A concept of fuzzy input mix-efficiency in fuzzy DEA and its application in banking sector Jolly Puri ⇑ , Shiv Prasad Yadav Department of Mathematics, I.I.T. Roorkee, Roorkee 247667, India article info Keywords: Fuzzy data envelopment analysis Fuzzy CCR input efficiency Fuzzy SBM input efficiency Fuzzy input mix-efficiency Fuzzy correlation coefficient Fuzzy ranking approach Banking performance evaluation abstract Data envelopment analysis (DEA) is a linear programming based non-parametric technique for evaluating the relative efficiency of homogeneous decision making units (DMUs) on the basis of multiple inputs and multiple outputs. There exist radial and non-radial models in DEA. Radial models only deal with propor- tional changes of inputs/outputs and neglect the input/output slacks. On the other hand, non-radial mod- els directly deal with the input/output slacks. The slack-based measure (SBM) model is a non-radial model in which the SBM efficiency can be decomposed into radial, scale and mix-efficiency. The mix- efficiency is a measure to estimate how well the set of inputs are used (or outputs are produced) together. The conventional mix-efficiency measure requires crisp data which may not always be available in real world applications. In real world problems, data may be imprecise or fuzzy. In this paper, we propose (i) a concept of fuzzy input mix-efficiency and evaluate the fuzzy input mix-efficiency using a – cut approach, (ii) a fuzzy correlation coefficient method using expected value approach which calculates the expected intervals and expected values of fuzzy correlation coefficients between fuzzy inputs and fuzzy outputs, and (iii) a new method for ranking the DMUs on the basis of fuzzy input mix-efficiency. The proposed approaches are then applied to the State Bank of Patiala in the Punjab state of India with districts as the DMUs. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Data envelopment analysis (DEA), proposed by Charnes, Cooper, and Rhodes (1978), is a linear programming based non-parametric method for evaluating the relative efficiency of homogeneous deci- sion making units (DMUs) on the basis of multiple inputs and mul- tiple outputs. The popularity of DEA is due to its ability to measure relative efficiencies of DMUs without prior weights on the inputs and outputs. There are two types of models in DEA: radial and non-radial. Radial model is represented by the CCR model (Charnes et al., 1978), the first DEA model. Basically, it deals with propor- tional changes of inputs or outputs. The CCR efficiency score re- flects the proportional maximum input (output) reduction (expansion) rate which is common to all inputs (outputs). How- ever, in real world businesses, not all inputs (outputs) behave in the proportional way. Also a radial model neglects slacks in in- puts/outputs while reporting the efficiency score. In many cases, we find a lot of remaining non-radial slacks. So, if these slacks have an important role in evaluating the efficiency, the radial ap- proaches may mislead the decision when we utilize the efficiency score as the only index for evaluating performance of DMUs. In contrast, the non-radial model is represented by the slack-based measure (SBM) (Tone, 2001), which put aside the assumption of proportionate changes in inputs and outputs, and deal with slacks directly. Also the SBM model assesses the efficiency of the input or output mix as well as it assesses the overall level of efficiency. It has three variations (i) input-oriented, (ii) output-oriented, and (iii) non-oriented. For details of comparison between radial and non-radial measure, see Avkiran, Tone, and Tsutsui (2008) along with the shortcomings for both the CCR and the SBM models. Tone (1998) suggests that the results from both the CCR and the SBM models can be used to evaluate the mix-efficiency. The mix-effi- ciency is a measure to estimate how well the set of inputs are used (or outputs are produced) together (Herrero, Pascoe, & Mardle, 2006; Asbullah, 2010). Tone (1998) presented input mix-efficiency and output mix-efficiency by using input-oriented and output-ori- ented variations of both CCR and SBM models. Conventional mix-efficiency measure requires crisp input and output data, which may not always be available in real world applications. Actually, in real world problems, inputs and outputs are often imprecise or fuzzy. So, in order to calculate mix-efficiency with imprecise or fuzzy data, we propose the concept of fuzzy in- put mix-efficiency (FIME). For measuring FIME, we propose the in- put-oriented fuzzy CCR model (FCCR I ) and input-oriented fuzzy 0957-4174/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.eswa.2012.08.047 ⇑ Corresponding author. E-mail addresses: [email protected](J. Puri), [email protected](S.P. Yadav). Expert Systems with Applications 40 (2013) 1437–1450 Contents lists available at SciVerse ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Transcript
Expert Systems with Applications 40 (2013) 1437–1450
Contents lists available at SciVerse ScienceDirect
Expert Systems with Applications
journal homepage: www.elsevier .com/locate /eswa
A concept of fuzzy input mix-efficiency in fuzzy DEA and its applicationin banking sector
Jolly Puri ⇑, Shiv Prasad YadavDepartment of Mathematics, I.I.T. Roorkee, Roorkee 247667, India
Data envelopment analysis (DEA) is a linear programming based non-parametric technique for evaluatingthe relative efficiency of homogeneous decision making units (DMUs) on the basis of multiple inputs andmultiple outputs. There exist radial and non-radial models in DEA. Radial models only deal with propor-tional changes of inputs/outputs and neglect the input/output slacks. On the other hand, non-radial mod-els directly deal with the input/output slacks. The slack-based measure (SBM) model is a non-radialmodel in which the SBM efficiency can be decomposed into radial, scale and mix-efficiency. The mix-efficiency is a measure to estimate how well the set of inputs are used (or outputs are produced) together.The conventional mix-efficiency measure requires crisp data which may not always be available in realworld applications. In real world problems, data may be imprecise or fuzzy. In this paper, we propose (i) aconcept of fuzzy input mix-efficiency and evaluate the fuzzy input mix-efficiency using a – cut approach,(ii) a fuzzy correlation coefficient method using expected value approach which calculates the expectedintervals and expected values of fuzzy correlation coefficients between fuzzy inputs and fuzzy outputs,and (iii) a new method for ranking the DMUs on the basis of fuzzy input mix-efficiency. The proposedapproaches are then applied to the State Bank of Patiala in the Punjab state of India with districts asthe DMUs.
� 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Data envelopment analysis (DEA), proposed by Charnes, Cooper,and Rhodes (1978), is a linear programming based non-parametricmethod for evaluating the relative efficiency of homogeneous deci-sion making units (DMUs) on the basis of multiple inputs and mul-tiple outputs. The popularity of DEA is due to its ability to measurerelative efficiencies of DMUs without prior weights on the inputsand outputs. There are two types of models in DEA: radial andnon-radial. Radial model is represented by the CCR model (Charneset al., 1978), the first DEA model. Basically, it deals with propor-tional changes of inputs or outputs. The CCR efficiency score re-flects the proportional maximum input (output) reduction(expansion) rate which is common to all inputs (outputs). How-ever, in real world businesses, not all inputs (outputs) behave inthe proportional way. Also a radial model neglects slacks in in-puts/outputs while reporting the efficiency score. In many cases,we find a lot of remaining non-radial slacks. So, if these slacks havean important role in evaluating the efficiency, the radial ap-proaches may mislead the decision when we utilize the efficiency
score as the only index for evaluating performance of DMUs. Incontrast, the non-radial model is represented by the slack-basedmeasure (SBM) (Tone, 2001), which put aside the assumption ofproportionate changes in inputs and outputs, and deal with slacksdirectly. Also the SBM model assesses the efficiency of the input oroutput mix as well as it assesses the overall level of efficiency. Ithas three variations (i) input-oriented, (ii) output-oriented, and(iii) non-oriented. For details of comparison between radial andnon-radial measure, see Avkiran, Tone, and Tsutsui (2008) alongwith the shortcomings for both the CCR and the SBM models. Tone(1998) suggests that the results from both the CCR and the SBMmodels can be used to evaluate the mix-efficiency. The mix-effi-ciency is a measure to estimate how well the set of inputs are used(or outputs are produced) together (Herrero, Pascoe, & Mardle,2006; Asbullah, 2010). Tone (1998) presented input mix-efficiencyand output mix-efficiency by using input-oriented and output-ori-ented variations of both CCR and SBM models.
Conventional mix-efficiency measure requires crisp input andoutput data, which may not always be available in real worldapplications. Actually, in real world problems, inputs and outputsare often imprecise or fuzzy. So, in order to calculate mix-efficiencywith imprecise or fuzzy data, we propose the concept of fuzzy in-put mix-efficiency (FIME). For measuring FIME, we propose the in-put-oriented fuzzy CCR model (FCCRI) and input-oriented fuzzy
1438 J. Puri, S.P. Yadav / Expert Systems with Applications 40 (2013) 1437–1450
SBM model (FSBMI) with fuzzy input and fuzzy output data. Sev-eral approaches have been developed to deal with imprecise orfuzzy data in DEA. Sengupta (1992) applied principle of fuzzy settheory to introduce fuzziness in the objective function and theright-hand side vector of the conventional DEA model. Guo and Ta-naka (2001) used the ranking method and introduced a bi-levelprogramming model. Lertworasirikul (2001) developed a methodin which the inputs and outputs were firstly defuzzified and thenthe model was solved using a-cut approach. There are some otherapproaches based on a-cut which can be found in Meada, Entani,and Tanaka (1998), Kao and Liu (2000a) and Saati Mohtadi,Memariani, and Jahanshahloo (2002). Lertworasirikul, Fang, Jeffrey,Joines, and Nuttle (2003) proposed a possibility DEA model forfuzzy DEA (FDEA). Kao and Liu (2000a, 2000b, 2003, 2005) trans-formed fuzzy input and fuzzy output into intervals by using a-levelsets and built a family of crisp DEA models for the intervals. Liu(2008) and Liu and Chuang (2009) developed a fuzzy DEA/AR mod-el for the selection of flexible manufacturing systems and theassessment of university libraries respectively. Zhou, Lui, Ma, Liu,and Liu (2012) proposed a generalized fuzzy data envelopmentmodel with assurance regions, whose lower and upper bounds atgiven levels could be obtained. Entani, Maeda, and Tanaka (2002)
Table 3The expect intervals and the corresponding expected values of the fuzzy correlation coeffi
and Wang, Greatbanks, and Yang (2005) also changed fuzzy inputand fuzzy output data into intervals by usinga-level sets, but sug-gested two different interval DEA models. Dia (2004) proposed aFDEA model based on fuzzy arithmetic operations and fuzzy com-parisons between fuzzy numbers. The model requires the decisionmaker to specify a fuzzy aspiration level and a safety a-level so thatthe FDEA model could be transformed into a crisp DEA model forsolution. Wang, Luo, and Liang (2009) constructed two FDEA mod-els from the perspective of fuzzy arithmetic to deal with fuzzinessin input and output data in DEA. The two FDEA models were bothformulated as linear programs and could be solved to determinefuzzy efficiencies of DMUs. Jahanshahloo, Soleimani-damaneh,and Nasrabadi (2004) extended a slack-based measure (SBM) ofefficiency in DEA to fuzzy settings and developed a bi-objectivenonlinear DEA model for FDEA. Among all the approaches to solveFDEA, the most popular approach is a-cut approach. Hatami-Mar-bini, Saati, and Makui (2010) introduced two virtual DMUs calledideal DMU (IDMU) and anti-ideal DMU (ADMU) with fuzzy in-puts-outputs, and evaluated efficiency of DMUs by FDEA. Hatam-i-Marbini, Saati, and Tavana (2010) presented a four-phase FDEAframework based on the theory of displaced ideal. Wang and Chin(2011) proposed a ‘‘fuzzy expected value approach’’ for DEA inwhich fuzzy inputs and fuzzy outputs are first weighted respec-tively, and their expected values then used to measure the optimis-tic and pessimistic efficiencies of DMUs in fuzzy environments.Hsiao, Chern, Chiu, and Chiu (2011) proposed the fuzzy super-effi-ciency slack-based measure DEA model using a-cut approach andanalyze the operational performance of 24 commercial banks fac-ing problems on loan and investment parameters with vague char-acteristics. Majid Zerafat Angiz, Emrouznejad, and Mustafa (2012)introduced an alternative linear programming model that can in-clude some uncertainty information from the intervals within thea-cut approach and proposed the concept of ‘‘local a-level’’ to de-velop a multi-objective linear programming to measure the effi-ciency of DMUs under uncertainty.
In this paper, we use a-cut approach to solve FCCRI and FSBMI.Then, the results of these models are applied to calculate FIME.We propose a new method for calculating the fuzzy correlation
J. Puri, S.P. Yadav / Expert Systems with Applications 40 (2013) 1437–1450 1439
coefficients between fuzzy inputs and fuzzy outputs by using ex-pected value approach. Further, we also propose a new rankingmethod to rank the DMUs on the basis of FIME. All the proposedapproaches are then applied to the banking sector.
The paper is organized as follows: Section 2 presents an over-view of DEA with input-oriented CCR and SBM models, and inputmix-efficiency. Section 3 presents the description of FDEA withFCCRI and FSBMI. Section 4 presents the methodology for solvingFCCRI and FSBMI. Section 5 gives the definition of FIME. Section 6proposes a new method for evaluating the fuzzy correlation coeffi-cients between fuzzy inputs and fuzzy outputs by using expectedvalue approach. A numerical illustration is presented in Section7. Section 8 describes a new ranking method for DMUs. Section 9presents an application of the proposed approaches to the bankingsector. The last Section 10 concludes the findings of our study.
2. Data envelopment analysis (DEA)
Data envelopment analysis (DEA), proposed by Charnes et al.(1978), is a linear programming based non-parametric methodfor evaluating the relative efficiency of DMUs which uses multipleinputs to produce multiple outputs. Since 1978, it has got compre-hensive attention both in theory and applications. Based on theoriginal DEA model (Charnes et al., 1978), various theoreticalextensions have been developed (Banker, Charnes, & Cooper,1984; Charnes, Cooper, Seiford, & Stutz, 1982; Petersen, 1990;Tone, 2001; Cooper, Seiford, & Tone, 2007). DEA is a non-paramet-ric technique to construct a piecewise frontier (surface) over thedata. Efficiency measure is then calculated relative to this frontier.Using this frontier, DEA computes a maximal performance measurefor each DMU relative to that of all other DMUs with the restrictionthat each DMU lies on the efficient (extremal) frontier or is envel-oped by the frontier. There are two types of models in DEA: radialand non-radial. Radial model is represented by the CCR model(Charnes et al., 1978) and non-radial model is represented bySBM model (Tone, 2001). Both CCR and SBM models are orientedmodels in terms of input-oriented and output-oriented. In thispaper, we are taking input-oriented models. Assume that theperformance of a set of n homogeneous DMUs (DMUj; j = 1, . . . ,n)is to be measured. The performance of DMUj is characterized by aproduction process of m inputs (xij; i = 1, . . . , m) to yield s outputs(yrj;r = 1, . . . ,s). Let yrk be the amount of the rth output producedby the kth DMU and xik be the amount of the ith input used bythe kth DMU. Assume that input and output data is positive.
2.1. Input-oriented CCR model (CCRI model)
The CCRI model evaluates the CCR input efficiency of the DMUs.The CCR input efficiency of the kth DMU is denoted by hk
I and is de-fined as
hkI ¼min h
subject to hxik ¼Xn
j¼1
xijgjk þ s�ik 8i;
yrk ¼Xn
j¼1
yrjgjk � sþrk 8r;
gjk P 0; s�ik P 0; sþrk P 0 8i; r; j;
h unrestricted in sign;k ¼ 1; . . . ;n:
ð1Þ
where sþrk is slack in the rth output of the kth DMU; s�ik is slack in theith input of the kth DMU; gjk’s i.e. (gj1,gj2, . . . ,gjn) are non negativevariables forj = 1, 2, . . . ,n. Due to nonzero assumption of the data,we have 0 < hk
I 6 1 for k ¼ 1;2; . . . ;n (Cooper et al., 2007).
A DMUk is CCR input efficient if (i) hkI ¼ 1 and (ii) all slacks are
zero, i.e., s�ik ¼ 0 for i ¼ 1;2; . . . ;m and sþrk ¼ 0 for r ¼ 1;2; . . . ; s.
2.2. Input-oriented SBM model (SBMI model)
The SBMI model evaluates the SBM input efficiency of theDMUs. The SBM input efficiency of the kth DMU is denoted by qk
I
and is defined as
qkI ¼min 1� 1
m
Xm
i¼1
S�ikxik
subject to xik ¼Xn
j¼1
xijkjk þ S�ik 8i;
yrk ¼Xn
j¼1
yrjkjk � Sþrk 8r;
kjk P 0; S�ik P 0; Sþrk P 0 8i; r; j;
k ¼ 1; . . . ;n:
ð2Þ
where Sþrk is slack in the rth output of the kth DMU; S�ik is slack in theith input of the kth DMU; kjk’s i.e. (kj1,kj2, . . . ,kjn) are non negativevariables for j = 1, 2, . . . ,n.
A DMUk is SBM input efficient if (i) qkI ¼ 1 and (ii) all output
slacks are zero, i.e., Sþrk ¼ 0 for r ¼ 1;2; . . . ; s. However, input slacksmay be nonzero.
2.3. Input mix-efficiency (IME)
The IME of the kth DMU is defined as the ratio of SBM input effi-ciency of the kth DMU to CCR input efficiency of the kth DMU. TheIME of the kth DMU is denoted by wk
I and is defined by
wkI ¼
qkI
hkI
: ð3Þ
Due to nonzero assumption of the data, we have 0 < qkI 6 1
and 0 < hkI 6 1 for k ¼ 1;2; . . . ;n. Also qk
I 6 hkI (Tone, 1998). This
implies that 0 < wkI 6 1; and wk
I ¼ 1 if and only if qkI ¼ hk
I holds. Ifwk
I ¼ 1, it shows that DMUk has the most efficient combination of in-puts, even though it may be technically inefficient.
3. Fuzzy data envelopment analysis (FDEA)
Conventional DEA requires crisp input and output data, whichmay not always be available in real world applications. However,in real-world problems, inputs and outputs are often imprecise.To deal with imprecise data, the notion of fuzziness has been intro-duced. The DEA is extended to FDEA in which the imprecision isrepresented by fuzzy sets or fuzzy numbers. Various efforts havebeen made to handle fuzzy input and fuzzy output data in FDEA.Based on fuzzy input and fuzzy output data, both CCR and SBMmodels were extended to fuzzy CCR model (Guh, 2001; Guo &Tanaka, 2001) and fuzzy SBM model (Jahanshahloo et al., 2004;Saati & Memariani, 2009). Since in this paper, we are taking in-put-oriented models, we are defining FCCRI and FSBMI. Assumethat the performance of a set of n homogeneous DMUs (DMUj;j = 1,. . ., n) is to be measured. The performance of DMUj is charac-terized by a production process of m fuzzy inputsð~xij; i ¼ 1; . . . ;mÞ to yield s fuzzy outputs ð~yrj; r ¼ 1; . . . ; sÞ. Let ~yrk
be the amount of the rth fuzzy output produced by the kth DMUand ~xik be the amount of the ith fuzzy input used by the kthDMU. Assume that the fuzzy inputs and fuzzy outputs are positivefuzzy numbers (Nasseri, 2008). In this paper, we have taken thefuzzy inputs and fuzzy outputs as the triangular fuzzy numbers(TFNs) (Chen, 1994).
1440 J. Puri, S.P. Yadav / Expert Systems with Applications 40 (2013) 1437–1450
Definition 1. A TFN eA is denoted by (a1,a2,a3) and is defined by themembership function leAðxÞ given by
leAðxÞ ¼x�a1
a2�a1; a1 < x 6 a2;
1; x ¼ a2;x�a3
a2�a3; a2 6 x < a3;
0; otherwise:
8>>>><>>>>:
3.1. Input-oriented fuzzy CCR model (FCCRI model)
The FCCRI model evaluates the fuzzy CCR input efficiency of theDMUs. The fuzzy CCR input efficiency of the kth DMU, denoted by~hk
I ; is defined by~hk
I ¼ min ~h
subject to ~h~xik ¼Xn
j¼1
~xijgjk þ ~s�ik 8i;
~yrk ¼Xn
j¼1
~yrjgjk � ~sþrk 8r;
gjk P 0; ~s�ik P ~0; ~sþrk P ~0 8; i; r; j;k ¼ 1; . . . ;n:
ð4Þ
where ~sþrk is the fuzzy slack in the rth fuzzy output of the kth DMU;~s�ik is the fuzzy slack in the ith fuzzy input of the kth DMU;gjk’s, i.e.,(gj1,gj2, . . . ,gjn) are non-negative variables for j = 1,2, . . . ,n.
3.2. Input-oriented fuzzy SBM model (FSBMI model)
The FSBMI model evaluates the fuzzy SBM input efficiency ofthe DMUs. The fuzzy SBM input efficiency of the kth DMU, denotedby ~qk
I ; is defined by
~qkI ¼min 1� 1
m
Xm
i¼1
eS�i~xik
subject to ~xik ¼Xn
j¼1
~xijkjk þ ~S�ik 8i;
~yrk ¼Xn
j¼1
~yrjkjk � eSþrk 8r;
kjk P 0; eS�ik P ~0; eSþrk P ~0 8i; r; j;
k ¼ 1; . . . ;n:
ð5Þ
where eSþrk is the fuzzy slack in the rth fuzzy output of the kth DMU;eS�ik is the fuzzy slack in the ith fuzzy input of the kth DMU; kjk’s, i.e.,(kj1,kj2, . . . ,kjn) are non-negative variables for j = 1, 2, . . . ,n.
4. Methodology for solving FCCRI model and FSBMI model
Kao and Liu (2000a) developed a procedure to transform a FDEAmodel to a family of crisp DEA models by applying the a-cuts andZadeh’s extension principle (Zadeh, 1975). Now, we will describethe procedure to convert FCCRI and FSBMI models into a familyof crisp DEA models. Let Sð~xijÞ and Sð~yrjÞ be the support of m fuzzyinputs ð~xij; i ¼ 1;2; . . . ;mÞ and s fuzzy outputs ð~yrj; r ¼ 1;2; . . . ; sÞof DMUj (j = 1, . . . ,n) respectively, given bySð~xijÞ ¼ fxijjl~xij
ðxijÞ > 0g and Sð~yrjÞ ¼ fyrjjl~yrjðyrjÞ > 0g: ð6Þ
The a- cuts of ~xij and ~yrj are respectively defined as
ð~xijÞa ¼ fxij 2 Sð~xijÞjl~xijðxijÞP ag ¼ ðxijÞLa; ðxijÞUa
h i8i; j ð7aÞ
¼ minxij
fxij 2 Sð~xijÞjl~xijðxijÞP ag;
�max
xij
fxij 2 Sð~xijÞjl~xijðxijÞP ag
�8i; j ð7bÞ
and
ð~yrjÞa ¼ fyrj 2 Sð~yrjÞjl~yrjðyrjÞP ag ¼ ðyrjÞ
La; ðyrjÞ
Ua
h i8r; j ð8aÞ
¼ minyrj
fyrj 2 Sð~yrjÞjl~yrjðyrjÞP ag;
�max
yrj
fyrj 2 Sð~yrjÞjl~yrjðyrjÞP ag
�8r; j; ð8bÞ
where 0 < a 6 1.Further, FCCRI and FSBMI models can easily be transformed into
crisp models by using a-cuts given in (7b) and (8b). Owing to theinput and output data being fuzzy numbers, the efficiency scoresare also fuzzy numbers. Let them be represented by ~hk
I and ~qkI with
the membership functions l~hkI
and l~qkI
respectively. Let S ~hkI
� �and
S ~qkI
� �be the support of the fuzzy efficiency scores ~hk
I and ~qkI of
the kth DMU respectively, given by
S ~hkI
� �¼ hk
I jl~hkI
hkI
� �> 0
n oand S ~qk
I
� �¼ qk
I jl~qkI
qkI
� �> 0
n oð9Þ
The a-cuts of ~hkI and ~qk
I are respectively defined as
~hkI
� �a¼ hk
I 2 S ~hkI
� �jl~hk
Ihk
I
� �P a
n o¼ hk
I
� �L
a; hk
I
� �U
a
� �8i; j ð10aÞ
¼ minhk
I
hkI 2 S ~hk
I
� �jl~hk
Ihk
I
� �P a
n o;
"
maxhk
I
hkI 2 S ~hk
I
� �jl~hk
Ihk
I
� �P a
n o#8i; j ð10bÞ
and
~qkI
� �a ¼ qk
I 2 S ~qkI
� �jl~qk
Iqk
I
� �P a
n o¼ qk
I
� �L
a; qkI
� �U
a
h i8i; j ð11aÞ
¼ minqk
I
qkI 2 S ~qk
I
� �jl~qk
Iqk
I
� �P a
n o;
"
maxqk
I
qkI 2 S ~qk
I
� �jl~qk
Iqk
I
� �P a
n o#8i; j; ð11bÞ
where 0 < a 6 1.
hkI
� �L
a¼ min
ðxij ÞLa6xij6ðxij Þ
Ua
ðyrj ÞLa6yrj6ðyrj Þ
Ua
8i;r;j
min h
subject to hxik¼Xn
j¼1
xijgjkþs�ik 8i;
yrk¼Xn
j¼1
yrjgjk�sþrk 8r;
gjk P0; s�ik P0; sþrk P0 8i;r;j;
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;ð12aÞ
hkI
� �U
a¼ max
ðxij ÞLa6xij6ðxij Þ
Ua
ðyrj ÞLa6yrj6ðyrj Þ
Ua
8i;r;j
min h
subject to hxik¼Xn
j¼1
xijgjkþs�ik 8i;
yrk¼Xn
j¼1
yrjgjk�sþrk 8r;
gjk P0; s�ik P0; sþrk P0 8i;r;j
8>>>>>>>>><>>>>>>>>>:
9>>>>>>>>>=>>>>>>>>>;ð12bÞ
and
qkI
� �L
a¼ minðxij Þ
La6xij6ðxij Þ
Ua
ðyrj ÞLa6yrj6ðyrj Þ
Ua
8i;r;j
min 1� 1m
Xm
i¼1
S�ikxik
subject to xik¼Xn
j¼1
xijkjkþS�ik 8i;
yrk¼Xn
j¼1
yrjkjk�Sþrk 8r;
kjk P0; S�ik P0; Sþrk P0 8i;r;j;
8>>>>>>>>>>>>><>>>>>>>>>>>>>:
9>>>>>>>>>>>>>=>>>>>>>>>>>>>;ð13aÞ
J. Puri, S.P. Yadav / Expert Systems with Applications 40 (2013) 1437–1450 1441
qkI
� �U
a ¼ maxðxij Þ
La6xij6ðxij Þ
Ua
ðyrj ÞLa6yrj6ðyrjÞ
Ua
8i;r;j
min 1� 1m
Xm
i¼1
S�ikxik
subject to xik¼Xn
j¼1
xijkjkþS�ik 8i;
yrk¼Xn
j¼1
yrjkjk�Sþrk 8r;
kjk P0; S�ik P0; Sþrk P0 8i;r; j:
8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:
9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;
ð13bÞ
Further, following the Pareto’s efficiency concept, we can find the‘minimum efficiency’ of specific or targeted DMU. For this, we usetargeted DMU with lower bound outputs and other DMUs withupper bound outputs, and targeted DMU with upper bound inputsand other DMUs with lower bound inputs. Similarly, if we findthe targeted DMU to have the ‘maximum efficiency’, we use tar-geted DMU with upper bound outputs and other DMUs with lowerbound outputs, and targeted DMU with lower bound inputs andother DMUs with upper bound inputs. Thus, models 12a,12b,13aand 13b reduce to the following models:
hkI
� �L
a¼min h
subject to hðxikÞUa ¼Xn
j¼1;j–k
ðxijÞLagjk þ ðxikÞUagjk þ s�ik� �U 8i;
ðyrkÞLa ¼
Xn
j¼1;j–k
ðyrjÞUagjk þ ðyrkÞ
Lagjk � sþrk
� �L 8r;
gjk P 0; s�ik� �U P 0; sþrk
� �L P 0 8i; r; j;
ð14aÞ
hkI
� �U
a¼ min h
subject to hðxikÞLa ¼Xn
j¼1;j–k
ðxijÞUagjk þ ðxikÞLagjk þ s�ik� �L8i;
ðyrkÞUa ¼
Xn
j¼1;j–k
ðyrjÞLagjk þ ðyrkÞ
Uagjk � sþrk
� �U 8r;
gjk P 0; s�ik� �L P 0; sþrk
� �U P 0 8i; r; j
ð14bÞ
and
qkI
� �L
a ¼min 1� 1m
Xm
i¼1
S�ik� �U
ðxikÞUa
subject to ðxikÞUa ¼Xn
j¼1;j–k
ðxijÞLakjk þ ðxikÞUa kjk þ S�ik� �U 8i;
ðyrkÞLa ¼
Xn
j¼1;j–k
ðyrjÞUa kjk þ ðyrkÞ
Lakjk � Sþrk
� �L 8r;
kjk P 0; S�ik� �U P 0; Sþrk
� �LP 0 8i; r; j;
ð15aÞ
qkI
� �U
a ¼ min 1� 1m
Xm
i¼1
S�ik� �L
ðxikÞLa
subject to ðxikÞLa ¼Xn
j¼1;j–k
ðxijÞUa kjk þ ðxikÞLakjk þ S�ik� �L 8i;
ðyrkÞU ¼
Xn
j¼1;j–k
ðyrjÞLakjk þ ðyrkÞ
Ua kjk � Sþrk
� �U 8r;
kjk P 0; S�ik� �L P 0; Sþrk
� �UP 0 8i; r; j;
ð15bÞ
The sets of intervals hkI
� �L
a; hk
I
� �U
a
� �ja 2 ð0;1�; k ¼ 1;2; . . . ; n
� and
qkI
� �L
a; qkI
� �U
a
h ija 2 ð0;1�; k ¼ 1;2; . . . ;n
n oreveal the shape of l~hk
I
and l~qkI, respectively, although the exact form of the membership
functions are not known explicitly. In our study, we are taking fuzzyinputs ð~xij; i ¼ 1;2; . . . ;mÞ and fuzzy outputs ð~yrj; r ¼ 1;2; . . . ; sÞ asTFNs. Therefore, the membership functions l~hk
Iand l~qk
Ican be
approximated by the triangular membership functions whose a-cuts are represented by the sets of intervals
hkI
� �L
a; hk
I
� �U
a
� �ja 2 ð0;1�; k ¼ 1;2; . . . ; n
� and qk
I
� �L
a; qkI
� �U
a
h ija 2
nð0;1�; k ¼ 1;2; . . . ;ng; respectively. Thus, the fuzzy efficiencies ~hk
I
and ~qkI can be approximated by TFNs and are defined as:
Definition 2. The fuzzy CCR input efficiency ~hkI of the kth DMU is
defined by its a-cut ~hkI
� �a
which is given by
~hkI
� �a¼ hk
I
� �L
a; hk
I
� �U
a
� �; a 2 ð0;1�; k ¼ 1;2; . . . ;n:
where hkI
� �L
aand hk
I
� �U
aare obtained from the optimal values of
(14a) and (14b) respectively. Further, ~hkI by using a-cuts
~hkI
� �a; a 2 ð0;1� can be approximated by a TFN hk
a; hkb; h
kc
� �.
Definition 3. The fuzzy SBM input efficiency ~qkI of the kth DMU is
defined by its a-cut ~qkI
� �awhich is given by
~qkI
� �a ¼ qk
I
� �L
a; qkI
� �U
a
h i; a 2 ð0;1�; k ¼ 1;2; . . . ;n;
where qkI
� �L
a and qkI
� �U
a are obtained from the optimal values of (15a)and (15b) respectively. Further, ~qk
I by usinga-cuts ~qkI
� �a; a 2 ð0;1�
can be approximated by a TFN qka;qk
b;qkc
� �.
5. Fuzzy input mix-efficiency (FIME)
The FIME of the kth DMU, denoted by ~wkI ; is defined as the ratio
of fuzzy SBM input efficiency of the kth DMU to the fuzzy CCR in-put efficiency of the kth DMU. By using arithmetic operations onTFNs (Chen, 1994), ~wk
I can be defined as
~wkI ¼
~qkI
~hkI
; ~hkI – ~0 ð16Þ
¼qk
a;qkb;q
kc
� �hk
a; hkb; h
kc
� � ¼ qka;q
kb;q
kc
� �� hk
a; hkb; h
kc
� ��1
¼ qka;q
kb;q
kc
� �� 1
hkc
;1hk
b
;1hk
a
!; hk
a > 0
� qka
hkc
;qk
b
hkb
;qk
c
hka
!; hk
a > 0: ð17Þ
6. The proposed method for evaluating the fuzzy correlationcoefficients between fuzzy inputs and fuzzy outputs by usingexpected value approach
To ensure the validity of the crisp DEA model specification, thecorrelation coefficients between inputs and outputs are calculated.If positive inter-correlations are found, the inclusion of the inputsand outputs is justified. This test is called the isotonicity test (Avki-ran, 2006; Tsai, Chen, & Tzeng, 2006). It identifies whether increas-ing amounts of inputs lead to greater outputs. In the present study,we are concerned with the evaluation of ~hk
I ; ~qkI and ~wk
I on the basisof the fuzzy inputs and fuzzy outputs. To ensure the validity of thepresent FDEA models specification, it is mandatory to calculate thefuzzy correlation coefficients between the fuzzy inputs and fuzzyoutputs. However, in the literature of FDEA, nobody has thrownlight on this important part of the analysis. Therefore, we are pro-posing a method to calculate fuzzy correlation coefficients between
Table 4a- cuts ~hk
I
� �a
of the fuzzy CCR input efficiency ~hkI for different values of a 2 (0,1].
1442 J. Puri, S.P. Yadav / Expert Systems with Applications 40 (2013) 1437–1450
the fuzzy inputs and fuzzy outputs by using expected value ap-proach (Hung & Wu, 2001). In this approach, we calculate expectedintervals and expected values of fuzzy correlation coefficients.
6.1. Expected interval and expected value of a fuzzy number
Let eA be a fuzzy number with membership function leAðxÞ givenby
leAðxÞ ¼feAðxÞ; a1 6 x 6 a2;
1; a2 6 x 6 a3;
geAðxÞ; a3 6 x 6 a4;
0; otherwise;
8>>>><>>>>:
where feA is an increasing function and geA is a decreasing function.The expected interval of eA is a crisp interval EIðeAÞ given byEIðeAÞ ¼ ½ELðeAÞ; ERðeAÞ� (Hung and Wu, 2001), where
ELðeAÞ ¼ a2 �Z a2
a1
feAðxÞdx and ERðeAÞ ¼ a3 þZ a4
a3
geAðxÞdx:
The expected value is defined by
EVðeAÞ ¼ ELðeAÞ þ ERðeAÞ2
:
Let eA be a TFN denoted by (a1,a2,a3). The expected interval and ex-pected value of eA are given by
J. Puri, S.P. Yadav / Expert Systems with Applications 40 (2013) 1437–1450 1443
EIðeAÞ ¼ a1 þ a2
2;a2 þ a3
2
h iand
EVðeAÞ ¼ a1 þ 2a2 þ a3
4; respectively: ð18Þ
Fig. 1b. Shape of membership functions of ~hkI for k = 7 to 12.
Fig. 1c. Shape of membership functions of ~qkI for k =1 to 6.
Fig. 1d. Shape of membership functions of ~qkI for k =7 to 12.
6.2. Fuzzy correlation coefficient between fuzzy input ~x and fuzzyoutput ~y
The crisp correlation coefficient denoted by r(x,y) between twosets of crisp data x = {x1,x2, . . . ,xn} and y = {y1,y2, . . . ,yn} is calculatedby the following formula:
However, in real world applications due to non-availability of exactdata, the data may be imprecise or fuzzy. Hence, the fuzzy correla-tion coefficient between two sets ~x and ~y of fuzzy dataf~x1; ~x2; . . . ; ~xng and f~y1; ~y2; . . . ; ~yng respectively, denoted by ~rð~x; ~yÞ;can be calculated by the following formula:
With the assumption of positive fuzzy data and by using arithmeticoperations of addition, subtraction, multiplication and division onfuzzy numbers (Chen, 1994), we can find out the fuzzy correlationcoefficient between two sets of fuzzy data by solving (20). Bansal(2010) defined the square root of a TFN which is non linear arithme-tic operation. Thus, we can find out the square root operation in(20). As all the fuzzy inputs and fuzzy outputs are positive andare represented by TFNs, the square of a positive TFN eA ¼ ða; b; cÞcan be defined as
eA2 ¼ eA � eA ¼ ða; b; cÞ � ða; b; cÞ¼ ðminða2; ac; c2Þ; b2
;maxða2; ac; c2ÞÞ ¼ ða2; b2; c2Þ:
But, it is a difficult task to apply (20) if data set is large. So, in orderto obtain the fuzzy correlation coefficient between the sets of fuzzydata, we propose a method to calculate the fuzzy correlation coeffi-cient using the expected value approach. Firstly, find the expectedintervals of fuzzy data sets, i.e., find EIð~xiÞ ¼ xi
L; xiR
� �,
EIð~yiÞ ¼ yiL; y
iR
� �; i ¼ 1;2; . . . n and then by using these expected
intervals find the expected interval of the fuzzy correlation coeffi-cient ~rð~x; ~yÞ denoted by rEIð~x; ~yÞ ¼ ½rLð~x; ~yÞ; rRð~x; ~yÞ�; where
If the value of rEV ð~x; ~yÞ for each ~x and ~y is positive, then the FDEAmodel is said to be consistent and inclusion of fuzzy inputs andfuzzy outputs is justified.
Remark 1. If the data is crisp, the proposed approach gives thesimilar results as given by the conventional correlation coefficientformula defined in Eq. (19). In case of crisp data,
The CCR input efficiency hkI ; SBM input efficiency qk
I and Mix in-put efficiency wk
I of the twelve DMUs with crisp data are evaluatedand results are shown in Table 1.
The results show that DMU1, DMU2 and DMU4 are CCR input effi-cient, SBM input efficient and also Mix input efficient. There arenine DMUs (DMU3,DMU5,DMU6,DMU7,DMU8,DMU9,DMU10, -DMU11, and DMU12) which are input mix-inefficient. The inputmix-inefficiency represents the degree to which the input mixshould change to become fully efficient. The DMU6 is the mostmix-inefficient DMU with rank 12. The results shown in Table 1are for crisp data. However, in real world applications data maynot always be available crisply. Actually, in real world problems,inputs and outputs are often imprecise or fuzzy. Thus to deal withactual problems, we fuzzify the crisp data given in Table 1 and rep-resent the fuzzy data in terms of TFNs. The fuzzified data is shownin Table 2.
7.1. The fuzzy correlation coefficient between fuzzy inputs and fuzzyoutputs
To ensure the validity of the FDEA model specification, the ex-pected intervals and expected values of the fuzzy correlation coef-ficients between the fuzzy inputs and fuzzy outputs are calculatedby using the proposed method given in Section 6. The expectedinterval values and the corresponding expected values of the fuzzycorrelation coefficients between the sets of fuzzy data (given in Ta-ble 2) are shown in the matrix form in Table 3.
It can be seen from Table 3 that the left and right bounds of eachexpected interval and the corresponding expected value are posi-tive. Thus, the inclusion of the fuzzy inputs and fuzzy outputs is
J. Puri, S.P. Yadav / Expert Systems with Applications 40 (2013) 1437–1450 1445
value of ~wkI is obtained by using Eq. (17). Table 6 presents the val-
ues of ~hkI , ~qk
I and ~wkI approximated as TFNs.
Table 6 indicates that ~0 < ~hkI 6
~1 and ~0 < ~qkI 6
~1. But due to thedivision operation of TFNs, the right part of ~wk
I take the valuesgreater than or equal to 1. So, the FIME for each DMU does notlie between ~0 and ~1; and thus the ranking of the DMUs on the basisof FIME becomes very difficult.
8. A new ranking method for DMUs
For ranking the DMUs on the basis of ~hkI , ~qk
I and ~wkI ; we use
defuzzification method. The aim of the defuzzification is to deter-mine a real value which corresponds to the fuzzy number. In liter-ature, there are various methods to defuzzify a fuzzy number(Kataria, 2010).
Let eA be a fuzzy number defined on R. Then the height of a fuzzy
number eA denoted by hðeAÞ is defined as hðeAÞ ¼ supx2RleAðxÞ. Let
MðeAÞ ¼ x 2 RjleAðxÞ ¼ hðeAÞn obe the set of all those points for
which membership value is equal to the height of eA: Then someof the popularly known deffuzification methods (Kataria, 2010)are defined as follows:
(i) Middle/Mean of maximum (MOM) method – The MOMmethod gives the mean of all those points where member-ship value is maximum. Mathematically, the defuzzifiedvalue dMOMðeAÞ of eA is given by
Table 9Input an
DMU
123456789
1011121314151617
Source:
Table 1The exp
I1I2I3O1O2
dMOMðeAÞ¼X
x2MðeAÞxjMðeAÞj where j:j stands for the cardinality of the set:
d output data of SBOP in various districts of Punjab for the period 2010–2011.
(ii) Largest of maximum (LOM) method – The LOM methodresults into the largest of all those points where membershipvalue is maximum. Mathematically, the defuzzified valuedLOMðeAÞ of eA is given by
fficients
784935000787183
dLOMðeAÞ ¼maxfxjx 2 MðeAÞg:
(iii) Smallest of maximum (SOM) method – The SOM method
gives the minimum value of all those points where member-ship value is maximum. Mathematically, the defuzzifiedvalue dSOMðeAÞ of eA is given by
dSOMðeAÞ ¼minfxjx 2 MðeAÞg:
(iv) Bisector/Centre of area (COA) method – The bisector method
gives the value that divides the region into two sub-regionsof equal area. Mathematically, the defuzzified value dCOAðeAÞof eA is given by
Z dCOAðeAÞxmin
leAðxÞdx ¼Z xmax
dCOAðeAÞ leAðxÞdx:
(v) Centroid/Centre of gravity (COG) method – The COG methoddetermines the centre of gravity of a fuzzy number. Mathe-matically, COG method defines the defuzzified value dCOGðeAÞas
dCOGðeAÞ ¼R
x x � leAðxÞdxRx leAðxÞdx
:
The results of MOM, LOM and SOM are the same in case of thefuzzy numbers having unique maximum of the membership func-tion. In this paper, we are using TFNs which have a unique maxi-
1446 J. Puri, S.P. Yadav / Expert Systems with Applications 40 (2013) 1437–1450
Fig. 2a. Shape of membership functions of ~hkI for k = 1 to 9.
Fig. 2b. Shape of membership functions of ~hkI for k = 10 to 17.
Fig. 2c. Shape of membership functions of ~qkI for k =1 to 9.
Fig. 2d. Shape of membership functions of ~qkI for k =10 to 17.
J. Puri, S.P. Yadav / Expert Systems with Applications 40 (2013) 1437–1450 1447
mum at the modal value, so the results of MOM, LOM and SOM arethe same and are not very much useful. The results of the bisectormethod sometimes, but not always, coincide with the COG meth-od. The COG method is one of the best methods because it takesinto account all the information provided in the fuzzy numberand this method is popularly used in case of TFNs. Keeping in viewthe popularity of the COG method among all other defuzzificationmethods, we are using the COG method to defuzzify the fuzzy effi-ciencies and on the basis of these defuzzified values we are going
to rank the DMUs. Let dCOG~hk
I
� �; dCOG ~qk
I
� �and dCOG
~wkI
� �be the
defuzzified values of ~hkI , ~qk
I and ~wkI respectively which are shown
in Table 7.Table 7 indicates that the ranking of DMUs on the basis of ~hk
I and
~qkI is possible in conventional way, as the values of dCOG
~hkI
� �and
dCOG ~qkI
� �lie between 0 and 1. But there are some DMUs whose
dCOG~wk
I
� �> 1 and thus the ranking of DMUs on the basis of ~wk
I is
still unattainable. Therefore, for doing exact ranking of the DMUson the basis of ~wk
I ; we propose a new method of ranking based
on dCOG~wk
I
� �.
The proposed method for ranking the DMUs on the basis of ~wkI –
The algorithm of the new ranking method is as given below:Let J = {1, 2, . . . ,n} be the index set where n be the number of
DMUs; and let i = 1 and RKi¼ 1 be the initialization.
Step 1. Evaluate ~wkI for each DMU.
Step 2. Find the defuzzified value dCOG~wk
I
� �of ~wk
I for each DMU.
Step 3. If the value of dCOG~wk
I
� �2 ð0;1�for each DMU, STOP the
algorithm and evaluate the ranks of the DMUs in the con-ventional way starting from RKi
. Otherwise go to step 4.
Step 4. Let Ki ¼ j 2 J : dCOG~wj
I
� �¼ 1
n obe the set of all those indi-
ces j’s of DMUj’s for which the defuzzified value of ~wjI is
equal to 1. If Ki – /, give the rank RKito each DMUj such
that j 2 Kiand repeat the algorithm with J = J � {J \ Ki},i = i + 1 and RKi
¼ RKiþ jKij. Otherwise STOP the algorithm
and give the ranks to the remaining DMUs in the conven-tional way starting from RKi
.
Remark 2. The DMUj is known as the fuzzy input mix-efficient ifj 2 K1. This implies that such a DMU has the most efficient combi-nation of the fuzzy inputs, even though it may be technically fuzzyinefficient.
Using the above algorithm the ranking of the DMUs on the basisof ~wk
I is shown in Table 8.Table 8 shows that the FIME determines the ranking of the
DMUs at an order of R(1) � R(2) > R(4) � R(11) >R(3) > R(7) > R(12) > R(9) > R(5) > R(8) > R(10) > R(6) where R(k) =rank of DMUk on the basis of ~wk
I : Here K1 = {1,2}, so DMU1 andDMU2 are fuzzy input mix-efficient and have the most efficientcombination of fuzzy inputs.
9. An application to banking sector
Consider a performance assessment problem of the State Bankof Patiala (SBOP) in India where the performance of SBOP is evalu-ated in seventeen districts of the Punjab State in terms of three in-puts and two outputs. The DMUs in our study are the districts ofthe Punjab state. The inputs used in our study are (i) Labour, (ii)Fixed assets and (iii) Total expenses. Labour is the number ofemployees working in the bank in different districts. Fixed assetsare the physical capital of the bank in different districts. Total ex-penses include interest and non-interest expenses. Out of these
Note: Input mix-inefficiency in kth DMU (In %age form) = dIECOG
~wkI
� �¼ 1� dCOG
~wkI
� �� �� 100.
1448 J. Puri, S.P. Yadav / Expert Systems with Applications 40 (2013) 1437–1450
three inputs, two inputs, namely, labour and total expenses are ta-ken as fuzzy inputs and are represented as TFNs. The outputs usedin our study are (i) Interest income and (ii) Other income. Interestincome is the income earned by the bank in different districts fromadvances and investments. Other income accounts for the incomefrom off-balance sheet items such as commission, exchange andbrokerage etc. The objective of our study is to measure the FIMEof SBOP in various districts for the period 2010–2011 and to knowwhich districts are the best performers in terms of FIME and whichone are the worst performers. In the period 2010–2011, the totalnumbers of districts in the Punjab state were twenty. We areexcluding three districts in our study because these were estab-lished in the same year 2010. The data is shown in Table 9.
To ensure the validity of the FDEA model specification, the cor-relation coefficients between the fuzzy inputs and fuzzy outputsare calculated. The expected interval values and the correspondingexpected values of the fuzzy correlation coefficients between thesets of fuzzy data (given in Table 9) are shown in the matrix formin Table 10.
Table 10 indicates that the left and right bounds of each ex-pected interval and the corresponding expected value are positive.It means that all the inter-correlations between the inputs and out-puts are positive. Thus, the inclusion of the input and output data isjustified and the FDEA model which is taken in our present study isconsistent.
The a-cuts ~hkI
� �a
and ~qkI
� �a of ~hk
I and ~qkI of the seventeen DMUs
are evaluated by using models 14a, 14b, 15a and 15b at differentvalues of a and are shown in Tables 11 and 12, respectively. Thegraphical representations of the fuzzy efficiencies ~hk
I and ~qkI for
the kth DMU which are obtained by using a-cuts ~hkI
� �a
and ~qkI
� �a
are shown in Figs. 2a,2b, 2c and 2d. It can be seen from the figuresthat the shape of the membership functions of ~hk
I and ~qkI are
approximated as triangular membership functions. Further, Table13 presents the values of ~hk
I , ~qkI and ~wk
I approximated as TFNs.Let dCOG
~hkI
� �; dCOG ~qk
I
� �and dCOG
~wkI
� �be the defuzzified values
of ~hkI , ~qk
I and ~wkI respectively which are obtained by using COG
method of defuzzification and are shown in Table 14. The ranking
of DMUs on the basis of dCOG~hk
I
� �and dCOG ~qk
I
� �is done in a conven-
tional way. However, the ranking of DMUs on the basis of dCOG~wk
I
� �is done by applying the proposed method of ranking given in Sec-tion 8. In this case the algorithm terminates at the Step 3 since the
value of dCOG~wk
I
� �2 ð0;1� for each DMU.
Table 14 indicates that DMU10, i.e., Ludhiana district is the mostefficient district in terms of ~hk
I , ~qkI and ~wk
I . The order of performanceas well as level of inefficiency (in brackets) of the districts in termsof ~wk
I is given by DMU10 > DMU5 > DMU2 > DMU4 > DMU12 >
J. Puri, S.P. Yadav / Expert Systems with Applications 40 (2013) 1437–1450 1449
I Þ ¼ 0:675266. This means thatPatiala district is not utilizing its inputs (fuzzy as well as non-fuzzy)efficiently. Table 14 also indicates that Bathinda district is inefficient
in terms of ~hkI and ~qk
I with dCOG~hk
I
� �and dCOG ~qk
I
� �equal to 0.555538
and 0.602858 respectively. However, Bathinda district is a good per-
former in terms of ~wkI with dCOG
~wkI
� �¼ 0:925713. Table 14 also re-
veals that the highest and lowest levels of fuzzy mix-inefficiencyhave been seen for Patiala (32.4734%) and Ferozepur (1.579%)respectively.
10. Conclusions
In view of the fact that precise input and output data are not al-ways available in real world applications, we have developed, inthis paper FIME model. For measuring the FIME, we have proposedthe FCCRI and FSBMI. These two FDEA models have been formu-lated as linear programming models using a-cut approach for easeof solution and implementation. To ensure the validity of the FDEAmodel specification, we have proposed a fuzzy correlation coeffi-cient method using expected value approach which calculatesthe expected interval and expected value of fuzzy correlation coef-ficient between fuzzy inputs and fuzzy outputs. If positive inter-correlations are found, the inclusion of the fuzzy inputs and fuzzyoutputs is justified. Further, a new ranking method based ondefuzzification approach has been developed for comparing andranking DMUs in terms of FIME, which provides not only a fullranking but also the information that to what degree a FIME is big-ger than another one. All of the proposed approaches have been ap-plied to evaluate the performances of seventeen districts of SBOPin the Punjab State of India in terms of FIME. It is shown thatLudhiana district is the most efficient district in terms of ~hk
I , ~qkI
and ~wkI . The highest and lowest levels of fuzzy mix-inefficiency
have been seen for Patiala (32.4734 %) and Ferozepur (1.579 %)respectively. The input mix-inefficiency represents the degree towhich the input mix should change to become fully efficient.According to the findings of our study, all the fuzzy input mix-inef-ficient districts are suggested to decrease their input mix in orderto become fully efficient.
Acknowledgments
The first author is thankful to the University Grants Commission(UGC), Government of India for financial assistance. The authorsare thankful to Ms. M. A. Malini, Manager (Systems), State Bankof Patiala, IT Services Deptt, H. O., Patiala, India for providing therelevant data of the bank.
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