matrix for all options
Journal of Applied Mathematics 13
from the criteria values Considering this matrix it is possibleto show the frequencywithwhich one option is ranked higherthan the other options In the second step by triangularizingthe outranking matrix establish an implicit preordering orprovisional ordering of optionsTheoutrankingmatrix showsthe degree to which there is a complete overall order ofoptions In the third step based on information given in anadvantages-disadvantages table the provisional ordering issubjected to different screening and balancing operations
Chen [60] utilized a nonparametric approach DEA toestimate and rank the efficiency of association rules withmultiple criteria in following steps Proposed postprocessingapproach is as follows
Step 1 Input data for association rule mining
Step 2 Mine association rules by using the a priori algorithmwith minimum support and minimum confidence
Step 3 Determine subjective interestingness measures byfurther considering the domain related knowledge
Step 4 Calculate the preference scores of association rulesdiscovered in Step 2 by using Cook and Kressrsquos DEA model
Step 5 Discriminate the efficient association rules found inStep 3 by using Obata and Ishiirsquos [104] discriminate model
Step 6 Select rules for implementation by considering thereference scores generated in Step 5 and domain relatedknowledge
Jablonsky [61] presented two original models super SBMand AHP for ranking of efficient units in DEA As theauthormentioned thesemodels are based onmultiple criteriadecision-making techniques-goal programming and analytichierarchy process Super SBM model for ranking units is asfollows
min 120579119866
119902= 1 + 119905120574 + (1 minus 119905) (
sum119898
119894=1119904+
1119894
119909119894119902
+
sum119898
119894=1119904minus
2119896
119910119896119902
)
st119899
sum
119895=1119895 = 119902
120582119895119909119894119895+ 119904
minus
1119894minus 119904
+
1119894= 119909
119894119902 i = 1 119898
119899
sum
119895=1119895 = 119902
120582119895119910119896119895+ 119904
minus
2119896minus 119904
+
2119896= 119910
119896119902 119896 = 1 119903
119904+
1119894le 120574 119894 = 1 119898
119904minus
2119896le 120574 119896 = 1 119903
119905 isin 0 1 120582119895ge 0 119878
+
1 119878
+
2ge 0 119878
minus
1 119878
minus
2ge 0
119895 = 1 119899
(62)
Wang and Jiang [62] presented an alternative mixedinteger linear programming models in order to identifythe most efficient units in DEA technique As the authors
mentioned presented models can make full use of input-output information with no need to specify any assuranceregions for input and output weights to avoid zero weights
min119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) minus
119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895)
st119904
sum
119903=1
119906119903119910119903119895minus
119898
sum
119894=1
V119894119909119894119895le 119868
119895 119895 = 1 119899
119899
sum
119895=1
119868119895= 1
119868119895isin 0 1 119895 = 1 119899
119906119903ge
1
(119898 + 119904)max119895119910
119903119895
119903 = 1 119904
V119894ge
1
(119898 + 119904)max119895119909
119894119895
119894 = 1 119898
(63)
Hosseinzadeh Lotfi et al [63] provided an improved three-stage method for ranking alternatives in multiple criteriadecision analysis In the first stage based on the bestand worst weights in the optimistic and pessimistic casesobtain the rank position of each alternative respectively Theobtained weight in the first stage is not unique thus it seemsnecessary to introduce a secondary goal that is used in thesecond stage Finally in the third stage the ranks of thealternatives compute in the optimistic or pessimistic case Itis mentionable that the model proposed in the third stageis a multicriteria decision-making (MCDM) model and itis solved by mixed integer programming As the authorsmentioned and provided in their paper this model can beconverted into an LP problem
The previous model with 119897 = 1 is the original CCR modelNote that DMUs in 119864
1 show the first level efficient frontier119897 = 2 indicates the second level efficient frontier when thefirst level efficient frontier is omittedThe following algorithmfinds these efficient frontiers
Step 1 Set 119897 = 1 and evaluate the entire set of DMUs 1198691 Inthis way the first level efficient DMU 1198641 is identified
Step 2 Exclude the efficient DMUs from future DEA runsand set 119869119897+1 = 119869
119897minus 119864
119897 (If 119869119897+1 = oslash stop)
Step 3 Evaluate the new subset of ldquoinefficientrdquo DMUs 119869119897+1to obtain a new set of efficient DMUs 119864119897+1
Step 4 Let l = l + 1 Go to Step 2
When 119869119897+1 = oslash the algorithm stopsAs the authors proved in this way it is possible to rank
the DMUs in the first efficient frontier based upon theirattractiveness scores and identify the best one
Jahanshahloo et al [65] provided a paper for rankingunits using gradient line As the authors mentioned theadvantage of this model is stability and robustness
supports on 119879119888 the obtained PPS by omitting theDMUunder
assessment in 119879119888 As the authors proved a unit is efficient
iff the optimal objective function of the following model isgreater than zero Consider
1198750= (119883 119884) 119883 = 120572119883
119900 119884 = 120573119884
119900
1198780= (119883 119884) isin 1198750
119883 = 120572119883119900 119884 = 120573119884
119900 120572 ge 0 120573 ge 0
(67)
Intersection of 1198780and efficient surface of
119879119888119888 is a half line
where its equation is (minus119881lowast119879119883
119900)120572 + (119880
lowast119879119884119900)120573 = 0 To rank
DMU119900the length of connecting arc DMU
119900with intersection
point of line and previous ellipse is calculated in (120572 120573) spaceThis intersection is as follows
120572lowast= (
1198702
120572119870
2
120573
1198702
120573+ (119881
lowast119879119883
119900119880
lowast119879119884119900)2119870
2
120572
)
12
120573lowast= (
119881lowast119879119883
119900
119880lowast119879119884119900
)120572lowast
(68)
Now the length of connecting arc DMU119900to the point
corresponding to 120572lowast 120573lowast in (120572 120573) plan is calculated as follows119868 = int
120572lowast
1(1 + 120573)
12119889120572 where
119870120572= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119899
119895=11199092
119894119900
)
12
119870120573= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119904
119903=11199102
119903119900
)
12
(69)
Jahanshahloo et al [66] also provided a paper with theconcept of advantage in data envelopment analysis
In their paper Jahanshahloo et al [67] consideringMonteCarlo method presented a new method of ranking
Step 1 Generate a uniformly distributed sequence of 1198801198952119899
119895=1
on(0 1)
Step 2 Random numbers should be classified into 119873 pairslike (119880
1 119880
1015840
1) (119880
119873 119880
1015840
119873) in a way that each number is used
just one time
Step 3 Compute119883119894= 119886 + 119880
119894(119887 minus 119886) and 119891(119883
119894) gt 119888119880
1015840
119894
Step 4 Estimate the integral 119868 by 120579119868= 119888(119887 minus 119886)(119873
119867119873)
Now consider DMU119900as an efficient unit measure those
units that are dominated DMU119900 As for dome DMUs this
would be unbounded thus the authors for each unitbounded the region Then for DMU
119900if (minus119883
119900 119884
119900) ge (minus119883 119884)
then (minus119883 119884) is in the RED of DMU119900 Now by using 119881
119901=
119881lowast(119873
119867119873) all the hits (the above condition) can be counted
where 119881lowast is the measure of the whole region JahanshahlooandAfzalinejad [68] presented amethod based on distance of
Journal of Applied Mathematics 15
the unit under evaluation to the full inefficient frontier Theypresented two models radian and nonradial
min 120601
st119899
sum
119895=1
120582119895119909119894119895= 120601119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
120582119895ge 0 119895 = 1 119898
max119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903
st sum
119895isin119869minus119887
120582119895119909119894119895minus 119904
minus
119894= 119909
119894119900 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895+ 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
119904minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(70)
Amirteimoori [69] based on the same idea considered bothefficient and antiefficient frontiers for efficiency analysis andranking units
Kao [70] mentioned that determining the weights ofindividual criteria in multiple criteria decision analysis in away that all alternatives can be compared according to theaggregate performance of all criteria is of great importanceAs Kao noted this problem relates to search for alternativeswith a shorter and longer distance respectively to the idealand anti-ideal units He proposed a measure that consideredthe calculation of the relative position of an alternativebetween the ideal and anti-ideal for finding an appropriaterankings
Khodabakhshi and Aryavash [71] presented a method forranking all units using DEA concept
First Compute the minimum and maximum efficiencyvalues of eachDMU in regard to this assumption that the sumof efficiency values of all DMUs equals to 1
Second determine the rank of each DMU in relation to acombination of itsminimumandmaximumefficiency values
Zerafat Angiz et al [72] proposed a technique in orderto aggregate the opinions of experts in voting system Asthe authors mentioned the presented method uses fuzzyconcept and it is computationally efficient and can fullyrank alternatives At first number of votes given to a rankposition was grouped to construct fuzzy numbers and thenthe artificial ideal alternative introduced Furthermore byperforming DEA the efficiency measure of alternatives was
obtained considering artificial ideal alternative comparedby each of the alternatives pair by pair Thus alternativesare ranked in accordance with their efficiency scores Ifthis method cannot completely rank alternatives weightrestrictions based on fuzzy concept are imposed into theanalysis
10 Different Applications in Ranking Units
As discussed formerly there exist a variety of ranking meth-ods and applications in theliterature Nowadays DEAmodelsare widely used in different areas for efficiency evaluationbenchmarking and target setting ranking entities and soforth Ranking units as one of the important issues in DEAhas been performed in different areas Jahanshahloo et al [42]ranked cities in Iran to find the best place for creating a datafactory Hosseinzadeh Lotfi et al [28] used a new methodfor ranking in order to find the best place for power plantlocation Ali andNakosteen [80] Amirteimoori et al [37] andAlirezaee and Afsharian [24] Soltanifar and HosseinzadehLotfi [105] Zerafat Angiz et al [72] Hosseinzadeh Lotfiet al [28] Jahanshahloo [50] Chen and Deng [52] andJablonsky [61] used different ranking methods in bankingsystem Sadjadi [40] ranked provincial gas companies in IranMehrabian et al [32] Li et al [39] Orkcu and Bal [12] andWu et al [19] ranked different departments of universitiesJahanshahloo et al [35] Jahanshahloo and Afzalinejad [68]utilized the presented models in ranking 28 Chinese citiesJahanshahloo at al [35] provided an application to burdensharing amongst NATOmember nations Jahanshahloo et al[14] utilized the presented model for ranking nursery homesContreras [18] andWang et al [23] used the provided rankingtechniques in ranking candidates Lu and Lo [51] applied theirmethod on an application to financial holding companiesHosseinzadeh Lotfi et al [63] consider an empirical exampleinwhich voters are asked to rank twoout of seven alternativesWang and Jiang [62] utilized the provided method in facilitylayout design in manufacturing systems and performanceevaluation of 30OECD countries Chen [60] used an exampleof market basket data in order to illustrate the providedapproach
11 Application
In this section some of the reviewed models are applied onthe example used in Soltanifar and Hosseinzadeh Lotfi [105]Consider twenty commercial banks of Iran with input-outputdata tabulated in Table 1 and summarized as follows Also theresults of CCRmodel are listed in this table As it can be seenseven units are efficient DMUS 14 7 12 15 17 and 20 Inputsare staff computer terminal and spaceOutputs are depositsloans granted and charge
Consider some of the important ranking methods inliterature as follows
RM1 AP model [31] is based upon the idea of leaveunit evaluation out and measuring the distance of theunit under evaluation from the production possibilityset constructed by the remaining DMUs
RM2MAJmodel [32] presented for ranking efficientwhich is always stable but might be infeasible in somecasesRM3 Modified MAJ model [38] overcomes theproblem which might occurr in MAJ modelRM4 A new model based on the idea of alterationsin the reference set of the inefficient units [50]RM5 A model presented by Li et al [39] whichis a super-efficiency method that does not have thesuffering in previous methodsRM6 Slack-basedmodel [33] is based upon the inputand output variables at the same timeRM7 SA DEAmodel [49] overcomes the problem ofinfeasibility which existed in AP modelRM8 Cross-efficiency [10] is provided based onusing weights of each unit under evaluation in opti-mality for other unitsRM9 A model based on finding common set ofweights [25] which determine the common set ofweights for DMUs and ranked DMUs based this ideaRM10 A model based on finding common set ofweights [22 67] for ranking efficient unitsRM11 L1-norm model [34 35 65 66] the idea isbased upon the leave-one-out efficient unit and l1-norm which is always feasible and stableRM12 119871
infin-norm model Rezai Balf et al [46] pro-
vided a method with more ability over other existingmethods based on Tchebycheff normRM13 An enhanced Russel measure of super-efficiency model for ranking units [44]RM14 A rankingmodel which considers the distanceof unit from the full inefficient frontier [38]RM15 Amodified super-efficiencymodel [45] whichovercomes the infeasibility that may happen in prob-lem This model is based on simultaneous projectionof input output
In accordance with properties of different ranking models inorder to rank efficient units consider Table 2
1199011 Feasibility
1199012 Ranking extreme efficient units
1199013 Complexity in computation
1199014 Instability
1199015 Absence of multiple optimal solution
1199016 Dependency to 120579 and slacks
1199017 Dependency to the number of efficient and ineffi-
cient units
In Tables 3 and 4 the rank of efficient units consideringthe above mentioned methods is listed Note that ED showsefficient DMUs (ED) and RM
119895 (119895 = 1 15) are those
explained previously
12 Conclusion
In this paper the DEA ranking was reviewed and classifiedinto seven general groups In the first group those papersbased on a cross-efficiencymatrix were reviewed In this fieldDMUs have been evaluated by self- and peer pressure Thesecond group of papers is based on those papers lookingfor optimal weights in DEA analysis The third one is thesuper-efficiency method By omitting the under evaluationunit and constructing a new frontier by the remaining unitsthe unit under evaluation can get an equal or greater scoreThe fourth group is based on benchmarking idea In thisclass the effect of an efficient unit considered as a targetfor inefficient units is investigated This idea is very usefulfor the managers in decision making Another class thefifth one involves the application of multivariate statisticaltools The sixth section discusses the ranking methods basedon multicriteria decision-making (MCDM) methodologiesand DEA The final section includes some various rankingmethods presented in the literature Finally in an applicationthe result of some of the above-mentioned ranking methodsis presented
References
[1] M J Farrell ldquoThe measurement of productive efficiencyrdquoJournal of the Royal Statistical Society A vol 120 pp 253ndash2811957
[2] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
18 Journal of Applied Mathematics
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
119896+1ge 119889 (119896 120576) 119896 = 1 119899 minus 1
120583119899ge 119889 (119899 120576)
(19)
where 120579119895119896obtained from Step 5 and 120573lowast
119895is the optimal solution
of the following model Finally the DMUs are ranked basedon their 119911lowast
119895= sum
119899
119896=1120583lowast
119896120579119895119896values
Washio and Yamada [21] discussed that in real casesfinding the best ranking is more significant than acquiringthe most advantage weight and maximizing the efficiencyThus they presented a model called rank-based measure(RBM) for evaluating units from different viewpoint Thusthey suggested a method for acquiring those weight resultedfrom the best ranking as long as calculating those weightthat maximizes the efficiency score Finally they applied thepresented model to the cross-efficiency assessment
4 Ranking Techniques Based on FindingOptimal Weights in DEA Analysis
Jahanshahloo et al [22] gave a note on some of the DEAmodels for complete ranking using common set of weightsThey proved that by solving only one problem it is possibleto determine the common set of weights
where 119860 is the set of efficient units of the following model
max
sum119904
119903=11199061199031199101199031+ 119906
0
sum119898
119894=1V1198941199091198941
sdot sdot sdot
sum119904
119903=1119906119903119910119903119899+ 119906
0
sum119898
119894=1V119894119909119894119899
stsum
119904
119903=1119906119903119910119903119895+ 119906
0
sum119898
119894=1V119894119909119894119895
le 1 119895 = 1 119899
119880 119881 120578 ge 120576 1199060free
(21)
Note that DMUs can be ranked based on the evaluation oftheir efficiencies
Wang et al [23] provided an aggregating preference rank-ing In this paper use of ordered weighted averaging (OWA)operator is proposed for aggregating preference rankings Let
119908119895be the relative importance weight given to the jth ranking
place and V119894119895the vote candidate i receives in the jth ranking
place The total score of each candidate is defined as
119911119894=
119898
sum
119894=1
V119894119895119882
119895 119894 = 1 119898 (22)
Alirezaee and Afsharian [24] discussed multiplier model inwhich the variables are considered as shadow prices note thatsum
119894=1V119894119909119894119895are total revenue and cost of DMU
119895
which are considered in optimization problem The authorsclaimed that sum119904
119903=1119906119903119910119903119895minus sum
119898
119894=1V119894119909119894119895le 0 119895 = 1 119899 is the
profit restriction for DMU119895 If 119865(119909 119910) = 0 is considered to be
the efficient production function then
sum
119894
120597119865
120597119909119894
119909119894+sum
119895
120597119865
120597119910119895
119910119895= 0 (23)
They mentioned that the connected profit of the DMU iszero when shadow prices are derived from the technologyand called this situation as balance situation As the authorsmentioned therefore in the case that DMU
1is efficient but
DMU2is inefficient or the efficiency score of both DMUs is
the same and it obtains more negative quantity in balanceindex it can be concluded that DMU
1has a better rank than
DMU2
Liu and Hsuan Peng [25] in their paper proposed amethod for determining the common set of weights forranking units In common weights analysis methodologythey provided the following model
Δlowast= min sum
119895isin119864
Δ119900
119895+ Δ
119894
119895
stsum
119904
119903=1119910119903119895119880119903+ Δ
119900
119895
sum119898
119894=1119909119894119895119881119894minus Δ
119894
119895
= 1 119895 isin 119864
Δ119900
119895 Δ
119894
119895ge 0 forall119895 isin 119864
119880119903ge 120576 119903 = 1 119904
119881119894ge 120576 119894 = 1 119898
(24)
The mentioned ratio form the linear equationsWang et al [26] proposed a paper for ranking decision-
making units by imposing a minimum weight restrictionin DEA The authors noted that using data envelopmentanalysis it is not possible to distinguish between DEA effi-cient units Thus they presented a method for ranking unitsusing imposing minimum weight restriction for the input-output data As they mentioned these weights restrictionsare decided by a decision maker (DM) or an assessor asregards the solutions to a series of LP models considered fordetermining a maximin weight for each efficient DMU
Hatefi and Torabi [27] proposed a common weightmulti criteria decision analysis (MCDA)-data envelopmentanalysis (DEA) for constructing composite indicators (CIs)As the authors proved the presented model can discrimi-nate between efficient units The obtained common weightshave discriminating power more than those obtained from
Journal of Applied Mathematics 7
previous models Finally they studied the robustness anddiscriminating power of the proposed method by Spearmanrsquosrank correlation coefficient
Hosseinzadeh Lotfi et al [28] proposed one DEA rankingmodel based on applying aggregate units In doing so artificialunits called aggregate units are defined as follows Theaggregate unit is shown by DMU
119886
119909119901
119894119886= sum
119896isin119877119901
119909119894119896 119910
119901
119903119886= sum
119896isin119877119901
119910119903119896
119894 = 1 119898 119903 = 1 119904
(25)
where 119877119901= 119895 | DMU
119895isin 119864
119901 119864
119901= 119864DMU
119901 Note that 119864
is the set of efficient unitsFirst it is tried to maximize the efficiency score of the
DMU119886and then to maximize the efficiency score of the
DMU119901
119886 For resolving the existence of alternative solutions
the authors presented an approach comprising (119898+119904) simplelinear problems to achieve the most appropriate optimalsolutions among all alternative optimal solutions Finallythey proposed theRI index for ranking all efficientDMUs Let119880119886 119881
119875119886 and119880119901
119886 119881
119901
119886be the optimal solutions of the multiplier
model for assentDMU119886andDMU119901
119886 Consider 120578
119886= sum
119904
119903=1119906119903119886minus
sum119898
119894=1V119894119886 120578
119901
119886= sum
119904
119903=1119906119901
119903119886minus sum
119898
119894=1V119901119894119886 thus RI
119901= 120578
119900
119886minus 120578
119886
Wang et al [29] presented two nonlinear regressionmodels for deriving common set of weights for fully rankunits
min 119911 =
119899
sum
119895=1
(120579lowast
119895minus
sum119904
119903=1119906119903119910119903119895
sum119898
119894=1V119894119909119894119895
)
2
st 119880 119881 ge 0
min119899
sum
119895=1
(
119904
sum
119903=1
119906119903119910119903119895minus 120579
lowast
119895
119898
sum
119894=1
V119894119909119894119895)
2
st119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895) +
119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) = 119899
119880 119881 ge 0
(26)
Ramon et al [30] aimed at deriving a common set of weightsfor ranking units As the authors mentioned the idea is basedupon minimization of the deviations of the common weightsfrom the nonzero weights obtained fromDEA Furthermoreseveral norms are used for measuring such differences
5 Super-Efficiency Ranking Techniques
Super efficiency models introduced in DEA technique arebased upon the idea of leave one out and assessing this unittrough the remanding units
Andersen and Petersen [31] introduced a model for rank-ing efficient units The proposed model is as follows
Although this idea is useful for further discriminatingefficient units it has been shown in the literature that itmay be infeasible and nonstable Thrall [7] mentioned theinfeasibility of super-efficiency CCRmodel Also a conditionunder which infeasibility occurred in super-efficiency DEAmodels is mentioned by Zhu [95] Seiford and Zhu [96] andDula and Hickman [97]
Hashimoto [98] provided a model based on the idea ofone leave out and assurance region for ranking units
Mehrabian et al [32] presented a complete ranking forefficiency units in DEA context As the authors mentionedthis model does not have difficulties of AP model
min 119908119901+ 1
st119899
sum
119895=1 119895 = 119901
120582119895119909119894119895le 119909
119894119901+ 119908
1199011 119894 = 1 119898
119899
sum
119895=1 119895 = 119901
120582119895119910119903119895ge 119910
119903119901 119903 = 1 119904
120582119895ge 0 119895 = 119901
(28)
With this method it is not possible to rank nonextremeefficient units
Tone [33] presented super efficiency of SBM model Thismodel has the advantages of nonradial models and it isalways feasible and stable
minsum
119898
119894=1119909119894119909
119894119900
sum119904
119903=1119910119903119910
119903119900
st119899
sum
119895=1119895 = 119900
120582119895119909119894119895le 119909
119894 119894 = 1 119898
119899
sum
119895=1119895 = 119900
120582119895119910119903119895ge 119910
119903 119903 = 1 119904
119909119894ge 119909
119894119900 0 le 119910
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119900
(29)
8 Journal of Applied Mathematics
Jahanshahloo et al [34] added some ratio constraints to themultiplier form of AP model and introduced a new methodfor ranking DMUs
In their paper Chen and Sherman [36] presented a non-radial super-efficiency method and discussed the advantageof it They verified that this model is invariant to units ofinputoutput measurement Let 119869119900 = 119869DMU
119900
Step 1 Solve the followingmodel to find the extreme efficientunits in 119869119900
This method cannot rank nonextreme efficient units
Jahanshahloo et al [38] presented modified MAJ modelfor ranking efficiency units in DEA technique
min 119908119901+ 1
st119899
sum
119895=1119895 = 119901
120582119895
119909119894119895
119872119894
le
119909119894119901
119872119894
+ 1199081199011 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895ge 119910
119903119901 119903 = 1 119904
120582119895ge 0 119895 = 119901
(39)
where 119872119894= Max 119909
119894119895| DMU
119895is efficient It cannot rank
nonextreme efficient unitsLi et al [39] presented a new method for ranking which
does not have difficulties of earlier methods The presentedmodel is always feasible and stable
min 1 +
1
119898
119898
sum
119894=1
119904+
1198942
119877minus
119894
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895+ 119904
minus
1198941minus 119904
+
1198942= 119909
119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895minus 119904
+
119903= 119910
119903119901 119903 = 1 119904
119904minus
1198942ge 0 119904
minus
1198941ge 0 119904
+
119903ge 0 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 119901
(40)
In this case extreme efficient units cannot be ranked
In a paper Khodabakhshi [99] addresses super efficiencyon improved outputs He mentioned that as AP modelmay be infeasible under variable returns to scale technol-ogy using the presented model gives a complete rankingwhen getting an input combination for improving outputs issuitable
Sadjadi et al [40] presented a robust super-efficiencyDEA for ranking efficient units They noted that as inmost of the times exact data do not exist and the sto-chastic super-efficiency model presented in their paperincorporates the robust counterpart of super-efficiencyDEA
where 119883119884 are input-output data This model does notrank nonextreme efficient units It may be unstable andinfeasible
Gholam Abri et al [41] proposed a model for rankingefficient units They used representation theory and rep-resented the DMU under assessment as a convex combi-nation of extreme efficient units As the authors noted itis expected that the performance of DMU
119900is the same
as the performance of convex combination of extremeefficient units Thus it is possible to represent DMU
119900as
follows
(119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119899
sum
119895=1
120582119895= 1 119895 = 1 119904 (42)
As regards representation theorem this system has119898 + 119904 minus 1
constraints and 119904 variables (1205821 120582
119904) If this system has a
10 Journal of Applied Mathematics
unique solution we will have 120579lowast119900= sum
119899
119895=1= 1 120582
lowast
119895120579119895 otherwise
two models should be considered
min119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
max119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
(43)
Consider 1205791and 120579
2as the optimal solution of the above-
mentioned models respectively If 1205791= 120579
2 this is the same
as what has been mentioned previously If 1205791lt 120579
2 then
mentionedmodels provide an interval which helps rank unitsfrom the worst to the best Sometimes it will obtain theranking score with a bounded interval [120579
1 120579
2]
Jahanshahloo et al [42] presented models for rankingefficient units The presented models are somehow a modifi-cation of cross-efficiency model that overcomes the difficultyof alternative optimal weightsThe authors in their paperwithregard to the changes and also utilizing TOPSIS techniquepresented a new super-efficientmethod for ranking unitsThepresented model is as follows
120579119894119895=
119880119905119884119894
119881119905119883
119894
|
119880119905119884119897
119881119905119883
119897
le 1
119880119905119884119895
119881119905119883
119895
= 120579119895119895
119897 = 1 119899 119897 = 119894 119880 ge 0 119881 ge 0
(44)
where 120579119895119895is the efficiency score of DMU
119869using correspond-
ing weights Also 120579119894119895is the efficiency of DMU
119894using optimal
weights of DMU119895 Noura et al [43] provided a method for
ranking efficient units based on this idea that more effectiveand useful units in society should have better rank
Step 1 For each efficient unit choose the lower and upperlimit for each inputs and outputs Let 119864 be the set of efficientunits
where 120585 is representative of a small and nonzero number usedfor not dividing by zero Now consider119863minus119895 as follows whichshows that more successful DMU
119895will be if the larger value
of the119863119895is
119863119895= sum
119894isin119868
119889119894119895+ sum
119903isin119877
119889119903119895 (48)
where 119868 = 119863+
119894cup 119863
minus
119894 119877 = 119863
+
119903cup 119863
minus
119903
Ashrafi et al [44] introduced an enhanced Russell mea-sure of super efficiency for ranking efficient units inDEAThelinear counterpart of the proposed model is as follows
119900 le 120573 le V119903ge 120573 119903 = 1 119904
(49)
It cannot rank nonextreme efficient units
Journal of Applied Mathematics 11
Chen et al [45] proposed a modified super-efficiencymethod for ranking units based on simultaneous input-output projectionThe presentedmodel overcomes the infea-sibility problem
1199011= min
120579119904119903
119900
120601119904119903
119900
st sum
119895=1
119895 = 119900
120582119895119909119894119895le 120579
119904119903
119900119909119894119900 119894 = 1 119898
sum
119895=1
119895 = 119900
120582119895119910119903119895ge 120601
119904119903
119900119910119903119900 119903 = 1 119904
sum
119895=1
119895 = 119900
120582119895= 1
0 lt 120579119904119903
119900le 1 120601
119904119903
119900ge 1 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(50)
Rezai Balf et al [46] provide a model for ranking units basedon Tchebycheff norm As proved that this model is alwaysfeasible and stable it seems to have superiority over othermodels
max 119881119901
st 119881119901ge
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901 119894 = 1 119898
119881119901ge 119910
119903119901minus
119899
sum
119895=1
119895 = 119900
120582119895119910119903119895 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(51)
where
119881119901= Max
(
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901)119894 = 1 119898
(119910119903119901minus
119899
sum
119895=1
119895 = 119901
120582119895119910119903119895)119903 = 1 119904
(52)
Chen et al [47] for overcoming the infeasibility problem thatoccurred in variable returns to scale super-efficiency DEAmodel according to a directional distance function developedNerlove-Luenberger (N-L) measure of super-efficiency
6 Benchmarking Ranking Techniques
Sueyoshi et al [49] proposes a ldquobenchmark approachrdquo forbaseball evaluation This method is the combination of DEA
and (Offensive earned-run average) OERA As the authorsnoted using this method it is possible to select best unitsand also their ranking orders They mentioned that usingonly DEAmay result in a shortcoming in assessment as manyefficient units can be identifiedThus the authors used slack-adjusted DEA model and OERA to overcome this difficulty
Jahanshahloo et al [50] presented a new model forranking DMUs based on alteration in reference set The ideais based on this fact that efficient units can be the target unitfor inefficient units
min 120597119886119887
= 120579 minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st sum
119895isin119869minus119887
120582119895119909119894119895+ 119904
minus
119894minus 120579119909
119894119886 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
120579 free 119904minus119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 isin 119869 minus 119887
(53)
Finally the ranking order for each efficient unit b can becomputing by
119887= sum
119886isin119869119899
120597119886119887
119899
(54)
Note that this method cannot rank nonextreme efficientunits
Lu and Lo [51] provided an interactive benchmark modelfor ranking units The idea is based upon considering a fixedunit as a benchmark and calculating the efficiency of otherunits pair by pair to this unitThis procedure continueswhenall units are accounted for as a benchmark unit ConsiderDMU
119900as a unit under evaluation andDMU
119887as a benchmark
Assume
119909119894= 119909
119894119900(1 + 120601
119894) 119910
119903= 119910
119903119900(1 minus 120593
119903)
min 120579lowast119887
119900=
1 + (1119898)sum119898
119894=1120601119894
1 minus (1119904)sum119904
119903=1120593119903
st 120582119887119909119894119887minus 119909
119894119900120601119894le 119909
119894119900 119894 = 1 119898
120582119887119910119903119887minus 119910
119903119900120593119903ge 119910
119903119900 119903 = 1 119904
120601119894ge 119900 120593
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(55)
Then using (sum119899
119887=1120579119887lowast
119900)119899 119900 = 1 119899 the efficiency of
benchmark DMU119900can be obtained Using the following
index which indicates the increment in efficiency of a unitby moving from peer appraisal to self-appraisal it is nowpossible to rank units Note that the less the magnitude of
12 Journal of Applied Mathematics
this index is the better rank for corresponding unit will beobtained
FPIIBM119870
=
(TEBCC119896
minus STDIBM119896
)
(STDIBM119896
)
(56)
where TEBCC119896
and STDIBM119896
are respectively efficiency in BCCmodel and normalization of TEIBM of DMU
119896 Chen [52]
provided a paper for ranking efficient and inefficient unitsin DEA They noted that the evaluation of efficient unitsis based upon the alterations in efficiency of all inefficientunits by omitting in reference set At first solve the followingmodel which measures the efficiency of DMU
119886when DMU
119887
is eliminated from the reference set
min 120588119886119887
= 120579119904
119886minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1119895 = 119887
120582119895119909119894119895+ 119904
minus
119894= 120579
119904
119886119909119894119886 119894 = 1 119898
119899
sum
119895=1119895 = 119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
119899
sum
119895=1119895 = 119887
= 1
120579119904
119886ge 0 119904
minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119887
(57)
Then again solve the previous model this time for calculat-ing the efficiency score of each inefficient unit when each ofefficient units in turn is eliminated from the reference set 120578lowast
119886
and calculate the efficiency change
120591119886119887
= 120588lowast
119886119887minus 120578
lowast
119886 (58)
Then calculate the following index called as ldquoMCDErdquo indexfor each efficient and inefficient unit
119864119886= sum
119887isin119881119864
119882lowast
119887120588lowast
119886119887 119864
119887= sum
119887isin119881119868
119882lowast
119887120588lowast
119886119887 (59)
Now in accordance with themagnitude of the acquired indexit is possible to rank unitsThose units with higher score havebetter ranking order
7 Ranking Techniques by MultivariateStatistics in the DEA
As described in DEA literature in DEA technique frontier istaken into consideration rather than central tendency consid-ered in regression analysis DEA technique considers that anenvelope encompasses through all the observations as tight aspossible and does not try to fit regression planes in center ofdata In DEA methodology each unit is considered initiallyand compared to the efficient frontier but in regressionanalysis a procedure is considered in which a single functionfits to the data DEA uses different weights for differentunits but does not let the units use weights of other units
Canonical correlation analysis as Friedman and Sinuany-Stern [53] noted can be used for ranking units This methodis somehow the extension of regression analysis Adler et al[89] The aim in canonical correlation analysis is to find asingle vector commonweight for the inputs and outputs of allunits Consider119885
119895119882
119895as the composite input and output and
as corresponding weights respectively The presentedmodel by Tatsuoka and Lohnes [100] is as follows
max 119903119911119908
=
119878119909119910119880
(119878119909119909119881) (119878
119910119910119880)
(12)
st 119909119909119881 = 1
119910119910119880 = 1
(60)
where 119878119909119909 119878
119910119910 and 119878
119909119910are respectively defined as the matri-
ces of the sums of squares and sums of products of thevariables
Friedman and Sinuany-Stern [53] while defining the ratioof linear combinations of the inputs and outputs 119879
119895=
119882119895119885
119895used canonical correlation analysis As they noted that
scaling ratio 119879119895of the canonical correlation analysisDEA is
unboundedSinuany-Stern et al [101] used linear discriminant analy-
sis for ranking units They defined
119863119895=
119904
sum
119903=1
119906119903119910119903119895+
119898
sum
119894=1
V119903(minus119909
119894119895) (61)
DMU119895is said to be efficient if 119863
119895gt 119863
119888where 119863
119888is a critical
value based on themidpoint of themeans of the discriminantfunction value of the two groups Morrison [102] The largerthe amount of119863
119895is the better rank DMU
119895will have
Friedman and Sinuany-Stern [53] noted that as cross-efficiencyDEA canonical correlation analysisDEA and dis-criminant analysisDEA ranking orders may vary from eachother thus it seems necessary to introduce the combinedranking (CODEA) Combined ranking for each unit con-sidered all the ranks obtained from the above-mentionedrankings Moreover statistical tests are one for goodness offit between DEA and a specific ranking and the other fortesting correlation between variety of ranking orders Siegeland Castellan [103]
8 Ranking with MulticriteriaDecision-Making (MCDM) Methodologiesand DEA
Li and Reeves [56] for increasing discrimination power ofDEA presented amultiple-objective linear program (MOLP)As the authors mentioned using minimax and minsumefficiency in addition to the standard DEA objective functionhelp to increase discrimination power of DEA
Strassert and Prato [59] presented the balancing andranking method which uses a three-step procedure forderiving an overall complete or partial final order of optionsIn the first step derive an outranking matrix for all options
Journal of Applied Mathematics 13
from the criteria values Considering this matrix it is possibleto show the frequencywithwhich one option is ranked higherthan the other options In the second step by triangularizingthe outranking matrix establish an implicit preordering orprovisional ordering of optionsTheoutrankingmatrix showsthe degree to which there is a complete overall order ofoptions In the third step based on information given in anadvantages-disadvantages table the provisional ordering issubjected to different screening and balancing operations
Chen [60] utilized a nonparametric approach DEA toestimate and rank the efficiency of association rules withmultiple criteria in following steps Proposed postprocessingapproach is as follows
Step 1 Input data for association rule mining
Step 2 Mine association rules by using the a priori algorithmwith minimum support and minimum confidence
Step 3 Determine subjective interestingness measures byfurther considering the domain related knowledge
Step 4 Calculate the preference scores of association rulesdiscovered in Step 2 by using Cook and Kressrsquos DEA model
Step 5 Discriminate the efficient association rules found inStep 3 by using Obata and Ishiirsquos [104] discriminate model
Step 6 Select rules for implementation by considering thereference scores generated in Step 5 and domain relatedknowledge
Jablonsky [61] presented two original models super SBMand AHP for ranking of efficient units in DEA As theauthormentioned thesemodels are based onmultiple criteriadecision-making techniques-goal programming and analytichierarchy process Super SBM model for ranking units is asfollows
min 120579119866
119902= 1 + 119905120574 + (1 minus 119905) (
sum119898
119894=1119904+
1119894
119909119894119902
+
sum119898
119894=1119904minus
2119896
119910119896119902
)
st119899
sum
119895=1119895 = 119902
120582119895119909119894119895+ 119904
minus
1119894minus 119904
+
1119894= 119909
119894119902 i = 1 119898
119899
sum
119895=1119895 = 119902
120582119895119910119896119895+ 119904
minus
2119896minus 119904
+
2119896= 119910
119896119902 119896 = 1 119903
119904+
1119894le 120574 119894 = 1 119898
119904minus
2119896le 120574 119896 = 1 119903
119905 isin 0 1 120582119895ge 0 119878
+
1 119878
+
2ge 0 119878
minus
1 119878
minus
2ge 0
119895 = 1 119899
(62)
Wang and Jiang [62] presented an alternative mixedinteger linear programming models in order to identifythe most efficient units in DEA technique As the authors
mentioned presented models can make full use of input-output information with no need to specify any assuranceregions for input and output weights to avoid zero weights
min119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) minus
119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895)
st119904
sum
119903=1
119906119903119910119903119895minus
119898
sum
119894=1
V119894119909119894119895le 119868
119895 119895 = 1 119899
119899
sum
119895=1
119868119895= 1
119868119895isin 0 1 119895 = 1 119899
119906119903ge
1
(119898 + 119904)max119895119910
119903119895
119903 = 1 119904
V119894ge
1
(119898 + 119904)max119895119909
119894119895
119894 = 1 119898
(63)
Hosseinzadeh Lotfi et al [63] provided an improved three-stage method for ranking alternatives in multiple criteriadecision analysis In the first stage based on the bestand worst weights in the optimistic and pessimistic casesobtain the rank position of each alternative respectively Theobtained weight in the first stage is not unique thus it seemsnecessary to introduce a secondary goal that is used in thesecond stage Finally in the third stage the ranks of thealternatives compute in the optimistic or pessimistic case Itis mentionable that the model proposed in the third stageis a multicriteria decision-making (MCDM) model and itis solved by mixed integer programming As the authorsmentioned and provided in their paper this model can beconverted into an LP problem
The previous model with 119897 = 1 is the original CCR modelNote that DMUs in 119864
1 show the first level efficient frontier119897 = 2 indicates the second level efficient frontier when thefirst level efficient frontier is omittedThe following algorithmfinds these efficient frontiers
Step 1 Set 119897 = 1 and evaluate the entire set of DMUs 1198691 Inthis way the first level efficient DMU 1198641 is identified
Step 2 Exclude the efficient DMUs from future DEA runsand set 119869119897+1 = 119869
119897minus 119864
119897 (If 119869119897+1 = oslash stop)
Step 3 Evaluate the new subset of ldquoinefficientrdquo DMUs 119869119897+1to obtain a new set of efficient DMUs 119864119897+1
Step 4 Let l = l + 1 Go to Step 2
When 119869119897+1 = oslash the algorithm stopsAs the authors proved in this way it is possible to rank
the DMUs in the first efficient frontier based upon theirattractiveness scores and identify the best one
Jahanshahloo et al [65] provided a paper for rankingunits using gradient line As the authors mentioned theadvantage of this model is stability and robustness
supports on 119879119888 the obtained PPS by omitting theDMUunder
assessment in 119879119888 As the authors proved a unit is efficient
iff the optimal objective function of the following model isgreater than zero Consider
1198750= (119883 119884) 119883 = 120572119883
119900 119884 = 120573119884
119900
1198780= (119883 119884) isin 1198750
119883 = 120572119883119900 119884 = 120573119884
119900 120572 ge 0 120573 ge 0
(67)
Intersection of 1198780and efficient surface of
119879119888119888 is a half line
where its equation is (minus119881lowast119879119883
119900)120572 + (119880
lowast119879119884119900)120573 = 0 To rank
DMU119900the length of connecting arc DMU
119900with intersection
point of line and previous ellipse is calculated in (120572 120573) spaceThis intersection is as follows
120572lowast= (
1198702
120572119870
2
120573
1198702
120573+ (119881
lowast119879119883
119900119880
lowast119879119884119900)2119870
2
120572
)
12
120573lowast= (
119881lowast119879119883
119900
119880lowast119879119884119900
)120572lowast
(68)
Now the length of connecting arc DMU119900to the point
corresponding to 120572lowast 120573lowast in (120572 120573) plan is calculated as follows119868 = int
120572lowast
1(1 + 120573)
12119889120572 where
119870120572= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119899
119895=11199092
119894119900
)
12
119870120573= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119904
119903=11199102
119903119900
)
12
(69)
Jahanshahloo et al [66] also provided a paper with theconcept of advantage in data envelopment analysis
In their paper Jahanshahloo et al [67] consideringMonteCarlo method presented a new method of ranking
Step 1 Generate a uniformly distributed sequence of 1198801198952119899
119895=1
on(0 1)
Step 2 Random numbers should be classified into 119873 pairslike (119880
1 119880
1015840
1) (119880
119873 119880
1015840
119873) in a way that each number is used
just one time
Step 3 Compute119883119894= 119886 + 119880
119894(119887 minus 119886) and 119891(119883
119894) gt 119888119880
1015840
119894
Step 4 Estimate the integral 119868 by 120579119868= 119888(119887 minus 119886)(119873
119867119873)
Now consider DMU119900as an efficient unit measure those
units that are dominated DMU119900 As for dome DMUs this
would be unbounded thus the authors for each unitbounded the region Then for DMU
119900if (minus119883
119900 119884
119900) ge (minus119883 119884)
then (minus119883 119884) is in the RED of DMU119900 Now by using 119881
119901=
119881lowast(119873
119867119873) all the hits (the above condition) can be counted
where 119881lowast is the measure of the whole region JahanshahlooandAfzalinejad [68] presented amethod based on distance of
Journal of Applied Mathematics 15
the unit under evaluation to the full inefficient frontier Theypresented two models radian and nonradial
min 120601
st119899
sum
119895=1
120582119895119909119894119895= 120601119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
120582119895ge 0 119895 = 1 119898
max119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903
st sum
119895isin119869minus119887
120582119895119909119894119895minus 119904
minus
119894= 119909
119894119900 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895+ 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
119904minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(70)
Amirteimoori [69] based on the same idea considered bothefficient and antiefficient frontiers for efficiency analysis andranking units
Kao [70] mentioned that determining the weights ofindividual criteria in multiple criteria decision analysis in away that all alternatives can be compared according to theaggregate performance of all criteria is of great importanceAs Kao noted this problem relates to search for alternativeswith a shorter and longer distance respectively to the idealand anti-ideal units He proposed a measure that consideredthe calculation of the relative position of an alternativebetween the ideal and anti-ideal for finding an appropriaterankings
Khodabakhshi and Aryavash [71] presented a method forranking all units using DEA concept
First Compute the minimum and maximum efficiencyvalues of eachDMU in regard to this assumption that the sumof efficiency values of all DMUs equals to 1
Second determine the rank of each DMU in relation to acombination of itsminimumandmaximumefficiency values
Zerafat Angiz et al [72] proposed a technique in orderto aggregate the opinions of experts in voting system Asthe authors mentioned the presented method uses fuzzyconcept and it is computationally efficient and can fullyrank alternatives At first number of votes given to a rankposition was grouped to construct fuzzy numbers and thenthe artificial ideal alternative introduced Furthermore byperforming DEA the efficiency measure of alternatives was
obtained considering artificial ideal alternative comparedby each of the alternatives pair by pair Thus alternativesare ranked in accordance with their efficiency scores Ifthis method cannot completely rank alternatives weightrestrictions based on fuzzy concept are imposed into theanalysis
10 Different Applications in Ranking Units
As discussed formerly there exist a variety of ranking meth-ods and applications in theliterature Nowadays DEAmodelsare widely used in different areas for efficiency evaluationbenchmarking and target setting ranking entities and soforth Ranking units as one of the important issues in DEAhas been performed in different areas Jahanshahloo et al [42]ranked cities in Iran to find the best place for creating a datafactory Hosseinzadeh Lotfi et al [28] used a new methodfor ranking in order to find the best place for power plantlocation Ali andNakosteen [80] Amirteimoori et al [37] andAlirezaee and Afsharian [24] Soltanifar and HosseinzadehLotfi [105] Zerafat Angiz et al [72] Hosseinzadeh Lotfiet al [28] Jahanshahloo [50] Chen and Deng [52] andJablonsky [61] used different ranking methods in bankingsystem Sadjadi [40] ranked provincial gas companies in IranMehrabian et al [32] Li et al [39] Orkcu and Bal [12] andWu et al [19] ranked different departments of universitiesJahanshahloo et al [35] Jahanshahloo and Afzalinejad [68]utilized the presented models in ranking 28 Chinese citiesJahanshahloo at al [35] provided an application to burdensharing amongst NATOmember nations Jahanshahloo et al[14] utilized the presented model for ranking nursery homesContreras [18] andWang et al [23] used the provided rankingtechniques in ranking candidates Lu and Lo [51] applied theirmethod on an application to financial holding companiesHosseinzadeh Lotfi et al [63] consider an empirical exampleinwhich voters are asked to rank twoout of seven alternativesWang and Jiang [62] utilized the provided method in facilitylayout design in manufacturing systems and performanceevaluation of 30OECD countries Chen [60] used an exampleof market basket data in order to illustrate the providedapproach
11 Application
In this section some of the reviewed models are applied onthe example used in Soltanifar and Hosseinzadeh Lotfi [105]Consider twenty commercial banks of Iran with input-outputdata tabulated in Table 1 and summarized as follows Also theresults of CCRmodel are listed in this table As it can be seenseven units are efficient DMUS 14 7 12 15 17 and 20 Inputsare staff computer terminal and spaceOutputs are depositsloans granted and charge
Consider some of the important ranking methods inliterature as follows
RM1 AP model [31] is based upon the idea of leaveunit evaluation out and measuring the distance of theunit under evaluation from the production possibilityset constructed by the remaining DMUs
RM2MAJmodel [32] presented for ranking efficientwhich is always stable but might be infeasible in somecasesRM3 Modified MAJ model [38] overcomes theproblem which might occurr in MAJ modelRM4 A new model based on the idea of alterationsin the reference set of the inefficient units [50]RM5 A model presented by Li et al [39] whichis a super-efficiency method that does not have thesuffering in previous methodsRM6 Slack-basedmodel [33] is based upon the inputand output variables at the same timeRM7 SA DEAmodel [49] overcomes the problem ofinfeasibility which existed in AP modelRM8 Cross-efficiency [10] is provided based onusing weights of each unit under evaluation in opti-mality for other unitsRM9 A model based on finding common set ofweights [25] which determine the common set ofweights for DMUs and ranked DMUs based this ideaRM10 A model based on finding common set ofweights [22 67] for ranking efficient unitsRM11 L1-norm model [34 35 65 66] the idea isbased upon the leave-one-out efficient unit and l1-norm which is always feasible and stableRM12 119871
infin-norm model Rezai Balf et al [46] pro-
vided a method with more ability over other existingmethods based on Tchebycheff normRM13 An enhanced Russel measure of super-efficiency model for ranking units [44]RM14 A rankingmodel which considers the distanceof unit from the full inefficient frontier [38]RM15 Amodified super-efficiencymodel [45] whichovercomes the infeasibility that may happen in prob-lem This model is based on simultaneous projectionof input output
In accordance with properties of different ranking models inorder to rank efficient units consider Table 2
1199011 Feasibility
1199012 Ranking extreme efficient units
1199013 Complexity in computation
1199014 Instability
1199015 Absence of multiple optimal solution
1199016 Dependency to 120579 and slacks
1199017 Dependency to the number of efficient and ineffi-
cient units
In Tables 3 and 4 the rank of efficient units consideringthe above mentioned methods is listed Note that ED showsefficient DMUs (ED) and RM
119895 (119895 = 1 15) are those
explained previously
12 Conclusion
In this paper the DEA ranking was reviewed and classifiedinto seven general groups In the first group those papersbased on a cross-efficiencymatrix were reviewed In this fieldDMUs have been evaluated by self- and peer pressure Thesecond group of papers is based on those papers lookingfor optimal weights in DEA analysis The third one is thesuper-efficiency method By omitting the under evaluationunit and constructing a new frontier by the remaining unitsthe unit under evaluation can get an equal or greater scoreThe fourth group is based on benchmarking idea In thisclass the effect of an efficient unit considered as a targetfor inefficient units is investigated This idea is very usefulfor the managers in decision making Another class thefifth one involves the application of multivariate statisticaltools The sixth section discusses the ranking methods basedon multicriteria decision-making (MCDM) methodologiesand DEA The final section includes some various rankingmethods presented in the literature Finally in an applicationthe result of some of the above-mentioned ranking methodsis presented
References
[1] M J Farrell ldquoThe measurement of productive efficiencyrdquoJournal of the Royal Statistical Society A vol 120 pp 253ndash2811957
[2] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
18 Journal of Applied Mathematics
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
119896+1ge 119889 (119896 120576) 119896 = 1 119899 minus 1
120583119899ge 119889 (119899 120576)
(19)
where 120579119895119896obtained from Step 5 and 120573lowast
119895is the optimal solution
of the following model Finally the DMUs are ranked basedon their 119911lowast
119895= sum
119899
119896=1120583lowast
119896120579119895119896values
Washio and Yamada [21] discussed that in real casesfinding the best ranking is more significant than acquiringthe most advantage weight and maximizing the efficiencyThus they presented a model called rank-based measure(RBM) for evaluating units from different viewpoint Thusthey suggested a method for acquiring those weight resultedfrom the best ranking as long as calculating those weightthat maximizes the efficiency score Finally they applied thepresented model to the cross-efficiency assessment
4 Ranking Techniques Based on FindingOptimal Weights in DEA Analysis
Jahanshahloo et al [22] gave a note on some of the DEAmodels for complete ranking using common set of weightsThey proved that by solving only one problem it is possibleto determine the common set of weights
where 119860 is the set of efficient units of the following model
max
sum119904
119903=11199061199031199101199031+ 119906
0
sum119898
119894=1V1198941199091198941
sdot sdot sdot
sum119904
119903=1119906119903119910119903119899+ 119906
0
sum119898
119894=1V119894119909119894119899
stsum
119904
119903=1119906119903119910119903119895+ 119906
0
sum119898
119894=1V119894119909119894119895
le 1 119895 = 1 119899
119880 119881 120578 ge 120576 1199060free
(21)
Note that DMUs can be ranked based on the evaluation oftheir efficiencies
Wang et al [23] provided an aggregating preference rank-ing In this paper use of ordered weighted averaging (OWA)operator is proposed for aggregating preference rankings Let
119908119895be the relative importance weight given to the jth ranking
place and V119894119895the vote candidate i receives in the jth ranking
place The total score of each candidate is defined as
119911119894=
119898
sum
119894=1
V119894119895119882
119895 119894 = 1 119898 (22)
Alirezaee and Afsharian [24] discussed multiplier model inwhich the variables are considered as shadow prices note thatsum
119894=1V119894119909119894119895are total revenue and cost of DMU
119895
which are considered in optimization problem The authorsclaimed that sum119904
119903=1119906119903119910119903119895minus sum
119898
119894=1V119894119909119894119895le 0 119895 = 1 119899 is the
profit restriction for DMU119895 If 119865(119909 119910) = 0 is considered to be
the efficient production function then
sum
119894
120597119865
120597119909119894
119909119894+sum
119895
120597119865
120597119910119895
119910119895= 0 (23)
They mentioned that the connected profit of the DMU iszero when shadow prices are derived from the technologyand called this situation as balance situation As the authorsmentioned therefore in the case that DMU
1is efficient but
DMU2is inefficient or the efficiency score of both DMUs is
the same and it obtains more negative quantity in balanceindex it can be concluded that DMU
1has a better rank than
DMU2
Liu and Hsuan Peng [25] in their paper proposed amethod for determining the common set of weights forranking units In common weights analysis methodologythey provided the following model
Δlowast= min sum
119895isin119864
Δ119900
119895+ Δ
119894
119895
stsum
119904
119903=1119910119903119895119880119903+ Δ
119900
119895
sum119898
119894=1119909119894119895119881119894minus Δ
119894
119895
= 1 119895 isin 119864
Δ119900
119895 Δ
119894
119895ge 0 forall119895 isin 119864
119880119903ge 120576 119903 = 1 119904
119881119894ge 120576 119894 = 1 119898
(24)
The mentioned ratio form the linear equationsWang et al [26] proposed a paper for ranking decision-
making units by imposing a minimum weight restrictionin DEA The authors noted that using data envelopmentanalysis it is not possible to distinguish between DEA effi-cient units Thus they presented a method for ranking unitsusing imposing minimum weight restriction for the input-output data As they mentioned these weights restrictionsare decided by a decision maker (DM) or an assessor asregards the solutions to a series of LP models considered fordetermining a maximin weight for each efficient DMU
Hatefi and Torabi [27] proposed a common weightmulti criteria decision analysis (MCDA)-data envelopmentanalysis (DEA) for constructing composite indicators (CIs)As the authors proved the presented model can discrimi-nate between efficient units The obtained common weightshave discriminating power more than those obtained from
Journal of Applied Mathematics 7
previous models Finally they studied the robustness anddiscriminating power of the proposed method by Spearmanrsquosrank correlation coefficient
Hosseinzadeh Lotfi et al [28] proposed one DEA rankingmodel based on applying aggregate units In doing so artificialunits called aggregate units are defined as follows Theaggregate unit is shown by DMU
119886
119909119901
119894119886= sum
119896isin119877119901
119909119894119896 119910
119901
119903119886= sum
119896isin119877119901
119910119903119896
119894 = 1 119898 119903 = 1 119904
(25)
where 119877119901= 119895 | DMU
119895isin 119864
119901 119864
119901= 119864DMU
119901 Note that 119864
is the set of efficient unitsFirst it is tried to maximize the efficiency score of the
DMU119886and then to maximize the efficiency score of the
DMU119901
119886 For resolving the existence of alternative solutions
the authors presented an approach comprising (119898+119904) simplelinear problems to achieve the most appropriate optimalsolutions among all alternative optimal solutions Finallythey proposed theRI index for ranking all efficientDMUs Let119880119886 119881
119875119886 and119880119901
119886 119881
119901
119886be the optimal solutions of the multiplier
model for assentDMU119886andDMU119901
119886 Consider 120578
119886= sum
119904
119903=1119906119903119886minus
sum119898
119894=1V119894119886 120578
119901
119886= sum
119904
119903=1119906119901
119903119886minus sum
119898
119894=1V119901119894119886 thus RI
119901= 120578
119900
119886minus 120578
119886
Wang et al [29] presented two nonlinear regressionmodels for deriving common set of weights for fully rankunits
min 119911 =
119899
sum
119895=1
(120579lowast
119895minus
sum119904
119903=1119906119903119910119903119895
sum119898
119894=1V119894119909119894119895
)
2
st 119880 119881 ge 0
min119899
sum
119895=1
(
119904
sum
119903=1
119906119903119910119903119895minus 120579
lowast
119895
119898
sum
119894=1
V119894119909119894119895)
2
st119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895) +
119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) = 119899
119880 119881 ge 0
(26)
Ramon et al [30] aimed at deriving a common set of weightsfor ranking units As the authors mentioned the idea is basedupon minimization of the deviations of the common weightsfrom the nonzero weights obtained fromDEA Furthermoreseveral norms are used for measuring such differences
5 Super-Efficiency Ranking Techniques
Super efficiency models introduced in DEA technique arebased upon the idea of leave one out and assessing this unittrough the remanding units
Andersen and Petersen [31] introduced a model for rank-ing efficient units The proposed model is as follows
Although this idea is useful for further discriminatingefficient units it has been shown in the literature that itmay be infeasible and nonstable Thrall [7] mentioned theinfeasibility of super-efficiency CCRmodel Also a conditionunder which infeasibility occurred in super-efficiency DEAmodels is mentioned by Zhu [95] Seiford and Zhu [96] andDula and Hickman [97]
Hashimoto [98] provided a model based on the idea ofone leave out and assurance region for ranking units
Mehrabian et al [32] presented a complete ranking forefficiency units in DEA context As the authors mentionedthis model does not have difficulties of AP model
min 119908119901+ 1
st119899
sum
119895=1 119895 = 119901
120582119895119909119894119895le 119909
119894119901+ 119908
1199011 119894 = 1 119898
119899
sum
119895=1 119895 = 119901
120582119895119910119903119895ge 119910
119903119901 119903 = 1 119904
120582119895ge 0 119895 = 119901
(28)
With this method it is not possible to rank nonextremeefficient units
Tone [33] presented super efficiency of SBM model Thismodel has the advantages of nonradial models and it isalways feasible and stable
minsum
119898
119894=1119909119894119909
119894119900
sum119904
119903=1119910119903119910
119903119900
st119899
sum
119895=1119895 = 119900
120582119895119909119894119895le 119909
119894 119894 = 1 119898
119899
sum
119895=1119895 = 119900
120582119895119910119903119895ge 119910
119903 119903 = 1 119904
119909119894ge 119909
119894119900 0 le 119910
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119900
(29)
8 Journal of Applied Mathematics
Jahanshahloo et al [34] added some ratio constraints to themultiplier form of AP model and introduced a new methodfor ranking DMUs
In their paper Chen and Sherman [36] presented a non-radial super-efficiency method and discussed the advantageof it They verified that this model is invariant to units ofinputoutput measurement Let 119869119900 = 119869DMU
119900
Step 1 Solve the followingmodel to find the extreme efficientunits in 119869119900
This method cannot rank nonextreme efficient units
Jahanshahloo et al [38] presented modified MAJ modelfor ranking efficiency units in DEA technique
min 119908119901+ 1
st119899
sum
119895=1119895 = 119901
120582119895
119909119894119895
119872119894
le
119909119894119901
119872119894
+ 1199081199011 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895ge 119910
119903119901 119903 = 1 119904
120582119895ge 0 119895 = 119901
(39)
where 119872119894= Max 119909
119894119895| DMU
119895is efficient It cannot rank
nonextreme efficient unitsLi et al [39] presented a new method for ranking which
does not have difficulties of earlier methods The presentedmodel is always feasible and stable
min 1 +
1
119898
119898
sum
119894=1
119904+
1198942
119877minus
119894
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895+ 119904
minus
1198941minus 119904
+
1198942= 119909
119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895minus 119904
+
119903= 119910
119903119901 119903 = 1 119904
119904minus
1198942ge 0 119904
minus
1198941ge 0 119904
+
119903ge 0 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 119901
(40)
In this case extreme efficient units cannot be ranked
In a paper Khodabakhshi [99] addresses super efficiencyon improved outputs He mentioned that as AP modelmay be infeasible under variable returns to scale technol-ogy using the presented model gives a complete rankingwhen getting an input combination for improving outputs issuitable
Sadjadi et al [40] presented a robust super-efficiencyDEA for ranking efficient units They noted that as inmost of the times exact data do not exist and the sto-chastic super-efficiency model presented in their paperincorporates the robust counterpart of super-efficiencyDEA
where 119883119884 are input-output data This model does notrank nonextreme efficient units It may be unstable andinfeasible
Gholam Abri et al [41] proposed a model for rankingefficient units They used representation theory and rep-resented the DMU under assessment as a convex combi-nation of extreme efficient units As the authors noted itis expected that the performance of DMU
119900is the same
as the performance of convex combination of extremeefficient units Thus it is possible to represent DMU
119900as
follows
(119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119899
sum
119895=1
120582119895= 1 119895 = 1 119904 (42)
As regards representation theorem this system has119898 + 119904 minus 1
constraints and 119904 variables (1205821 120582
119904) If this system has a
10 Journal of Applied Mathematics
unique solution we will have 120579lowast119900= sum
119899
119895=1= 1 120582
lowast
119895120579119895 otherwise
two models should be considered
min119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
max119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
(43)
Consider 1205791and 120579
2as the optimal solution of the above-
mentioned models respectively If 1205791= 120579
2 this is the same
as what has been mentioned previously If 1205791lt 120579
2 then
mentionedmodels provide an interval which helps rank unitsfrom the worst to the best Sometimes it will obtain theranking score with a bounded interval [120579
1 120579
2]
Jahanshahloo et al [42] presented models for rankingefficient units The presented models are somehow a modifi-cation of cross-efficiency model that overcomes the difficultyof alternative optimal weightsThe authors in their paperwithregard to the changes and also utilizing TOPSIS techniquepresented a new super-efficientmethod for ranking unitsThepresented model is as follows
120579119894119895=
119880119905119884119894
119881119905119883
119894
|
119880119905119884119897
119881119905119883
119897
le 1
119880119905119884119895
119881119905119883
119895
= 120579119895119895
119897 = 1 119899 119897 = 119894 119880 ge 0 119881 ge 0
(44)
where 120579119895119895is the efficiency score of DMU
119869using correspond-
ing weights Also 120579119894119895is the efficiency of DMU
119894using optimal
weights of DMU119895 Noura et al [43] provided a method for
ranking efficient units based on this idea that more effectiveand useful units in society should have better rank
Step 1 For each efficient unit choose the lower and upperlimit for each inputs and outputs Let 119864 be the set of efficientunits
where 120585 is representative of a small and nonzero number usedfor not dividing by zero Now consider119863minus119895 as follows whichshows that more successful DMU
119895will be if the larger value
of the119863119895is
119863119895= sum
119894isin119868
119889119894119895+ sum
119903isin119877
119889119903119895 (48)
where 119868 = 119863+
119894cup 119863
minus
119894 119877 = 119863
+
119903cup 119863
minus
119903
Ashrafi et al [44] introduced an enhanced Russell mea-sure of super efficiency for ranking efficient units inDEAThelinear counterpart of the proposed model is as follows
119900 le 120573 le V119903ge 120573 119903 = 1 119904
(49)
It cannot rank nonextreme efficient units
Journal of Applied Mathematics 11
Chen et al [45] proposed a modified super-efficiencymethod for ranking units based on simultaneous input-output projectionThe presentedmodel overcomes the infea-sibility problem
1199011= min
120579119904119903
119900
120601119904119903
119900
st sum
119895=1
119895 = 119900
120582119895119909119894119895le 120579
119904119903
119900119909119894119900 119894 = 1 119898
sum
119895=1
119895 = 119900
120582119895119910119903119895ge 120601
119904119903
119900119910119903119900 119903 = 1 119904
sum
119895=1
119895 = 119900
120582119895= 1
0 lt 120579119904119903
119900le 1 120601
119904119903
119900ge 1 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(50)
Rezai Balf et al [46] provide a model for ranking units basedon Tchebycheff norm As proved that this model is alwaysfeasible and stable it seems to have superiority over othermodels
max 119881119901
st 119881119901ge
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901 119894 = 1 119898
119881119901ge 119910
119903119901minus
119899
sum
119895=1
119895 = 119900
120582119895119910119903119895 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(51)
where
119881119901= Max
(
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901)119894 = 1 119898
(119910119903119901minus
119899
sum
119895=1
119895 = 119901
120582119895119910119903119895)119903 = 1 119904
(52)
Chen et al [47] for overcoming the infeasibility problem thatoccurred in variable returns to scale super-efficiency DEAmodel according to a directional distance function developedNerlove-Luenberger (N-L) measure of super-efficiency
6 Benchmarking Ranking Techniques
Sueyoshi et al [49] proposes a ldquobenchmark approachrdquo forbaseball evaluation This method is the combination of DEA
and (Offensive earned-run average) OERA As the authorsnoted using this method it is possible to select best unitsand also their ranking orders They mentioned that usingonly DEAmay result in a shortcoming in assessment as manyefficient units can be identifiedThus the authors used slack-adjusted DEA model and OERA to overcome this difficulty
Jahanshahloo et al [50] presented a new model forranking DMUs based on alteration in reference set The ideais based on this fact that efficient units can be the target unitfor inefficient units
min 120597119886119887
= 120579 minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st sum
119895isin119869minus119887
120582119895119909119894119895+ 119904
minus
119894minus 120579119909
119894119886 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
120579 free 119904minus119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 isin 119869 minus 119887
(53)
Finally the ranking order for each efficient unit b can becomputing by
119887= sum
119886isin119869119899
120597119886119887
119899
(54)
Note that this method cannot rank nonextreme efficientunits
Lu and Lo [51] provided an interactive benchmark modelfor ranking units The idea is based upon considering a fixedunit as a benchmark and calculating the efficiency of otherunits pair by pair to this unitThis procedure continueswhenall units are accounted for as a benchmark unit ConsiderDMU
119900as a unit under evaluation andDMU
119887as a benchmark
Assume
119909119894= 119909
119894119900(1 + 120601
119894) 119910
119903= 119910
119903119900(1 minus 120593
119903)
min 120579lowast119887
119900=
1 + (1119898)sum119898
119894=1120601119894
1 minus (1119904)sum119904
119903=1120593119903
st 120582119887119909119894119887minus 119909
119894119900120601119894le 119909
119894119900 119894 = 1 119898
120582119887119910119903119887minus 119910
119903119900120593119903ge 119910
119903119900 119903 = 1 119904
120601119894ge 119900 120593
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(55)
Then using (sum119899
119887=1120579119887lowast
119900)119899 119900 = 1 119899 the efficiency of
benchmark DMU119900can be obtained Using the following
index which indicates the increment in efficiency of a unitby moving from peer appraisal to self-appraisal it is nowpossible to rank units Note that the less the magnitude of
12 Journal of Applied Mathematics
this index is the better rank for corresponding unit will beobtained
FPIIBM119870
=
(TEBCC119896
minus STDIBM119896
)
(STDIBM119896
)
(56)
where TEBCC119896
and STDIBM119896
are respectively efficiency in BCCmodel and normalization of TEIBM of DMU
119896 Chen [52]
provided a paper for ranking efficient and inefficient unitsin DEA They noted that the evaluation of efficient unitsis based upon the alterations in efficiency of all inefficientunits by omitting in reference set At first solve the followingmodel which measures the efficiency of DMU
119886when DMU
119887
is eliminated from the reference set
min 120588119886119887
= 120579119904
119886minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1119895 = 119887
120582119895119909119894119895+ 119904
minus
119894= 120579
119904
119886119909119894119886 119894 = 1 119898
119899
sum
119895=1119895 = 119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
119899
sum
119895=1119895 = 119887
= 1
120579119904
119886ge 0 119904
minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119887
(57)
Then again solve the previous model this time for calculat-ing the efficiency score of each inefficient unit when each ofefficient units in turn is eliminated from the reference set 120578lowast
119886
and calculate the efficiency change
120591119886119887
= 120588lowast
119886119887minus 120578
lowast
119886 (58)
Then calculate the following index called as ldquoMCDErdquo indexfor each efficient and inefficient unit
119864119886= sum
119887isin119881119864
119882lowast
119887120588lowast
119886119887 119864
119887= sum
119887isin119881119868
119882lowast
119887120588lowast
119886119887 (59)
Now in accordance with themagnitude of the acquired indexit is possible to rank unitsThose units with higher score havebetter ranking order
7 Ranking Techniques by MultivariateStatistics in the DEA
As described in DEA literature in DEA technique frontier istaken into consideration rather than central tendency consid-ered in regression analysis DEA technique considers that anenvelope encompasses through all the observations as tight aspossible and does not try to fit regression planes in center ofdata In DEA methodology each unit is considered initiallyand compared to the efficient frontier but in regressionanalysis a procedure is considered in which a single functionfits to the data DEA uses different weights for differentunits but does not let the units use weights of other units
Canonical correlation analysis as Friedman and Sinuany-Stern [53] noted can be used for ranking units This methodis somehow the extension of regression analysis Adler et al[89] The aim in canonical correlation analysis is to find asingle vector commonweight for the inputs and outputs of allunits Consider119885
119895119882
119895as the composite input and output and
as corresponding weights respectively The presentedmodel by Tatsuoka and Lohnes [100] is as follows
max 119903119911119908
=
119878119909119910119880
(119878119909119909119881) (119878
119910119910119880)
(12)
st 119909119909119881 = 1
119910119910119880 = 1
(60)
where 119878119909119909 119878
119910119910 and 119878
119909119910are respectively defined as the matri-
ces of the sums of squares and sums of products of thevariables
Friedman and Sinuany-Stern [53] while defining the ratioof linear combinations of the inputs and outputs 119879
119895=
119882119895119885
119895used canonical correlation analysis As they noted that
scaling ratio 119879119895of the canonical correlation analysisDEA is
unboundedSinuany-Stern et al [101] used linear discriminant analy-
sis for ranking units They defined
119863119895=
119904
sum
119903=1
119906119903119910119903119895+
119898
sum
119894=1
V119903(minus119909
119894119895) (61)
DMU119895is said to be efficient if 119863
119895gt 119863
119888where 119863
119888is a critical
value based on themidpoint of themeans of the discriminantfunction value of the two groups Morrison [102] The largerthe amount of119863
119895is the better rank DMU
119895will have
Friedman and Sinuany-Stern [53] noted that as cross-efficiencyDEA canonical correlation analysisDEA and dis-criminant analysisDEA ranking orders may vary from eachother thus it seems necessary to introduce the combinedranking (CODEA) Combined ranking for each unit con-sidered all the ranks obtained from the above-mentionedrankings Moreover statistical tests are one for goodness offit between DEA and a specific ranking and the other fortesting correlation between variety of ranking orders Siegeland Castellan [103]
8 Ranking with MulticriteriaDecision-Making (MCDM) Methodologiesand DEA
Li and Reeves [56] for increasing discrimination power ofDEA presented amultiple-objective linear program (MOLP)As the authors mentioned using minimax and minsumefficiency in addition to the standard DEA objective functionhelp to increase discrimination power of DEA
Strassert and Prato [59] presented the balancing andranking method which uses a three-step procedure forderiving an overall complete or partial final order of optionsIn the first step derive an outranking matrix for all options
Journal of Applied Mathematics 13
from the criteria values Considering this matrix it is possibleto show the frequencywithwhich one option is ranked higherthan the other options In the second step by triangularizingthe outranking matrix establish an implicit preordering orprovisional ordering of optionsTheoutrankingmatrix showsthe degree to which there is a complete overall order ofoptions In the third step based on information given in anadvantages-disadvantages table the provisional ordering issubjected to different screening and balancing operations
Chen [60] utilized a nonparametric approach DEA toestimate and rank the efficiency of association rules withmultiple criteria in following steps Proposed postprocessingapproach is as follows
Step 1 Input data for association rule mining
Step 2 Mine association rules by using the a priori algorithmwith minimum support and minimum confidence
Step 3 Determine subjective interestingness measures byfurther considering the domain related knowledge
Step 4 Calculate the preference scores of association rulesdiscovered in Step 2 by using Cook and Kressrsquos DEA model
Step 5 Discriminate the efficient association rules found inStep 3 by using Obata and Ishiirsquos [104] discriminate model
Step 6 Select rules for implementation by considering thereference scores generated in Step 5 and domain relatedknowledge
Jablonsky [61] presented two original models super SBMand AHP for ranking of efficient units in DEA As theauthormentioned thesemodels are based onmultiple criteriadecision-making techniques-goal programming and analytichierarchy process Super SBM model for ranking units is asfollows
min 120579119866
119902= 1 + 119905120574 + (1 minus 119905) (
sum119898
119894=1119904+
1119894
119909119894119902
+
sum119898
119894=1119904minus
2119896
119910119896119902
)
st119899
sum
119895=1119895 = 119902
120582119895119909119894119895+ 119904
minus
1119894minus 119904
+
1119894= 119909
119894119902 i = 1 119898
119899
sum
119895=1119895 = 119902
120582119895119910119896119895+ 119904
minus
2119896minus 119904
+
2119896= 119910
119896119902 119896 = 1 119903
119904+
1119894le 120574 119894 = 1 119898
119904minus
2119896le 120574 119896 = 1 119903
119905 isin 0 1 120582119895ge 0 119878
+
1 119878
+
2ge 0 119878
minus
1 119878
minus
2ge 0
119895 = 1 119899
(62)
Wang and Jiang [62] presented an alternative mixedinteger linear programming models in order to identifythe most efficient units in DEA technique As the authors
mentioned presented models can make full use of input-output information with no need to specify any assuranceregions for input and output weights to avoid zero weights
min119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) minus
119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895)
st119904
sum
119903=1
119906119903119910119903119895minus
119898
sum
119894=1
V119894119909119894119895le 119868
119895 119895 = 1 119899
119899
sum
119895=1
119868119895= 1
119868119895isin 0 1 119895 = 1 119899
119906119903ge
1
(119898 + 119904)max119895119910
119903119895
119903 = 1 119904
V119894ge
1
(119898 + 119904)max119895119909
119894119895
119894 = 1 119898
(63)
Hosseinzadeh Lotfi et al [63] provided an improved three-stage method for ranking alternatives in multiple criteriadecision analysis In the first stage based on the bestand worst weights in the optimistic and pessimistic casesobtain the rank position of each alternative respectively Theobtained weight in the first stage is not unique thus it seemsnecessary to introduce a secondary goal that is used in thesecond stage Finally in the third stage the ranks of thealternatives compute in the optimistic or pessimistic case Itis mentionable that the model proposed in the third stageis a multicriteria decision-making (MCDM) model and itis solved by mixed integer programming As the authorsmentioned and provided in their paper this model can beconverted into an LP problem
The previous model with 119897 = 1 is the original CCR modelNote that DMUs in 119864
1 show the first level efficient frontier119897 = 2 indicates the second level efficient frontier when thefirst level efficient frontier is omittedThe following algorithmfinds these efficient frontiers
Step 1 Set 119897 = 1 and evaluate the entire set of DMUs 1198691 Inthis way the first level efficient DMU 1198641 is identified
Step 2 Exclude the efficient DMUs from future DEA runsand set 119869119897+1 = 119869
119897minus 119864
119897 (If 119869119897+1 = oslash stop)
Step 3 Evaluate the new subset of ldquoinefficientrdquo DMUs 119869119897+1to obtain a new set of efficient DMUs 119864119897+1
Step 4 Let l = l + 1 Go to Step 2
When 119869119897+1 = oslash the algorithm stopsAs the authors proved in this way it is possible to rank
the DMUs in the first efficient frontier based upon theirattractiveness scores and identify the best one
Jahanshahloo et al [65] provided a paper for rankingunits using gradient line As the authors mentioned theadvantage of this model is stability and robustness
supports on 119879119888 the obtained PPS by omitting theDMUunder
assessment in 119879119888 As the authors proved a unit is efficient
iff the optimal objective function of the following model isgreater than zero Consider
1198750= (119883 119884) 119883 = 120572119883
119900 119884 = 120573119884
119900
1198780= (119883 119884) isin 1198750
119883 = 120572119883119900 119884 = 120573119884
119900 120572 ge 0 120573 ge 0
(67)
Intersection of 1198780and efficient surface of
119879119888119888 is a half line
where its equation is (minus119881lowast119879119883
119900)120572 + (119880
lowast119879119884119900)120573 = 0 To rank
DMU119900the length of connecting arc DMU
119900with intersection
point of line and previous ellipse is calculated in (120572 120573) spaceThis intersection is as follows
120572lowast= (
1198702
120572119870
2
120573
1198702
120573+ (119881
lowast119879119883
119900119880
lowast119879119884119900)2119870
2
120572
)
12
120573lowast= (
119881lowast119879119883
119900
119880lowast119879119884119900
)120572lowast
(68)
Now the length of connecting arc DMU119900to the point
corresponding to 120572lowast 120573lowast in (120572 120573) plan is calculated as follows119868 = int
120572lowast
1(1 + 120573)
12119889120572 where
119870120572= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119899
119895=11199092
119894119900
)
12
119870120573= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119904
119903=11199102
119903119900
)
12
(69)
Jahanshahloo et al [66] also provided a paper with theconcept of advantage in data envelopment analysis
In their paper Jahanshahloo et al [67] consideringMonteCarlo method presented a new method of ranking
Step 1 Generate a uniformly distributed sequence of 1198801198952119899
119895=1
on(0 1)
Step 2 Random numbers should be classified into 119873 pairslike (119880
1 119880
1015840
1) (119880
119873 119880
1015840
119873) in a way that each number is used
just one time
Step 3 Compute119883119894= 119886 + 119880
119894(119887 minus 119886) and 119891(119883
119894) gt 119888119880
1015840
119894
Step 4 Estimate the integral 119868 by 120579119868= 119888(119887 minus 119886)(119873
119867119873)
Now consider DMU119900as an efficient unit measure those
units that are dominated DMU119900 As for dome DMUs this
would be unbounded thus the authors for each unitbounded the region Then for DMU
119900if (minus119883
119900 119884
119900) ge (minus119883 119884)
then (minus119883 119884) is in the RED of DMU119900 Now by using 119881
119901=
119881lowast(119873
119867119873) all the hits (the above condition) can be counted
where 119881lowast is the measure of the whole region JahanshahlooandAfzalinejad [68] presented amethod based on distance of
Journal of Applied Mathematics 15
the unit under evaluation to the full inefficient frontier Theypresented two models radian and nonradial
min 120601
st119899
sum
119895=1
120582119895119909119894119895= 120601119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
120582119895ge 0 119895 = 1 119898
max119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903
st sum
119895isin119869minus119887
120582119895119909119894119895minus 119904
minus
119894= 119909
119894119900 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895+ 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
119904minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(70)
Amirteimoori [69] based on the same idea considered bothefficient and antiefficient frontiers for efficiency analysis andranking units
Kao [70] mentioned that determining the weights ofindividual criteria in multiple criteria decision analysis in away that all alternatives can be compared according to theaggregate performance of all criteria is of great importanceAs Kao noted this problem relates to search for alternativeswith a shorter and longer distance respectively to the idealand anti-ideal units He proposed a measure that consideredthe calculation of the relative position of an alternativebetween the ideal and anti-ideal for finding an appropriaterankings
Khodabakhshi and Aryavash [71] presented a method forranking all units using DEA concept
First Compute the minimum and maximum efficiencyvalues of eachDMU in regard to this assumption that the sumof efficiency values of all DMUs equals to 1
Second determine the rank of each DMU in relation to acombination of itsminimumandmaximumefficiency values
Zerafat Angiz et al [72] proposed a technique in orderto aggregate the opinions of experts in voting system Asthe authors mentioned the presented method uses fuzzyconcept and it is computationally efficient and can fullyrank alternatives At first number of votes given to a rankposition was grouped to construct fuzzy numbers and thenthe artificial ideal alternative introduced Furthermore byperforming DEA the efficiency measure of alternatives was
obtained considering artificial ideal alternative comparedby each of the alternatives pair by pair Thus alternativesare ranked in accordance with their efficiency scores Ifthis method cannot completely rank alternatives weightrestrictions based on fuzzy concept are imposed into theanalysis
10 Different Applications in Ranking Units
As discussed formerly there exist a variety of ranking meth-ods and applications in theliterature Nowadays DEAmodelsare widely used in different areas for efficiency evaluationbenchmarking and target setting ranking entities and soforth Ranking units as one of the important issues in DEAhas been performed in different areas Jahanshahloo et al [42]ranked cities in Iran to find the best place for creating a datafactory Hosseinzadeh Lotfi et al [28] used a new methodfor ranking in order to find the best place for power plantlocation Ali andNakosteen [80] Amirteimoori et al [37] andAlirezaee and Afsharian [24] Soltanifar and HosseinzadehLotfi [105] Zerafat Angiz et al [72] Hosseinzadeh Lotfiet al [28] Jahanshahloo [50] Chen and Deng [52] andJablonsky [61] used different ranking methods in bankingsystem Sadjadi [40] ranked provincial gas companies in IranMehrabian et al [32] Li et al [39] Orkcu and Bal [12] andWu et al [19] ranked different departments of universitiesJahanshahloo et al [35] Jahanshahloo and Afzalinejad [68]utilized the presented models in ranking 28 Chinese citiesJahanshahloo at al [35] provided an application to burdensharing amongst NATOmember nations Jahanshahloo et al[14] utilized the presented model for ranking nursery homesContreras [18] andWang et al [23] used the provided rankingtechniques in ranking candidates Lu and Lo [51] applied theirmethod on an application to financial holding companiesHosseinzadeh Lotfi et al [63] consider an empirical exampleinwhich voters are asked to rank twoout of seven alternativesWang and Jiang [62] utilized the provided method in facilitylayout design in manufacturing systems and performanceevaluation of 30OECD countries Chen [60] used an exampleof market basket data in order to illustrate the providedapproach
11 Application
In this section some of the reviewed models are applied onthe example used in Soltanifar and Hosseinzadeh Lotfi [105]Consider twenty commercial banks of Iran with input-outputdata tabulated in Table 1 and summarized as follows Also theresults of CCRmodel are listed in this table As it can be seenseven units are efficient DMUS 14 7 12 15 17 and 20 Inputsare staff computer terminal and spaceOutputs are depositsloans granted and charge
Consider some of the important ranking methods inliterature as follows
RM1 AP model [31] is based upon the idea of leaveunit evaluation out and measuring the distance of theunit under evaluation from the production possibilityset constructed by the remaining DMUs
RM2MAJmodel [32] presented for ranking efficientwhich is always stable but might be infeasible in somecasesRM3 Modified MAJ model [38] overcomes theproblem which might occurr in MAJ modelRM4 A new model based on the idea of alterationsin the reference set of the inefficient units [50]RM5 A model presented by Li et al [39] whichis a super-efficiency method that does not have thesuffering in previous methodsRM6 Slack-basedmodel [33] is based upon the inputand output variables at the same timeRM7 SA DEAmodel [49] overcomes the problem ofinfeasibility which existed in AP modelRM8 Cross-efficiency [10] is provided based onusing weights of each unit under evaluation in opti-mality for other unitsRM9 A model based on finding common set ofweights [25] which determine the common set ofweights for DMUs and ranked DMUs based this ideaRM10 A model based on finding common set ofweights [22 67] for ranking efficient unitsRM11 L1-norm model [34 35 65 66] the idea isbased upon the leave-one-out efficient unit and l1-norm which is always feasible and stableRM12 119871
infin-norm model Rezai Balf et al [46] pro-
vided a method with more ability over other existingmethods based on Tchebycheff normRM13 An enhanced Russel measure of super-efficiency model for ranking units [44]RM14 A rankingmodel which considers the distanceof unit from the full inefficient frontier [38]RM15 Amodified super-efficiencymodel [45] whichovercomes the infeasibility that may happen in prob-lem This model is based on simultaneous projectionof input output
In accordance with properties of different ranking models inorder to rank efficient units consider Table 2
1199011 Feasibility
1199012 Ranking extreme efficient units
1199013 Complexity in computation
1199014 Instability
1199015 Absence of multiple optimal solution
1199016 Dependency to 120579 and slacks
1199017 Dependency to the number of efficient and ineffi-
cient units
In Tables 3 and 4 the rank of efficient units consideringthe above mentioned methods is listed Note that ED showsefficient DMUs (ED) and RM
119895 (119895 = 1 15) are those
explained previously
12 Conclusion
In this paper the DEA ranking was reviewed and classifiedinto seven general groups In the first group those papersbased on a cross-efficiencymatrix were reviewed In this fieldDMUs have been evaluated by self- and peer pressure Thesecond group of papers is based on those papers lookingfor optimal weights in DEA analysis The third one is thesuper-efficiency method By omitting the under evaluationunit and constructing a new frontier by the remaining unitsthe unit under evaluation can get an equal or greater scoreThe fourth group is based on benchmarking idea In thisclass the effect of an efficient unit considered as a targetfor inefficient units is investigated This idea is very usefulfor the managers in decision making Another class thefifth one involves the application of multivariate statisticaltools The sixth section discusses the ranking methods basedon multicriteria decision-making (MCDM) methodologiesand DEA The final section includes some various rankingmethods presented in the literature Finally in an applicationthe result of some of the above-mentioned ranking methodsis presented
References
[1] M J Farrell ldquoThe measurement of productive efficiencyrdquoJournal of the Royal Statistical Society A vol 120 pp 253ndash2811957
[2] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
18 Journal of Applied Mathematics
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
previous models Finally they studied the robustness anddiscriminating power of the proposed method by Spearmanrsquosrank correlation coefficient
Hosseinzadeh Lotfi et al [28] proposed one DEA rankingmodel based on applying aggregate units In doing so artificialunits called aggregate units are defined as follows Theaggregate unit is shown by DMU
119886
119909119901
119894119886= sum
119896isin119877119901
119909119894119896 119910
119901
119903119886= sum
119896isin119877119901
119910119903119896
119894 = 1 119898 119903 = 1 119904
(25)
where 119877119901= 119895 | DMU
119895isin 119864
119901 119864
119901= 119864DMU
119901 Note that 119864
is the set of efficient unitsFirst it is tried to maximize the efficiency score of the
DMU119886and then to maximize the efficiency score of the
DMU119901
119886 For resolving the existence of alternative solutions
the authors presented an approach comprising (119898+119904) simplelinear problems to achieve the most appropriate optimalsolutions among all alternative optimal solutions Finallythey proposed theRI index for ranking all efficientDMUs Let119880119886 119881
119875119886 and119880119901
119886 119881
119901
119886be the optimal solutions of the multiplier
model for assentDMU119886andDMU119901
119886 Consider 120578
119886= sum
119904
119903=1119906119903119886minus
sum119898
119894=1V119894119886 120578
119901
119886= sum
119904
119903=1119906119901
119903119886minus sum
119898
119894=1V119901119894119886 thus RI
119901= 120578
119900
119886minus 120578
119886
Wang et al [29] presented two nonlinear regressionmodels for deriving common set of weights for fully rankunits
min 119911 =
119899
sum
119895=1
(120579lowast
119895minus
sum119904
119903=1119906119903119910119903119895
sum119898
119894=1V119894119909119894119895
)
2
st 119880 119881 ge 0
min119899
sum
119895=1
(
119904
sum
119903=1
119906119903119910119903119895minus 120579
lowast
119895
119898
sum
119894=1
V119894119909119894119895)
2
st119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895) +
119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) = 119899
119880 119881 ge 0
(26)
Ramon et al [30] aimed at deriving a common set of weightsfor ranking units As the authors mentioned the idea is basedupon minimization of the deviations of the common weightsfrom the nonzero weights obtained fromDEA Furthermoreseveral norms are used for measuring such differences
5 Super-Efficiency Ranking Techniques
Super efficiency models introduced in DEA technique arebased upon the idea of leave one out and assessing this unittrough the remanding units
Andersen and Petersen [31] introduced a model for rank-ing efficient units The proposed model is as follows
Although this idea is useful for further discriminatingefficient units it has been shown in the literature that itmay be infeasible and nonstable Thrall [7] mentioned theinfeasibility of super-efficiency CCRmodel Also a conditionunder which infeasibility occurred in super-efficiency DEAmodels is mentioned by Zhu [95] Seiford and Zhu [96] andDula and Hickman [97]
Hashimoto [98] provided a model based on the idea ofone leave out and assurance region for ranking units
Mehrabian et al [32] presented a complete ranking forefficiency units in DEA context As the authors mentionedthis model does not have difficulties of AP model
min 119908119901+ 1
st119899
sum
119895=1 119895 = 119901
120582119895119909119894119895le 119909
119894119901+ 119908
1199011 119894 = 1 119898
119899
sum
119895=1 119895 = 119901
120582119895119910119903119895ge 119910
119903119901 119903 = 1 119904
120582119895ge 0 119895 = 119901
(28)
With this method it is not possible to rank nonextremeefficient units
Tone [33] presented super efficiency of SBM model Thismodel has the advantages of nonradial models and it isalways feasible and stable
minsum
119898
119894=1119909119894119909
119894119900
sum119904
119903=1119910119903119910
119903119900
st119899
sum
119895=1119895 = 119900
120582119895119909119894119895le 119909
119894 119894 = 1 119898
119899
sum
119895=1119895 = 119900
120582119895119910119903119895ge 119910
119903 119903 = 1 119904
119909119894ge 119909
119894119900 0 le 119910
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119900
(29)
8 Journal of Applied Mathematics
Jahanshahloo et al [34] added some ratio constraints to themultiplier form of AP model and introduced a new methodfor ranking DMUs
In their paper Chen and Sherman [36] presented a non-radial super-efficiency method and discussed the advantageof it They verified that this model is invariant to units ofinputoutput measurement Let 119869119900 = 119869DMU
119900
Step 1 Solve the followingmodel to find the extreme efficientunits in 119869119900
This method cannot rank nonextreme efficient units
Jahanshahloo et al [38] presented modified MAJ modelfor ranking efficiency units in DEA technique
min 119908119901+ 1
st119899
sum
119895=1119895 = 119901
120582119895
119909119894119895
119872119894
le
119909119894119901
119872119894
+ 1199081199011 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895ge 119910
119903119901 119903 = 1 119904
120582119895ge 0 119895 = 119901
(39)
where 119872119894= Max 119909
119894119895| DMU
119895is efficient It cannot rank
nonextreme efficient unitsLi et al [39] presented a new method for ranking which
does not have difficulties of earlier methods The presentedmodel is always feasible and stable
min 1 +
1
119898
119898
sum
119894=1
119904+
1198942
119877minus
119894
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895+ 119904
minus
1198941minus 119904
+
1198942= 119909
119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895minus 119904
+
119903= 119910
119903119901 119903 = 1 119904
119904minus
1198942ge 0 119904
minus
1198941ge 0 119904
+
119903ge 0 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 119901
(40)
In this case extreme efficient units cannot be ranked
In a paper Khodabakhshi [99] addresses super efficiencyon improved outputs He mentioned that as AP modelmay be infeasible under variable returns to scale technol-ogy using the presented model gives a complete rankingwhen getting an input combination for improving outputs issuitable
Sadjadi et al [40] presented a robust super-efficiencyDEA for ranking efficient units They noted that as inmost of the times exact data do not exist and the sto-chastic super-efficiency model presented in their paperincorporates the robust counterpart of super-efficiencyDEA
where 119883119884 are input-output data This model does notrank nonextreme efficient units It may be unstable andinfeasible
Gholam Abri et al [41] proposed a model for rankingefficient units They used representation theory and rep-resented the DMU under assessment as a convex combi-nation of extreme efficient units As the authors noted itis expected that the performance of DMU
119900is the same
as the performance of convex combination of extremeefficient units Thus it is possible to represent DMU
119900as
follows
(119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119899
sum
119895=1
120582119895= 1 119895 = 1 119904 (42)
As regards representation theorem this system has119898 + 119904 minus 1
constraints and 119904 variables (1205821 120582
119904) If this system has a
10 Journal of Applied Mathematics
unique solution we will have 120579lowast119900= sum
119899
119895=1= 1 120582
lowast
119895120579119895 otherwise
two models should be considered
min119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
max119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
(43)
Consider 1205791and 120579
2as the optimal solution of the above-
mentioned models respectively If 1205791= 120579
2 this is the same
as what has been mentioned previously If 1205791lt 120579
2 then
mentionedmodels provide an interval which helps rank unitsfrom the worst to the best Sometimes it will obtain theranking score with a bounded interval [120579
1 120579
2]
Jahanshahloo et al [42] presented models for rankingefficient units The presented models are somehow a modifi-cation of cross-efficiency model that overcomes the difficultyof alternative optimal weightsThe authors in their paperwithregard to the changes and also utilizing TOPSIS techniquepresented a new super-efficientmethod for ranking unitsThepresented model is as follows
120579119894119895=
119880119905119884119894
119881119905119883
119894
|
119880119905119884119897
119881119905119883
119897
le 1
119880119905119884119895
119881119905119883
119895
= 120579119895119895
119897 = 1 119899 119897 = 119894 119880 ge 0 119881 ge 0
(44)
where 120579119895119895is the efficiency score of DMU
119869using correspond-
ing weights Also 120579119894119895is the efficiency of DMU
119894using optimal
weights of DMU119895 Noura et al [43] provided a method for
ranking efficient units based on this idea that more effectiveand useful units in society should have better rank
Step 1 For each efficient unit choose the lower and upperlimit for each inputs and outputs Let 119864 be the set of efficientunits
where 120585 is representative of a small and nonzero number usedfor not dividing by zero Now consider119863minus119895 as follows whichshows that more successful DMU
119895will be if the larger value
of the119863119895is
119863119895= sum
119894isin119868
119889119894119895+ sum
119903isin119877
119889119903119895 (48)
where 119868 = 119863+
119894cup 119863
minus
119894 119877 = 119863
+
119903cup 119863
minus
119903
Ashrafi et al [44] introduced an enhanced Russell mea-sure of super efficiency for ranking efficient units inDEAThelinear counterpart of the proposed model is as follows
119900 le 120573 le V119903ge 120573 119903 = 1 119904
(49)
It cannot rank nonextreme efficient units
Journal of Applied Mathematics 11
Chen et al [45] proposed a modified super-efficiencymethod for ranking units based on simultaneous input-output projectionThe presentedmodel overcomes the infea-sibility problem
1199011= min
120579119904119903
119900
120601119904119903
119900
st sum
119895=1
119895 = 119900
120582119895119909119894119895le 120579
119904119903
119900119909119894119900 119894 = 1 119898
sum
119895=1
119895 = 119900
120582119895119910119903119895ge 120601
119904119903
119900119910119903119900 119903 = 1 119904
sum
119895=1
119895 = 119900
120582119895= 1
0 lt 120579119904119903
119900le 1 120601
119904119903
119900ge 1 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(50)
Rezai Balf et al [46] provide a model for ranking units basedon Tchebycheff norm As proved that this model is alwaysfeasible and stable it seems to have superiority over othermodels
max 119881119901
st 119881119901ge
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901 119894 = 1 119898
119881119901ge 119910
119903119901minus
119899
sum
119895=1
119895 = 119900
120582119895119910119903119895 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(51)
where
119881119901= Max
(
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901)119894 = 1 119898
(119910119903119901minus
119899
sum
119895=1
119895 = 119901
120582119895119910119903119895)119903 = 1 119904
(52)
Chen et al [47] for overcoming the infeasibility problem thatoccurred in variable returns to scale super-efficiency DEAmodel according to a directional distance function developedNerlove-Luenberger (N-L) measure of super-efficiency
6 Benchmarking Ranking Techniques
Sueyoshi et al [49] proposes a ldquobenchmark approachrdquo forbaseball evaluation This method is the combination of DEA
and (Offensive earned-run average) OERA As the authorsnoted using this method it is possible to select best unitsand also their ranking orders They mentioned that usingonly DEAmay result in a shortcoming in assessment as manyefficient units can be identifiedThus the authors used slack-adjusted DEA model and OERA to overcome this difficulty
Jahanshahloo et al [50] presented a new model forranking DMUs based on alteration in reference set The ideais based on this fact that efficient units can be the target unitfor inefficient units
min 120597119886119887
= 120579 minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st sum
119895isin119869minus119887
120582119895119909119894119895+ 119904
minus
119894minus 120579119909
119894119886 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
120579 free 119904minus119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 isin 119869 minus 119887
(53)
Finally the ranking order for each efficient unit b can becomputing by
119887= sum
119886isin119869119899
120597119886119887
119899
(54)
Note that this method cannot rank nonextreme efficientunits
Lu and Lo [51] provided an interactive benchmark modelfor ranking units The idea is based upon considering a fixedunit as a benchmark and calculating the efficiency of otherunits pair by pair to this unitThis procedure continueswhenall units are accounted for as a benchmark unit ConsiderDMU
119900as a unit under evaluation andDMU
119887as a benchmark
Assume
119909119894= 119909
119894119900(1 + 120601
119894) 119910
119903= 119910
119903119900(1 minus 120593
119903)
min 120579lowast119887
119900=
1 + (1119898)sum119898
119894=1120601119894
1 minus (1119904)sum119904
119903=1120593119903
st 120582119887119909119894119887minus 119909
119894119900120601119894le 119909
119894119900 119894 = 1 119898
120582119887119910119903119887minus 119910
119903119900120593119903ge 119910
119903119900 119903 = 1 119904
120601119894ge 119900 120593
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(55)
Then using (sum119899
119887=1120579119887lowast
119900)119899 119900 = 1 119899 the efficiency of
benchmark DMU119900can be obtained Using the following
index which indicates the increment in efficiency of a unitby moving from peer appraisal to self-appraisal it is nowpossible to rank units Note that the less the magnitude of
12 Journal of Applied Mathematics
this index is the better rank for corresponding unit will beobtained
FPIIBM119870
=
(TEBCC119896
minus STDIBM119896
)
(STDIBM119896
)
(56)
where TEBCC119896
and STDIBM119896
are respectively efficiency in BCCmodel and normalization of TEIBM of DMU
119896 Chen [52]
provided a paper for ranking efficient and inefficient unitsin DEA They noted that the evaluation of efficient unitsis based upon the alterations in efficiency of all inefficientunits by omitting in reference set At first solve the followingmodel which measures the efficiency of DMU
119886when DMU
119887
is eliminated from the reference set
min 120588119886119887
= 120579119904
119886minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1119895 = 119887
120582119895119909119894119895+ 119904
minus
119894= 120579
119904
119886119909119894119886 119894 = 1 119898
119899
sum
119895=1119895 = 119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
119899
sum
119895=1119895 = 119887
= 1
120579119904
119886ge 0 119904
minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119887
(57)
Then again solve the previous model this time for calculat-ing the efficiency score of each inefficient unit when each ofefficient units in turn is eliminated from the reference set 120578lowast
119886
and calculate the efficiency change
120591119886119887
= 120588lowast
119886119887minus 120578
lowast
119886 (58)
Then calculate the following index called as ldquoMCDErdquo indexfor each efficient and inefficient unit
119864119886= sum
119887isin119881119864
119882lowast
119887120588lowast
119886119887 119864
119887= sum
119887isin119881119868
119882lowast
119887120588lowast
119886119887 (59)
Now in accordance with themagnitude of the acquired indexit is possible to rank unitsThose units with higher score havebetter ranking order
7 Ranking Techniques by MultivariateStatistics in the DEA
As described in DEA literature in DEA technique frontier istaken into consideration rather than central tendency consid-ered in regression analysis DEA technique considers that anenvelope encompasses through all the observations as tight aspossible and does not try to fit regression planes in center ofdata In DEA methodology each unit is considered initiallyand compared to the efficient frontier but in regressionanalysis a procedure is considered in which a single functionfits to the data DEA uses different weights for differentunits but does not let the units use weights of other units
Canonical correlation analysis as Friedman and Sinuany-Stern [53] noted can be used for ranking units This methodis somehow the extension of regression analysis Adler et al[89] The aim in canonical correlation analysis is to find asingle vector commonweight for the inputs and outputs of allunits Consider119885
119895119882
119895as the composite input and output and
as corresponding weights respectively The presentedmodel by Tatsuoka and Lohnes [100] is as follows
max 119903119911119908
=
119878119909119910119880
(119878119909119909119881) (119878
119910119910119880)
(12)
st 119909119909119881 = 1
119910119910119880 = 1
(60)
where 119878119909119909 119878
119910119910 and 119878
119909119910are respectively defined as the matri-
ces of the sums of squares and sums of products of thevariables
Friedman and Sinuany-Stern [53] while defining the ratioof linear combinations of the inputs and outputs 119879
119895=
119882119895119885
119895used canonical correlation analysis As they noted that
scaling ratio 119879119895of the canonical correlation analysisDEA is
unboundedSinuany-Stern et al [101] used linear discriminant analy-
sis for ranking units They defined
119863119895=
119904
sum
119903=1
119906119903119910119903119895+
119898
sum
119894=1
V119903(minus119909
119894119895) (61)
DMU119895is said to be efficient if 119863
119895gt 119863
119888where 119863
119888is a critical
value based on themidpoint of themeans of the discriminantfunction value of the two groups Morrison [102] The largerthe amount of119863
119895is the better rank DMU
119895will have
Friedman and Sinuany-Stern [53] noted that as cross-efficiencyDEA canonical correlation analysisDEA and dis-criminant analysisDEA ranking orders may vary from eachother thus it seems necessary to introduce the combinedranking (CODEA) Combined ranking for each unit con-sidered all the ranks obtained from the above-mentionedrankings Moreover statistical tests are one for goodness offit between DEA and a specific ranking and the other fortesting correlation between variety of ranking orders Siegeland Castellan [103]
8 Ranking with MulticriteriaDecision-Making (MCDM) Methodologiesand DEA
Li and Reeves [56] for increasing discrimination power ofDEA presented amultiple-objective linear program (MOLP)As the authors mentioned using minimax and minsumefficiency in addition to the standard DEA objective functionhelp to increase discrimination power of DEA
Strassert and Prato [59] presented the balancing andranking method which uses a three-step procedure forderiving an overall complete or partial final order of optionsIn the first step derive an outranking matrix for all options
Journal of Applied Mathematics 13
from the criteria values Considering this matrix it is possibleto show the frequencywithwhich one option is ranked higherthan the other options In the second step by triangularizingthe outranking matrix establish an implicit preordering orprovisional ordering of optionsTheoutrankingmatrix showsthe degree to which there is a complete overall order ofoptions In the third step based on information given in anadvantages-disadvantages table the provisional ordering issubjected to different screening and balancing operations
Chen [60] utilized a nonparametric approach DEA toestimate and rank the efficiency of association rules withmultiple criteria in following steps Proposed postprocessingapproach is as follows
Step 1 Input data for association rule mining
Step 2 Mine association rules by using the a priori algorithmwith minimum support and minimum confidence
Step 3 Determine subjective interestingness measures byfurther considering the domain related knowledge
Step 4 Calculate the preference scores of association rulesdiscovered in Step 2 by using Cook and Kressrsquos DEA model
Step 5 Discriminate the efficient association rules found inStep 3 by using Obata and Ishiirsquos [104] discriminate model
Step 6 Select rules for implementation by considering thereference scores generated in Step 5 and domain relatedknowledge
Jablonsky [61] presented two original models super SBMand AHP for ranking of efficient units in DEA As theauthormentioned thesemodels are based onmultiple criteriadecision-making techniques-goal programming and analytichierarchy process Super SBM model for ranking units is asfollows
min 120579119866
119902= 1 + 119905120574 + (1 minus 119905) (
sum119898
119894=1119904+
1119894
119909119894119902
+
sum119898
119894=1119904minus
2119896
119910119896119902
)
st119899
sum
119895=1119895 = 119902
120582119895119909119894119895+ 119904
minus
1119894minus 119904
+
1119894= 119909
119894119902 i = 1 119898
119899
sum
119895=1119895 = 119902
120582119895119910119896119895+ 119904
minus
2119896minus 119904
+
2119896= 119910
119896119902 119896 = 1 119903
119904+
1119894le 120574 119894 = 1 119898
119904minus
2119896le 120574 119896 = 1 119903
119905 isin 0 1 120582119895ge 0 119878
+
1 119878
+
2ge 0 119878
minus
1 119878
minus
2ge 0
119895 = 1 119899
(62)
Wang and Jiang [62] presented an alternative mixedinteger linear programming models in order to identifythe most efficient units in DEA technique As the authors
mentioned presented models can make full use of input-output information with no need to specify any assuranceregions for input and output weights to avoid zero weights
min119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) minus
119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895)
st119904
sum
119903=1
119906119903119910119903119895minus
119898
sum
119894=1
V119894119909119894119895le 119868
119895 119895 = 1 119899
119899
sum
119895=1
119868119895= 1
119868119895isin 0 1 119895 = 1 119899
119906119903ge
1
(119898 + 119904)max119895119910
119903119895
119903 = 1 119904
V119894ge
1
(119898 + 119904)max119895119909
119894119895
119894 = 1 119898
(63)
Hosseinzadeh Lotfi et al [63] provided an improved three-stage method for ranking alternatives in multiple criteriadecision analysis In the first stage based on the bestand worst weights in the optimistic and pessimistic casesobtain the rank position of each alternative respectively Theobtained weight in the first stage is not unique thus it seemsnecessary to introduce a secondary goal that is used in thesecond stage Finally in the third stage the ranks of thealternatives compute in the optimistic or pessimistic case Itis mentionable that the model proposed in the third stageis a multicriteria decision-making (MCDM) model and itis solved by mixed integer programming As the authorsmentioned and provided in their paper this model can beconverted into an LP problem
The previous model with 119897 = 1 is the original CCR modelNote that DMUs in 119864
1 show the first level efficient frontier119897 = 2 indicates the second level efficient frontier when thefirst level efficient frontier is omittedThe following algorithmfinds these efficient frontiers
Step 1 Set 119897 = 1 and evaluate the entire set of DMUs 1198691 Inthis way the first level efficient DMU 1198641 is identified
Step 2 Exclude the efficient DMUs from future DEA runsand set 119869119897+1 = 119869
119897minus 119864
119897 (If 119869119897+1 = oslash stop)
Step 3 Evaluate the new subset of ldquoinefficientrdquo DMUs 119869119897+1to obtain a new set of efficient DMUs 119864119897+1
Step 4 Let l = l + 1 Go to Step 2
When 119869119897+1 = oslash the algorithm stopsAs the authors proved in this way it is possible to rank
the DMUs in the first efficient frontier based upon theirattractiveness scores and identify the best one
Jahanshahloo et al [65] provided a paper for rankingunits using gradient line As the authors mentioned theadvantage of this model is stability and robustness
supports on 119879119888 the obtained PPS by omitting theDMUunder
assessment in 119879119888 As the authors proved a unit is efficient
iff the optimal objective function of the following model isgreater than zero Consider
1198750= (119883 119884) 119883 = 120572119883
119900 119884 = 120573119884
119900
1198780= (119883 119884) isin 1198750
119883 = 120572119883119900 119884 = 120573119884
119900 120572 ge 0 120573 ge 0
(67)
Intersection of 1198780and efficient surface of
119879119888119888 is a half line
where its equation is (minus119881lowast119879119883
119900)120572 + (119880
lowast119879119884119900)120573 = 0 To rank
DMU119900the length of connecting arc DMU
119900with intersection
point of line and previous ellipse is calculated in (120572 120573) spaceThis intersection is as follows
120572lowast= (
1198702
120572119870
2
120573
1198702
120573+ (119881
lowast119879119883
119900119880
lowast119879119884119900)2119870
2
120572
)
12
120573lowast= (
119881lowast119879119883
119900
119880lowast119879119884119900
)120572lowast
(68)
Now the length of connecting arc DMU119900to the point
corresponding to 120572lowast 120573lowast in (120572 120573) plan is calculated as follows119868 = int
120572lowast
1(1 + 120573)
12119889120572 where
119870120572= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119899
119895=11199092
119894119900
)
12
119870120573= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119904
119903=11199102
119903119900
)
12
(69)
Jahanshahloo et al [66] also provided a paper with theconcept of advantage in data envelopment analysis
In their paper Jahanshahloo et al [67] consideringMonteCarlo method presented a new method of ranking
Step 1 Generate a uniformly distributed sequence of 1198801198952119899
119895=1
on(0 1)
Step 2 Random numbers should be classified into 119873 pairslike (119880
1 119880
1015840
1) (119880
119873 119880
1015840
119873) in a way that each number is used
just one time
Step 3 Compute119883119894= 119886 + 119880
119894(119887 minus 119886) and 119891(119883
119894) gt 119888119880
1015840
119894
Step 4 Estimate the integral 119868 by 120579119868= 119888(119887 minus 119886)(119873
119867119873)
Now consider DMU119900as an efficient unit measure those
units that are dominated DMU119900 As for dome DMUs this
would be unbounded thus the authors for each unitbounded the region Then for DMU
119900if (minus119883
119900 119884
119900) ge (minus119883 119884)
then (minus119883 119884) is in the RED of DMU119900 Now by using 119881
119901=
119881lowast(119873
119867119873) all the hits (the above condition) can be counted
where 119881lowast is the measure of the whole region JahanshahlooandAfzalinejad [68] presented amethod based on distance of
Journal of Applied Mathematics 15
the unit under evaluation to the full inefficient frontier Theypresented two models radian and nonradial
min 120601
st119899
sum
119895=1
120582119895119909119894119895= 120601119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
120582119895ge 0 119895 = 1 119898
max119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903
st sum
119895isin119869minus119887
120582119895119909119894119895minus 119904
minus
119894= 119909
119894119900 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895+ 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
119904minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(70)
Amirteimoori [69] based on the same idea considered bothefficient and antiefficient frontiers for efficiency analysis andranking units
Kao [70] mentioned that determining the weights ofindividual criteria in multiple criteria decision analysis in away that all alternatives can be compared according to theaggregate performance of all criteria is of great importanceAs Kao noted this problem relates to search for alternativeswith a shorter and longer distance respectively to the idealand anti-ideal units He proposed a measure that consideredthe calculation of the relative position of an alternativebetween the ideal and anti-ideal for finding an appropriaterankings
Khodabakhshi and Aryavash [71] presented a method forranking all units using DEA concept
First Compute the minimum and maximum efficiencyvalues of eachDMU in regard to this assumption that the sumof efficiency values of all DMUs equals to 1
Second determine the rank of each DMU in relation to acombination of itsminimumandmaximumefficiency values
Zerafat Angiz et al [72] proposed a technique in orderto aggregate the opinions of experts in voting system Asthe authors mentioned the presented method uses fuzzyconcept and it is computationally efficient and can fullyrank alternatives At first number of votes given to a rankposition was grouped to construct fuzzy numbers and thenthe artificial ideal alternative introduced Furthermore byperforming DEA the efficiency measure of alternatives was
obtained considering artificial ideal alternative comparedby each of the alternatives pair by pair Thus alternativesare ranked in accordance with their efficiency scores Ifthis method cannot completely rank alternatives weightrestrictions based on fuzzy concept are imposed into theanalysis
10 Different Applications in Ranking Units
As discussed formerly there exist a variety of ranking meth-ods and applications in theliterature Nowadays DEAmodelsare widely used in different areas for efficiency evaluationbenchmarking and target setting ranking entities and soforth Ranking units as one of the important issues in DEAhas been performed in different areas Jahanshahloo et al [42]ranked cities in Iran to find the best place for creating a datafactory Hosseinzadeh Lotfi et al [28] used a new methodfor ranking in order to find the best place for power plantlocation Ali andNakosteen [80] Amirteimoori et al [37] andAlirezaee and Afsharian [24] Soltanifar and HosseinzadehLotfi [105] Zerafat Angiz et al [72] Hosseinzadeh Lotfiet al [28] Jahanshahloo [50] Chen and Deng [52] andJablonsky [61] used different ranking methods in bankingsystem Sadjadi [40] ranked provincial gas companies in IranMehrabian et al [32] Li et al [39] Orkcu and Bal [12] andWu et al [19] ranked different departments of universitiesJahanshahloo et al [35] Jahanshahloo and Afzalinejad [68]utilized the presented models in ranking 28 Chinese citiesJahanshahloo at al [35] provided an application to burdensharing amongst NATOmember nations Jahanshahloo et al[14] utilized the presented model for ranking nursery homesContreras [18] andWang et al [23] used the provided rankingtechniques in ranking candidates Lu and Lo [51] applied theirmethod on an application to financial holding companiesHosseinzadeh Lotfi et al [63] consider an empirical exampleinwhich voters are asked to rank twoout of seven alternativesWang and Jiang [62] utilized the provided method in facilitylayout design in manufacturing systems and performanceevaluation of 30OECD countries Chen [60] used an exampleof market basket data in order to illustrate the providedapproach
11 Application
In this section some of the reviewed models are applied onthe example used in Soltanifar and Hosseinzadeh Lotfi [105]Consider twenty commercial banks of Iran with input-outputdata tabulated in Table 1 and summarized as follows Also theresults of CCRmodel are listed in this table As it can be seenseven units are efficient DMUS 14 7 12 15 17 and 20 Inputsare staff computer terminal and spaceOutputs are depositsloans granted and charge
Consider some of the important ranking methods inliterature as follows
RM1 AP model [31] is based upon the idea of leaveunit evaluation out and measuring the distance of theunit under evaluation from the production possibilityset constructed by the remaining DMUs
RM2MAJmodel [32] presented for ranking efficientwhich is always stable but might be infeasible in somecasesRM3 Modified MAJ model [38] overcomes theproblem which might occurr in MAJ modelRM4 A new model based on the idea of alterationsin the reference set of the inefficient units [50]RM5 A model presented by Li et al [39] whichis a super-efficiency method that does not have thesuffering in previous methodsRM6 Slack-basedmodel [33] is based upon the inputand output variables at the same timeRM7 SA DEAmodel [49] overcomes the problem ofinfeasibility which existed in AP modelRM8 Cross-efficiency [10] is provided based onusing weights of each unit under evaluation in opti-mality for other unitsRM9 A model based on finding common set ofweights [25] which determine the common set ofweights for DMUs and ranked DMUs based this ideaRM10 A model based on finding common set ofweights [22 67] for ranking efficient unitsRM11 L1-norm model [34 35 65 66] the idea isbased upon the leave-one-out efficient unit and l1-norm which is always feasible and stableRM12 119871
infin-norm model Rezai Balf et al [46] pro-
vided a method with more ability over other existingmethods based on Tchebycheff normRM13 An enhanced Russel measure of super-efficiency model for ranking units [44]RM14 A rankingmodel which considers the distanceof unit from the full inefficient frontier [38]RM15 Amodified super-efficiencymodel [45] whichovercomes the infeasibility that may happen in prob-lem This model is based on simultaneous projectionof input output
In accordance with properties of different ranking models inorder to rank efficient units consider Table 2
1199011 Feasibility
1199012 Ranking extreme efficient units
1199013 Complexity in computation
1199014 Instability
1199015 Absence of multiple optimal solution
1199016 Dependency to 120579 and slacks
1199017 Dependency to the number of efficient and ineffi-
cient units
In Tables 3 and 4 the rank of efficient units consideringthe above mentioned methods is listed Note that ED showsefficient DMUs (ED) and RM
119895 (119895 = 1 15) are those
explained previously
12 Conclusion
In this paper the DEA ranking was reviewed and classifiedinto seven general groups In the first group those papersbased on a cross-efficiencymatrix were reviewed In this fieldDMUs have been evaluated by self- and peer pressure Thesecond group of papers is based on those papers lookingfor optimal weights in DEA analysis The third one is thesuper-efficiency method By omitting the under evaluationunit and constructing a new frontier by the remaining unitsthe unit under evaluation can get an equal or greater scoreThe fourth group is based on benchmarking idea In thisclass the effect of an efficient unit considered as a targetfor inefficient units is investigated This idea is very usefulfor the managers in decision making Another class thefifth one involves the application of multivariate statisticaltools The sixth section discusses the ranking methods basedon multicriteria decision-making (MCDM) methodologiesand DEA The final section includes some various rankingmethods presented in the literature Finally in an applicationthe result of some of the above-mentioned ranking methodsis presented
References
[1] M J Farrell ldquoThe measurement of productive efficiencyrdquoJournal of the Royal Statistical Society A vol 120 pp 253ndash2811957
[2] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
18 Journal of Applied Mathematics
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
In their paper Chen and Sherman [36] presented a non-radial super-efficiency method and discussed the advantageof it They verified that this model is invariant to units ofinputoutput measurement Let 119869119900 = 119869DMU
119900
Step 1 Solve the followingmodel to find the extreme efficientunits in 119869119900
This method cannot rank nonextreme efficient units
Jahanshahloo et al [38] presented modified MAJ modelfor ranking efficiency units in DEA technique
min 119908119901+ 1
st119899
sum
119895=1119895 = 119901
120582119895
119909119894119895
119872119894
le
119909119894119901
119872119894
+ 1199081199011 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895ge 119910
119903119901 119903 = 1 119904
120582119895ge 0 119895 = 119901
(39)
where 119872119894= Max 119909
119894119895| DMU
119895is efficient It cannot rank
nonextreme efficient unitsLi et al [39] presented a new method for ranking which
does not have difficulties of earlier methods The presentedmodel is always feasible and stable
min 1 +
1
119898
119898
sum
119894=1
119904+
1198942
119877minus
119894
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895+ 119904
minus
1198941minus 119904
+
1198942= 119909
119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895minus 119904
+
119903= 119910
119903119901 119903 = 1 119904
119904minus
1198942ge 0 119904
minus
1198941ge 0 119904
+
119903ge 0 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 119901
(40)
In this case extreme efficient units cannot be ranked
In a paper Khodabakhshi [99] addresses super efficiencyon improved outputs He mentioned that as AP modelmay be infeasible under variable returns to scale technol-ogy using the presented model gives a complete rankingwhen getting an input combination for improving outputs issuitable
Sadjadi et al [40] presented a robust super-efficiencyDEA for ranking efficient units They noted that as inmost of the times exact data do not exist and the sto-chastic super-efficiency model presented in their paperincorporates the robust counterpart of super-efficiencyDEA
where 119883119884 are input-output data This model does notrank nonextreme efficient units It may be unstable andinfeasible
Gholam Abri et al [41] proposed a model for rankingefficient units They used representation theory and rep-resented the DMU under assessment as a convex combi-nation of extreme efficient units As the authors noted itis expected that the performance of DMU
119900is the same
as the performance of convex combination of extremeefficient units Thus it is possible to represent DMU
119900as
follows
(119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119899
sum
119895=1
120582119895= 1 119895 = 1 119904 (42)
As regards representation theorem this system has119898 + 119904 minus 1
constraints and 119904 variables (1205821 120582
119904) If this system has a
10 Journal of Applied Mathematics
unique solution we will have 120579lowast119900= sum
119899
119895=1= 1 120582
lowast
119895120579119895 otherwise
two models should be considered
min119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
max119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
(43)
Consider 1205791and 120579
2as the optimal solution of the above-
mentioned models respectively If 1205791= 120579
2 this is the same
as what has been mentioned previously If 1205791lt 120579
2 then
mentionedmodels provide an interval which helps rank unitsfrom the worst to the best Sometimes it will obtain theranking score with a bounded interval [120579
1 120579
2]
Jahanshahloo et al [42] presented models for rankingefficient units The presented models are somehow a modifi-cation of cross-efficiency model that overcomes the difficultyof alternative optimal weightsThe authors in their paperwithregard to the changes and also utilizing TOPSIS techniquepresented a new super-efficientmethod for ranking unitsThepresented model is as follows
120579119894119895=
119880119905119884119894
119881119905119883
119894
|
119880119905119884119897
119881119905119883
119897
le 1
119880119905119884119895
119881119905119883
119895
= 120579119895119895
119897 = 1 119899 119897 = 119894 119880 ge 0 119881 ge 0
(44)
where 120579119895119895is the efficiency score of DMU
119869using correspond-
ing weights Also 120579119894119895is the efficiency of DMU
119894using optimal
weights of DMU119895 Noura et al [43] provided a method for
ranking efficient units based on this idea that more effectiveand useful units in society should have better rank
Step 1 For each efficient unit choose the lower and upperlimit for each inputs and outputs Let 119864 be the set of efficientunits
where 120585 is representative of a small and nonzero number usedfor not dividing by zero Now consider119863minus119895 as follows whichshows that more successful DMU
119895will be if the larger value
of the119863119895is
119863119895= sum
119894isin119868
119889119894119895+ sum
119903isin119877
119889119903119895 (48)
where 119868 = 119863+
119894cup 119863
minus
119894 119877 = 119863
+
119903cup 119863
minus
119903
Ashrafi et al [44] introduced an enhanced Russell mea-sure of super efficiency for ranking efficient units inDEAThelinear counterpart of the proposed model is as follows
119900 le 120573 le V119903ge 120573 119903 = 1 119904
(49)
It cannot rank nonextreme efficient units
Journal of Applied Mathematics 11
Chen et al [45] proposed a modified super-efficiencymethod for ranking units based on simultaneous input-output projectionThe presentedmodel overcomes the infea-sibility problem
1199011= min
120579119904119903
119900
120601119904119903
119900
st sum
119895=1
119895 = 119900
120582119895119909119894119895le 120579
119904119903
119900119909119894119900 119894 = 1 119898
sum
119895=1
119895 = 119900
120582119895119910119903119895ge 120601
119904119903
119900119910119903119900 119903 = 1 119904
sum
119895=1
119895 = 119900
120582119895= 1
0 lt 120579119904119903
119900le 1 120601
119904119903
119900ge 1 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(50)
Rezai Balf et al [46] provide a model for ranking units basedon Tchebycheff norm As proved that this model is alwaysfeasible and stable it seems to have superiority over othermodels
max 119881119901
st 119881119901ge
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901 119894 = 1 119898
119881119901ge 119910
119903119901minus
119899
sum
119895=1
119895 = 119900
120582119895119910119903119895 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(51)
where
119881119901= Max
(
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901)119894 = 1 119898
(119910119903119901minus
119899
sum
119895=1
119895 = 119901
120582119895119910119903119895)119903 = 1 119904
(52)
Chen et al [47] for overcoming the infeasibility problem thatoccurred in variable returns to scale super-efficiency DEAmodel according to a directional distance function developedNerlove-Luenberger (N-L) measure of super-efficiency
6 Benchmarking Ranking Techniques
Sueyoshi et al [49] proposes a ldquobenchmark approachrdquo forbaseball evaluation This method is the combination of DEA
and (Offensive earned-run average) OERA As the authorsnoted using this method it is possible to select best unitsand also their ranking orders They mentioned that usingonly DEAmay result in a shortcoming in assessment as manyefficient units can be identifiedThus the authors used slack-adjusted DEA model and OERA to overcome this difficulty
Jahanshahloo et al [50] presented a new model forranking DMUs based on alteration in reference set The ideais based on this fact that efficient units can be the target unitfor inefficient units
min 120597119886119887
= 120579 minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st sum
119895isin119869minus119887
120582119895119909119894119895+ 119904
minus
119894minus 120579119909
119894119886 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
120579 free 119904minus119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 isin 119869 minus 119887
(53)
Finally the ranking order for each efficient unit b can becomputing by
119887= sum
119886isin119869119899
120597119886119887
119899
(54)
Note that this method cannot rank nonextreme efficientunits
Lu and Lo [51] provided an interactive benchmark modelfor ranking units The idea is based upon considering a fixedunit as a benchmark and calculating the efficiency of otherunits pair by pair to this unitThis procedure continueswhenall units are accounted for as a benchmark unit ConsiderDMU
119900as a unit under evaluation andDMU
119887as a benchmark
Assume
119909119894= 119909
119894119900(1 + 120601
119894) 119910
119903= 119910
119903119900(1 minus 120593
119903)
min 120579lowast119887
119900=
1 + (1119898)sum119898
119894=1120601119894
1 minus (1119904)sum119904
119903=1120593119903
st 120582119887119909119894119887minus 119909
119894119900120601119894le 119909
119894119900 119894 = 1 119898
120582119887119910119903119887minus 119910
119903119900120593119903ge 119910
119903119900 119903 = 1 119904
120601119894ge 119900 120593
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(55)
Then using (sum119899
119887=1120579119887lowast
119900)119899 119900 = 1 119899 the efficiency of
benchmark DMU119900can be obtained Using the following
index which indicates the increment in efficiency of a unitby moving from peer appraisal to self-appraisal it is nowpossible to rank units Note that the less the magnitude of
12 Journal of Applied Mathematics
this index is the better rank for corresponding unit will beobtained
FPIIBM119870
=
(TEBCC119896
minus STDIBM119896
)
(STDIBM119896
)
(56)
where TEBCC119896
and STDIBM119896
are respectively efficiency in BCCmodel and normalization of TEIBM of DMU
119896 Chen [52]
provided a paper for ranking efficient and inefficient unitsin DEA They noted that the evaluation of efficient unitsis based upon the alterations in efficiency of all inefficientunits by omitting in reference set At first solve the followingmodel which measures the efficiency of DMU
119886when DMU
119887
is eliminated from the reference set
min 120588119886119887
= 120579119904
119886minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1119895 = 119887
120582119895119909119894119895+ 119904
minus
119894= 120579
119904
119886119909119894119886 119894 = 1 119898
119899
sum
119895=1119895 = 119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
119899
sum
119895=1119895 = 119887
= 1
120579119904
119886ge 0 119904
minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119887
(57)
Then again solve the previous model this time for calculat-ing the efficiency score of each inefficient unit when each ofefficient units in turn is eliminated from the reference set 120578lowast
119886
and calculate the efficiency change
120591119886119887
= 120588lowast
119886119887minus 120578
lowast
119886 (58)
Then calculate the following index called as ldquoMCDErdquo indexfor each efficient and inefficient unit
119864119886= sum
119887isin119881119864
119882lowast
119887120588lowast
119886119887 119864
119887= sum
119887isin119881119868
119882lowast
119887120588lowast
119886119887 (59)
Now in accordance with themagnitude of the acquired indexit is possible to rank unitsThose units with higher score havebetter ranking order
7 Ranking Techniques by MultivariateStatistics in the DEA
As described in DEA literature in DEA technique frontier istaken into consideration rather than central tendency consid-ered in regression analysis DEA technique considers that anenvelope encompasses through all the observations as tight aspossible and does not try to fit regression planes in center ofdata In DEA methodology each unit is considered initiallyand compared to the efficient frontier but in regressionanalysis a procedure is considered in which a single functionfits to the data DEA uses different weights for differentunits but does not let the units use weights of other units
Canonical correlation analysis as Friedman and Sinuany-Stern [53] noted can be used for ranking units This methodis somehow the extension of regression analysis Adler et al[89] The aim in canonical correlation analysis is to find asingle vector commonweight for the inputs and outputs of allunits Consider119885
119895119882
119895as the composite input and output and
as corresponding weights respectively The presentedmodel by Tatsuoka and Lohnes [100] is as follows
max 119903119911119908
=
119878119909119910119880
(119878119909119909119881) (119878
119910119910119880)
(12)
st 119909119909119881 = 1
119910119910119880 = 1
(60)
where 119878119909119909 119878
119910119910 and 119878
119909119910are respectively defined as the matri-
ces of the sums of squares and sums of products of thevariables
Friedman and Sinuany-Stern [53] while defining the ratioof linear combinations of the inputs and outputs 119879
119895=
119882119895119885
119895used canonical correlation analysis As they noted that
scaling ratio 119879119895of the canonical correlation analysisDEA is
unboundedSinuany-Stern et al [101] used linear discriminant analy-
sis for ranking units They defined
119863119895=
119904
sum
119903=1
119906119903119910119903119895+
119898
sum
119894=1
V119903(minus119909
119894119895) (61)
DMU119895is said to be efficient if 119863
119895gt 119863
119888where 119863
119888is a critical
value based on themidpoint of themeans of the discriminantfunction value of the two groups Morrison [102] The largerthe amount of119863
119895is the better rank DMU
119895will have
Friedman and Sinuany-Stern [53] noted that as cross-efficiencyDEA canonical correlation analysisDEA and dis-criminant analysisDEA ranking orders may vary from eachother thus it seems necessary to introduce the combinedranking (CODEA) Combined ranking for each unit con-sidered all the ranks obtained from the above-mentionedrankings Moreover statistical tests are one for goodness offit between DEA and a specific ranking and the other fortesting correlation between variety of ranking orders Siegeland Castellan [103]
8 Ranking with MulticriteriaDecision-Making (MCDM) Methodologiesand DEA
Li and Reeves [56] for increasing discrimination power ofDEA presented amultiple-objective linear program (MOLP)As the authors mentioned using minimax and minsumefficiency in addition to the standard DEA objective functionhelp to increase discrimination power of DEA
Strassert and Prato [59] presented the balancing andranking method which uses a three-step procedure forderiving an overall complete or partial final order of optionsIn the first step derive an outranking matrix for all options
Journal of Applied Mathematics 13
from the criteria values Considering this matrix it is possibleto show the frequencywithwhich one option is ranked higherthan the other options In the second step by triangularizingthe outranking matrix establish an implicit preordering orprovisional ordering of optionsTheoutrankingmatrix showsthe degree to which there is a complete overall order ofoptions In the third step based on information given in anadvantages-disadvantages table the provisional ordering issubjected to different screening and balancing operations
Chen [60] utilized a nonparametric approach DEA toestimate and rank the efficiency of association rules withmultiple criteria in following steps Proposed postprocessingapproach is as follows
Step 1 Input data for association rule mining
Step 2 Mine association rules by using the a priori algorithmwith minimum support and minimum confidence
Step 3 Determine subjective interestingness measures byfurther considering the domain related knowledge
Step 4 Calculate the preference scores of association rulesdiscovered in Step 2 by using Cook and Kressrsquos DEA model
Step 5 Discriminate the efficient association rules found inStep 3 by using Obata and Ishiirsquos [104] discriminate model
Step 6 Select rules for implementation by considering thereference scores generated in Step 5 and domain relatedknowledge
Jablonsky [61] presented two original models super SBMand AHP for ranking of efficient units in DEA As theauthormentioned thesemodels are based onmultiple criteriadecision-making techniques-goal programming and analytichierarchy process Super SBM model for ranking units is asfollows
min 120579119866
119902= 1 + 119905120574 + (1 minus 119905) (
sum119898
119894=1119904+
1119894
119909119894119902
+
sum119898
119894=1119904minus
2119896
119910119896119902
)
st119899
sum
119895=1119895 = 119902
120582119895119909119894119895+ 119904
minus
1119894minus 119904
+
1119894= 119909
119894119902 i = 1 119898
119899
sum
119895=1119895 = 119902
120582119895119910119896119895+ 119904
minus
2119896minus 119904
+
2119896= 119910
119896119902 119896 = 1 119903
119904+
1119894le 120574 119894 = 1 119898
119904minus
2119896le 120574 119896 = 1 119903
119905 isin 0 1 120582119895ge 0 119878
+
1 119878
+
2ge 0 119878
minus
1 119878
minus
2ge 0
119895 = 1 119899
(62)
Wang and Jiang [62] presented an alternative mixedinteger linear programming models in order to identifythe most efficient units in DEA technique As the authors
mentioned presented models can make full use of input-output information with no need to specify any assuranceregions for input and output weights to avoid zero weights
min119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) minus
119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895)
st119904
sum
119903=1
119906119903119910119903119895minus
119898
sum
119894=1
V119894119909119894119895le 119868
119895 119895 = 1 119899
119899
sum
119895=1
119868119895= 1
119868119895isin 0 1 119895 = 1 119899
119906119903ge
1
(119898 + 119904)max119895119910
119903119895
119903 = 1 119904
V119894ge
1
(119898 + 119904)max119895119909
119894119895
119894 = 1 119898
(63)
Hosseinzadeh Lotfi et al [63] provided an improved three-stage method for ranking alternatives in multiple criteriadecision analysis In the first stage based on the bestand worst weights in the optimistic and pessimistic casesobtain the rank position of each alternative respectively Theobtained weight in the first stage is not unique thus it seemsnecessary to introduce a secondary goal that is used in thesecond stage Finally in the third stage the ranks of thealternatives compute in the optimistic or pessimistic case Itis mentionable that the model proposed in the third stageis a multicriteria decision-making (MCDM) model and itis solved by mixed integer programming As the authorsmentioned and provided in their paper this model can beconverted into an LP problem
The previous model with 119897 = 1 is the original CCR modelNote that DMUs in 119864
1 show the first level efficient frontier119897 = 2 indicates the second level efficient frontier when thefirst level efficient frontier is omittedThe following algorithmfinds these efficient frontiers
Step 1 Set 119897 = 1 and evaluate the entire set of DMUs 1198691 Inthis way the first level efficient DMU 1198641 is identified
Step 2 Exclude the efficient DMUs from future DEA runsand set 119869119897+1 = 119869
119897minus 119864
119897 (If 119869119897+1 = oslash stop)
Step 3 Evaluate the new subset of ldquoinefficientrdquo DMUs 119869119897+1to obtain a new set of efficient DMUs 119864119897+1
Step 4 Let l = l + 1 Go to Step 2
When 119869119897+1 = oslash the algorithm stopsAs the authors proved in this way it is possible to rank
the DMUs in the first efficient frontier based upon theirattractiveness scores and identify the best one
Jahanshahloo et al [65] provided a paper for rankingunits using gradient line As the authors mentioned theadvantage of this model is stability and robustness
supports on 119879119888 the obtained PPS by omitting theDMUunder
assessment in 119879119888 As the authors proved a unit is efficient
iff the optimal objective function of the following model isgreater than zero Consider
1198750= (119883 119884) 119883 = 120572119883
119900 119884 = 120573119884
119900
1198780= (119883 119884) isin 1198750
119883 = 120572119883119900 119884 = 120573119884
119900 120572 ge 0 120573 ge 0
(67)
Intersection of 1198780and efficient surface of
119879119888119888 is a half line
where its equation is (minus119881lowast119879119883
119900)120572 + (119880
lowast119879119884119900)120573 = 0 To rank
DMU119900the length of connecting arc DMU
119900with intersection
point of line and previous ellipse is calculated in (120572 120573) spaceThis intersection is as follows
120572lowast= (
1198702
120572119870
2
120573
1198702
120573+ (119881
lowast119879119883
119900119880
lowast119879119884119900)2119870
2
120572
)
12
120573lowast= (
119881lowast119879119883
119900
119880lowast119879119884119900
)120572lowast
(68)
Now the length of connecting arc DMU119900to the point
corresponding to 120572lowast 120573lowast in (120572 120573) plan is calculated as follows119868 = int
120572lowast
1(1 + 120573)
12119889120572 where
119870120572= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119899
119895=11199092
119894119900
)
12
119870120573= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119904
119903=11199102
119903119900
)
12
(69)
Jahanshahloo et al [66] also provided a paper with theconcept of advantage in data envelopment analysis
In their paper Jahanshahloo et al [67] consideringMonteCarlo method presented a new method of ranking
Step 1 Generate a uniformly distributed sequence of 1198801198952119899
119895=1
on(0 1)
Step 2 Random numbers should be classified into 119873 pairslike (119880
1 119880
1015840
1) (119880
119873 119880
1015840
119873) in a way that each number is used
just one time
Step 3 Compute119883119894= 119886 + 119880
119894(119887 minus 119886) and 119891(119883
119894) gt 119888119880
1015840
119894
Step 4 Estimate the integral 119868 by 120579119868= 119888(119887 minus 119886)(119873
119867119873)
Now consider DMU119900as an efficient unit measure those
units that are dominated DMU119900 As for dome DMUs this
would be unbounded thus the authors for each unitbounded the region Then for DMU
119900if (minus119883
119900 119884
119900) ge (minus119883 119884)
then (minus119883 119884) is in the RED of DMU119900 Now by using 119881
119901=
119881lowast(119873
119867119873) all the hits (the above condition) can be counted
where 119881lowast is the measure of the whole region JahanshahlooandAfzalinejad [68] presented amethod based on distance of
Journal of Applied Mathematics 15
the unit under evaluation to the full inefficient frontier Theypresented two models radian and nonradial
min 120601
st119899
sum
119895=1
120582119895119909119894119895= 120601119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
120582119895ge 0 119895 = 1 119898
max119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903
st sum
119895isin119869minus119887
120582119895119909119894119895minus 119904
minus
119894= 119909
119894119900 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895+ 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
119904minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(70)
Amirteimoori [69] based on the same idea considered bothefficient and antiefficient frontiers for efficiency analysis andranking units
Kao [70] mentioned that determining the weights ofindividual criteria in multiple criteria decision analysis in away that all alternatives can be compared according to theaggregate performance of all criteria is of great importanceAs Kao noted this problem relates to search for alternativeswith a shorter and longer distance respectively to the idealand anti-ideal units He proposed a measure that consideredthe calculation of the relative position of an alternativebetween the ideal and anti-ideal for finding an appropriaterankings
Khodabakhshi and Aryavash [71] presented a method forranking all units using DEA concept
First Compute the minimum and maximum efficiencyvalues of eachDMU in regard to this assumption that the sumof efficiency values of all DMUs equals to 1
Second determine the rank of each DMU in relation to acombination of itsminimumandmaximumefficiency values
Zerafat Angiz et al [72] proposed a technique in orderto aggregate the opinions of experts in voting system Asthe authors mentioned the presented method uses fuzzyconcept and it is computationally efficient and can fullyrank alternatives At first number of votes given to a rankposition was grouped to construct fuzzy numbers and thenthe artificial ideal alternative introduced Furthermore byperforming DEA the efficiency measure of alternatives was
obtained considering artificial ideal alternative comparedby each of the alternatives pair by pair Thus alternativesare ranked in accordance with their efficiency scores Ifthis method cannot completely rank alternatives weightrestrictions based on fuzzy concept are imposed into theanalysis
10 Different Applications in Ranking Units
As discussed formerly there exist a variety of ranking meth-ods and applications in theliterature Nowadays DEAmodelsare widely used in different areas for efficiency evaluationbenchmarking and target setting ranking entities and soforth Ranking units as one of the important issues in DEAhas been performed in different areas Jahanshahloo et al [42]ranked cities in Iran to find the best place for creating a datafactory Hosseinzadeh Lotfi et al [28] used a new methodfor ranking in order to find the best place for power plantlocation Ali andNakosteen [80] Amirteimoori et al [37] andAlirezaee and Afsharian [24] Soltanifar and HosseinzadehLotfi [105] Zerafat Angiz et al [72] Hosseinzadeh Lotfiet al [28] Jahanshahloo [50] Chen and Deng [52] andJablonsky [61] used different ranking methods in bankingsystem Sadjadi [40] ranked provincial gas companies in IranMehrabian et al [32] Li et al [39] Orkcu and Bal [12] andWu et al [19] ranked different departments of universitiesJahanshahloo et al [35] Jahanshahloo and Afzalinejad [68]utilized the presented models in ranking 28 Chinese citiesJahanshahloo at al [35] provided an application to burdensharing amongst NATOmember nations Jahanshahloo et al[14] utilized the presented model for ranking nursery homesContreras [18] andWang et al [23] used the provided rankingtechniques in ranking candidates Lu and Lo [51] applied theirmethod on an application to financial holding companiesHosseinzadeh Lotfi et al [63] consider an empirical exampleinwhich voters are asked to rank twoout of seven alternativesWang and Jiang [62] utilized the provided method in facilitylayout design in manufacturing systems and performanceevaluation of 30OECD countries Chen [60] used an exampleof market basket data in order to illustrate the providedapproach
11 Application
In this section some of the reviewed models are applied onthe example used in Soltanifar and Hosseinzadeh Lotfi [105]Consider twenty commercial banks of Iran with input-outputdata tabulated in Table 1 and summarized as follows Also theresults of CCRmodel are listed in this table As it can be seenseven units are efficient DMUS 14 7 12 15 17 and 20 Inputsare staff computer terminal and spaceOutputs are depositsloans granted and charge
Consider some of the important ranking methods inliterature as follows
RM1 AP model [31] is based upon the idea of leaveunit evaluation out and measuring the distance of theunit under evaluation from the production possibilityset constructed by the remaining DMUs
RM2MAJmodel [32] presented for ranking efficientwhich is always stable but might be infeasible in somecasesRM3 Modified MAJ model [38] overcomes theproblem which might occurr in MAJ modelRM4 A new model based on the idea of alterationsin the reference set of the inefficient units [50]RM5 A model presented by Li et al [39] whichis a super-efficiency method that does not have thesuffering in previous methodsRM6 Slack-basedmodel [33] is based upon the inputand output variables at the same timeRM7 SA DEAmodel [49] overcomes the problem ofinfeasibility which existed in AP modelRM8 Cross-efficiency [10] is provided based onusing weights of each unit under evaluation in opti-mality for other unitsRM9 A model based on finding common set ofweights [25] which determine the common set ofweights for DMUs and ranked DMUs based this ideaRM10 A model based on finding common set ofweights [22 67] for ranking efficient unitsRM11 L1-norm model [34 35 65 66] the idea isbased upon the leave-one-out efficient unit and l1-norm which is always feasible and stableRM12 119871
infin-norm model Rezai Balf et al [46] pro-
vided a method with more ability over other existingmethods based on Tchebycheff normRM13 An enhanced Russel measure of super-efficiency model for ranking units [44]RM14 A rankingmodel which considers the distanceof unit from the full inefficient frontier [38]RM15 Amodified super-efficiencymodel [45] whichovercomes the infeasibility that may happen in prob-lem This model is based on simultaneous projectionof input output
In accordance with properties of different ranking models inorder to rank efficient units consider Table 2
1199011 Feasibility
1199012 Ranking extreme efficient units
1199013 Complexity in computation
1199014 Instability
1199015 Absence of multiple optimal solution
1199016 Dependency to 120579 and slacks
1199017 Dependency to the number of efficient and ineffi-
cient units
In Tables 3 and 4 the rank of efficient units consideringthe above mentioned methods is listed Note that ED showsefficient DMUs (ED) and RM
119895 (119895 = 1 15) are those
explained previously
12 Conclusion
In this paper the DEA ranking was reviewed and classifiedinto seven general groups In the first group those papersbased on a cross-efficiencymatrix were reviewed In this fieldDMUs have been evaluated by self- and peer pressure Thesecond group of papers is based on those papers lookingfor optimal weights in DEA analysis The third one is thesuper-efficiency method By omitting the under evaluationunit and constructing a new frontier by the remaining unitsthe unit under evaluation can get an equal or greater scoreThe fourth group is based on benchmarking idea In thisclass the effect of an efficient unit considered as a targetfor inefficient units is investigated This idea is very usefulfor the managers in decision making Another class thefifth one involves the application of multivariate statisticaltools The sixth section discusses the ranking methods basedon multicriteria decision-making (MCDM) methodologiesand DEA The final section includes some various rankingmethods presented in the literature Finally in an applicationthe result of some of the above-mentioned ranking methodsis presented
References
[1] M J Farrell ldquoThe measurement of productive efficiencyrdquoJournal of the Royal Statistical Society A vol 120 pp 253ndash2811957
[2] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
18 Journal of Applied Mathematics
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
This method cannot rank nonextreme efficient units
Jahanshahloo et al [38] presented modified MAJ modelfor ranking efficiency units in DEA technique
min 119908119901+ 1
st119899
sum
119895=1119895 = 119901
120582119895
119909119894119895
119872119894
le
119909119894119901
119872119894
+ 1199081199011 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895ge 119910
119903119901 119903 = 1 119904
120582119895ge 0 119895 = 119901
(39)
where 119872119894= Max 119909
119894119895| DMU
119895is efficient It cannot rank
nonextreme efficient unitsLi et al [39] presented a new method for ranking which
does not have difficulties of earlier methods The presentedmodel is always feasible and stable
min 1 +
1
119898
119898
sum
119894=1
119904+
1198942
119877minus
119894
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895+ 119904
minus
1198941minus 119904
+
1198942= 119909
119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895minus 119904
+
119903= 119910
119903119901 119903 = 1 119904
119904minus
1198942ge 0 119904
minus
1198941ge 0 119904
+
119903ge 0 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 119901
(40)
In this case extreme efficient units cannot be ranked
In a paper Khodabakhshi [99] addresses super efficiencyon improved outputs He mentioned that as AP modelmay be infeasible under variable returns to scale technol-ogy using the presented model gives a complete rankingwhen getting an input combination for improving outputs issuitable
Sadjadi et al [40] presented a robust super-efficiencyDEA for ranking efficient units They noted that as inmost of the times exact data do not exist and the sto-chastic super-efficiency model presented in their paperincorporates the robust counterpart of super-efficiencyDEA
where 119883119884 are input-output data This model does notrank nonextreme efficient units It may be unstable andinfeasible
Gholam Abri et al [41] proposed a model for rankingefficient units They used representation theory and rep-resented the DMU under assessment as a convex combi-nation of extreme efficient units As the authors noted itis expected that the performance of DMU
119900is the same
as the performance of convex combination of extremeefficient units Thus it is possible to represent DMU
119900as
follows
(119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119899
sum
119895=1
120582119895= 1 119895 = 1 119904 (42)
As regards representation theorem this system has119898 + 119904 minus 1
constraints and 119904 variables (1205821 120582
119904) If this system has a
10 Journal of Applied Mathematics
unique solution we will have 120579lowast119900= sum
119899
119895=1= 1 120582
lowast
119895120579119895 otherwise
two models should be considered
min119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
max119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
(43)
Consider 1205791and 120579
2as the optimal solution of the above-
mentioned models respectively If 1205791= 120579
2 this is the same
as what has been mentioned previously If 1205791lt 120579
2 then
mentionedmodels provide an interval which helps rank unitsfrom the worst to the best Sometimes it will obtain theranking score with a bounded interval [120579
1 120579
2]
Jahanshahloo et al [42] presented models for rankingefficient units The presented models are somehow a modifi-cation of cross-efficiency model that overcomes the difficultyof alternative optimal weightsThe authors in their paperwithregard to the changes and also utilizing TOPSIS techniquepresented a new super-efficientmethod for ranking unitsThepresented model is as follows
120579119894119895=
119880119905119884119894
119881119905119883
119894
|
119880119905119884119897
119881119905119883
119897
le 1
119880119905119884119895
119881119905119883
119895
= 120579119895119895
119897 = 1 119899 119897 = 119894 119880 ge 0 119881 ge 0
(44)
where 120579119895119895is the efficiency score of DMU
119869using correspond-
ing weights Also 120579119894119895is the efficiency of DMU
119894using optimal
weights of DMU119895 Noura et al [43] provided a method for
ranking efficient units based on this idea that more effectiveand useful units in society should have better rank
Step 1 For each efficient unit choose the lower and upperlimit for each inputs and outputs Let 119864 be the set of efficientunits
where 120585 is representative of a small and nonzero number usedfor not dividing by zero Now consider119863minus119895 as follows whichshows that more successful DMU
119895will be if the larger value
of the119863119895is
119863119895= sum
119894isin119868
119889119894119895+ sum
119903isin119877
119889119903119895 (48)
where 119868 = 119863+
119894cup 119863
minus
119894 119877 = 119863
+
119903cup 119863
minus
119903
Ashrafi et al [44] introduced an enhanced Russell mea-sure of super efficiency for ranking efficient units inDEAThelinear counterpart of the proposed model is as follows
119900 le 120573 le V119903ge 120573 119903 = 1 119904
(49)
It cannot rank nonextreme efficient units
Journal of Applied Mathematics 11
Chen et al [45] proposed a modified super-efficiencymethod for ranking units based on simultaneous input-output projectionThe presentedmodel overcomes the infea-sibility problem
1199011= min
120579119904119903
119900
120601119904119903
119900
st sum
119895=1
119895 = 119900
120582119895119909119894119895le 120579
119904119903
119900119909119894119900 119894 = 1 119898
sum
119895=1
119895 = 119900
120582119895119910119903119895ge 120601
119904119903
119900119910119903119900 119903 = 1 119904
sum
119895=1
119895 = 119900
120582119895= 1
0 lt 120579119904119903
119900le 1 120601
119904119903
119900ge 1 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(50)
Rezai Balf et al [46] provide a model for ranking units basedon Tchebycheff norm As proved that this model is alwaysfeasible and stable it seems to have superiority over othermodels
max 119881119901
st 119881119901ge
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901 119894 = 1 119898
119881119901ge 119910
119903119901minus
119899
sum
119895=1
119895 = 119900
120582119895119910119903119895 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(51)
where
119881119901= Max
(
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901)119894 = 1 119898
(119910119903119901minus
119899
sum
119895=1
119895 = 119901
120582119895119910119903119895)119903 = 1 119904
(52)
Chen et al [47] for overcoming the infeasibility problem thatoccurred in variable returns to scale super-efficiency DEAmodel according to a directional distance function developedNerlove-Luenberger (N-L) measure of super-efficiency
6 Benchmarking Ranking Techniques
Sueyoshi et al [49] proposes a ldquobenchmark approachrdquo forbaseball evaluation This method is the combination of DEA
and (Offensive earned-run average) OERA As the authorsnoted using this method it is possible to select best unitsand also their ranking orders They mentioned that usingonly DEAmay result in a shortcoming in assessment as manyefficient units can be identifiedThus the authors used slack-adjusted DEA model and OERA to overcome this difficulty
Jahanshahloo et al [50] presented a new model forranking DMUs based on alteration in reference set The ideais based on this fact that efficient units can be the target unitfor inefficient units
min 120597119886119887
= 120579 minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st sum
119895isin119869minus119887
120582119895119909119894119895+ 119904
minus
119894minus 120579119909
119894119886 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
120579 free 119904minus119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 isin 119869 minus 119887
(53)
Finally the ranking order for each efficient unit b can becomputing by
119887= sum
119886isin119869119899
120597119886119887
119899
(54)
Note that this method cannot rank nonextreme efficientunits
Lu and Lo [51] provided an interactive benchmark modelfor ranking units The idea is based upon considering a fixedunit as a benchmark and calculating the efficiency of otherunits pair by pair to this unitThis procedure continueswhenall units are accounted for as a benchmark unit ConsiderDMU
119900as a unit under evaluation andDMU
119887as a benchmark
Assume
119909119894= 119909
119894119900(1 + 120601
119894) 119910
119903= 119910
119903119900(1 minus 120593
119903)
min 120579lowast119887
119900=
1 + (1119898)sum119898
119894=1120601119894
1 minus (1119904)sum119904
119903=1120593119903
st 120582119887119909119894119887minus 119909
119894119900120601119894le 119909
119894119900 119894 = 1 119898
120582119887119910119903119887minus 119910
119903119900120593119903ge 119910
119903119900 119903 = 1 119904
120601119894ge 119900 120593
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(55)
Then using (sum119899
119887=1120579119887lowast
119900)119899 119900 = 1 119899 the efficiency of
benchmark DMU119900can be obtained Using the following
index which indicates the increment in efficiency of a unitby moving from peer appraisal to self-appraisal it is nowpossible to rank units Note that the less the magnitude of
12 Journal of Applied Mathematics
this index is the better rank for corresponding unit will beobtained
FPIIBM119870
=
(TEBCC119896
minus STDIBM119896
)
(STDIBM119896
)
(56)
where TEBCC119896
and STDIBM119896
are respectively efficiency in BCCmodel and normalization of TEIBM of DMU
119896 Chen [52]
provided a paper for ranking efficient and inefficient unitsin DEA They noted that the evaluation of efficient unitsis based upon the alterations in efficiency of all inefficientunits by omitting in reference set At first solve the followingmodel which measures the efficiency of DMU
119886when DMU
119887
is eliminated from the reference set
min 120588119886119887
= 120579119904
119886minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1119895 = 119887
120582119895119909119894119895+ 119904
minus
119894= 120579
119904
119886119909119894119886 119894 = 1 119898
119899
sum
119895=1119895 = 119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
119899
sum
119895=1119895 = 119887
= 1
120579119904
119886ge 0 119904
minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119887
(57)
Then again solve the previous model this time for calculat-ing the efficiency score of each inefficient unit when each ofefficient units in turn is eliminated from the reference set 120578lowast
119886
and calculate the efficiency change
120591119886119887
= 120588lowast
119886119887minus 120578
lowast
119886 (58)
Then calculate the following index called as ldquoMCDErdquo indexfor each efficient and inefficient unit
119864119886= sum
119887isin119881119864
119882lowast
119887120588lowast
119886119887 119864
119887= sum
119887isin119881119868
119882lowast
119887120588lowast
119886119887 (59)
Now in accordance with themagnitude of the acquired indexit is possible to rank unitsThose units with higher score havebetter ranking order
7 Ranking Techniques by MultivariateStatistics in the DEA
As described in DEA literature in DEA technique frontier istaken into consideration rather than central tendency consid-ered in regression analysis DEA technique considers that anenvelope encompasses through all the observations as tight aspossible and does not try to fit regression planes in center ofdata In DEA methodology each unit is considered initiallyand compared to the efficient frontier but in regressionanalysis a procedure is considered in which a single functionfits to the data DEA uses different weights for differentunits but does not let the units use weights of other units
Canonical correlation analysis as Friedman and Sinuany-Stern [53] noted can be used for ranking units This methodis somehow the extension of regression analysis Adler et al[89] The aim in canonical correlation analysis is to find asingle vector commonweight for the inputs and outputs of allunits Consider119885
119895119882
119895as the composite input and output and
as corresponding weights respectively The presentedmodel by Tatsuoka and Lohnes [100] is as follows
max 119903119911119908
=
119878119909119910119880
(119878119909119909119881) (119878
119910119910119880)
(12)
st 119909119909119881 = 1
119910119910119880 = 1
(60)
where 119878119909119909 119878
119910119910 and 119878
119909119910are respectively defined as the matri-
ces of the sums of squares and sums of products of thevariables
Friedman and Sinuany-Stern [53] while defining the ratioof linear combinations of the inputs and outputs 119879
119895=
119882119895119885
119895used canonical correlation analysis As they noted that
scaling ratio 119879119895of the canonical correlation analysisDEA is
unboundedSinuany-Stern et al [101] used linear discriminant analy-
sis for ranking units They defined
119863119895=
119904
sum
119903=1
119906119903119910119903119895+
119898
sum
119894=1
V119903(minus119909
119894119895) (61)
DMU119895is said to be efficient if 119863
119895gt 119863
119888where 119863
119888is a critical
value based on themidpoint of themeans of the discriminantfunction value of the two groups Morrison [102] The largerthe amount of119863
119895is the better rank DMU
119895will have
Friedman and Sinuany-Stern [53] noted that as cross-efficiencyDEA canonical correlation analysisDEA and dis-criminant analysisDEA ranking orders may vary from eachother thus it seems necessary to introduce the combinedranking (CODEA) Combined ranking for each unit con-sidered all the ranks obtained from the above-mentionedrankings Moreover statistical tests are one for goodness offit between DEA and a specific ranking and the other fortesting correlation between variety of ranking orders Siegeland Castellan [103]
8 Ranking with MulticriteriaDecision-Making (MCDM) Methodologiesand DEA
Li and Reeves [56] for increasing discrimination power ofDEA presented amultiple-objective linear program (MOLP)As the authors mentioned using minimax and minsumefficiency in addition to the standard DEA objective functionhelp to increase discrimination power of DEA
Strassert and Prato [59] presented the balancing andranking method which uses a three-step procedure forderiving an overall complete or partial final order of optionsIn the first step derive an outranking matrix for all options
Journal of Applied Mathematics 13
from the criteria values Considering this matrix it is possibleto show the frequencywithwhich one option is ranked higherthan the other options In the second step by triangularizingthe outranking matrix establish an implicit preordering orprovisional ordering of optionsTheoutrankingmatrix showsthe degree to which there is a complete overall order ofoptions In the third step based on information given in anadvantages-disadvantages table the provisional ordering issubjected to different screening and balancing operations
Chen [60] utilized a nonparametric approach DEA toestimate and rank the efficiency of association rules withmultiple criteria in following steps Proposed postprocessingapproach is as follows
Step 1 Input data for association rule mining
Step 2 Mine association rules by using the a priori algorithmwith minimum support and minimum confidence
Step 3 Determine subjective interestingness measures byfurther considering the domain related knowledge
Step 4 Calculate the preference scores of association rulesdiscovered in Step 2 by using Cook and Kressrsquos DEA model
Step 5 Discriminate the efficient association rules found inStep 3 by using Obata and Ishiirsquos [104] discriminate model
Step 6 Select rules for implementation by considering thereference scores generated in Step 5 and domain relatedknowledge
Jablonsky [61] presented two original models super SBMand AHP for ranking of efficient units in DEA As theauthormentioned thesemodels are based onmultiple criteriadecision-making techniques-goal programming and analytichierarchy process Super SBM model for ranking units is asfollows
min 120579119866
119902= 1 + 119905120574 + (1 minus 119905) (
sum119898
119894=1119904+
1119894
119909119894119902
+
sum119898
119894=1119904minus
2119896
119910119896119902
)
st119899
sum
119895=1119895 = 119902
120582119895119909119894119895+ 119904
minus
1119894minus 119904
+
1119894= 119909
119894119902 i = 1 119898
119899
sum
119895=1119895 = 119902
120582119895119910119896119895+ 119904
minus
2119896minus 119904
+
2119896= 119910
119896119902 119896 = 1 119903
119904+
1119894le 120574 119894 = 1 119898
119904minus
2119896le 120574 119896 = 1 119903
119905 isin 0 1 120582119895ge 0 119878
+
1 119878
+
2ge 0 119878
minus
1 119878
minus
2ge 0
119895 = 1 119899
(62)
Wang and Jiang [62] presented an alternative mixedinteger linear programming models in order to identifythe most efficient units in DEA technique As the authors
mentioned presented models can make full use of input-output information with no need to specify any assuranceregions for input and output weights to avoid zero weights
min119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) minus
119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895)
st119904
sum
119903=1
119906119903119910119903119895minus
119898
sum
119894=1
V119894119909119894119895le 119868
119895 119895 = 1 119899
119899
sum
119895=1
119868119895= 1
119868119895isin 0 1 119895 = 1 119899
119906119903ge
1
(119898 + 119904)max119895119910
119903119895
119903 = 1 119904
V119894ge
1
(119898 + 119904)max119895119909
119894119895
119894 = 1 119898
(63)
Hosseinzadeh Lotfi et al [63] provided an improved three-stage method for ranking alternatives in multiple criteriadecision analysis In the first stage based on the bestand worst weights in the optimistic and pessimistic casesobtain the rank position of each alternative respectively Theobtained weight in the first stage is not unique thus it seemsnecessary to introduce a secondary goal that is used in thesecond stage Finally in the third stage the ranks of thealternatives compute in the optimistic or pessimistic case Itis mentionable that the model proposed in the third stageis a multicriteria decision-making (MCDM) model and itis solved by mixed integer programming As the authorsmentioned and provided in their paper this model can beconverted into an LP problem
The previous model with 119897 = 1 is the original CCR modelNote that DMUs in 119864
1 show the first level efficient frontier119897 = 2 indicates the second level efficient frontier when thefirst level efficient frontier is omittedThe following algorithmfinds these efficient frontiers
Step 1 Set 119897 = 1 and evaluate the entire set of DMUs 1198691 Inthis way the first level efficient DMU 1198641 is identified
Step 2 Exclude the efficient DMUs from future DEA runsand set 119869119897+1 = 119869
119897minus 119864
119897 (If 119869119897+1 = oslash stop)
Step 3 Evaluate the new subset of ldquoinefficientrdquo DMUs 119869119897+1to obtain a new set of efficient DMUs 119864119897+1
Step 4 Let l = l + 1 Go to Step 2
When 119869119897+1 = oslash the algorithm stopsAs the authors proved in this way it is possible to rank
the DMUs in the first efficient frontier based upon theirattractiveness scores and identify the best one
Jahanshahloo et al [65] provided a paper for rankingunits using gradient line As the authors mentioned theadvantage of this model is stability and robustness
supports on 119879119888 the obtained PPS by omitting theDMUunder
assessment in 119879119888 As the authors proved a unit is efficient
iff the optimal objective function of the following model isgreater than zero Consider
1198750= (119883 119884) 119883 = 120572119883
119900 119884 = 120573119884
119900
1198780= (119883 119884) isin 1198750
119883 = 120572119883119900 119884 = 120573119884
119900 120572 ge 0 120573 ge 0
(67)
Intersection of 1198780and efficient surface of
119879119888119888 is a half line
where its equation is (minus119881lowast119879119883
119900)120572 + (119880
lowast119879119884119900)120573 = 0 To rank
DMU119900the length of connecting arc DMU
119900with intersection
point of line and previous ellipse is calculated in (120572 120573) spaceThis intersection is as follows
120572lowast= (
1198702
120572119870
2
120573
1198702
120573+ (119881
lowast119879119883
119900119880
lowast119879119884119900)2119870
2
120572
)
12
120573lowast= (
119881lowast119879119883
119900
119880lowast119879119884119900
)120572lowast
(68)
Now the length of connecting arc DMU119900to the point
corresponding to 120572lowast 120573lowast in (120572 120573) plan is calculated as follows119868 = int
120572lowast
1(1 + 120573)
12119889120572 where
119870120572= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119899
119895=11199092
119894119900
)
12
119870120573= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119904
119903=11199102
119903119900
)
12
(69)
Jahanshahloo et al [66] also provided a paper with theconcept of advantage in data envelopment analysis
In their paper Jahanshahloo et al [67] consideringMonteCarlo method presented a new method of ranking
Step 1 Generate a uniformly distributed sequence of 1198801198952119899
119895=1
on(0 1)
Step 2 Random numbers should be classified into 119873 pairslike (119880
1 119880
1015840
1) (119880
119873 119880
1015840
119873) in a way that each number is used
just one time
Step 3 Compute119883119894= 119886 + 119880
119894(119887 minus 119886) and 119891(119883
119894) gt 119888119880
1015840
119894
Step 4 Estimate the integral 119868 by 120579119868= 119888(119887 minus 119886)(119873
119867119873)
Now consider DMU119900as an efficient unit measure those
units that are dominated DMU119900 As for dome DMUs this
would be unbounded thus the authors for each unitbounded the region Then for DMU
119900if (minus119883
119900 119884
119900) ge (minus119883 119884)
then (minus119883 119884) is in the RED of DMU119900 Now by using 119881
119901=
119881lowast(119873
119867119873) all the hits (the above condition) can be counted
where 119881lowast is the measure of the whole region JahanshahlooandAfzalinejad [68] presented amethod based on distance of
Journal of Applied Mathematics 15
the unit under evaluation to the full inefficient frontier Theypresented two models radian and nonradial
min 120601
st119899
sum
119895=1
120582119895119909119894119895= 120601119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
120582119895ge 0 119895 = 1 119898
max119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903
st sum
119895isin119869minus119887
120582119895119909119894119895minus 119904
minus
119894= 119909
119894119900 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895+ 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
119904minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(70)
Amirteimoori [69] based on the same idea considered bothefficient and antiefficient frontiers for efficiency analysis andranking units
Kao [70] mentioned that determining the weights ofindividual criteria in multiple criteria decision analysis in away that all alternatives can be compared according to theaggregate performance of all criteria is of great importanceAs Kao noted this problem relates to search for alternativeswith a shorter and longer distance respectively to the idealand anti-ideal units He proposed a measure that consideredthe calculation of the relative position of an alternativebetween the ideal and anti-ideal for finding an appropriaterankings
Khodabakhshi and Aryavash [71] presented a method forranking all units using DEA concept
First Compute the minimum and maximum efficiencyvalues of eachDMU in regard to this assumption that the sumof efficiency values of all DMUs equals to 1
Second determine the rank of each DMU in relation to acombination of itsminimumandmaximumefficiency values
Zerafat Angiz et al [72] proposed a technique in orderto aggregate the opinions of experts in voting system Asthe authors mentioned the presented method uses fuzzyconcept and it is computationally efficient and can fullyrank alternatives At first number of votes given to a rankposition was grouped to construct fuzzy numbers and thenthe artificial ideal alternative introduced Furthermore byperforming DEA the efficiency measure of alternatives was
obtained considering artificial ideal alternative comparedby each of the alternatives pair by pair Thus alternativesare ranked in accordance with their efficiency scores Ifthis method cannot completely rank alternatives weightrestrictions based on fuzzy concept are imposed into theanalysis
10 Different Applications in Ranking Units
As discussed formerly there exist a variety of ranking meth-ods and applications in theliterature Nowadays DEAmodelsare widely used in different areas for efficiency evaluationbenchmarking and target setting ranking entities and soforth Ranking units as one of the important issues in DEAhas been performed in different areas Jahanshahloo et al [42]ranked cities in Iran to find the best place for creating a datafactory Hosseinzadeh Lotfi et al [28] used a new methodfor ranking in order to find the best place for power plantlocation Ali andNakosteen [80] Amirteimoori et al [37] andAlirezaee and Afsharian [24] Soltanifar and HosseinzadehLotfi [105] Zerafat Angiz et al [72] Hosseinzadeh Lotfiet al [28] Jahanshahloo [50] Chen and Deng [52] andJablonsky [61] used different ranking methods in bankingsystem Sadjadi [40] ranked provincial gas companies in IranMehrabian et al [32] Li et al [39] Orkcu and Bal [12] andWu et al [19] ranked different departments of universitiesJahanshahloo et al [35] Jahanshahloo and Afzalinejad [68]utilized the presented models in ranking 28 Chinese citiesJahanshahloo at al [35] provided an application to burdensharing amongst NATOmember nations Jahanshahloo et al[14] utilized the presented model for ranking nursery homesContreras [18] andWang et al [23] used the provided rankingtechniques in ranking candidates Lu and Lo [51] applied theirmethod on an application to financial holding companiesHosseinzadeh Lotfi et al [63] consider an empirical exampleinwhich voters are asked to rank twoout of seven alternativesWang and Jiang [62] utilized the provided method in facilitylayout design in manufacturing systems and performanceevaluation of 30OECD countries Chen [60] used an exampleof market basket data in order to illustrate the providedapproach
11 Application
In this section some of the reviewed models are applied onthe example used in Soltanifar and Hosseinzadeh Lotfi [105]Consider twenty commercial banks of Iran with input-outputdata tabulated in Table 1 and summarized as follows Also theresults of CCRmodel are listed in this table As it can be seenseven units are efficient DMUS 14 7 12 15 17 and 20 Inputsare staff computer terminal and spaceOutputs are depositsloans granted and charge
Consider some of the important ranking methods inliterature as follows
RM1 AP model [31] is based upon the idea of leaveunit evaluation out and measuring the distance of theunit under evaluation from the production possibilityset constructed by the remaining DMUs
RM2MAJmodel [32] presented for ranking efficientwhich is always stable but might be infeasible in somecasesRM3 Modified MAJ model [38] overcomes theproblem which might occurr in MAJ modelRM4 A new model based on the idea of alterationsin the reference set of the inefficient units [50]RM5 A model presented by Li et al [39] whichis a super-efficiency method that does not have thesuffering in previous methodsRM6 Slack-basedmodel [33] is based upon the inputand output variables at the same timeRM7 SA DEAmodel [49] overcomes the problem ofinfeasibility which existed in AP modelRM8 Cross-efficiency [10] is provided based onusing weights of each unit under evaluation in opti-mality for other unitsRM9 A model based on finding common set ofweights [25] which determine the common set ofweights for DMUs and ranked DMUs based this ideaRM10 A model based on finding common set ofweights [22 67] for ranking efficient unitsRM11 L1-norm model [34 35 65 66] the idea isbased upon the leave-one-out efficient unit and l1-norm which is always feasible and stableRM12 119871
infin-norm model Rezai Balf et al [46] pro-
vided a method with more ability over other existingmethods based on Tchebycheff normRM13 An enhanced Russel measure of super-efficiency model for ranking units [44]RM14 A rankingmodel which considers the distanceof unit from the full inefficient frontier [38]RM15 Amodified super-efficiencymodel [45] whichovercomes the infeasibility that may happen in prob-lem This model is based on simultaneous projectionof input output
In accordance with properties of different ranking models inorder to rank efficient units consider Table 2
1199011 Feasibility
1199012 Ranking extreme efficient units
1199013 Complexity in computation
1199014 Instability
1199015 Absence of multiple optimal solution
1199016 Dependency to 120579 and slacks
1199017 Dependency to the number of efficient and ineffi-
cient units
In Tables 3 and 4 the rank of efficient units consideringthe above mentioned methods is listed Note that ED showsefficient DMUs (ED) and RM
119895 (119895 = 1 15) are those
explained previously
12 Conclusion
In this paper the DEA ranking was reviewed and classifiedinto seven general groups In the first group those papersbased on a cross-efficiencymatrix were reviewed In this fieldDMUs have been evaluated by self- and peer pressure Thesecond group of papers is based on those papers lookingfor optimal weights in DEA analysis The third one is thesuper-efficiency method By omitting the under evaluationunit and constructing a new frontier by the remaining unitsthe unit under evaluation can get an equal or greater scoreThe fourth group is based on benchmarking idea In thisclass the effect of an efficient unit considered as a targetfor inefficient units is investigated This idea is very usefulfor the managers in decision making Another class thefifth one involves the application of multivariate statisticaltools The sixth section discusses the ranking methods basedon multicriteria decision-making (MCDM) methodologiesand DEA The final section includes some various rankingmethods presented in the literature Finally in an applicationthe result of some of the above-mentioned ranking methodsis presented
References
[1] M J Farrell ldquoThe measurement of productive efficiencyrdquoJournal of the Royal Statistical Society A vol 120 pp 253ndash2811957
[2] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
18 Journal of Applied Mathematics
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
unique solution we will have 120579lowast119900= sum
119899
119895=1= 1 120582
lowast
119895120579119895 otherwise
two models should be considered
min119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
max119904
sum
119895=1
= 1 120582119895120579119895
st (119883119900 119884
119900) =
119904
sum
119895=1
120582119895(119883
119895 119884
119895)
119904
sum
119895=1
120582119895= 1
120582119895ge 0 119895 = 1 119904
(43)
Consider 1205791and 120579
2as the optimal solution of the above-
mentioned models respectively If 1205791= 120579
2 this is the same
as what has been mentioned previously If 1205791lt 120579
2 then
mentionedmodels provide an interval which helps rank unitsfrom the worst to the best Sometimes it will obtain theranking score with a bounded interval [120579
1 120579
2]
Jahanshahloo et al [42] presented models for rankingefficient units The presented models are somehow a modifi-cation of cross-efficiency model that overcomes the difficultyof alternative optimal weightsThe authors in their paperwithregard to the changes and also utilizing TOPSIS techniquepresented a new super-efficientmethod for ranking unitsThepresented model is as follows
120579119894119895=
119880119905119884119894
119881119905119883
119894
|
119880119905119884119897
119881119905119883
119897
le 1
119880119905119884119895
119881119905119883
119895
= 120579119895119895
119897 = 1 119899 119897 = 119894 119880 ge 0 119881 ge 0
(44)
where 120579119895119895is the efficiency score of DMU
119869using correspond-
ing weights Also 120579119894119895is the efficiency of DMU
119894using optimal
weights of DMU119895 Noura et al [43] provided a method for
ranking efficient units based on this idea that more effectiveand useful units in society should have better rank
Step 1 For each efficient unit choose the lower and upperlimit for each inputs and outputs Let 119864 be the set of efficientunits
where 120585 is representative of a small and nonzero number usedfor not dividing by zero Now consider119863minus119895 as follows whichshows that more successful DMU
119895will be if the larger value
of the119863119895is
119863119895= sum
119894isin119868
119889119894119895+ sum
119903isin119877
119889119903119895 (48)
where 119868 = 119863+
119894cup 119863
minus
119894 119877 = 119863
+
119903cup 119863
minus
119903
Ashrafi et al [44] introduced an enhanced Russell mea-sure of super efficiency for ranking efficient units inDEAThelinear counterpart of the proposed model is as follows
119900 le 120573 le V119903ge 120573 119903 = 1 119904
(49)
It cannot rank nonextreme efficient units
Journal of Applied Mathematics 11
Chen et al [45] proposed a modified super-efficiencymethod for ranking units based on simultaneous input-output projectionThe presentedmodel overcomes the infea-sibility problem
1199011= min
120579119904119903
119900
120601119904119903
119900
st sum
119895=1
119895 = 119900
120582119895119909119894119895le 120579
119904119903
119900119909119894119900 119894 = 1 119898
sum
119895=1
119895 = 119900
120582119895119910119903119895ge 120601
119904119903
119900119910119903119900 119903 = 1 119904
sum
119895=1
119895 = 119900
120582119895= 1
0 lt 120579119904119903
119900le 1 120601
119904119903
119900ge 1 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(50)
Rezai Balf et al [46] provide a model for ranking units basedon Tchebycheff norm As proved that this model is alwaysfeasible and stable it seems to have superiority over othermodels
max 119881119901
st 119881119901ge
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901 119894 = 1 119898
119881119901ge 119910
119903119901minus
119899
sum
119895=1
119895 = 119900
120582119895119910119903119895 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(51)
where
119881119901= Max
(
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901)119894 = 1 119898
(119910119903119901minus
119899
sum
119895=1
119895 = 119901
120582119895119910119903119895)119903 = 1 119904
(52)
Chen et al [47] for overcoming the infeasibility problem thatoccurred in variable returns to scale super-efficiency DEAmodel according to a directional distance function developedNerlove-Luenberger (N-L) measure of super-efficiency
6 Benchmarking Ranking Techniques
Sueyoshi et al [49] proposes a ldquobenchmark approachrdquo forbaseball evaluation This method is the combination of DEA
and (Offensive earned-run average) OERA As the authorsnoted using this method it is possible to select best unitsand also their ranking orders They mentioned that usingonly DEAmay result in a shortcoming in assessment as manyefficient units can be identifiedThus the authors used slack-adjusted DEA model and OERA to overcome this difficulty
Jahanshahloo et al [50] presented a new model forranking DMUs based on alteration in reference set The ideais based on this fact that efficient units can be the target unitfor inefficient units
min 120597119886119887
= 120579 minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st sum
119895isin119869minus119887
120582119895119909119894119895+ 119904
minus
119894minus 120579119909
119894119886 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
120579 free 119904minus119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 isin 119869 minus 119887
(53)
Finally the ranking order for each efficient unit b can becomputing by
119887= sum
119886isin119869119899
120597119886119887
119899
(54)
Note that this method cannot rank nonextreme efficientunits
Lu and Lo [51] provided an interactive benchmark modelfor ranking units The idea is based upon considering a fixedunit as a benchmark and calculating the efficiency of otherunits pair by pair to this unitThis procedure continueswhenall units are accounted for as a benchmark unit ConsiderDMU
119900as a unit under evaluation andDMU
119887as a benchmark
Assume
119909119894= 119909
119894119900(1 + 120601
119894) 119910
119903= 119910
119903119900(1 minus 120593
119903)
min 120579lowast119887
119900=
1 + (1119898)sum119898
119894=1120601119894
1 minus (1119904)sum119904
119903=1120593119903
st 120582119887119909119894119887minus 119909
119894119900120601119894le 119909
119894119900 119894 = 1 119898
120582119887119910119903119887minus 119910
119903119900120593119903ge 119910
119903119900 119903 = 1 119904
120601119894ge 119900 120593
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(55)
Then using (sum119899
119887=1120579119887lowast
119900)119899 119900 = 1 119899 the efficiency of
benchmark DMU119900can be obtained Using the following
index which indicates the increment in efficiency of a unitby moving from peer appraisal to self-appraisal it is nowpossible to rank units Note that the less the magnitude of
12 Journal of Applied Mathematics
this index is the better rank for corresponding unit will beobtained
FPIIBM119870
=
(TEBCC119896
minus STDIBM119896
)
(STDIBM119896
)
(56)
where TEBCC119896
and STDIBM119896
are respectively efficiency in BCCmodel and normalization of TEIBM of DMU
119896 Chen [52]
provided a paper for ranking efficient and inefficient unitsin DEA They noted that the evaluation of efficient unitsis based upon the alterations in efficiency of all inefficientunits by omitting in reference set At first solve the followingmodel which measures the efficiency of DMU
119886when DMU
119887
is eliminated from the reference set
min 120588119886119887
= 120579119904
119886minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1119895 = 119887
120582119895119909119894119895+ 119904
minus
119894= 120579
119904
119886119909119894119886 119894 = 1 119898
119899
sum
119895=1119895 = 119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
119899
sum
119895=1119895 = 119887
= 1
120579119904
119886ge 0 119904
minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119887
(57)
Then again solve the previous model this time for calculat-ing the efficiency score of each inefficient unit when each ofefficient units in turn is eliminated from the reference set 120578lowast
119886
and calculate the efficiency change
120591119886119887
= 120588lowast
119886119887minus 120578
lowast
119886 (58)
Then calculate the following index called as ldquoMCDErdquo indexfor each efficient and inefficient unit
119864119886= sum
119887isin119881119864
119882lowast
119887120588lowast
119886119887 119864
119887= sum
119887isin119881119868
119882lowast
119887120588lowast
119886119887 (59)
Now in accordance with themagnitude of the acquired indexit is possible to rank unitsThose units with higher score havebetter ranking order
7 Ranking Techniques by MultivariateStatistics in the DEA
As described in DEA literature in DEA technique frontier istaken into consideration rather than central tendency consid-ered in regression analysis DEA technique considers that anenvelope encompasses through all the observations as tight aspossible and does not try to fit regression planes in center ofdata In DEA methodology each unit is considered initiallyand compared to the efficient frontier but in regressionanalysis a procedure is considered in which a single functionfits to the data DEA uses different weights for differentunits but does not let the units use weights of other units
Canonical correlation analysis as Friedman and Sinuany-Stern [53] noted can be used for ranking units This methodis somehow the extension of regression analysis Adler et al[89] The aim in canonical correlation analysis is to find asingle vector commonweight for the inputs and outputs of allunits Consider119885
119895119882
119895as the composite input and output and
as corresponding weights respectively The presentedmodel by Tatsuoka and Lohnes [100] is as follows
max 119903119911119908
=
119878119909119910119880
(119878119909119909119881) (119878
119910119910119880)
(12)
st 119909119909119881 = 1
119910119910119880 = 1
(60)
where 119878119909119909 119878
119910119910 and 119878
119909119910are respectively defined as the matri-
ces of the sums of squares and sums of products of thevariables
Friedman and Sinuany-Stern [53] while defining the ratioof linear combinations of the inputs and outputs 119879
119895=
119882119895119885
119895used canonical correlation analysis As they noted that
scaling ratio 119879119895of the canonical correlation analysisDEA is
unboundedSinuany-Stern et al [101] used linear discriminant analy-
sis for ranking units They defined
119863119895=
119904
sum
119903=1
119906119903119910119903119895+
119898
sum
119894=1
V119903(minus119909
119894119895) (61)
DMU119895is said to be efficient if 119863
119895gt 119863
119888where 119863
119888is a critical
value based on themidpoint of themeans of the discriminantfunction value of the two groups Morrison [102] The largerthe amount of119863
119895is the better rank DMU
119895will have
Friedman and Sinuany-Stern [53] noted that as cross-efficiencyDEA canonical correlation analysisDEA and dis-criminant analysisDEA ranking orders may vary from eachother thus it seems necessary to introduce the combinedranking (CODEA) Combined ranking for each unit con-sidered all the ranks obtained from the above-mentionedrankings Moreover statistical tests are one for goodness offit between DEA and a specific ranking and the other fortesting correlation between variety of ranking orders Siegeland Castellan [103]
8 Ranking with MulticriteriaDecision-Making (MCDM) Methodologiesand DEA
Li and Reeves [56] for increasing discrimination power ofDEA presented amultiple-objective linear program (MOLP)As the authors mentioned using minimax and minsumefficiency in addition to the standard DEA objective functionhelp to increase discrimination power of DEA
Strassert and Prato [59] presented the balancing andranking method which uses a three-step procedure forderiving an overall complete or partial final order of optionsIn the first step derive an outranking matrix for all options
Journal of Applied Mathematics 13
from the criteria values Considering this matrix it is possibleto show the frequencywithwhich one option is ranked higherthan the other options In the second step by triangularizingthe outranking matrix establish an implicit preordering orprovisional ordering of optionsTheoutrankingmatrix showsthe degree to which there is a complete overall order ofoptions In the third step based on information given in anadvantages-disadvantages table the provisional ordering issubjected to different screening and balancing operations
Chen [60] utilized a nonparametric approach DEA toestimate and rank the efficiency of association rules withmultiple criteria in following steps Proposed postprocessingapproach is as follows
Step 1 Input data for association rule mining
Step 2 Mine association rules by using the a priori algorithmwith minimum support and minimum confidence
Step 3 Determine subjective interestingness measures byfurther considering the domain related knowledge
Step 4 Calculate the preference scores of association rulesdiscovered in Step 2 by using Cook and Kressrsquos DEA model
Step 5 Discriminate the efficient association rules found inStep 3 by using Obata and Ishiirsquos [104] discriminate model
Step 6 Select rules for implementation by considering thereference scores generated in Step 5 and domain relatedknowledge
Jablonsky [61] presented two original models super SBMand AHP for ranking of efficient units in DEA As theauthormentioned thesemodels are based onmultiple criteriadecision-making techniques-goal programming and analytichierarchy process Super SBM model for ranking units is asfollows
min 120579119866
119902= 1 + 119905120574 + (1 minus 119905) (
sum119898
119894=1119904+
1119894
119909119894119902
+
sum119898
119894=1119904minus
2119896
119910119896119902
)
st119899
sum
119895=1119895 = 119902
120582119895119909119894119895+ 119904
minus
1119894minus 119904
+
1119894= 119909
119894119902 i = 1 119898
119899
sum
119895=1119895 = 119902
120582119895119910119896119895+ 119904
minus
2119896minus 119904
+
2119896= 119910
119896119902 119896 = 1 119903
119904+
1119894le 120574 119894 = 1 119898
119904minus
2119896le 120574 119896 = 1 119903
119905 isin 0 1 120582119895ge 0 119878
+
1 119878
+
2ge 0 119878
minus
1 119878
minus
2ge 0
119895 = 1 119899
(62)
Wang and Jiang [62] presented an alternative mixedinteger linear programming models in order to identifythe most efficient units in DEA technique As the authors
mentioned presented models can make full use of input-output information with no need to specify any assuranceregions for input and output weights to avoid zero weights
min119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) minus
119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895)
st119904
sum
119903=1
119906119903119910119903119895minus
119898
sum
119894=1
V119894119909119894119895le 119868
119895 119895 = 1 119899
119899
sum
119895=1
119868119895= 1
119868119895isin 0 1 119895 = 1 119899
119906119903ge
1
(119898 + 119904)max119895119910
119903119895
119903 = 1 119904
V119894ge
1
(119898 + 119904)max119895119909
119894119895
119894 = 1 119898
(63)
Hosseinzadeh Lotfi et al [63] provided an improved three-stage method for ranking alternatives in multiple criteriadecision analysis In the first stage based on the bestand worst weights in the optimistic and pessimistic casesobtain the rank position of each alternative respectively Theobtained weight in the first stage is not unique thus it seemsnecessary to introduce a secondary goal that is used in thesecond stage Finally in the third stage the ranks of thealternatives compute in the optimistic or pessimistic case Itis mentionable that the model proposed in the third stageis a multicriteria decision-making (MCDM) model and itis solved by mixed integer programming As the authorsmentioned and provided in their paper this model can beconverted into an LP problem
The previous model with 119897 = 1 is the original CCR modelNote that DMUs in 119864
1 show the first level efficient frontier119897 = 2 indicates the second level efficient frontier when thefirst level efficient frontier is omittedThe following algorithmfinds these efficient frontiers
Step 1 Set 119897 = 1 and evaluate the entire set of DMUs 1198691 Inthis way the first level efficient DMU 1198641 is identified
Step 2 Exclude the efficient DMUs from future DEA runsand set 119869119897+1 = 119869
119897minus 119864
119897 (If 119869119897+1 = oslash stop)
Step 3 Evaluate the new subset of ldquoinefficientrdquo DMUs 119869119897+1to obtain a new set of efficient DMUs 119864119897+1
Step 4 Let l = l + 1 Go to Step 2
When 119869119897+1 = oslash the algorithm stopsAs the authors proved in this way it is possible to rank
the DMUs in the first efficient frontier based upon theirattractiveness scores and identify the best one
Jahanshahloo et al [65] provided a paper for rankingunits using gradient line As the authors mentioned theadvantage of this model is stability and robustness
supports on 119879119888 the obtained PPS by omitting theDMUunder
assessment in 119879119888 As the authors proved a unit is efficient
iff the optimal objective function of the following model isgreater than zero Consider
1198750= (119883 119884) 119883 = 120572119883
119900 119884 = 120573119884
119900
1198780= (119883 119884) isin 1198750
119883 = 120572119883119900 119884 = 120573119884
119900 120572 ge 0 120573 ge 0
(67)
Intersection of 1198780and efficient surface of
119879119888119888 is a half line
where its equation is (minus119881lowast119879119883
119900)120572 + (119880
lowast119879119884119900)120573 = 0 To rank
DMU119900the length of connecting arc DMU
119900with intersection
point of line and previous ellipse is calculated in (120572 120573) spaceThis intersection is as follows
120572lowast= (
1198702
120572119870
2
120573
1198702
120573+ (119881
lowast119879119883
119900119880
lowast119879119884119900)2119870
2
120572
)
12
120573lowast= (
119881lowast119879119883
119900
119880lowast119879119884119900
)120572lowast
(68)
Now the length of connecting arc DMU119900to the point
corresponding to 120572lowast 120573lowast in (120572 120573) plan is calculated as follows119868 = int
120572lowast
1(1 + 120573)
12119889120572 where
119870120572= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119899
119895=11199092
119894119900
)
12
119870120573= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119904
119903=11199102
119903119900
)
12
(69)
Jahanshahloo et al [66] also provided a paper with theconcept of advantage in data envelopment analysis
In their paper Jahanshahloo et al [67] consideringMonteCarlo method presented a new method of ranking
Step 1 Generate a uniformly distributed sequence of 1198801198952119899
119895=1
on(0 1)
Step 2 Random numbers should be classified into 119873 pairslike (119880
1 119880
1015840
1) (119880
119873 119880
1015840
119873) in a way that each number is used
just one time
Step 3 Compute119883119894= 119886 + 119880
119894(119887 minus 119886) and 119891(119883
119894) gt 119888119880
1015840
119894
Step 4 Estimate the integral 119868 by 120579119868= 119888(119887 minus 119886)(119873
119867119873)
Now consider DMU119900as an efficient unit measure those
units that are dominated DMU119900 As for dome DMUs this
would be unbounded thus the authors for each unitbounded the region Then for DMU
119900if (minus119883
119900 119884
119900) ge (minus119883 119884)
then (minus119883 119884) is in the RED of DMU119900 Now by using 119881
119901=
119881lowast(119873
119867119873) all the hits (the above condition) can be counted
where 119881lowast is the measure of the whole region JahanshahlooandAfzalinejad [68] presented amethod based on distance of
Journal of Applied Mathematics 15
the unit under evaluation to the full inefficient frontier Theypresented two models radian and nonradial
min 120601
st119899
sum
119895=1
120582119895119909119894119895= 120601119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
120582119895ge 0 119895 = 1 119898
max119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903
st sum
119895isin119869minus119887
120582119895119909119894119895minus 119904
minus
119894= 119909
119894119900 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895+ 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
119904minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(70)
Amirteimoori [69] based on the same idea considered bothefficient and antiefficient frontiers for efficiency analysis andranking units
Kao [70] mentioned that determining the weights ofindividual criteria in multiple criteria decision analysis in away that all alternatives can be compared according to theaggregate performance of all criteria is of great importanceAs Kao noted this problem relates to search for alternativeswith a shorter and longer distance respectively to the idealand anti-ideal units He proposed a measure that consideredthe calculation of the relative position of an alternativebetween the ideal and anti-ideal for finding an appropriaterankings
Khodabakhshi and Aryavash [71] presented a method forranking all units using DEA concept
First Compute the minimum and maximum efficiencyvalues of eachDMU in regard to this assumption that the sumof efficiency values of all DMUs equals to 1
Second determine the rank of each DMU in relation to acombination of itsminimumandmaximumefficiency values
Zerafat Angiz et al [72] proposed a technique in orderto aggregate the opinions of experts in voting system Asthe authors mentioned the presented method uses fuzzyconcept and it is computationally efficient and can fullyrank alternatives At first number of votes given to a rankposition was grouped to construct fuzzy numbers and thenthe artificial ideal alternative introduced Furthermore byperforming DEA the efficiency measure of alternatives was
obtained considering artificial ideal alternative comparedby each of the alternatives pair by pair Thus alternativesare ranked in accordance with their efficiency scores Ifthis method cannot completely rank alternatives weightrestrictions based on fuzzy concept are imposed into theanalysis
10 Different Applications in Ranking Units
As discussed formerly there exist a variety of ranking meth-ods and applications in theliterature Nowadays DEAmodelsare widely used in different areas for efficiency evaluationbenchmarking and target setting ranking entities and soforth Ranking units as one of the important issues in DEAhas been performed in different areas Jahanshahloo et al [42]ranked cities in Iran to find the best place for creating a datafactory Hosseinzadeh Lotfi et al [28] used a new methodfor ranking in order to find the best place for power plantlocation Ali andNakosteen [80] Amirteimoori et al [37] andAlirezaee and Afsharian [24] Soltanifar and HosseinzadehLotfi [105] Zerafat Angiz et al [72] Hosseinzadeh Lotfiet al [28] Jahanshahloo [50] Chen and Deng [52] andJablonsky [61] used different ranking methods in bankingsystem Sadjadi [40] ranked provincial gas companies in IranMehrabian et al [32] Li et al [39] Orkcu and Bal [12] andWu et al [19] ranked different departments of universitiesJahanshahloo et al [35] Jahanshahloo and Afzalinejad [68]utilized the presented models in ranking 28 Chinese citiesJahanshahloo at al [35] provided an application to burdensharing amongst NATOmember nations Jahanshahloo et al[14] utilized the presented model for ranking nursery homesContreras [18] andWang et al [23] used the provided rankingtechniques in ranking candidates Lu and Lo [51] applied theirmethod on an application to financial holding companiesHosseinzadeh Lotfi et al [63] consider an empirical exampleinwhich voters are asked to rank twoout of seven alternativesWang and Jiang [62] utilized the provided method in facilitylayout design in manufacturing systems and performanceevaluation of 30OECD countries Chen [60] used an exampleof market basket data in order to illustrate the providedapproach
11 Application
In this section some of the reviewed models are applied onthe example used in Soltanifar and Hosseinzadeh Lotfi [105]Consider twenty commercial banks of Iran with input-outputdata tabulated in Table 1 and summarized as follows Also theresults of CCRmodel are listed in this table As it can be seenseven units are efficient DMUS 14 7 12 15 17 and 20 Inputsare staff computer terminal and spaceOutputs are depositsloans granted and charge
Consider some of the important ranking methods inliterature as follows
RM1 AP model [31] is based upon the idea of leaveunit evaluation out and measuring the distance of theunit under evaluation from the production possibilityset constructed by the remaining DMUs
RM2MAJmodel [32] presented for ranking efficientwhich is always stable but might be infeasible in somecasesRM3 Modified MAJ model [38] overcomes theproblem which might occurr in MAJ modelRM4 A new model based on the idea of alterationsin the reference set of the inefficient units [50]RM5 A model presented by Li et al [39] whichis a super-efficiency method that does not have thesuffering in previous methodsRM6 Slack-basedmodel [33] is based upon the inputand output variables at the same timeRM7 SA DEAmodel [49] overcomes the problem ofinfeasibility which existed in AP modelRM8 Cross-efficiency [10] is provided based onusing weights of each unit under evaluation in opti-mality for other unitsRM9 A model based on finding common set ofweights [25] which determine the common set ofweights for DMUs and ranked DMUs based this ideaRM10 A model based on finding common set ofweights [22 67] for ranking efficient unitsRM11 L1-norm model [34 35 65 66] the idea isbased upon the leave-one-out efficient unit and l1-norm which is always feasible and stableRM12 119871
infin-norm model Rezai Balf et al [46] pro-
vided a method with more ability over other existingmethods based on Tchebycheff normRM13 An enhanced Russel measure of super-efficiency model for ranking units [44]RM14 A rankingmodel which considers the distanceof unit from the full inefficient frontier [38]RM15 Amodified super-efficiencymodel [45] whichovercomes the infeasibility that may happen in prob-lem This model is based on simultaneous projectionof input output
In accordance with properties of different ranking models inorder to rank efficient units consider Table 2
1199011 Feasibility
1199012 Ranking extreme efficient units
1199013 Complexity in computation
1199014 Instability
1199015 Absence of multiple optimal solution
1199016 Dependency to 120579 and slacks
1199017 Dependency to the number of efficient and ineffi-
cient units
In Tables 3 and 4 the rank of efficient units consideringthe above mentioned methods is listed Note that ED showsefficient DMUs (ED) and RM
119895 (119895 = 1 15) are those
explained previously
12 Conclusion
In this paper the DEA ranking was reviewed and classifiedinto seven general groups In the first group those papersbased on a cross-efficiencymatrix were reviewed In this fieldDMUs have been evaluated by self- and peer pressure Thesecond group of papers is based on those papers lookingfor optimal weights in DEA analysis The third one is thesuper-efficiency method By omitting the under evaluationunit and constructing a new frontier by the remaining unitsthe unit under evaluation can get an equal or greater scoreThe fourth group is based on benchmarking idea In thisclass the effect of an efficient unit considered as a targetfor inefficient units is investigated This idea is very usefulfor the managers in decision making Another class thefifth one involves the application of multivariate statisticaltools The sixth section discusses the ranking methods basedon multicriteria decision-making (MCDM) methodologiesand DEA The final section includes some various rankingmethods presented in the literature Finally in an applicationthe result of some of the above-mentioned ranking methodsis presented
References
[1] M J Farrell ldquoThe measurement of productive efficiencyrdquoJournal of the Royal Statistical Society A vol 120 pp 253ndash2811957
[2] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
18 Journal of Applied Mathematics
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
Chen et al [45] proposed a modified super-efficiencymethod for ranking units based on simultaneous input-output projectionThe presentedmodel overcomes the infea-sibility problem
1199011= min
120579119904119903
119900
120601119904119903
119900
st sum
119895=1
119895 = 119900
120582119895119909119894119895le 120579
119904119903
119900119909119894119900 119894 = 1 119898
sum
119895=1
119895 = 119900
120582119895119910119903119895ge 120601
119904119903
119900119910119903119900 119903 = 1 119904
sum
119895=1
119895 = 119900
120582119895= 1
0 lt 120579119904119903
119900le 1 120601
119904119903
119900ge 1 119894 = 1 119898
119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(50)
Rezai Balf et al [46] provide a model for ranking units basedon Tchebycheff norm As proved that this model is alwaysfeasible and stable it seems to have superiority over othermodels
max 119881119901
st 119881119901ge
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901 119894 = 1 119898
119881119901ge 119910
119903119901minus
119899
sum
119895=1
119895 = 119900
120582119895119910119903119895 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(51)
where
119881119901= Max
(
119899
sum
119895=1
119895 = 119900
120582119895119909119894119895minus 119909
119894119901)119894 = 1 119898
(119910119903119901minus
119899
sum
119895=1
119895 = 119901
120582119895119910119903119895)119903 = 1 119904
(52)
Chen et al [47] for overcoming the infeasibility problem thatoccurred in variable returns to scale super-efficiency DEAmodel according to a directional distance function developedNerlove-Luenberger (N-L) measure of super-efficiency
6 Benchmarking Ranking Techniques
Sueyoshi et al [49] proposes a ldquobenchmark approachrdquo forbaseball evaluation This method is the combination of DEA
and (Offensive earned-run average) OERA As the authorsnoted using this method it is possible to select best unitsand also their ranking orders They mentioned that usingonly DEAmay result in a shortcoming in assessment as manyefficient units can be identifiedThus the authors used slack-adjusted DEA model and OERA to overcome this difficulty
Jahanshahloo et al [50] presented a new model forranking DMUs based on alteration in reference set The ideais based on this fact that efficient units can be the target unitfor inefficient units
min 120597119886119887
= 120579 minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st sum
119895isin119869minus119887
120582119895119909119894119895+ 119904
minus
119894minus 120579119909
119894119886 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
120579 free 119904minus119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 isin 119869 minus 119887
(53)
Finally the ranking order for each efficient unit b can becomputing by
119887= sum
119886isin119869119899
120597119886119887
119899
(54)
Note that this method cannot rank nonextreme efficientunits
Lu and Lo [51] provided an interactive benchmark modelfor ranking units The idea is based upon considering a fixedunit as a benchmark and calculating the efficiency of otherunits pair by pair to this unitThis procedure continueswhenall units are accounted for as a benchmark unit ConsiderDMU
119900as a unit under evaluation andDMU
119887as a benchmark
Assume
119909119894= 119909
119894119900(1 + 120601
119894) 119910
119903= 119910
119903119900(1 minus 120593
119903)
min 120579lowast119887
119900=
1 + (1119898)sum119898
119894=1120601119894
1 minus (1119904)sum119904
119903=1120593119903
st 120582119887119909119894119887minus 119909
119894119900120601119894le 119909
119894119900 119894 = 1 119898
120582119887119910119903119887minus 119910
119903119900120593119903ge 119910
119903119900 119903 = 1 119904
120601119894ge 119900 120593
119903le 119910
119903119900 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(55)
Then using (sum119899
119887=1120579119887lowast
119900)119899 119900 = 1 119899 the efficiency of
benchmark DMU119900can be obtained Using the following
index which indicates the increment in efficiency of a unitby moving from peer appraisal to self-appraisal it is nowpossible to rank units Note that the less the magnitude of
12 Journal of Applied Mathematics
this index is the better rank for corresponding unit will beobtained
FPIIBM119870
=
(TEBCC119896
minus STDIBM119896
)
(STDIBM119896
)
(56)
where TEBCC119896
and STDIBM119896
are respectively efficiency in BCCmodel and normalization of TEIBM of DMU
119896 Chen [52]
provided a paper for ranking efficient and inefficient unitsin DEA They noted that the evaluation of efficient unitsis based upon the alterations in efficiency of all inefficientunits by omitting in reference set At first solve the followingmodel which measures the efficiency of DMU
119886when DMU
119887
is eliminated from the reference set
min 120588119886119887
= 120579119904
119886minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1119895 = 119887
120582119895119909119894119895+ 119904
minus
119894= 120579
119904
119886119909119894119886 119894 = 1 119898
119899
sum
119895=1119895 = 119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
119899
sum
119895=1119895 = 119887
= 1
120579119904
119886ge 0 119904
minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119887
(57)
Then again solve the previous model this time for calculat-ing the efficiency score of each inefficient unit when each ofefficient units in turn is eliminated from the reference set 120578lowast
119886
and calculate the efficiency change
120591119886119887
= 120588lowast
119886119887minus 120578
lowast
119886 (58)
Then calculate the following index called as ldquoMCDErdquo indexfor each efficient and inefficient unit
119864119886= sum
119887isin119881119864
119882lowast
119887120588lowast
119886119887 119864
119887= sum
119887isin119881119868
119882lowast
119887120588lowast
119886119887 (59)
Now in accordance with themagnitude of the acquired indexit is possible to rank unitsThose units with higher score havebetter ranking order
7 Ranking Techniques by MultivariateStatistics in the DEA
As described in DEA literature in DEA technique frontier istaken into consideration rather than central tendency consid-ered in regression analysis DEA technique considers that anenvelope encompasses through all the observations as tight aspossible and does not try to fit regression planes in center ofdata In DEA methodology each unit is considered initiallyand compared to the efficient frontier but in regressionanalysis a procedure is considered in which a single functionfits to the data DEA uses different weights for differentunits but does not let the units use weights of other units
Canonical correlation analysis as Friedman and Sinuany-Stern [53] noted can be used for ranking units This methodis somehow the extension of regression analysis Adler et al[89] The aim in canonical correlation analysis is to find asingle vector commonweight for the inputs and outputs of allunits Consider119885
119895119882
119895as the composite input and output and
as corresponding weights respectively The presentedmodel by Tatsuoka and Lohnes [100] is as follows
max 119903119911119908
=
119878119909119910119880
(119878119909119909119881) (119878
119910119910119880)
(12)
st 119909119909119881 = 1
119910119910119880 = 1
(60)
where 119878119909119909 119878
119910119910 and 119878
119909119910are respectively defined as the matri-
ces of the sums of squares and sums of products of thevariables
Friedman and Sinuany-Stern [53] while defining the ratioof linear combinations of the inputs and outputs 119879
119895=
119882119895119885
119895used canonical correlation analysis As they noted that
scaling ratio 119879119895of the canonical correlation analysisDEA is
unboundedSinuany-Stern et al [101] used linear discriminant analy-
sis for ranking units They defined
119863119895=
119904
sum
119903=1
119906119903119910119903119895+
119898
sum
119894=1
V119903(minus119909
119894119895) (61)
DMU119895is said to be efficient if 119863
119895gt 119863
119888where 119863
119888is a critical
value based on themidpoint of themeans of the discriminantfunction value of the two groups Morrison [102] The largerthe amount of119863
119895is the better rank DMU
119895will have
Friedman and Sinuany-Stern [53] noted that as cross-efficiencyDEA canonical correlation analysisDEA and dis-criminant analysisDEA ranking orders may vary from eachother thus it seems necessary to introduce the combinedranking (CODEA) Combined ranking for each unit con-sidered all the ranks obtained from the above-mentionedrankings Moreover statistical tests are one for goodness offit between DEA and a specific ranking and the other fortesting correlation between variety of ranking orders Siegeland Castellan [103]
8 Ranking with MulticriteriaDecision-Making (MCDM) Methodologiesand DEA
Li and Reeves [56] for increasing discrimination power ofDEA presented amultiple-objective linear program (MOLP)As the authors mentioned using minimax and minsumefficiency in addition to the standard DEA objective functionhelp to increase discrimination power of DEA
Strassert and Prato [59] presented the balancing andranking method which uses a three-step procedure forderiving an overall complete or partial final order of optionsIn the first step derive an outranking matrix for all options
Journal of Applied Mathematics 13
from the criteria values Considering this matrix it is possibleto show the frequencywithwhich one option is ranked higherthan the other options In the second step by triangularizingthe outranking matrix establish an implicit preordering orprovisional ordering of optionsTheoutrankingmatrix showsthe degree to which there is a complete overall order ofoptions In the third step based on information given in anadvantages-disadvantages table the provisional ordering issubjected to different screening and balancing operations
Chen [60] utilized a nonparametric approach DEA toestimate and rank the efficiency of association rules withmultiple criteria in following steps Proposed postprocessingapproach is as follows
Step 1 Input data for association rule mining
Step 2 Mine association rules by using the a priori algorithmwith minimum support and minimum confidence
Step 3 Determine subjective interestingness measures byfurther considering the domain related knowledge
Step 4 Calculate the preference scores of association rulesdiscovered in Step 2 by using Cook and Kressrsquos DEA model
Step 5 Discriminate the efficient association rules found inStep 3 by using Obata and Ishiirsquos [104] discriminate model
Step 6 Select rules for implementation by considering thereference scores generated in Step 5 and domain relatedknowledge
Jablonsky [61] presented two original models super SBMand AHP for ranking of efficient units in DEA As theauthormentioned thesemodels are based onmultiple criteriadecision-making techniques-goal programming and analytichierarchy process Super SBM model for ranking units is asfollows
min 120579119866
119902= 1 + 119905120574 + (1 minus 119905) (
sum119898
119894=1119904+
1119894
119909119894119902
+
sum119898
119894=1119904minus
2119896
119910119896119902
)
st119899
sum
119895=1119895 = 119902
120582119895119909119894119895+ 119904
minus
1119894minus 119904
+
1119894= 119909
119894119902 i = 1 119898
119899
sum
119895=1119895 = 119902
120582119895119910119896119895+ 119904
minus
2119896minus 119904
+
2119896= 119910
119896119902 119896 = 1 119903
119904+
1119894le 120574 119894 = 1 119898
119904minus
2119896le 120574 119896 = 1 119903
119905 isin 0 1 120582119895ge 0 119878
+
1 119878
+
2ge 0 119878
minus
1 119878
minus
2ge 0
119895 = 1 119899
(62)
Wang and Jiang [62] presented an alternative mixedinteger linear programming models in order to identifythe most efficient units in DEA technique As the authors
mentioned presented models can make full use of input-output information with no need to specify any assuranceregions for input and output weights to avoid zero weights
min119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) minus
119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895)
st119904
sum
119903=1
119906119903119910119903119895minus
119898
sum
119894=1
V119894119909119894119895le 119868
119895 119895 = 1 119899
119899
sum
119895=1
119868119895= 1
119868119895isin 0 1 119895 = 1 119899
119906119903ge
1
(119898 + 119904)max119895119910
119903119895
119903 = 1 119904
V119894ge
1
(119898 + 119904)max119895119909
119894119895
119894 = 1 119898
(63)
Hosseinzadeh Lotfi et al [63] provided an improved three-stage method for ranking alternatives in multiple criteriadecision analysis In the first stage based on the bestand worst weights in the optimistic and pessimistic casesobtain the rank position of each alternative respectively Theobtained weight in the first stage is not unique thus it seemsnecessary to introduce a secondary goal that is used in thesecond stage Finally in the third stage the ranks of thealternatives compute in the optimistic or pessimistic case Itis mentionable that the model proposed in the third stageis a multicriteria decision-making (MCDM) model and itis solved by mixed integer programming As the authorsmentioned and provided in their paper this model can beconverted into an LP problem
The previous model with 119897 = 1 is the original CCR modelNote that DMUs in 119864
1 show the first level efficient frontier119897 = 2 indicates the second level efficient frontier when thefirst level efficient frontier is omittedThe following algorithmfinds these efficient frontiers
Step 1 Set 119897 = 1 and evaluate the entire set of DMUs 1198691 Inthis way the first level efficient DMU 1198641 is identified
Step 2 Exclude the efficient DMUs from future DEA runsand set 119869119897+1 = 119869
119897minus 119864
119897 (If 119869119897+1 = oslash stop)
Step 3 Evaluate the new subset of ldquoinefficientrdquo DMUs 119869119897+1to obtain a new set of efficient DMUs 119864119897+1
Step 4 Let l = l + 1 Go to Step 2
When 119869119897+1 = oslash the algorithm stopsAs the authors proved in this way it is possible to rank
the DMUs in the first efficient frontier based upon theirattractiveness scores and identify the best one
Jahanshahloo et al [65] provided a paper for rankingunits using gradient line As the authors mentioned theadvantage of this model is stability and robustness
supports on 119879119888 the obtained PPS by omitting theDMUunder
assessment in 119879119888 As the authors proved a unit is efficient
iff the optimal objective function of the following model isgreater than zero Consider
1198750= (119883 119884) 119883 = 120572119883
119900 119884 = 120573119884
119900
1198780= (119883 119884) isin 1198750
119883 = 120572119883119900 119884 = 120573119884
119900 120572 ge 0 120573 ge 0
(67)
Intersection of 1198780and efficient surface of
119879119888119888 is a half line
where its equation is (minus119881lowast119879119883
119900)120572 + (119880
lowast119879119884119900)120573 = 0 To rank
DMU119900the length of connecting arc DMU
119900with intersection
point of line and previous ellipse is calculated in (120572 120573) spaceThis intersection is as follows
120572lowast= (
1198702
120572119870
2
120573
1198702
120573+ (119881
lowast119879119883
119900119880
lowast119879119884119900)2119870
2
120572
)
12
120573lowast= (
119881lowast119879119883
119900
119880lowast119879119884119900
)120572lowast
(68)
Now the length of connecting arc DMU119900to the point
corresponding to 120572lowast 120573lowast in (120572 120573) plan is calculated as follows119868 = int
120572lowast
1(1 + 120573)
12119889120572 where
119870120572= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119899
119895=11199092
119894119900
)
12
119870120573= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119904
119903=11199102
119903119900
)
12
(69)
Jahanshahloo et al [66] also provided a paper with theconcept of advantage in data envelopment analysis
In their paper Jahanshahloo et al [67] consideringMonteCarlo method presented a new method of ranking
Step 1 Generate a uniformly distributed sequence of 1198801198952119899
119895=1
on(0 1)
Step 2 Random numbers should be classified into 119873 pairslike (119880
1 119880
1015840
1) (119880
119873 119880
1015840
119873) in a way that each number is used
just one time
Step 3 Compute119883119894= 119886 + 119880
119894(119887 minus 119886) and 119891(119883
119894) gt 119888119880
1015840
119894
Step 4 Estimate the integral 119868 by 120579119868= 119888(119887 minus 119886)(119873
119867119873)
Now consider DMU119900as an efficient unit measure those
units that are dominated DMU119900 As for dome DMUs this
would be unbounded thus the authors for each unitbounded the region Then for DMU
119900if (minus119883
119900 119884
119900) ge (minus119883 119884)
then (minus119883 119884) is in the RED of DMU119900 Now by using 119881
119901=
119881lowast(119873
119867119873) all the hits (the above condition) can be counted
where 119881lowast is the measure of the whole region JahanshahlooandAfzalinejad [68] presented amethod based on distance of
Journal of Applied Mathematics 15
the unit under evaluation to the full inefficient frontier Theypresented two models radian and nonradial
min 120601
st119899
sum
119895=1
120582119895119909119894119895= 120601119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
120582119895ge 0 119895 = 1 119898
max119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903
st sum
119895isin119869minus119887
120582119895119909119894119895minus 119904
minus
119894= 119909
119894119900 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895+ 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
119904minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(70)
Amirteimoori [69] based on the same idea considered bothefficient and antiefficient frontiers for efficiency analysis andranking units
Kao [70] mentioned that determining the weights ofindividual criteria in multiple criteria decision analysis in away that all alternatives can be compared according to theaggregate performance of all criteria is of great importanceAs Kao noted this problem relates to search for alternativeswith a shorter and longer distance respectively to the idealand anti-ideal units He proposed a measure that consideredthe calculation of the relative position of an alternativebetween the ideal and anti-ideal for finding an appropriaterankings
Khodabakhshi and Aryavash [71] presented a method forranking all units using DEA concept
First Compute the minimum and maximum efficiencyvalues of eachDMU in regard to this assumption that the sumof efficiency values of all DMUs equals to 1
Second determine the rank of each DMU in relation to acombination of itsminimumandmaximumefficiency values
Zerafat Angiz et al [72] proposed a technique in orderto aggregate the opinions of experts in voting system Asthe authors mentioned the presented method uses fuzzyconcept and it is computationally efficient and can fullyrank alternatives At first number of votes given to a rankposition was grouped to construct fuzzy numbers and thenthe artificial ideal alternative introduced Furthermore byperforming DEA the efficiency measure of alternatives was
obtained considering artificial ideal alternative comparedby each of the alternatives pair by pair Thus alternativesare ranked in accordance with their efficiency scores Ifthis method cannot completely rank alternatives weightrestrictions based on fuzzy concept are imposed into theanalysis
10 Different Applications in Ranking Units
As discussed formerly there exist a variety of ranking meth-ods and applications in theliterature Nowadays DEAmodelsare widely used in different areas for efficiency evaluationbenchmarking and target setting ranking entities and soforth Ranking units as one of the important issues in DEAhas been performed in different areas Jahanshahloo et al [42]ranked cities in Iran to find the best place for creating a datafactory Hosseinzadeh Lotfi et al [28] used a new methodfor ranking in order to find the best place for power plantlocation Ali andNakosteen [80] Amirteimoori et al [37] andAlirezaee and Afsharian [24] Soltanifar and HosseinzadehLotfi [105] Zerafat Angiz et al [72] Hosseinzadeh Lotfiet al [28] Jahanshahloo [50] Chen and Deng [52] andJablonsky [61] used different ranking methods in bankingsystem Sadjadi [40] ranked provincial gas companies in IranMehrabian et al [32] Li et al [39] Orkcu and Bal [12] andWu et al [19] ranked different departments of universitiesJahanshahloo et al [35] Jahanshahloo and Afzalinejad [68]utilized the presented models in ranking 28 Chinese citiesJahanshahloo at al [35] provided an application to burdensharing amongst NATOmember nations Jahanshahloo et al[14] utilized the presented model for ranking nursery homesContreras [18] andWang et al [23] used the provided rankingtechniques in ranking candidates Lu and Lo [51] applied theirmethod on an application to financial holding companiesHosseinzadeh Lotfi et al [63] consider an empirical exampleinwhich voters are asked to rank twoout of seven alternativesWang and Jiang [62] utilized the provided method in facilitylayout design in manufacturing systems and performanceevaluation of 30OECD countries Chen [60] used an exampleof market basket data in order to illustrate the providedapproach
11 Application
In this section some of the reviewed models are applied onthe example used in Soltanifar and Hosseinzadeh Lotfi [105]Consider twenty commercial banks of Iran with input-outputdata tabulated in Table 1 and summarized as follows Also theresults of CCRmodel are listed in this table As it can be seenseven units are efficient DMUS 14 7 12 15 17 and 20 Inputsare staff computer terminal and spaceOutputs are depositsloans granted and charge
Consider some of the important ranking methods inliterature as follows
RM1 AP model [31] is based upon the idea of leaveunit evaluation out and measuring the distance of theunit under evaluation from the production possibilityset constructed by the remaining DMUs
RM2MAJmodel [32] presented for ranking efficientwhich is always stable but might be infeasible in somecasesRM3 Modified MAJ model [38] overcomes theproblem which might occurr in MAJ modelRM4 A new model based on the idea of alterationsin the reference set of the inefficient units [50]RM5 A model presented by Li et al [39] whichis a super-efficiency method that does not have thesuffering in previous methodsRM6 Slack-basedmodel [33] is based upon the inputand output variables at the same timeRM7 SA DEAmodel [49] overcomes the problem ofinfeasibility which existed in AP modelRM8 Cross-efficiency [10] is provided based onusing weights of each unit under evaluation in opti-mality for other unitsRM9 A model based on finding common set ofweights [25] which determine the common set ofweights for DMUs and ranked DMUs based this ideaRM10 A model based on finding common set ofweights [22 67] for ranking efficient unitsRM11 L1-norm model [34 35 65 66] the idea isbased upon the leave-one-out efficient unit and l1-norm which is always feasible and stableRM12 119871
infin-norm model Rezai Balf et al [46] pro-
vided a method with more ability over other existingmethods based on Tchebycheff normRM13 An enhanced Russel measure of super-efficiency model for ranking units [44]RM14 A rankingmodel which considers the distanceof unit from the full inefficient frontier [38]RM15 Amodified super-efficiencymodel [45] whichovercomes the infeasibility that may happen in prob-lem This model is based on simultaneous projectionof input output
In accordance with properties of different ranking models inorder to rank efficient units consider Table 2
1199011 Feasibility
1199012 Ranking extreme efficient units
1199013 Complexity in computation
1199014 Instability
1199015 Absence of multiple optimal solution
1199016 Dependency to 120579 and slacks
1199017 Dependency to the number of efficient and ineffi-
cient units
In Tables 3 and 4 the rank of efficient units consideringthe above mentioned methods is listed Note that ED showsefficient DMUs (ED) and RM
119895 (119895 = 1 15) are those
explained previously
12 Conclusion
In this paper the DEA ranking was reviewed and classifiedinto seven general groups In the first group those papersbased on a cross-efficiencymatrix were reviewed In this fieldDMUs have been evaluated by self- and peer pressure Thesecond group of papers is based on those papers lookingfor optimal weights in DEA analysis The third one is thesuper-efficiency method By omitting the under evaluationunit and constructing a new frontier by the remaining unitsthe unit under evaluation can get an equal or greater scoreThe fourth group is based on benchmarking idea In thisclass the effect of an efficient unit considered as a targetfor inefficient units is investigated This idea is very usefulfor the managers in decision making Another class thefifth one involves the application of multivariate statisticaltools The sixth section discusses the ranking methods basedon multicriteria decision-making (MCDM) methodologiesand DEA The final section includes some various rankingmethods presented in the literature Finally in an applicationthe result of some of the above-mentioned ranking methodsis presented
References
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[2] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
18 Journal of Applied Mathematics
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
this index is the better rank for corresponding unit will beobtained
FPIIBM119870
=
(TEBCC119896
minus STDIBM119896
)
(STDIBM119896
)
(56)
where TEBCC119896
and STDIBM119896
are respectively efficiency in BCCmodel and normalization of TEIBM of DMU
119896 Chen [52]
provided a paper for ranking efficient and inefficient unitsin DEA They noted that the evaluation of efficient unitsis based upon the alterations in efficiency of all inefficientunits by omitting in reference set At first solve the followingmodel which measures the efficiency of DMU
119886when DMU
119887
is eliminated from the reference set
min 120588119886119887
= 120579119904
119886minus 120576(
119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903)
st119899
sum
119895=1119895 = 119887
120582119895119909119894119895+ 119904
minus
119894= 120579
119904
119886119909119894119886 119894 = 1 119898
119899
sum
119895=1119895 = 119887
120582119895119910119903119895minus 119904
+
119903= 119910
119903119886 119903 = 1 119904
119899
sum
119895=1119895 = 119887
= 1
120579119904
119886ge 0 119904
minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 119887
(57)
Then again solve the previous model this time for calculat-ing the efficiency score of each inefficient unit when each ofefficient units in turn is eliminated from the reference set 120578lowast
119886
and calculate the efficiency change
120591119886119887
= 120588lowast
119886119887minus 120578
lowast
119886 (58)
Then calculate the following index called as ldquoMCDErdquo indexfor each efficient and inefficient unit
119864119886= sum
119887isin119881119864
119882lowast
119887120588lowast
119886119887 119864
119887= sum
119887isin119881119868
119882lowast
119887120588lowast
119886119887 (59)
Now in accordance with themagnitude of the acquired indexit is possible to rank unitsThose units with higher score havebetter ranking order
7 Ranking Techniques by MultivariateStatistics in the DEA
As described in DEA literature in DEA technique frontier istaken into consideration rather than central tendency consid-ered in regression analysis DEA technique considers that anenvelope encompasses through all the observations as tight aspossible and does not try to fit regression planes in center ofdata In DEA methodology each unit is considered initiallyand compared to the efficient frontier but in regressionanalysis a procedure is considered in which a single functionfits to the data DEA uses different weights for differentunits but does not let the units use weights of other units
Canonical correlation analysis as Friedman and Sinuany-Stern [53] noted can be used for ranking units This methodis somehow the extension of regression analysis Adler et al[89] The aim in canonical correlation analysis is to find asingle vector commonweight for the inputs and outputs of allunits Consider119885
119895119882
119895as the composite input and output and
as corresponding weights respectively The presentedmodel by Tatsuoka and Lohnes [100] is as follows
max 119903119911119908
=
119878119909119910119880
(119878119909119909119881) (119878
119910119910119880)
(12)
st 119909119909119881 = 1
119910119910119880 = 1
(60)
where 119878119909119909 119878
119910119910 and 119878
119909119910are respectively defined as the matri-
ces of the sums of squares and sums of products of thevariables
Friedman and Sinuany-Stern [53] while defining the ratioof linear combinations of the inputs and outputs 119879
119895=
119882119895119885
119895used canonical correlation analysis As they noted that
scaling ratio 119879119895of the canonical correlation analysisDEA is
unboundedSinuany-Stern et al [101] used linear discriminant analy-
sis for ranking units They defined
119863119895=
119904
sum
119903=1
119906119903119910119903119895+
119898
sum
119894=1
V119903(minus119909
119894119895) (61)
DMU119895is said to be efficient if 119863
119895gt 119863
119888where 119863
119888is a critical
value based on themidpoint of themeans of the discriminantfunction value of the two groups Morrison [102] The largerthe amount of119863
119895is the better rank DMU
119895will have
Friedman and Sinuany-Stern [53] noted that as cross-efficiencyDEA canonical correlation analysisDEA and dis-criminant analysisDEA ranking orders may vary from eachother thus it seems necessary to introduce the combinedranking (CODEA) Combined ranking for each unit con-sidered all the ranks obtained from the above-mentionedrankings Moreover statistical tests are one for goodness offit between DEA and a specific ranking and the other fortesting correlation between variety of ranking orders Siegeland Castellan [103]
8 Ranking with MulticriteriaDecision-Making (MCDM) Methodologiesand DEA
Li and Reeves [56] for increasing discrimination power ofDEA presented amultiple-objective linear program (MOLP)As the authors mentioned using minimax and minsumefficiency in addition to the standard DEA objective functionhelp to increase discrimination power of DEA
Strassert and Prato [59] presented the balancing andranking method which uses a three-step procedure forderiving an overall complete or partial final order of optionsIn the first step derive an outranking matrix for all options
Journal of Applied Mathematics 13
from the criteria values Considering this matrix it is possibleto show the frequencywithwhich one option is ranked higherthan the other options In the second step by triangularizingthe outranking matrix establish an implicit preordering orprovisional ordering of optionsTheoutrankingmatrix showsthe degree to which there is a complete overall order ofoptions In the third step based on information given in anadvantages-disadvantages table the provisional ordering issubjected to different screening and balancing operations
Chen [60] utilized a nonparametric approach DEA toestimate and rank the efficiency of association rules withmultiple criteria in following steps Proposed postprocessingapproach is as follows
Step 1 Input data for association rule mining
Step 2 Mine association rules by using the a priori algorithmwith minimum support and minimum confidence
Step 3 Determine subjective interestingness measures byfurther considering the domain related knowledge
Step 4 Calculate the preference scores of association rulesdiscovered in Step 2 by using Cook and Kressrsquos DEA model
Step 5 Discriminate the efficient association rules found inStep 3 by using Obata and Ishiirsquos [104] discriminate model
Step 6 Select rules for implementation by considering thereference scores generated in Step 5 and domain relatedknowledge
Jablonsky [61] presented two original models super SBMand AHP for ranking of efficient units in DEA As theauthormentioned thesemodels are based onmultiple criteriadecision-making techniques-goal programming and analytichierarchy process Super SBM model for ranking units is asfollows
min 120579119866
119902= 1 + 119905120574 + (1 minus 119905) (
sum119898
119894=1119904+
1119894
119909119894119902
+
sum119898
119894=1119904minus
2119896
119910119896119902
)
st119899
sum
119895=1119895 = 119902
120582119895119909119894119895+ 119904
minus
1119894minus 119904
+
1119894= 119909
119894119902 i = 1 119898
119899
sum
119895=1119895 = 119902
120582119895119910119896119895+ 119904
minus
2119896minus 119904
+
2119896= 119910
119896119902 119896 = 1 119903
119904+
1119894le 120574 119894 = 1 119898
119904minus
2119896le 120574 119896 = 1 119903
119905 isin 0 1 120582119895ge 0 119878
+
1 119878
+
2ge 0 119878
minus
1 119878
minus
2ge 0
119895 = 1 119899
(62)
Wang and Jiang [62] presented an alternative mixedinteger linear programming models in order to identifythe most efficient units in DEA technique As the authors
mentioned presented models can make full use of input-output information with no need to specify any assuranceregions for input and output weights to avoid zero weights
min119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) minus
119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895)
st119904
sum
119903=1
119906119903119910119903119895minus
119898
sum
119894=1
V119894119909119894119895le 119868
119895 119895 = 1 119899
119899
sum
119895=1
119868119895= 1
119868119895isin 0 1 119895 = 1 119899
119906119903ge
1
(119898 + 119904)max119895119910
119903119895
119903 = 1 119904
V119894ge
1
(119898 + 119904)max119895119909
119894119895
119894 = 1 119898
(63)
Hosseinzadeh Lotfi et al [63] provided an improved three-stage method for ranking alternatives in multiple criteriadecision analysis In the first stage based on the bestand worst weights in the optimistic and pessimistic casesobtain the rank position of each alternative respectively Theobtained weight in the first stage is not unique thus it seemsnecessary to introduce a secondary goal that is used in thesecond stage Finally in the third stage the ranks of thealternatives compute in the optimistic or pessimistic case Itis mentionable that the model proposed in the third stageis a multicriteria decision-making (MCDM) model and itis solved by mixed integer programming As the authorsmentioned and provided in their paper this model can beconverted into an LP problem
The previous model with 119897 = 1 is the original CCR modelNote that DMUs in 119864
1 show the first level efficient frontier119897 = 2 indicates the second level efficient frontier when thefirst level efficient frontier is omittedThe following algorithmfinds these efficient frontiers
Step 1 Set 119897 = 1 and evaluate the entire set of DMUs 1198691 Inthis way the first level efficient DMU 1198641 is identified
Step 2 Exclude the efficient DMUs from future DEA runsand set 119869119897+1 = 119869
119897minus 119864
119897 (If 119869119897+1 = oslash stop)
Step 3 Evaluate the new subset of ldquoinefficientrdquo DMUs 119869119897+1to obtain a new set of efficient DMUs 119864119897+1
Step 4 Let l = l + 1 Go to Step 2
When 119869119897+1 = oslash the algorithm stopsAs the authors proved in this way it is possible to rank
the DMUs in the first efficient frontier based upon theirattractiveness scores and identify the best one
Jahanshahloo et al [65] provided a paper for rankingunits using gradient line As the authors mentioned theadvantage of this model is stability and robustness
supports on 119879119888 the obtained PPS by omitting theDMUunder
assessment in 119879119888 As the authors proved a unit is efficient
iff the optimal objective function of the following model isgreater than zero Consider
1198750= (119883 119884) 119883 = 120572119883
119900 119884 = 120573119884
119900
1198780= (119883 119884) isin 1198750
119883 = 120572119883119900 119884 = 120573119884
119900 120572 ge 0 120573 ge 0
(67)
Intersection of 1198780and efficient surface of
119879119888119888 is a half line
where its equation is (minus119881lowast119879119883
119900)120572 + (119880
lowast119879119884119900)120573 = 0 To rank
DMU119900the length of connecting arc DMU
119900with intersection
point of line and previous ellipse is calculated in (120572 120573) spaceThis intersection is as follows
120572lowast= (
1198702
120572119870
2
120573
1198702
120573+ (119881
lowast119879119883
119900119880
lowast119879119884119900)2119870
2
120572
)
12
120573lowast= (
119881lowast119879119883
119900
119880lowast119879119884119900
)120572lowast
(68)
Now the length of connecting arc DMU119900to the point
corresponding to 120572lowast 120573lowast in (120572 120573) plan is calculated as follows119868 = int
120572lowast
1(1 + 120573)
12119889120572 where
119870120572= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119899
119895=11199092
119894119900
)
12
119870120573= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119904
119903=11199102
119903119900
)
12
(69)
Jahanshahloo et al [66] also provided a paper with theconcept of advantage in data envelopment analysis
In their paper Jahanshahloo et al [67] consideringMonteCarlo method presented a new method of ranking
Step 1 Generate a uniformly distributed sequence of 1198801198952119899
119895=1
on(0 1)
Step 2 Random numbers should be classified into 119873 pairslike (119880
1 119880
1015840
1) (119880
119873 119880
1015840
119873) in a way that each number is used
just one time
Step 3 Compute119883119894= 119886 + 119880
119894(119887 minus 119886) and 119891(119883
119894) gt 119888119880
1015840
119894
Step 4 Estimate the integral 119868 by 120579119868= 119888(119887 minus 119886)(119873
119867119873)
Now consider DMU119900as an efficient unit measure those
units that are dominated DMU119900 As for dome DMUs this
would be unbounded thus the authors for each unitbounded the region Then for DMU
119900if (minus119883
119900 119884
119900) ge (minus119883 119884)
then (minus119883 119884) is in the RED of DMU119900 Now by using 119881
119901=
119881lowast(119873
119867119873) all the hits (the above condition) can be counted
where 119881lowast is the measure of the whole region JahanshahlooandAfzalinejad [68] presented amethod based on distance of
Journal of Applied Mathematics 15
the unit under evaluation to the full inefficient frontier Theypresented two models radian and nonradial
min 120601
st119899
sum
119895=1
120582119895119909119894119895= 120601119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
120582119895ge 0 119895 = 1 119898
max119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903
st sum
119895isin119869minus119887
120582119895119909119894119895minus 119904
minus
119894= 119909
119894119900 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895+ 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
119904minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(70)
Amirteimoori [69] based on the same idea considered bothefficient and antiefficient frontiers for efficiency analysis andranking units
Kao [70] mentioned that determining the weights ofindividual criteria in multiple criteria decision analysis in away that all alternatives can be compared according to theaggregate performance of all criteria is of great importanceAs Kao noted this problem relates to search for alternativeswith a shorter and longer distance respectively to the idealand anti-ideal units He proposed a measure that consideredthe calculation of the relative position of an alternativebetween the ideal and anti-ideal for finding an appropriaterankings
Khodabakhshi and Aryavash [71] presented a method forranking all units using DEA concept
First Compute the minimum and maximum efficiencyvalues of eachDMU in regard to this assumption that the sumof efficiency values of all DMUs equals to 1
Second determine the rank of each DMU in relation to acombination of itsminimumandmaximumefficiency values
Zerafat Angiz et al [72] proposed a technique in orderto aggregate the opinions of experts in voting system Asthe authors mentioned the presented method uses fuzzyconcept and it is computationally efficient and can fullyrank alternatives At first number of votes given to a rankposition was grouped to construct fuzzy numbers and thenthe artificial ideal alternative introduced Furthermore byperforming DEA the efficiency measure of alternatives was
obtained considering artificial ideal alternative comparedby each of the alternatives pair by pair Thus alternativesare ranked in accordance with their efficiency scores Ifthis method cannot completely rank alternatives weightrestrictions based on fuzzy concept are imposed into theanalysis
10 Different Applications in Ranking Units
As discussed formerly there exist a variety of ranking meth-ods and applications in theliterature Nowadays DEAmodelsare widely used in different areas for efficiency evaluationbenchmarking and target setting ranking entities and soforth Ranking units as one of the important issues in DEAhas been performed in different areas Jahanshahloo et al [42]ranked cities in Iran to find the best place for creating a datafactory Hosseinzadeh Lotfi et al [28] used a new methodfor ranking in order to find the best place for power plantlocation Ali andNakosteen [80] Amirteimoori et al [37] andAlirezaee and Afsharian [24] Soltanifar and HosseinzadehLotfi [105] Zerafat Angiz et al [72] Hosseinzadeh Lotfiet al [28] Jahanshahloo [50] Chen and Deng [52] andJablonsky [61] used different ranking methods in bankingsystem Sadjadi [40] ranked provincial gas companies in IranMehrabian et al [32] Li et al [39] Orkcu and Bal [12] andWu et al [19] ranked different departments of universitiesJahanshahloo et al [35] Jahanshahloo and Afzalinejad [68]utilized the presented models in ranking 28 Chinese citiesJahanshahloo at al [35] provided an application to burdensharing amongst NATOmember nations Jahanshahloo et al[14] utilized the presented model for ranking nursery homesContreras [18] andWang et al [23] used the provided rankingtechniques in ranking candidates Lu and Lo [51] applied theirmethod on an application to financial holding companiesHosseinzadeh Lotfi et al [63] consider an empirical exampleinwhich voters are asked to rank twoout of seven alternativesWang and Jiang [62] utilized the provided method in facilitylayout design in manufacturing systems and performanceevaluation of 30OECD countries Chen [60] used an exampleof market basket data in order to illustrate the providedapproach
11 Application
In this section some of the reviewed models are applied onthe example used in Soltanifar and Hosseinzadeh Lotfi [105]Consider twenty commercial banks of Iran with input-outputdata tabulated in Table 1 and summarized as follows Also theresults of CCRmodel are listed in this table As it can be seenseven units are efficient DMUS 14 7 12 15 17 and 20 Inputsare staff computer terminal and spaceOutputs are depositsloans granted and charge
Consider some of the important ranking methods inliterature as follows
RM1 AP model [31] is based upon the idea of leaveunit evaluation out and measuring the distance of theunit under evaluation from the production possibilityset constructed by the remaining DMUs
RM2MAJmodel [32] presented for ranking efficientwhich is always stable but might be infeasible in somecasesRM3 Modified MAJ model [38] overcomes theproblem which might occurr in MAJ modelRM4 A new model based on the idea of alterationsin the reference set of the inefficient units [50]RM5 A model presented by Li et al [39] whichis a super-efficiency method that does not have thesuffering in previous methodsRM6 Slack-basedmodel [33] is based upon the inputand output variables at the same timeRM7 SA DEAmodel [49] overcomes the problem ofinfeasibility which existed in AP modelRM8 Cross-efficiency [10] is provided based onusing weights of each unit under evaluation in opti-mality for other unitsRM9 A model based on finding common set ofweights [25] which determine the common set ofweights for DMUs and ranked DMUs based this ideaRM10 A model based on finding common set ofweights [22 67] for ranking efficient unitsRM11 L1-norm model [34 35 65 66] the idea isbased upon the leave-one-out efficient unit and l1-norm which is always feasible and stableRM12 119871
infin-norm model Rezai Balf et al [46] pro-
vided a method with more ability over other existingmethods based on Tchebycheff normRM13 An enhanced Russel measure of super-efficiency model for ranking units [44]RM14 A rankingmodel which considers the distanceof unit from the full inefficient frontier [38]RM15 Amodified super-efficiencymodel [45] whichovercomes the infeasibility that may happen in prob-lem This model is based on simultaneous projectionof input output
In accordance with properties of different ranking models inorder to rank efficient units consider Table 2
1199011 Feasibility
1199012 Ranking extreme efficient units
1199013 Complexity in computation
1199014 Instability
1199015 Absence of multiple optimal solution
1199016 Dependency to 120579 and slacks
1199017 Dependency to the number of efficient and ineffi-
cient units
In Tables 3 and 4 the rank of efficient units consideringthe above mentioned methods is listed Note that ED showsefficient DMUs (ED) and RM
119895 (119895 = 1 15) are those
explained previously
12 Conclusion
In this paper the DEA ranking was reviewed and classifiedinto seven general groups In the first group those papersbased on a cross-efficiencymatrix were reviewed In this fieldDMUs have been evaluated by self- and peer pressure Thesecond group of papers is based on those papers lookingfor optimal weights in DEA analysis The third one is thesuper-efficiency method By omitting the under evaluationunit and constructing a new frontier by the remaining unitsthe unit under evaluation can get an equal or greater scoreThe fourth group is based on benchmarking idea In thisclass the effect of an efficient unit considered as a targetfor inefficient units is investigated This idea is very usefulfor the managers in decision making Another class thefifth one involves the application of multivariate statisticaltools The sixth section discusses the ranking methods basedon multicriteria decision-making (MCDM) methodologiesand DEA The final section includes some various rankingmethods presented in the literature Finally in an applicationthe result of some of the above-mentioned ranking methodsis presented
References
[1] M J Farrell ldquoThe measurement of productive efficiencyrdquoJournal of the Royal Statistical Society A vol 120 pp 253ndash2811957
[2] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
18 Journal of Applied Mathematics
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
from the criteria values Considering this matrix it is possibleto show the frequencywithwhich one option is ranked higherthan the other options In the second step by triangularizingthe outranking matrix establish an implicit preordering orprovisional ordering of optionsTheoutrankingmatrix showsthe degree to which there is a complete overall order ofoptions In the third step based on information given in anadvantages-disadvantages table the provisional ordering issubjected to different screening and balancing operations
Chen [60] utilized a nonparametric approach DEA toestimate and rank the efficiency of association rules withmultiple criteria in following steps Proposed postprocessingapproach is as follows
Step 1 Input data for association rule mining
Step 2 Mine association rules by using the a priori algorithmwith minimum support and minimum confidence
Step 3 Determine subjective interestingness measures byfurther considering the domain related knowledge
Step 4 Calculate the preference scores of association rulesdiscovered in Step 2 by using Cook and Kressrsquos DEA model
Step 5 Discriminate the efficient association rules found inStep 3 by using Obata and Ishiirsquos [104] discriminate model
Step 6 Select rules for implementation by considering thereference scores generated in Step 5 and domain relatedknowledge
Jablonsky [61] presented two original models super SBMand AHP for ranking of efficient units in DEA As theauthormentioned thesemodels are based onmultiple criteriadecision-making techniques-goal programming and analytichierarchy process Super SBM model for ranking units is asfollows
min 120579119866
119902= 1 + 119905120574 + (1 minus 119905) (
sum119898
119894=1119904+
1119894
119909119894119902
+
sum119898
119894=1119904minus
2119896
119910119896119902
)
st119899
sum
119895=1119895 = 119902
120582119895119909119894119895+ 119904
minus
1119894minus 119904
+
1119894= 119909
119894119902 i = 1 119898
119899
sum
119895=1119895 = 119902
120582119895119910119896119895+ 119904
minus
2119896minus 119904
+
2119896= 119910
119896119902 119896 = 1 119903
119904+
1119894le 120574 119894 = 1 119898
119904minus
2119896le 120574 119896 = 1 119903
119905 isin 0 1 120582119895ge 0 119878
+
1 119878
+
2ge 0 119878
minus
1 119878
minus
2ge 0
119895 = 1 119899
(62)
Wang and Jiang [62] presented an alternative mixedinteger linear programming models in order to identifythe most efficient units in DEA technique As the authors
mentioned presented models can make full use of input-output information with no need to specify any assuranceregions for input and output weights to avoid zero weights
min119898
sum
119894=1
V119894(
119899
sum
119895=1
119909119894119895) minus
119904
sum
119903=1
119906119903(
119899
sum
119895=1
119910119903119895)
st119904
sum
119903=1
119906119903119910119903119895minus
119898
sum
119894=1
V119894119909119894119895le 119868
119895 119895 = 1 119899
119899
sum
119895=1
119868119895= 1
119868119895isin 0 1 119895 = 1 119899
119906119903ge
1
(119898 + 119904)max119895119910
119903119895
119903 = 1 119904
V119894ge
1
(119898 + 119904)max119895119909
119894119895
119894 = 1 119898
(63)
Hosseinzadeh Lotfi et al [63] provided an improved three-stage method for ranking alternatives in multiple criteriadecision analysis In the first stage based on the bestand worst weights in the optimistic and pessimistic casesobtain the rank position of each alternative respectively Theobtained weight in the first stage is not unique thus it seemsnecessary to introduce a secondary goal that is used in thesecond stage Finally in the third stage the ranks of thealternatives compute in the optimistic or pessimistic case Itis mentionable that the model proposed in the third stageis a multicriteria decision-making (MCDM) model and itis solved by mixed integer programming As the authorsmentioned and provided in their paper this model can beconverted into an LP problem
The previous model with 119897 = 1 is the original CCR modelNote that DMUs in 119864
1 show the first level efficient frontier119897 = 2 indicates the second level efficient frontier when thefirst level efficient frontier is omittedThe following algorithmfinds these efficient frontiers
Step 1 Set 119897 = 1 and evaluate the entire set of DMUs 1198691 Inthis way the first level efficient DMU 1198641 is identified
Step 2 Exclude the efficient DMUs from future DEA runsand set 119869119897+1 = 119869
119897minus 119864
119897 (If 119869119897+1 = oslash stop)
Step 3 Evaluate the new subset of ldquoinefficientrdquo DMUs 119869119897+1to obtain a new set of efficient DMUs 119864119897+1
Step 4 Let l = l + 1 Go to Step 2
When 119869119897+1 = oslash the algorithm stopsAs the authors proved in this way it is possible to rank
the DMUs in the first efficient frontier based upon theirattractiveness scores and identify the best one
Jahanshahloo et al [65] provided a paper for rankingunits using gradient line As the authors mentioned theadvantage of this model is stability and robustness
supports on 119879119888 the obtained PPS by omitting theDMUunder
assessment in 119879119888 As the authors proved a unit is efficient
iff the optimal objective function of the following model isgreater than zero Consider
1198750= (119883 119884) 119883 = 120572119883
119900 119884 = 120573119884
119900
1198780= (119883 119884) isin 1198750
119883 = 120572119883119900 119884 = 120573119884
119900 120572 ge 0 120573 ge 0
(67)
Intersection of 1198780and efficient surface of
119879119888119888 is a half line
where its equation is (minus119881lowast119879119883
119900)120572 + (119880
lowast119879119884119900)120573 = 0 To rank
DMU119900the length of connecting arc DMU
119900with intersection
point of line and previous ellipse is calculated in (120572 120573) spaceThis intersection is as follows
120572lowast= (
1198702
120572119870
2
120573
1198702
120573+ (119881
lowast119879119883
119900119880
lowast119879119884119900)2119870
2
120572
)
12
120573lowast= (
119881lowast119879119883
119900
119880lowast119879119884119900
)120572lowast
(68)
Now the length of connecting arc DMU119900to the point
corresponding to 120572lowast 120573lowast in (120572 120573) plan is calculated as follows119868 = int
120572lowast
1(1 + 120573)
12119889120572 where
119870120572= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119899
119895=11199092
119894119900
)
12
119870120573= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119904
119903=11199102
119903119900
)
12
(69)
Jahanshahloo et al [66] also provided a paper with theconcept of advantage in data envelopment analysis
In their paper Jahanshahloo et al [67] consideringMonteCarlo method presented a new method of ranking
Step 1 Generate a uniformly distributed sequence of 1198801198952119899
119895=1
on(0 1)
Step 2 Random numbers should be classified into 119873 pairslike (119880
1 119880
1015840
1) (119880
119873 119880
1015840
119873) in a way that each number is used
just one time
Step 3 Compute119883119894= 119886 + 119880
119894(119887 minus 119886) and 119891(119883
119894) gt 119888119880
1015840
119894
Step 4 Estimate the integral 119868 by 120579119868= 119888(119887 minus 119886)(119873
119867119873)
Now consider DMU119900as an efficient unit measure those
units that are dominated DMU119900 As for dome DMUs this
would be unbounded thus the authors for each unitbounded the region Then for DMU
119900if (minus119883
119900 119884
119900) ge (minus119883 119884)
then (minus119883 119884) is in the RED of DMU119900 Now by using 119881
119901=
119881lowast(119873
119867119873) all the hits (the above condition) can be counted
where 119881lowast is the measure of the whole region JahanshahlooandAfzalinejad [68] presented amethod based on distance of
Journal of Applied Mathematics 15
the unit under evaluation to the full inefficient frontier Theypresented two models radian and nonradial
min 120601
st119899
sum
119895=1
120582119895119909119894119895= 120601119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
120582119895ge 0 119895 = 1 119898
max119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903
st sum
119895isin119869minus119887
120582119895119909119894119895minus 119904
minus
119894= 119909
119894119900 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895+ 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
119904minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(70)
Amirteimoori [69] based on the same idea considered bothefficient and antiefficient frontiers for efficiency analysis andranking units
Kao [70] mentioned that determining the weights ofindividual criteria in multiple criteria decision analysis in away that all alternatives can be compared according to theaggregate performance of all criteria is of great importanceAs Kao noted this problem relates to search for alternativeswith a shorter and longer distance respectively to the idealand anti-ideal units He proposed a measure that consideredthe calculation of the relative position of an alternativebetween the ideal and anti-ideal for finding an appropriaterankings
Khodabakhshi and Aryavash [71] presented a method forranking all units using DEA concept
First Compute the minimum and maximum efficiencyvalues of eachDMU in regard to this assumption that the sumof efficiency values of all DMUs equals to 1
Second determine the rank of each DMU in relation to acombination of itsminimumandmaximumefficiency values
Zerafat Angiz et al [72] proposed a technique in orderto aggregate the opinions of experts in voting system Asthe authors mentioned the presented method uses fuzzyconcept and it is computationally efficient and can fullyrank alternatives At first number of votes given to a rankposition was grouped to construct fuzzy numbers and thenthe artificial ideal alternative introduced Furthermore byperforming DEA the efficiency measure of alternatives was
obtained considering artificial ideal alternative comparedby each of the alternatives pair by pair Thus alternativesare ranked in accordance with their efficiency scores Ifthis method cannot completely rank alternatives weightrestrictions based on fuzzy concept are imposed into theanalysis
10 Different Applications in Ranking Units
As discussed formerly there exist a variety of ranking meth-ods and applications in theliterature Nowadays DEAmodelsare widely used in different areas for efficiency evaluationbenchmarking and target setting ranking entities and soforth Ranking units as one of the important issues in DEAhas been performed in different areas Jahanshahloo et al [42]ranked cities in Iran to find the best place for creating a datafactory Hosseinzadeh Lotfi et al [28] used a new methodfor ranking in order to find the best place for power plantlocation Ali andNakosteen [80] Amirteimoori et al [37] andAlirezaee and Afsharian [24] Soltanifar and HosseinzadehLotfi [105] Zerafat Angiz et al [72] Hosseinzadeh Lotfiet al [28] Jahanshahloo [50] Chen and Deng [52] andJablonsky [61] used different ranking methods in bankingsystem Sadjadi [40] ranked provincial gas companies in IranMehrabian et al [32] Li et al [39] Orkcu and Bal [12] andWu et al [19] ranked different departments of universitiesJahanshahloo et al [35] Jahanshahloo and Afzalinejad [68]utilized the presented models in ranking 28 Chinese citiesJahanshahloo at al [35] provided an application to burdensharing amongst NATOmember nations Jahanshahloo et al[14] utilized the presented model for ranking nursery homesContreras [18] andWang et al [23] used the provided rankingtechniques in ranking candidates Lu and Lo [51] applied theirmethod on an application to financial holding companiesHosseinzadeh Lotfi et al [63] consider an empirical exampleinwhich voters are asked to rank twoout of seven alternativesWang and Jiang [62] utilized the provided method in facilitylayout design in manufacturing systems and performanceevaluation of 30OECD countries Chen [60] used an exampleof market basket data in order to illustrate the providedapproach
11 Application
In this section some of the reviewed models are applied onthe example used in Soltanifar and Hosseinzadeh Lotfi [105]Consider twenty commercial banks of Iran with input-outputdata tabulated in Table 1 and summarized as follows Also theresults of CCRmodel are listed in this table As it can be seenseven units are efficient DMUS 14 7 12 15 17 and 20 Inputsare staff computer terminal and spaceOutputs are depositsloans granted and charge
Consider some of the important ranking methods inliterature as follows
RM1 AP model [31] is based upon the idea of leaveunit evaluation out and measuring the distance of theunit under evaluation from the production possibilityset constructed by the remaining DMUs
RM2MAJmodel [32] presented for ranking efficientwhich is always stable but might be infeasible in somecasesRM3 Modified MAJ model [38] overcomes theproblem which might occurr in MAJ modelRM4 A new model based on the idea of alterationsin the reference set of the inefficient units [50]RM5 A model presented by Li et al [39] whichis a super-efficiency method that does not have thesuffering in previous methodsRM6 Slack-basedmodel [33] is based upon the inputand output variables at the same timeRM7 SA DEAmodel [49] overcomes the problem ofinfeasibility which existed in AP modelRM8 Cross-efficiency [10] is provided based onusing weights of each unit under evaluation in opti-mality for other unitsRM9 A model based on finding common set ofweights [25] which determine the common set ofweights for DMUs and ranked DMUs based this ideaRM10 A model based on finding common set ofweights [22 67] for ranking efficient unitsRM11 L1-norm model [34 35 65 66] the idea isbased upon the leave-one-out efficient unit and l1-norm which is always feasible and stableRM12 119871
infin-norm model Rezai Balf et al [46] pro-
vided a method with more ability over other existingmethods based on Tchebycheff normRM13 An enhanced Russel measure of super-efficiency model for ranking units [44]RM14 A rankingmodel which considers the distanceof unit from the full inefficient frontier [38]RM15 Amodified super-efficiencymodel [45] whichovercomes the infeasibility that may happen in prob-lem This model is based on simultaneous projectionof input output
In accordance with properties of different ranking models inorder to rank efficient units consider Table 2
1199011 Feasibility
1199012 Ranking extreme efficient units
1199013 Complexity in computation
1199014 Instability
1199015 Absence of multiple optimal solution
1199016 Dependency to 120579 and slacks
1199017 Dependency to the number of efficient and ineffi-
cient units
In Tables 3 and 4 the rank of efficient units consideringthe above mentioned methods is listed Note that ED showsefficient DMUs (ED) and RM
119895 (119895 = 1 15) are those
explained previously
12 Conclusion
In this paper the DEA ranking was reviewed and classifiedinto seven general groups In the first group those papersbased on a cross-efficiencymatrix were reviewed In this fieldDMUs have been evaluated by self- and peer pressure Thesecond group of papers is based on those papers lookingfor optimal weights in DEA analysis The third one is thesuper-efficiency method By omitting the under evaluationunit and constructing a new frontier by the remaining unitsthe unit under evaluation can get an equal or greater scoreThe fourth group is based on benchmarking idea In thisclass the effect of an efficient unit considered as a targetfor inefficient units is investigated This idea is very usefulfor the managers in decision making Another class thefifth one involves the application of multivariate statisticaltools The sixth section discusses the ranking methods basedon multicriteria decision-making (MCDM) methodologiesand DEA The final section includes some various rankingmethods presented in the literature Finally in an applicationthe result of some of the above-mentioned ranking methodsis presented
References
[1] M J Farrell ldquoThe measurement of productive efficiencyrdquoJournal of the Royal Statistical Society A vol 120 pp 253ndash2811957
[2] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
18 Journal of Applied Mathematics
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
The previous model with 119897 = 1 is the original CCR modelNote that DMUs in 119864
1 show the first level efficient frontier119897 = 2 indicates the second level efficient frontier when thefirst level efficient frontier is omittedThe following algorithmfinds these efficient frontiers
Step 1 Set 119897 = 1 and evaluate the entire set of DMUs 1198691 Inthis way the first level efficient DMU 1198641 is identified
Step 2 Exclude the efficient DMUs from future DEA runsand set 119869119897+1 = 119869
119897minus 119864
119897 (If 119869119897+1 = oslash stop)
Step 3 Evaluate the new subset of ldquoinefficientrdquo DMUs 119869119897+1to obtain a new set of efficient DMUs 119864119897+1
Step 4 Let l = l + 1 Go to Step 2
When 119869119897+1 = oslash the algorithm stopsAs the authors proved in this way it is possible to rank
the DMUs in the first efficient frontier based upon theirattractiveness scores and identify the best one
Jahanshahloo et al [65] provided a paper for rankingunits using gradient line As the authors mentioned theadvantage of this model is stability and robustness
supports on 119879119888 the obtained PPS by omitting theDMUunder
assessment in 119879119888 As the authors proved a unit is efficient
iff the optimal objective function of the following model isgreater than zero Consider
1198750= (119883 119884) 119883 = 120572119883
119900 119884 = 120573119884
119900
1198780= (119883 119884) isin 1198750
119883 = 120572119883119900 119884 = 120573119884
119900 120572 ge 0 120573 ge 0
(67)
Intersection of 1198780and efficient surface of
119879119888119888 is a half line
where its equation is (minus119881lowast119879119883
119900)120572 + (119880
lowast119879119884119900)120573 = 0 To rank
DMU119900the length of connecting arc DMU
119900with intersection
point of line and previous ellipse is calculated in (120572 120573) spaceThis intersection is as follows
120572lowast= (
1198702
120572119870
2
120573
1198702
120573+ (119881
lowast119879119883
119900119880
lowast119879119884119900)2119870
2
120572
)
12
120573lowast= (
119881lowast119879119883
119900
119880lowast119879119884119900
)120572lowast
(68)
Now the length of connecting arc DMU119900to the point
corresponding to 120572lowast 120573lowast in (120572 120573) plan is calculated as follows119868 = int
120572lowast
1(1 + 120573)
12119889120572 where
119870120572= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119899
119895=11199092
119894119900
)
12
119870120573= (
sum119899
119895=11199092
119894119900+ sum
119904
119903=11199102
119894119900
sum119904
119903=11199102
119903119900
)
12
(69)
Jahanshahloo et al [66] also provided a paper with theconcept of advantage in data envelopment analysis
In their paper Jahanshahloo et al [67] consideringMonteCarlo method presented a new method of ranking
Step 1 Generate a uniformly distributed sequence of 1198801198952119899
119895=1
on(0 1)
Step 2 Random numbers should be classified into 119873 pairslike (119880
1 119880
1015840
1) (119880
119873 119880
1015840
119873) in a way that each number is used
just one time
Step 3 Compute119883119894= 119886 + 119880
119894(119887 minus 119886) and 119891(119883
119894) gt 119888119880
1015840
119894
Step 4 Estimate the integral 119868 by 120579119868= 119888(119887 minus 119886)(119873
119867119873)
Now consider DMU119900as an efficient unit measure those
units that are dominated DMU119900 As for dome DMUs this
would be unbounded thus the authors for each unitbounded the region Then for DMU
119900if (minus119883
119900 119884
119900) ge (minus119883 119884)
then (minus119883 119884) is in the RED of DMU119900 Now by using 119881
119901=
119881lowast(119873
119867119873) all the hits (the above condition) can be counted
where 119881lowast is the measure of the whole region JahanshahlooandAfzalinejad [68] presented amethod based on distance of
Journal of Applied Mathematics 15
the unit under evaluation to the full inefficient frontier Theypresented two models radian and nonradial
min 120601
st119899
sum
119895=1
120582119895119909119894119895= 120601119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
120582119895ge 0 119895 = 1 119898
max119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903
st sum
119895isin119869minus119887
120582119895119909119894119895minus 119904
minus
119894= 119909
119894119900 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895+ 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
119904minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(70)
Amirteimoori [69] based on the same idea considered bothefficient and antiefficient frontiers for efficiency analysis andranking units
Kao [70] mentioned that determining the weights ofindividual criteria in multiple criteria decision analysis in away that all alternatives can be compared according to theaggregate performance of all criteria is of great importanceAs Kao noted this problem relates to search for alternativeswith a shorter and longer distance respectively to the idealand anti-ideal units He proposed a measure that consideredthe calculation of the relative position of an alternativebetween the ideal and anti-ideal for finding an appropriaterankings
Khodabakhshi and Aryavash [71] presented a method forranking all units using DEA concept
First Compute the minimum and maximum efficiencyvalues of eachDMU in regard to this assumption that the sumof efficiency values of all DMUs equals to 1
Second determine the rank of each DMU in relation to acombination of itsminimumandmaximumefficiency values
Zerafat Angiz et al [72] proposed a technique in orderto aggregate the opinions of experts in voting system Asthe authors mentioned the presented method uses fuzzyconcept and it is computationally efficient and can fullyrank alternatives At first number of votes given to a rankposition was grouped to construct fuzzy numbers and thenthe artificial ideal alternative introduced Furthermore byperforming DEA the efficiency measure of alternatives was
obtained considering artificial ideal alternative comparedby each of the alternatives pair by pair Thus alternativesare ranked in accordance with their efficiency scores Ifthis method cannot completely rank alternatives weightrestrictions based on fuzzy concept are imposed into theanalysis
10 Different Applications in Ranking Units
As discussed formerly there exist a variety of ranking meth-ods and applications in theliterature Nowadays DEAmodelsare widely used in different areas for efficiency evaluationbenchmarking and target setting ranking entities and soforth Ranking units as one of the important issues in DEAhas been performed in different areas Jahanshahloo et al [42]ranked cities in Iran to find the best place for creating a datafactory Hosseinzadeh Lotfi et al [28] used a new methodfor ranking in order to find the best place for power plantlocation Ali andNakosteen [80] Amirteimoori et al [37] andAlirezaee and Afsharian [24] Soltanifar and HosseinzadehLotfi [105] Zerafat Angiz et al [72] Hosseinzadeh Lotfiet al [28] Jahanshahloo [50] Chen and Deng [52] andJablonsky [61] used different ranking methods in bankingsystem Sadjadi [40] ranked provincial gas companies in IranMehrabian et al [32] Li et al [39] Orkcu and Bal [12] andWu et al [19] ranked different departments of universitiesJahanshahloo et al [35] Jahanshahloo and Afzalinejad [68]utilized the presented models in ranking 28 Chinese citiesJahanshahloo at al [35] provided an application to burdensharing amongst NATOmember nations Jahanshahloo et al[14] utilized the presented model for ranking nursery homesContreras [18] andWang et al [23] used the provided rankingtechniques in ranking candidates Lu and Lo [51] applied theirmethod on an application to financial holding companiesHosseinzadeh Lotfi et al [63] consider an empirical exampleinwhich voters are asked to rank twoout of seven alternativesWang and Jiang [62] utilized the provided method in facilitylayout design in manufacturing systems and performanceevaluation of 30OECD countries Chen [60] used an exampleof market basket data in order to illustrate the providedapproach
11 Application
In this section some of the reviewed models are applied onthe example used in Soltanifar and Hosseinzadeh Lotfi [105]Consider twenty commercial banks of Iran with input-outputdata tabulated in Table 1 and summarized as follows Also theresults of CCRmodel are listed in this table As it can be seenseven units are efficient DMUS 14 7 12 15 17 and 20 Inputsare staff computer terminal and spaceOutputs are depositsloans granted and charge
Consider some of the important ranking methods inliterature as follows
RM1 AP model [31] is based upon the idea of leaveunit evaluation out and measuring the distance of theunit under evaluation from the production possibilityset constructed by the remaining DMUs
RM2MAJmodel [32] presented for ranking efficientwhich is always stable but might be infeasible in somecasesRM3 Modified MAJ model [38] overcomes theproblem which might occurr in MAJ modelRM4 A new model based on the idea of alterationsin the reference set of the inefficient units [50]RM5 A model presented by Li et al [39] whichis a super-efficiency method that does not have thesuffering in previous methodsRM6 Slack-basedmodel [33] is based upon the inputand output variables at the same timeRM7 SA DEAmodel [49] overcomes the problem ofinfeasibility which existed in AP modelRM8 Cross-efficiency [10] is provided based onusing weights of each unit under evaluation in opti-mality for other unitsRM9 A model based on finding common set ofweights [25] which determine the common set ofweights for DMUs and ranked DMUs based this ideaRM10 A model based on finding common set ofweights [22 67] for ranking efficient unitsRM11 L1-norm model [34 35 65 66] the idea isbased upon the leave-one-out efficient unit and l1-norm which is always feasible and stableRM12 119871
infin-norm model Rezai Balf et al [46] pro-
vided a method with more ability over other existingmethods based on Tchebycheff normRM13 An enhanced Russel measure of super-efficiency model for ranking units [44]RM14 A rankingmodel which considers the distanceof unit from the full inefficient frontier [38]RM15 Amodified super-efficiencymodel [45] whichovercomes the infeasibility that may happen in prob-lem This model is based on simultaneous projectionof input output
In accordance with properties of different ranking models inorder to rank efficient units consider Table 2
1199011 Feasibility
1199012 Ranking extreme efficient units
1199013 Complexity in computation
1199014 Instability
1199015 Absence of multiple optimal solution
1199016 Dependency to 120579 and slacks
1199017 Dependency to the number of efficient and ineffi-
cient units
In Tables 3 and 4 the rank of efficient units consideringthe above mentioned methods is listed Note that ED showsefficient DMUs (ED) and RM
119895 (119895 = 1 15) are those
explained previously
12 Conclusion
In this paper the DEA ranking was reviewed and classifiedinto seven general groups In the first group those papersbased on a cross-efficiencymatrix were reviewed In this fieldDMUs have been evaluated by self- and peer pressure Thesecond group of papers is based on those papers lookingfor optimal weights in DEA analysis The third one is thesuper-efficiency method By omitting the under evaluationunit and constructing a new frontier by the remaining unitsthe unit under evaluation can get an equal or greater scoreThe fourth group is based on benchmarking idea In thisclass the effect of an efficient unit considered as a targetfor inefficient units is investigated This idea is very usefulfor the managers in decision making Another class thefifth one involves the application of multivariate statisticaltools The sixth section discusses the ranking methods basedon multicriteria decision-making (MCDM) methodologiesand DEA The final section includes some various rankingmethods presented in the literature Finally in an applicationthe result of some of the above-mentioned ranking methodsis presented
References
[1] M J Farrell ldquoThe measurement of productive efficiencyrdquoJournal of the Royal Statistical Society A vol 120 pp 253ndash2811957
[2] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
18 Journal of Applied Mathematics
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
the unit under evaluation to the full inefficient frontier Theypresented two models radian and nonradial
min 120601
st119899
sum
119895=1
120582119895119909119894119895= 120601119909
119894119900 119894 = 1 119898
119899
sum
119895=1
120582119895119910119903119895= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
120582119895ge 0 119895 = 1 119898
max119898
sum
119894=1
119904minus
119894+
119904
sum
119903=1
119904+
119903
st sum
119895isin119869minus119887
120582119895119909119894119895minus 119904
minus
119894= 119909
119894119900 119894 = 1 119898
sum
119895isin119869minus119887
120582119895119910119903119895+ 119904
+
119903= 119910
119903119900 119903 = 1 119904
119899
sum
119895=1
= 1
119904minus
119894ge 0 119904
+
119903ge 0 119894 = 1 119898 119903 = 1 119904
120582119895ge 0 119895 = 1 119899
(70)
Amirteimoori [69] based on the same idea considered bothefficient and antiefficient frontiers for efficiency analysis andranking units
Kao [70] mentioned that determining the weights ofindividual criteria in multiple criteria decision analysis in away that all alternatives can be compared according to theaggregate performance of all criteria is of great importanceAs Kao noted this problem relates to search for alternativeswith a shorter and longer distance respectively to the idealand anti-ideal units He proposed a measure that consideredthe calculation of the relative position of an alternativebetween the ideal and anti-ideal for finding an appropriaterankings
Khodabakhshi and Aryavash [71] presented a method forranking all units using DEA concept
First Compute the minimum and maximum efficiencyvalues of eachDMU in regard to this assumption that the sumof efficiency values of all DMUs equals to 1
Second determine the rank of each DMU in relation to acombination of itsminimumandmaximumefficiency values
Zerafat Angiz et al [72] proposed a technique in orderto aggregate the opinions of experts in voting system Asthe authors mentioned the presented method uses fuzzyconcept and it is computationally efficient and can fullyrank alternatives At first number of votes given to a rankposition was grouped to construct fuzzy numbers and thenthe artificial ideal alternative introduced Furthermore byperforming DEA the efficiency measure of alternatives was
obtained considering artificial ideal alternative comparedby each of the alternatives pair by pair Thus alternativesare ranked in accordance with their efficiency scores Ifthis method cannot completely rank alternatives weightrestrictions based on fuzzy concept are imposed into theanalysis
10 Different Applications in Ranking Units
As discussed formerly there exist a variety of ranking meth-ods and applications in theliterature Nowadays DEAmodelsare widely used in different areas for efficiency evaluationbenchmarking and target setting ranking entities and soforth Ranking units as one of the important issues in DEAhas been performed in different areas Jahanshahloo et al [42]ranked cities in Iran to find the best place for creating a datafactory Hosseinzadeh Lotfi et al [28] used a new methodfor ranking in order to find the best place for power plantlocation Ali andNakosteen [80] Amirteimoori et al [37] andAlirezaee and Afsharian [24] Soltanifar and HosseinzadehLotfi [105] Zerafat Angiz et al [72] Hosseinzadeh Lotfiet al [28] Jahanshahloo [50] Chen and Deng [52] andJablonsky [61] used different ranking methods in bankingsystem Sadjadi [40] ranked provincial gas companies in IranMehrabian et al [32] Li et al [39] Orkcu and Bal [12] andWu et al [19] ranked different departments of universitiesJahanshahloo et al [35] Jahanshahloo and Afzalinejad [68]utilized the presented models in ranking 28 Chinese citiesJahanshahloo at al [35] provided an application to burdensharing amongst NATOmember nations Jahanshahloo et al[14] utilized the presented model for ranking nursery homesContreras [18] andWang et al [23] used the provided rankingtechniques in ranking candidates Lu and Lo [51] applied theirmethod on an application to financial holding companiesHosseinzadeh Lotfi et al [63] consider an empirical exampleinwhich voters are asked to rank twoout of seven alternativesWang and Jiang [62] utilized the provided method in facilitylayout design in manufacturing systems and performanceevaluation of 30OECD countries Chen [60] used an exampleof market basket data in order to illustrate the providedapproach
11 Application
In this section some of the reviewed models are applied onthe example used in Soltanifar and Hosseinzadeh Lotfi [105]Consider twenty commercial banks of Iran with input-outputdata tabulated in Table 1 and summarized as follows Also theresults of CCRmodel are listed in this table As it can be seenseven units are efficient DMUS 14 7 12 15 17 and 20 Inputsare staff computer terminal and spaceOutputs are depositsloans granted and charge
Consider some of the important ranking methods inliterature as follows
RM1 AP model [31] is based upon the idea of leaveunit evaluation out and measuring the distance of theunit under evaluation from the production possibilityset constructed by the remaining DMUs
RM2MAJmodel [32] presented for ranking efficientwhich is always stable but might be infeasible in somecasesRM3 Modified MAJ model [38] overcomes theproblem which might occurr in MAJ modelRM4 A new model based on the idea of alterationsin the reference set of the inefficient units [50]RM5 A model presented by Li et al [39] whichis a super-efficiency method that does not have thesuffering in previous methodsRM6 Slack-basedmodel [33] is based upon the inputand output variables at the same timeRM7 SA DEAmodel [49] overcomes the problem ofinfeasibility which existed in AP modelRM8 Cross-efficiency [10] is provided based onusing weights of each unit under evaluation in opti-mality for other unitsRM9 A model based on finding common set ofweights [25] which determine the common set ofweights for DMUs and ranked DMUs based this ideaRM10 A model based on finding common set ofweights [22 67] for ranking efficient unitsRM11 L1-norm model [34 35 65 66] the idea isbased upon the leave-one-out efficient unit and l1-norm which is always feasible and stableRM12 119871
infin-norm model Rezai Balf et al [46] pro-
vided a method with more ability over other existingmethods based on Tchebycheff normRM13 An enhanced Russel measure of super-efficiency model for ranking units [44]RM14 A rankingmodel which considers the distanceof unit from the full inefficient frontier [38]RM15 Amodified super-efficiencymodel [45] whichovercomes the infeasibility that may happen in prob-lem This model is based on simultaneous projectionof input output
In accordance with properties of different ranking models inorder to rank efficient units consider Table 2
1199011 Feasibility
1199012 Ranking extreme efficient units
1199013 Complexity in computation
1199014 Instability
1199015 Absence of multiple optimal solution
1199016 Dependency to 120579 and slacks
1199017 Dependency to the number of efficient and ineffi-
cient units
In Tables 3 and 4 the rank of efficient units consideringthe above mentioned methods is listed Note that ED showsefficient DMUs (ED) and RM
119895 (119895 = 1 15) are those
explained previously
12 Conclusion
In this paper the DEA ranking was reviewed and classifiedinto seven general groups In the first group those papersbased on a cross-efficiencymatrix were reviewed In this fieldDMUs have been evaluated by self- and peer pressure Thesecond group of papers is based on those papers lookingfor optimal weights in DEA analysis The third one is thesuper-efficiency method By omitting the under evaluationunit and constructing a new frontier by the remaining unitsthe unit under evaluation can get an equal or greater scoreThe fourth group is based on benchmarking idea In thisclass the effect of an efficient unit considered as a targetfor inefficient units is investigated This idea is very usefulfor the managers in decision making Another class thefifth one involves the application of multivariate statisticaltools The sixth section discusses the ranking methods basedon multicriteria decision-making (MCDM) methodologiesand DEA The final section includes some various rankingmethods presented in the literature Finally in an applicationthe result of some of the above-mentioned ranking methodsis presented
References
[1] M J Farrell ldquoThe measurement of productive efficiencyrdquoJournal of the Royal Statistical Society A vol 120 pp 253ndash2811957
[2] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
18 Journal of Applied Mathematics
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
RM2MAJmodel [32] presented for ranking efficientwhich is always stable but might be infeasible in somecasesRM3 Modified MAJ model [38] overcomes theproblem which might occurr in MAJ modelRM4 A new model based on the idea of alterationsin the reference set of the inefficient units [50]RM5 A model presented by Li et al [39] whichis a super-efficiency method that does not have thesuffering in previous methodsRM6 Slack-basedmodel [33] is based upon the inputand output variables at the same timeRM7 SA DEAmodel [49] overcomes the problem ofinfeasibility which existed in AP modelRM8 Cross-efficiency [10] is provided based onusing weights of each unit under evaluation in opti-mality for other unitsRM9 A model based on finding common set ofweights [25] which determine the common set ofweights for DMUs and ranked DMUs based this ideaRM10 A model based on finding common set ofweights [22 67] for ranking efficient unitsRM11 L1-norm model [34 35 65 66] the idea isbased upon the leave-one-out efficient unit and l1-norm which is always feasible and stableRM12 119871
infin-norm model Rezai Balf et al [46] pro-
vided a method with more ability over other existingmethods based on Tchebycheff normRM13 An enhanced Russel measure of super-efficiency model for ranking units [44]RM14 A rankingmodel which considers the distanceof unit from the full inefficient frontier [38]RM15 Amodified super-efficiencymodel [45] whichovercomes the infeasibility that may happen in prob-lem This model is based on simultaneous projectionof input output
In accordance with properties of different ranking models inorder to rank efficient units consider Table 2
1199011 Feasibility
1199012 Ranking extreme efficient units
1199013 Complexity in computation
1199014 Instability
1199015 Absence of multiple optimal solution
1199016 Dependency to 120579 and slacks
1199017 Dependency to the number of efficient and ineffi-
cient units
In Tables 3 and 4 the rank of efficient units consideringthe above mentioned methods is listed Note that ED showsefficient DMUs (ED) and RM
119895 (119895 = 1 15) are those
explained previously
12 Conclusion
In this paper the DEA ranking was reviewed and classifiedinto seven general groups In the first group those papersbased on a cross-efficiencymatrix were reviewed In this fieldDMUs have been evaluated by self- and peer pressure Thesecond group of papers is based on those papers lookingfor optimal weights in DEA analysis The third one is thesuper-efficiency method By omitting the under evaluationunit and constructing a new frontier by the remaining unitsthe unit under evaluation can get an equal or greater scoreThe fourth group is based on benchmarking idea In thisclass the effect of an efficient unit considered as a targetfor inefficient units is investigated This idea is very usefulfor the managers in decision making Another class thefifth one involves the application of multivariate statisticaltools The sixth section discusses the ranking methods basedon multicriteria decision-making (MCDM) methodologiesand DEA The final section includes some various rankingmethods presented in the literature Finally in an applicationthe result of some of the above-mentioned ranking methodsis presented
References
[1] M J Farrell ldquoThe measurement of productive efficiencyrdquoJournal of the Royal Statistical Society A vol 120 pp 253ndash2811957
[2] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
18 Journal of Applied Mathematics
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
RM2MAJmodel [32] presented for ranking efficientwhich is always stable but might be infeasible in somecasesRM3 Modified MAJ model [38] overcomes theproblem which might occurr in MAJ modelRM4 A new model based on the idea of alterationsin the reference set of the inefficient units [50]RM5 A model presented by Li et al [39] whichis a super-efficiency method that does not have thesuffering in previous methodsRM6 Slack-basedmodel [33] is based upon the inputand output variables at the same timeRM7 SA DEAmodel [49] overcomes the problem ofinfeasibility which existed in AP modelRM8 Cross-efficiency [10] is provided based onusing weights of each unit under evaluation in opti-mality for other unitsRM9 A model based on finding common set ofweights [25] which determine the common set ofweights for DMUs and ranked DMUs based this ideaRM10 A model based on finding common set ofweights [22 67] for ranking efficient unitsRM11 L1-norm model [34 35 65 66] the idea isbased upon the leave-one-out efficient unit and l1-norm which is always feasible and stableRM12 119871
infin-norm model Rezai Balf et al [46] pro-
vided a method with more ability over other existingmethods based on Tchebycheff normRM13 An enhanced Russel measure of super-efficiency model for ranking units [44]RM14 A rankingmodel which considers the distanceof unit from the full inefficient frontier [38]RM15 Amodified super-efficiencymodel [45] whichovercomes the infeasibility that may happen in prob-lem This model is based on simultaneous projectionof input output
In accordance with properties of different ranking models inorder to rank efficient units consider Table 2
1199011 Feasibility
1199012 Ranking extreme efficient units
1199013 Complexity in computation
1199014 Instability
1199015 Absence of multiple optimal solution
1199016 Dependency to 120579 and slacks
1199017 Dependency to the number of efficient and ineffi-
cient units
In Tables 3 and 4 the rank of efficient units consideringthe above mentioned methods is listed Note that ED showsefficient DMUs (ED) and RM
119895 (119895 = 1 15) are those
explained previously
12 Conclusion
In this paper the DEA ranking was reviewed and classifiedinto seven general groups In the first group those papersbased on a cross-efficiencymatrix were reviewed In this fieldDMUs have been evaluated by self- and peer pressure Thesecond group of papers is based on those papers lookingfor optimal weights in DEA analysis The third one is thesuper-efficiency method By omitting the under evaluationunit and constructing a new frontier by the remaining unitsthe unit under evaluation can get an equal or greater scoreThe fourth group is based on benchmarking idea In thisclass the effect of an efficient unit considered as a targetfor inefficient units is investigated This idea is very usefulfor the managers in decision making Another class thefifth one involves the application of multivariate statisticaltools The sixth section discusses the ranking methods basedon multicriteria decision-making (MCDM) methodologiesand DEA The final section includes some various rankingmethods presented in the literature Finally in an applicationthe result of some of the above-mentioned ranking methodsis presented
References
[1] M J Farrell ldquoThe measurement of productive efficiencyrdquoJournal of the Royal Statistical Society A vol 120 pp 253ndash2811957
[2] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
18 Journal of Applied Mathematics
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
[3] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[4] A CharnesWWCooper L Seiford and J Stutz ldquoAmultiplica-tive model for efficiency analysisrdquo Socio-Economic PlanningSciences vol 16 no 5 pp 223ndash224 1982
[5] A Charnes C T Clark W W Cooper and B Golany ldquoAdevelopmental study of data envelopment analysis inmeasuringthe efficiency ofmaintenance units in theUS air forcesrdquoAnnalsof Operations Research vol 2 no 1 pp 95ndash112 1984
[6] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985
[7] R MThrall ldquoDuality classification and slacks in DEArdquo Annalsof Operations Research vol 66 pp 109ndash138 1996
[8] N Adler and B Golany ldquoEvaluation of deregulated airlinenetworks using data envelopment analysis combined withprincipal component analysis with an application to WesternEuroperdquo European Journal of Operational Research vol 132 no2 pp 260ndash273 2001
[9] F W Young and R M Hamer Multidimensional ScalingHistory Theory and Applications Lawrence Erlbaum LondonUK 1987
[10] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo in Measuring Efficiency AnAssessment of Data Envelopment Analysis R H Silkman Edpp 73ndash105 Jossey-Bass San Francisco Calif USA 1986
[11] W Rodder and E Reucher ldquoA consensual peer-based DEA-model with optimized cross-efficiencies input allocationinstead of radial reductionrdquo European Journal of OperationalResearch vol 212 no 1 pp 148ndash154 2011
[12] H H Orkcu and H Bal ldquoGoal programming approaches fordata envelopment analysis cross efficiency evaluationrdquo AppliedMathematics andComputation vol 218 no 2 pp 346ndash356 2011
[13] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011
[14] G R Jahanshahloo F H Lotfi Y Jafari and R MaddahildquoSelecting symmetric weights as a secondary goal inDEA cross-efficiency evaluationrdquo Applied Mathematical Modelling vol 35no 1 pp 544ndash549 2011
[15] Y-M Wang K-S Chin and Y Luo ldquoCross-efficiency eval-uation based on ideal and anti-ideal decision making unitsrdquoExpert Systems with Applications vol 38 no 8 pp 10312ndash103192011
[16] N Ramon J L Ruiz and I Sirvent ldquoReducing differencesbetween profiles of weights a ldquopeer-restrictedrdquo cross-efficiencyevaluationrdquo Omega vol 39 no 6 pp 634ndash641 2011
[17] D Guo and J Wu ldquoA complete ranking of DMUs with undesir-able outputs using restrictions in DEA modelsrdquo Mathematicaland Computer Modelling 2012
[18] I Contreras ldquoOptimizing the rank position of the DMU assecondary goal inDEA cross-evaluationrdquoAppliedMathematicalModelling vol 36 no 6 pp 2642ndash2648 2012
[19] J Wu J Sun and L Liang ldquoCross efficiency evaluation methodbased on weight-balanced data envelopment analysis modelrdquoComputers and Industrial Engineering vol 63 pp 513ndash519 2012
[20] M Zerafat Angiz A Mustafa andM J Kamali ldquoCross-rankingof decision making units in data envelopment analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 398ndash405 2013
[21] SWashio and S Yamada ldquoEvaluationmethod based on rankingin data envelopment analysisrdquo Expert Systems with Applicationsvol 40 no 1 pp 257ndash262 2013
[22] G R Jahanshahloo A Memariani F H Lotfi and H ZRezai ldquoA note on some of DEA models and finding efficiencyand complete ranking using common set of weightsrdquo AppliedMathematics and Computation vol 166 no 2 pp 265ndash2812005
[23] Y-M Wang Y Luo and Z Hua ldquoAggregating preferencerankings using OWA operator weightsrdquo Information Sciencesvol 177 no 16 pp 3356ndash3363 2007
[24] M R Alirezaee and M Afsharian ldquoA complete ranking ofDMUs using restrictions in DEAmodelsrdquo Applied Mathematicsand Computation vol 189 no 2 pp 1550ndash1559 2007
[25] F-H F Liu and H Hsuan Peng ldquoRanking of units on the DEAfrontier with common weightsrdquo Computers and OperationsResearch vol 35 no 5 pp 1624ndash1637 2008
[26] Y-M Wang Y Luo and L Liang ldquoRanking decision makingunits by imposing a minimum weight restriction in the dataenvelopment analysisrdquo Journal of Computational and AppliedMathematics vol 223 no 1 pp 469ndash484 2009
[27] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010
[28] F Hosseinzadeh Lotfi A A Noora G R Jahanshahloo andM Reshadi ldquoOne DEA ranking method based on applyingaggregate unitsrdquo Expert Systems with Applications vol 38 no10 pp 13468ndash13471 2011
[29] Y-M Wang Y Luo and Y-X Lan ldquoCommon weights for fullyranking decision making units by regression analysisrdquo ExpertSystems with Applications vol 38 no 8 pp 9122ndash9128 2011
[30] N Ramon J L Ruiz and I Sirvent ldquoCommon sets of weightsas summaries of DEA profiles of weights with an application tothe ranking of professional tennis playersrdquo Expert Systems withApplications vol 39 no 5 pp 4882ndash4889 2012
[31] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1294 1993
[32] S Mehrabian M R Alirezaee and G R Jahanshahloo ldquoAcomplete efficiency ranking of decision making units in dataenvelopment analysisrdquo Computational Optimization and Appli-cations vol 14 no 2 pp 261ndash266 1999
[33] K Tone ldquoA slacks-based measure of super-efficiency indata envelopment analysisrdquo European Journal of OperationalResearch vol 143 no 1 pp 32ndash41 2002
[34] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[35] G R Jahanshahloo F Hosseinzadeh Lotfi N Shoja G Tohidiand S Razavyan ldquoRanking using 119897
1-norm in data envelopment
analysisrdquo Applied Mathematics and Computation vol 153 no 1pp 215ndash224 2004
[36] Y Chen and H D Sherman ldquoThe benefits of non-radial vsradial super-efficiency DEA an application to burden-sharingamongst NATO member nationsrdquo Socio-Economic PlanningSciences vol 38 no 4 pp 307ndash320 2004
[37] A Amirteimoori G Jahanshahloo and S Kordrostami ldquoRank-ing of decision making units in data envelopment analysis adistance-based approachrdquo Applied Mathematics and Computa-tion vol 171 no 1 pp 122ndash135 2005
Journal of Applied Mathematics 19
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
[38] G R Jahanshahloo L Pourkarimi and M Zarepisheh ldquoMod-ified MAJ model for ranking decision making units in dataenvelopment analysisrdquo Applied Mathematics and Computationvol 174 no 2 pp 1054ndash1059 2006
[39] S Li G R Jahanshahloo and M Khodabakhshi ldquoA super-efficiency model for ranking efficient units in data envelopmentanalysisrdquoAppliedMathematics and Computation vol 184 no 2pp 638ndash648 2007
[40] S J Sadjadi H Omrani S Abdollahzadeh M Alinaghian andH Mohammadi ldquoA robust super-efficiency data envelopmentanalysis model for ranking of provincial gas companies in IranrdquoExpert Systems with Applications vol 38 no 9 pp 10875ndash108812011
[41] A Gholam Abri G R Jahanshahloo F Hosseinzadeh LotfiN Shoja and M Fallah Jelodar ldquoA new method for rankingnon-extreme efficient units in data envelopment analysisrdquoOptimization Letters vol 7 no 2 pp 309ndash324 2011
[42] G R Jahanshahloo M Khodabakhshi F Hosseinzadeh Lotfiand M R Moazami Goudarzi ldquoA cross-efficiency model basedon super-efficiency for ranking units through the TOPSISapproach and its extension to the interval caserdquo Mathematicaland Computer Modelling vol 53 no 9-10 pp 1946ndash1955 2011
[43] A A Noura F Hosseinzadeh Lotfi G R Jahanshahloo andS Fanati Rashidi ldquoSuper-efficiency in DEA by effectiveness ofeach unit in societyrdquo Applied Mathematics Letters vol 24 no 5pp 623ndash626 2011
[44] A Ashrafi A B Jaafar L S Lee and M R A BakarldquoAn enhanced russell measure of super-efficiency for rankingefficient units in data envelopment analysisrdquo The AmericanJournal of Applied Sciences vol 8 no 1 pp 92ndash96 2011
[45] J-X Chen M Deng and S Gingras ldquoA modified super-efficiency measure based on simultaneous input-output pro-jection in data envelopment analysisrdquo Computers amp OperationsResearch vol 38 no 2 pp 496ndash504 2011
[46] F Rezai Balf H Zhiani Rezai G R Jahanshahloo and F Hos-seinzadeh Lotfi ldquoRanking efficientDMUsusing the Tchebycheffnormrdquo Applied Mathematical Modelling vol 36 no 1 pp 46ndash56 2012
[47] YChen JDu and JHuo ldquoSuper-efficiency based on amodifieddirectional distance functionrdquoOmega vol 41 no 3 pp 621ndash6252013
[48] A M Torgersen F R Fooslashrsund and S A C Kittelsen ldquoSlack-adjusted efficiency measures and ranking of efficient unitsrdquoJournal of Productivity Analysis vol 7 no 4 pp 379ndash398 1996
[49] T Sueyoshi ldquoDEA non-parametric ranking test and indexmea-surement slack-adjusted DEA and an application to Japaneseagriculture cooperativesrdquo Omega vol 27 no 3 pp 315ndash3261999
[50] G R Jahanshahloo H V Junior F H Lotfi and D AkbarianldquoA new DEA ranking system based on changing the referencesetrdquo European Journal of Operational Research vol 181 no 1pp 331ndash337 2007
[51] W-M Lu and S-F Lo ldquoAn interactive benchmark model rank-ing performers application to financial holding companiesrdquoMathematical and Computer Modelling vol 49 no 1-2 pp 172ndash179 2009
[52] J-X Chen and M Deng ldquoA cross-dependence based rankingsystem for efficient and inefficient units inDEArdquo Expert Systemswith Applications vol 38 no 8 pp 9648ndash9655 2011
[53] L Friedman and Z Sinuany-Stern ldquoScaling units via thecanonical correlation analysis in the DEA contextrdquo EuropeanJournal ofOperational Research vol 100 no 3 pp 629ndash637 1997
[54] E D Mecit and I Alp ldquoA new proposed model of restricteddata envelopment analysis by correlation coefficientsrdquo AppliedMathematical Modeling vol 3 no 5 pp 3407ndash3425 2013
[55] T Joro P Korhonen and J Wallenius ldquoStructural comparisonof data envelopment analysis and multiple objective linearprogrammingrdquoManagement Science vol 44 no 7 pp 962ndash9701998
[56] X-B Li and G R Reeves ldquoMultiple criteria approach todata envelopment analysisrdquo European Journal of OperationalResearch vol 115 no 3 pp 507ndash517 1999
[57] V Belton and T J Stewart ldquoDEA and MCDA Competing orcomplementary approachesrdquo in Advances in Decision AnalysisN Meskens and M Roubens Eds Kluwer Academic NorwellMass USA 1999
[58] Z Sinuany-Stern A Mehrez and Y Hadad ldquoAn AHPDEAmethodology for ranking decision making unitsrdquo InternationalTransactions in Operational Research vol 7 no 2 pp 109ndash1242000
[59] G Strassert and T Prato ldquoSelecting farming systems using anewmultiple criteria decisionmodel the balancing and rankingmethodrdquo Ecological Economics vol 40 no 2 pp 269ndash277 2002
[60] M-C Chen ldquoRanking discovered rules from data mining withmultiple criteria by data envelopment analysisrdquo Expert Systemswith Applications vol 33 no 4 pp 1110ndash1116 2007
[61] J Jablonsky ldquoMulticriteria approaches for ranking of efficientunits in DEA modelsrdquo Central European Journal of OperationsResearch vol 20 no 3 pp 435ndash449 2012
[62] Y-M Wang and P Jiang ldquoAlternative mixed integer linearprogrammingmodels for identifying themost efficient decisionmaking unit in data envelopment analysisrdquo Computers andIndustrial Engineering vol 62 no 2 pp 546ndash553 2012
[63] F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh N AghayiZ Ghelej Beigi and K Gholami ldquoAn improved method forranking alternatives in multiple criteria decision analysisrdquoAppliedMathematical Modelling vol 37 no 1-2 pp 25ndash33 2013
[64] LM Seiford and J Zhu ldquoContext-dependent data envelopmentanalysis measuring attractiveness and progressrdquoOmega vol 31no 5 pp 397ndash408 2003
[65] G R Jahanshahloo M Sanei F Hosseinzadeh Lotfi and NShoja ldquoUsing the gradient line for ranking DMUs in DEArdquoApplied Mathematics and Computation vol 151 no 1 pp 209ndash219 2004
[66] G R Jahanshahloo M Sanei and N Shoja ldquoModified rankingmodels using the concept of advantage in data envelopmentanalysisrdquo Working paper 2004
[67] G R Jahanshahloo F Hosseinzadeh Lotfi H Zhiani Rezai andF Rezai Balf ldquoUsing Monte Carlo method for ranking efficientDMUsrdquo Applied Mathematics and Computation vol 162 no 1pp 371ndash379 2005
[68] G R Jahanshahloo and M Afzalinejad ldquoA ranking methodbased on a full-inefficient frontierrdquo Applied Mathematical Mod-elling vol 30 no 3 pp 248ndash260 2006
[69] A Amirteimoori ldquoDEA efficiency analysis efficient and anti-efficient frontierrdquo Applied Mathematics and Computation vol186 no 1 pp 10ndash16 2007
[70] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling vol 34 no 7 pp 1779ndash1787 2010
[71] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
20 Journal of Applied Mathematics
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
[72] M Zerafat Angiz A Tajaddini AMustafa andM Jalal KamalildquoRanking alternatives in a preferential voting system usingfuzzy concepts and data envelopment analysisrdquo Computers andIndustrial Engineering vol 63 no 4 pp 784ndash790 2012
[73] A Charnes W W Cooper and S Li ldquoUsing data envelopmentanalysis to evaluate efficiency in the economic performance ofChinese citiesrdquo Socio-Economic Planning Sciences vol 23 no 6pp 325ndash344 1989
[74] W D Cook and M Kress ldquoAn mth generation model for weakranging of players in a tournamentrdquo Journal of the OperationalResearch Society vol 41 no 12 pp 1111ndash1119 1990
[75] W D Cook J Doyle R Green and M Kress ldquoRanking playersin multiple tournamentsrdquo Computers amp Operations Researchvol 23 no 9 pp 869ndash880 1996
[76] MMartic andG Savic ldquoAn application ofDEA for comparativeanalysis and ranking of regions in Serbia with regards tosocial-economic developmentrdquo European Journal of Opera-tional Research vol 132 no 2 pp 343ndash356 2001
[77] I De Leeneer and H Pastijn ldquoSelecting land mine detectionstrategies by means of outranking MCDM techniquesrdquo Euro-pean Journal of Operational Research vol 139 no 2 pp 327ndash3382002
[78] M P E Lins E G Gomes J C C B Soares de Mello and AJ R Soares de Mello ldquoOlympic ranking based on a zero sumgains DEA modelrdquo European Journal of Operational Researchvol 148 no 2 pp 312ndash322 2003
[79] A N Paralikas and A I Lygeros ldquoA multi-criteria and fuzzylogic based methodology for the relative ranking of the firehazard of chemical substances and installationsrdquo Process Safetyand Environmental Protection vol 83 no 2 B pp 122ndash134 2005
[80] A I Ali and R Nakosteen ldquoRanking industry performance inthe USrdquo Socio-Economic Planning Sciences vol 39 no 1 pp 11ndash24 2005
[81] J CMartin andCRoman ldquoAbenchmarking analysis of spanishcommercial airports A comparison between SMOP and DEAranking methodsrdquo Networks and Spatial Economics vol 6 no2 pp 111ndash134 2006
[82] R L Raab and E H Feroz ldquoA productivity growth accountingapproach to the ranking of developing and developed nationsrdquoInternational Journal of Accounting vol 42 no 4 pp 396ndash4152007
[83] R Williams and N Van Dyke ldquoMeasuring the internationalstanding of universities with an application to AustralianuniversitiesrdquoHigher Education vol 53 no 6 pp 819ndash841 2007
[84] H Jurges andK Schneider ldquoFair ranking of teachersrdquo EmpiricalEconomics vol 32 no 2-3 pp 411ndash431 2007
[85] D I Giokas and G C Pentzaropoulos ldquoEfficiency ranking ofthe OECD member states in the area of telecommunicationsa composite AHPDEA studyrdquo Telecommunications Policy vol32 no 9-10 pp 672ndash685 2008
[86] M Darvish M Yasaei and A Saeedi ldquoApplication of thegraph theory and matrix methods to contractor rankingrdquoInternational Journal of Project Management vol 27 no 6 pp610ndash619 2009
[87] E H Feroz R L Raab G T Ulleberg and K Alsharif ldquoGlobalwarming and environmental production efficiency ranking ofthe Kyoto Protocol nationsrdquo Journal of Environmental Manage-ment vol 90 no 2 pp 1178ndash1183 2009
[88] S Sitarz ldquoThe medal pointsrsquo incenter for rankings in sportrdquoApplied Mathematics Letters vol 26 no 4 pp 408ndash412 2013
[89] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002
[90] R G Thompson F D Singleton R MThrall and B A SmithldquoComparative site evaluations for locating a high energy physicslab in Texasrdquo Interfaces vol 16 pp 35ndash49 1986
[91] R GThompson L N Langemeier C-T Lee E Lee and R MThrall ldquoThe role of multiplier bounds in efficiency analysis withapplication to Kansas farmingrdquo Journal of Econometrics vol 46no 1-2 pp 93ndash108 1990
[92] R G Thompson E Lee and R MThrall ldquoDEAAR-efficiencyof US independent oilgas producers over timerdquo Computersand Operations Research vol 19 no 5 pp 377ndash391 1992
[93] R H Green J R Doyle and W D Cook ldquoPreference votingand project ranking usingDEA and cross-evaluationrdquo EuropeanJournal of Operational Research vol 90 no 3 pp 461ndash472 1996
[94] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[95] J Zhu ldquoRobustness of the efficient decision-making units indata envelopment analysisrdquo European Journal of OperationalResearch vol 90 pp 451ndash460 1996
[96] L M Seiford and J Zhu ldquoInfeasibility of super-efficiency dataenvelopment analysis modelsrdquo INFOR Journal vol 37 no 2 pp174ndash187 1999
[97] J H Dula and B L Hickman ldquoEffects of excluding the col-umn being scored from the DEA envelopment LP technologymatrixrdquo Journal of the Operational Research Society vol 48 no10 pp 1001ndash1012 1997
[98] A Hashimoto ldquoA ranked voting system using a DEAARexclusion model a noterdquo European Journal of OperationalResearch vol 97 no 3 pp 600ndash604 1997
[99] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007
[100] M M Tatsuoka and P R Lohnes Multivariant AnalysisTechniques For Educational and Psycologial ResearchMacmillanPublishing Company New York NY USA 2nd edition 1988
[101] Z Sinuany-Stern AMehrez and A Barboy ldquoAcademic depart-ments efficiency via DEArdquo Computers and Operations Researchvol 21 no 5 pp 543ndash556 1994
[102] D F Morrison Multivariate Statistical Methods McGraw-HillNew York NY USA 2nd edition 1976
[103] S Siegel and N J Castellan Nonparametric Statistics for theBehavioral Sciences McGraw-Hill New York NY USA 1998
[104] T Obata and H Ishii ldquoA method for discriminating efficientcandidates with ranked voting datardquo European Journal ofOperational Research vol 151 no 1 pp 233ndash237 2003
[105] M Soltanifar and F Hosseinzadeh Lotfi ldquoThe voting analytichierarchy process method for discriminating among efficientdecisionmaking units in data envelopment analysisrdquoComputersand Industrial Engineering vol 60 no 4 pp 585ndash592 2011
