Research ArticleProfit Malmquist Index and Its Global Form inthe Presence of the Negative Data in DEA
Ghasem Tohidi,1 Shabnam Razavyan,2 and Simin Tohidnia1
1 Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran2Department of Mathematics, Islamic Azad University, South Tehran Branch, Tehran, Iran
Correspondence should be addressed to Ghasem Tohidi; [email protected]
Received 8 January 2014; Accepted 7 August 2014; Published 31 August 2014
This paper first introduces the allocative and profit efficiency in the presence of the negative data and then presents a new circularindex to measure the productivity change of decision making units (DMUs) for the case that the dataset contains the inputs and/oroutputs with the negative values in data envelopment analysis (DEA). The proposed index is decomposed into four componentsin the two stages. The range directional model (RDM) and the proposed efficiencies are used to compute the proposed index andits components. The interpretations of the components are presented. Finally, a numerical example is organized to illustrate theproposed index and its components at three successive periods of time.
1. Introduction
The Malmquist productivity index which measures the pro-ductivity change over time was introduced by Caves et al.[1]. Fare et al. [2] developed the Malmquist productivityindex that was based on data envelopment analysis (DEA).The Malmquist index can be applied in many fields [3, 4].By using Malmquist index, the productivity growth can bedecomposed into the efficiency change and technical changecomponents.The second componentmeasures the shift in thetechnology frontier.
Pastor and Lovell [5] suggested the global Malmquistproductivity index that is circular and can be decomposedinto the circular components.The global index and the meta-Malmquist index developed in Portela and Thanassoulis [6].In addition,Hosseinzadeh Lotfi et al. developed the new ideasabout Malmquist index in [7, 8].
Maniadakis and Thanassoulis [9] assumed the input costvector is known and suggested the cost Malmquist index.Tohidi et al. [10] extended the cost Malmquist index intothe profit Malmquist index that is used when the input costand the output price vector are available. Then, Tohidi andRazavyan [11] proposed the global profit Malmquist index.The cost Malmquist index was also developed in Tohidi et
al. [12] and is called the global cost Malmquist productivityindex.
Sometimes in the process of production negative inputsand/or outputs may occur. For example, DMUsmay generateundesirable outputs. In such a case, negative values canbe considered for these undesirable outputs. Instances ofsystems with negative inputs and/or outputs are explainedin [13, 14]. The traditional DEA models can be applied tocompute the Malmquist index with the nonnegative data.They cannot deal with the negative data. Portela et al. [15]presented an approach named the range directional model(RDM) and solved this problem. They calculated the meta-Malmquist index using the RDM model in Portela andThanassoulis [16].
To investigate the productivity change of DMUs with thenegative data, this paper defines the profit and allocativeefficiency and introduces an index to measure the productiv-ity changes of DMUs when some inputs and/or outputs arenegative and the costs of inputs and the prices of outputs areavailable.Then, the proposed index is decomposed into somecomponents in the two stages. The range directional model(RDM) and the proposed efficiencies are used to compute theproposed index and its components.
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014, Article ID 276092, 8 pageshttp://dx.doi.org/10.1155/2014/276092
2 Journal of Applied Mathematics
The current paper unfolds as follows. Section 2 describesthe RDM. In Section 3 the global Malmquist productivityindex is expressed. Section 4 defines the profit and the alloca-tive efficiency for the DMUs with some negative inputs andoutputs and then suggests a new index. The decompositionand the theorem of circularity property of the proposed indexand its components are presented in Section 5. Section 6prepares some DEA models to compute the proposed indexand its components. Section 7 provides a numerical example.Section 8 concludes.
2. The Range Directional Model
Portela et al. [15] introduced the RDM to compare DMUsunder the negative data. Assume that, in time period 𝑡 (𝑡 =1, . . . , 𝑇) and 𝑗th unit (𝑗 = 1, . . . , 𝑛) consumes an inputvector 𝑥𝑡
𝑗= (𝑥𝑡
1𝑗, 𝑥𝑡
2𝑗, . . . , 𝑥
𝑡
𝑚𝑗) to produce an output vector
𝑦𝑡
𝑗= (𝑦𝑡
1𝑗, 𝑦𝑡
2𝑗, . . . , 𝑦
𝑡
𝑠𝑗). We consider an ideal point (IP) for the
time period 𝑡 with the minimum inputs and the maximumoutputs observed in this time period, where for each input𝑖 (𝑖 = 1, . . . , 𝑚) and IP
𝑖is as min
𝑗{𝑥𝑡
𝑖𝑗} and for each output
𝑟 (𝑟 = 1, . . . , 𝑠) throughout IP𝑟is as max
𝑗{𝑦𝑡
𝑟𝑗} and the
directional vector is as (𝑔𝑥, 𝑔𝑦) = (𝑔
𝑥𝑡
1
, . . . , 𝑔𝑥𝑡
𝑚
, 𝑔𝑦𝑡
1
, . . . , 𝑔𝑦𝑡
𝑠
).In the RDM this directional vector for DMU
𝑘in time period
𝑡 is as 𝑔𝑥𝑡
𝑖𝑘
= 𝑅𝑥𝑡
𝑖𝑘
= 𝑥𝑡
𝑖𝑘− min
𝑗{𝑥𝑡
𝑖𝑗}; 𝑖 = 1, . . . , 𝑚 and
𝑔𝑦𝑡
𝑟𝑘
= 𝑅𝑦𝑡
𝑟𝑘
= max𝑗{𝑦𝑡
𝑟𝑗} − 𝑦𝑡
𝑟𝑘; 𝑟 = 1, . . . , 𝑠 that reflects the
ranges of possible improvement for this DMU. The RDM forDMU
𝑘in time period 𝑡 for the case of the VRS technology is
as follows (Portela et al., 2004):
𝛽∗
𝑘= max 𝛽
𝑘
s.t.𝑛
∑
𝑗=1
𝜆𝑗𝑥𝑡
𝑖𝑗≤ 𝑥𝑡
𝑖𝑘− 𝛽𝑘𝑅𝑥𝑡
𝑖𝑘
, 𝑖 = 1, . . . , 𝑚,
𝑛
∑
𝑗=1
𝜆𝑗𝑦𝑡
𝑟𝑗≥ 𝑦𝑡
𝑟𝑘+ 𝛽𝑘𝑅𝑦𝑡
𝑟𝑘
, 𝑟 = 1, . . . , 𝑠,
𝑛
∑
𝑗=1
𝜆𝑗= 1,
𝜆𝑗≥ 0, 𝑗 = 1, . . . , 𝑛.
(1)
The value of 𝛽∗𝑘
in model (1) is an inefficiencymeasure and the RDM efficiency measure of unit 𝑘is RDM𝑡(𝑥𝑡
𝑘, 𝑦𝑡
𝑘, 𝑅𝑥𝑡
𝑘
, 𝑅𝑦𝑡
𝑘
) = 1 − 𝛽∗
𝑘that is defined as
(𝑥𝑡
𝑖𝑘
∗
−min𝑗{𝑥𝑡
𝑖𝑗})/(𝑥𝑡
𝑖𝑘−min
𝑗{𝑥𝑡
𝑖𝑗}) if at the optimal solution
to model (1) a binding constraint corresponds to input 𝑖 or(max𝑗{𝑦𝑡
𝑟𝑗} − 𝑦𝑡
𝑟𝑘
∗
)/(max𝑗{𝑦𝑡
𝑟𝑗} − 𝑦𝑡
𝑟𝑘) if a binding constraint
corresponds to output 𝑟. 𝑥𝑡𝑖𝑘
∗ and 𝑦𝑡𝑟𝑘
∗ are the target inputsand target outputs in time 𝑡, respectively. It is clear that theupper bound of 1 − 𝛽∗
𝑘is 1.
3. The Global Profit MalmquistProductivity Index
The global profit Malmquist index (PM𝐺) is applied tomeasure the productivity changes of DMUs when the inputcosts and output prices are available. Assume that, in timeperiod 𝑡, (𝑡 = 1, . . . , 𝑇), the input cost vector is 𝑐𝑡 ∈ 𝑅𝑚
+and
the output price vector is 𝑝𝑡 ∈ 𝑅𝑠
+; by using these vectors
the common input cost vector 𝑐𝐺 ∈ 𝑅𝑚
+and the common
output price vector 𝑝𝐺 ∈ 𝑅𝑠
+can be defined, respectively,
as 𝑐𝐺 = ∑𝑇
𝑗=1𝜆𝑗𝑐𝑗, ∑𝑇𝑗=1𝜆𝑗= 1, 𝜆
𝑗≥ 0, and 𝑝𝐺 =
∑𝑇
𝑗=1𝜇𝑗𝑝𝑗, ∑𝑇
𝑗=1𝜇𝑗= 1, 𝜇
𝑗≥ 0, where the weights 𝜆
𝑗and
𝜇𝑗(𝑗 = 1, . . . , 𝑇) are the decision-makers’ preferences over 𝑐𝑗
and 𝑝𝑗 (𝑗 = 1, . . . , 𝑇), respectively [12].This paper uses the definition of profit [13] to compute
the global profit and the observed profit of DMU𝑘with
the negative data. The observed profit of DMU𝑗under the
cost and price vectors of time period 𝑡 can be calculatedas OP𝑡(𝑥𝑡
𝑗, 𝑦𝑡
𝑗) = 𝑝
𝑡𝑦𝑡
𝑗− 𝑐𝑡𝑥𝑡
𝑗and the maximum profit of
time 𝑡 under the price vector of this time is defined asMP𝑡(𝑥, 𝑦, 𝑐𝑡, 𝑝𝑡) = max{𝑝𝑡𝑦−𝑐𝑡𝑥 : (𝑥, 𝑦) ∈ 𝑇𝑡, 𝑐𝑡 > 0, 𝑝𝑡 > 0}.The set of activities (𝑥, 𝑦) ∈ 𝑇
𝑡 which corresponds to thescalar MP𝑡(𝑥, 𝑦, 𝑐𝑡, 𝑝𝑡) defines the profit boundary of time 𝑡as MP𝑡 = {(𝑥, 𝑦) : 𝑝𝑡𝑦 − 𝑐𝑡𝑥 = MP𝑡(𝑥, 𝑦, 𝑐𝑡, 𝑝𝑡)}. In this casethe PM𝐺 index for the DMUs with the nonnegative data isdefined as follows:
production technology [5]. The term OP𝐺(𝑥𝑡, 𝑦𝑡) = 𝑝𝐺𝑦𝑡 −𝑐𝐺𝑥𝑡 is the observed profit of (𝑥𝑡, 𝑦𝑡) under the common price
vectors 𝑐𝐺 and 𝑝𝐺. The ratio MP𝐺(𝑥𝑡, 𝑦𝑡, 𝑐𝐺, 𝑝𝐺)/OP𝐺(𝑥𝑡, 𝑦𝑡)in the denominator of PM𝐺 is the reciprocal to measureof the profit efficiency introduced in Cooper et al. [13] for(𝑥𝑡, 𝑦𝑡) under the common price vectors.This ratio measures
the distance between the observed profit OP𝐺(𝑥𝑡, 𝑦𝑡) andthe common profit boundary and will have a minimumvalue of 1. A value greater than 1 of PM𝐺 index indicatesthe productivity regress and a value less than 1 implies theproductivity progress between 𝑡 and 𝑡 + 1. A value of 1indicates that the productivity remains unchanged. Becauseof the limitations of the DEAmodels with the negative inputsand outputs [17, 18], we cannot investigate the productivitychange of DMUs with the negative data by using traditionMalmquist indices. To this end, the next section proposes anappropriate global profit Malmquist index when some inputsor/and outputs are negative.
Journal of Applied Mathematics 3
4. The Global Profit Malmquist Index withthe Negative Data
In this section we assume that there are some negativeinputs and/or negative outputs and evaluate the productivitychanges of DMUs by using the PM𝐺 index under the VRStechnology. We denote the profit of the ideal point of timeperiod 𝑡 by IP𝑡 that the superscript 𝑡 on IP𝑡 indicates the profitis computed under the cost and the price vectors of time 𝑡. Byusing the definition of the profit, we will have
OP𝑡 (𝑥𝑡, 𝑦𝑡) ≤ MP𝑡 (𝑥𝑡, 𝑦𝑡, 𝑐𝑡, 𝑝𝑡) ≤ IP𝑡. (3)
Now we define the profit and allocative efficiency of aDMU with the negative data.
Definition 1. The measure of the profit efficiency for (𝑥𝑡, 𝑦𝑡)under the price vectors of time 𝑡 is as
PE𝑡 (𝑦𝑡, 𝑥𝑡, 𝑐𝐺, 𝑝𝐺) =MP𝑡 (𝑥𝑡, 𝑦𝑡, 𝑐𝑡, 𝑝𝑡) − IP𝑡
OP𝑡 (𝑥𝑡, 𝑦𝑡) − IP𝑡. (4)
By using (3) it is clear that the value of PE𝑡(𝑦𝑡, 𝑥𝑡,𝑐𝐺, 𝑝𝐺) is equal to or less than one. As the profit efficiency
is less than 1 it may be because production takes place at thewrong input and/or output mix in light of the input costs andthe output prices; this is captured by using the measure of theallocative efficiency which is defined as follows.
Definition 2. Let (𝑥𝑡∗, 𝑦𝑡∗) be the target DMU of (𝑥𝑡, 𝑦𝑡) intime period 𝑡 using RDM.The allocative efficiency of (𝑥𝑡, 𝑦𝑡)can be defined as follows:
AE𝑡 (𝑦𝑡, 𝑥𝑡, 𝑤𝑡)
=MP𝑡 (𝑥𝑡, 𝑦𝑡, 𝑐𝑡, 𝑝𝑡) − IP𝑡
OP𝑡 (𝑥𝑡∗, 𝑦𝑡∗) − IP𝑡
=MP𝑡 (𝑥𝑡, 𝑦𝑡, 𝑐𝑡, 𝑝𝑡) − IP𝑡
(OP𝑡 (𝑥𝑡, 𝑦𝑡) − IP𝑡)RDM𝑡 (𝑥𝑡𝑘, 𝑦𝑡
𝑘, 𝑅𝑥𝑡
𝑘
, 𝑅𝑦𝑡
𝑘
)
=
PE𝑡 (𝑦𝑡, 𝑥𝑡, 𝑐𝐺, 𝑝𝐺)
RDM𝑡 (𝑥𝑡𝑘, 𝑦𝑡
𝑘, 𝑅𝑥𝑡
𝑘
, 𝑅𝑦𝑡
𝑘
)
.
(5)
If profit efficiency is less than 1 and it is because produc-tion is based on excessive input or shortage output usage, wecan capture it by using the RDM efficiency measure that wasdenoted by RDM𝑡(𝑥𝑡
𝑘, 𝑦𝑡
𝑘, 𝑅𝑥𝑡
𝑘
, 𝑅𝑦𝑡
𝑘
).To define the PM𝐺 index under theVRS technology in the
presence of the negative datawe first consider a global idealpoint (GIP) defined over the global technology that, for input𝑖 (𝑖 = 1, . . . , 𝑚), and output 𝑟 (𝑟 = 1, . . . , 𝑠), is as GIP
𝑖=
min𝑡{min𝑗{𝑥𝑡
𝑖𝑗}} and GIP
𝑟= max
𝑡{max𝑗{𝑦𝑡
𝑟𝑗}}, respectively.
This ideal point is used for computing the range 𝑅GF=
(𝑅GF𝑥𝑡 , 𝑅
GF𝑦𝑡 ) for DMU
𝑘observed in time 𝑡. The 𝑖th component
of 𝑅GF𝑥𝑡 is 𝑅GF
𝑥𝑡
𝑖𝑘
= 𝑥𝑡
𝑖𝑘− min
𝑡{min𝑗{𝑥𝑡
𝑖𝑗}}, 𝑖 = 1, . . . , 𝑚, and
𝑟th component of 𝑅GF𝑦𝑡 is 𝑅GF
𝑦𝑡
𝑟𝑘
= max𝑡{max𝑗{𝑦𝑡
𝑟𝑗}} − 𝑦
𝑡
𝑟𝑗, 𝑟 =
1, . . . , 𝑠. Now we define the PM𝐺 index in the presence of thenegative data as follows:
PM𝐺 = ((MP𝐺 (𝑥𝑡+1, 𝑦𝑡+1, 𝑐𝐺, 𝑝𝐺) − GIP𝐺)
(OP𝐺 (𝑥𝑡+1, 𝑦𝑡+1) − GIP𝐺))
× (
(MP𝐺 (𝑥𝑡, 𝑦𝑡, 𝑐𝐺, 𝑝𝐺) − GIP𝐺)
(OP𝐺 (𝑥𝑡, 𝑦𝑡) − GIP𝐺))
−1
,
(6)
where GIP𝐺 is the profit of the global ideal point andthe superscript 𝐺 indicates that the profit of this point iscomputed under the common price vectors 𝑐𝐺 and 𝑝𝐺. Theratio (MP𝐺(𝑥𝑡, 𝑦𝑡, 𝑐𝐺, 𝑝𝐺)−GIP𝐺)/(OP𝐺(𝑥𝑡, 𝑦𝑡)−GIP𝐺) is theprofit efficiency defined in Definition 1 under the commoncost and price vectors. When the PM𝐺 index has the valuegreater than 1, it means that the productivity of unit 𝑘 hasimproved from the time 𝑡 to 𝑡 + 1. The productivity hasdeclined when the value of the PM𝐺 index is below 1 andremains unchanged if PM𝐺 = 1. The PM𝐺 index is circularand it can be decomposed into four circular components asshown in the next section.
5. Decomposition of the PM𝐺 Index
In the first stage the PM𝐺 index is decomposed into two PEC𝐺
The termoutside the brackets in the right-hand side of (7)represents the profit efficiency change (PEC𝐺) component ofthe unit under evaluation from 𝑡 to 𝑡 + 1 and the term insidethe brackets provides the profit frontier shift (the technicalchange) between the periods 𝑡 and 𝑡 + 1 under the VRSproduction technologies oftwo times t and 𝑡 + 1 (PTC𝐺).
4 Journal of Applied Mathematics
The PEC𝐺 and PTC𝐺 components of the PM𝐺 index canthemselves be decomposed.
TheDecomposition of the𝑃𝐸𝐶𝐺.ThePEC𝐺 component can bedecomposed into two components as follows:
whereGIP𝑡 andGIP𝑡+1 are the profit of the global point underthe costs and prices of the periods 𝑡 and 𝑡 + 1, respectively.The first ratio in the right-hand side of (8) is the allocativeefficiency change (AEC𝐺) components of the PM𝐺 index; andthe second ratio is the RDMwithin-period-efficiency change
(REC𝐺) of the unit that is under evaluation from 𝑡 to 𝑡 + 1;the RDMwithin-period-efficiency change was introduced byPortela et al. [15].
The Decomposition of 𝑃𝑇𝐶𝐺. We can decompose the PTC𝐺into two components as follows:
The first term in the above decomposition measures thefrontier shift between the VRS frontiers of times 𝑡 and 𝑡 + 1and the technical change (TC𝐺), along the ray (𝑥𝑡+1
𝑗, 𝑦𝑡+1
𝑗)
[16]. The second term is a residual price effect (PE𝐺) part.The numerical values of the components of the PM𝐺 indexobtained in this section are interpreted in the similar manneras the index itself; a value below 1 indicates regress, greaterthan 1 indicates progress, and 1 indicates that performancestayed constant.
5.1. The Circularity Property of 𝑃𝑀𝐺 and All of Its Compo-nents. Circularity is a prominent property of the PM𝐺 indexand all of its components in the presence of the negative data.To show this property, the following theorem is stated.
Theorem3. For everyDMUj (𝑗 = 1, . . . , 𝑛), in three successiveperiods,
(1) 𝑃𝑀𝐺𝑡,𝑡+2
= 𝑃𝑀𝐺
𝑡,𝑡+1×𝑃𝑀𝐺
𝑡+1,𝑡+2(the circularity of PM𝐺),
(2) 𝑃𝐸𝐶𝐺𝑡,𝑡+2
= 𝑃𝐸𝐶𝐺
𝑡,𝑡+1× 𝑃𝐸𝐶
𝐺
𝑡+1,𝑡+2(the circularity of
PEC𝐺),(3) 𝑃𝑇𝐶𝐺
𝑡,𝑡+2= 𝑃𝑇𝐶
𝐺
𝑡,𝑡+1× 𝑃𝑇𝐶
𝐺
𝑡+1,𝑡+2(the circularity of
PTC𝐺),(4) 𝑅𝐸𝐶𝐺
𝑡,𝑡+2= 𝑅𝐸𝐶
𝐺
𝑡,𝑡+1× 𝑅𝐸𝐶
𝐺
𝑡+1,𝑡+2(the circularity of
REC𝐺),(5) 𝐴𝐸𝐶𝐺
𝑡,𝑡+2= 𝐴𝐸𝐶
𝐺
𝑡,𝑡+1× 𝐴𝐸𝐶
𝐺
𝑡+1,𝑡+2(the circularity of
AEC𝐺),(6) 𝑇𝐶𝐺
𝑡,𝑡+2= 𝑇𝐶𝐺
𝑡,𝑡+1× 𝑇𝐶𝐺
𝑡+1,𝑡+2(the circularity of TC𝐺),
(7) 𝑃𝐸𝐺𝑡,𝑡+2
= 𝑃𝐸𝐺
𝑡,𝑡+1× 𝑃𝐸𝐺
𝑡+1,𝑡+2(the circularity of PE𝐺),
Journal of Applied Mathematics 5
where, for instance, 𝑃𝑀𝐺𝑝,𝑞
(𝑝 = 𝑡, 𝑡 + 1, 𝑞 = 𝑡 + 1, 𝑡 + 2, 𝑝 = 𝑞)is the profit Malmquist changes between periods 𝑝 and 𝑞 forDMU
𝑗.
Proof. For instance, let PM𝐺𝑝,𝑞
be the profit Malmquistchanges between periods 𝑝 and 𝑞 for DMU
𝑗. Therefore,
PM𝐺𝑡,𝑡+1
× PM𝐺𝑡+1,𝑡+2
= ((
MP𝐺 (𝑥𝑡+1, 𝑦𝑡+1, 𝑐𝐺, 𝑝𝐺) − GIP𝐺
OP𝐺 (𝑥𝑡+1, 𝑦𝑡+1) − GIP𝐺)
×(
MP𝐺 (𝑥𝑡, 𝑦𝑡, 𝑐𝐺, 𝑝𝐺) − GIP𝐺
OP𝐺 (𝑥𝑡, 𝑦𝑡) − GIP𝐺)
−1
)
×((
MP𝐺 (𝑥𝑡+2, 𝑦𝑡+2, 𝑐𝐺, 𝑝𝐺) − GIP𝐺
OP𝐺 (𝑥𝑡+2, 𝑦𝑡+2) − GIP𝐺)
×(
MP𝐺 (𝑥𝑡+1, 𝑦𝑡+1, 𝑐𝐺, 𝑝𝐺) − GIP𝐺
OP𝐺 (𝑥𝑡+1, 𝑦𝑡+1) − GIP𝐺)
−1
)
= (
MP𝐺 (𝑥𝑡+2, 𝑦𝑡+2, 𝑐𝐺, 𝑝𝐺) − GIP𝐺
OP𝐺 (𝑥𝑡+2, 𝑦𝑡+2) − GIP𝐺)
× (
MP𝐺 (𝑥𝑡, 𝑦𝑡, 𝑐𝐺, 𝑝𝐺) − GIP𝐺
OP𝐺 (𝑥𝑡, 𝑦𝑡) − GIP𝐺)
−1
= MP𝐺𝑡,𝑡+2
.
(10)
In other words, the global profit Malmquist change fromperiod 𝑡 to 𝑡 + 2 is the product of the successive global profitMalmquist change from period 𝑡 to 𝑡 + 1 and from the period𝑡+1 to 𝑡+2. Similarly, we can show that all of the componentsare circular.
The above properties show that in the presence of the neg-ative data the PM𝐺 index and its components are appropriateindices in the fact that they link in clear way productivitychange indices over successive time periods.
6. The Computation of the Proposed Indexand Its Components
This section prepares someDEAmodels to compute the PM𝐺
index and its components. The observed profit OP𝐺(𝑥𝑡, 𝑦𝑡)is computed as ∑𝑠
, respectively. Similarly,we can compute the value of RDM𝑡+1(𝑥𝑡+1
𝑘, 𝑦𝑡+1
𝑘, 𝑅
GF𝑥𝑡+1
𝑘
, 𝑅GF𝑦𝑡+1
𝑘
)
after replacing the time period 𝑡with 𝑡+1. In order to computeRDM𝐺(𝑥𝑡
𝑘, 𝑦𝑡
𝑘, 𝑅
GF𝑥𝑡
𝑘
, 𝑅GF𝑦𝑡
𝑘
) = 1 − 𝛽∗
𝑘, we first extend model
(1) for the case that the production technology formed fromall DMUs observed in all time periods (𝑇𝐺). The proposedmodel is shown in (13). In fact, the optimal value of model(13),𝛽∗
𝑘, is an inefficiency of DMU
𝑘along direction (𝑔
𝑥, 𝑔𝑦) =
(𝑅GF𝑥𝑡
𝑘
, 𝑅GF𝑦𝑡
𝑘
) under the global production technology 𝑇𝐺:
𝛽∗
𝑘= max 𝛽
𝑘
s.t.𝑇
∑
𝑡=1
𝑛
∑
𝑗=1
𝜆𝑡
𝑗𝑥𝑡
𝑖𝑗≤ 𝑥𝑡
𝑖𝑘− 𝛽𝑘𝑅GF𝑥𝑡
𝑖𝑘
, 𝑖 = 1, . . . , 𝑚,
𝑇
∑
𝑡=1
𝑛
∑
𝑗=1
𝜆𝑡
𝑗𝑦𝑡
𝑟𝑗≥ 𝑦𝑡
𝑟𝑘+ 𝛽𝑘𝑅GF𝑦𝑡
𝑟𝑘
, 𝑟 = 1, . . . , 𝑠,
𝑇
∑
𝑡=1
𝑛
∑
𝑗=1
𝜆𝑡
𝑗= 1,
𝜆𝑡
𝑗≥ 0, 𝑗 = 1, . . . , 𝑛, 𝑡 = 1, . . . 𝑇.
(13)
RDM𝐺(𝑥𝑡+1𝑘, 𝑦𝑡+1
𝑘, 𝑅
GF𝑥𝑡+1
𝑘
, 𝑅GF𝑦𝑡+1
𝑘
) iscomputed using model(13) after replacing the time period 𝑡 with 𝑡 + 1.
By replacing the values obtained from the above modelsin the formulations of PM𝐺 index and its components, thevalues of them can be computed. A value greater than 1 ofPM𝐺 index and its components indicates the regress and avalue less than 1 implies the progress between periods 𝑡 and𝑡 + 1. A value equal to 1 indicates that there is not any changebetween two time periods.
7. Numerical Example
This section illustrates the property of PM𝐺 and its compo-nents, for example, the circularity property, using a numericalexample. Table 1 shows units DMU1–DMU4 with one input(I) and two outputs (O1 and O2) and their cost (𝑐𝑡) and prices(𝑝𝑡1and 𝑝𝑡
2) for 3 successive periods.
To compute the PM𝐺 indices and their components byusing the data in Table 1, suppose that the preferences aboutthe input costs and output prices are available and have beenspecified by decision-makers as 𝜆
𝑗= 𝜇𝑗= (1/3) (𝑗 = 1, 2, 3).
Therefore, the common costs and prices are obtained by usingthe decision-makers’ preferences:
𝑐𝐺=1
3𝑐1+1
3𝑐2+1
3𝑐3= 2.4333,
𝑝𝐺=1
3𝑝1+1
3𝑝2+1
3𝑝3= (1.6667, 3.4) .
(14)
By using the data of periods 1, 2, and 3, the inputs andthe outputs of the global ideal point will be 2, −1, and 7,respectively.
Table 2 shows PM𝐺𝑡,𝑠, PEC𝐺𝑡,𝑠, andTEC𝐺
𝑡,𝑠of all of theDMUs
for 𝑡 = 1, 2, 𝑠 = 2, 3, and 𝑠 = 𝑡. For instance, the columns 2, 3,and 4 in Table 2 show PM𝐺
1,2, PM𝐺2,3, and PM𝐺
1,3, respectively.
For DMU2, as an example, we have PMG1,2= 1.3049 > 1.
That is, the productivity of DMU2 in period 2 is more than itsproductivity in period 1 and hence the productivity has beenimproved from the time period 1 to 2.
According to PM𝐺2,3= 0.46951 < 1, the productivity of
DMU2 in period 3 is less than its productivity in period 2. Bythe circularity property of the profit Malmquist change, wecan conclude that PM𝐺
1,3= PM𝐺
1,2×PM𝐺2,3= 1.3049×0.4695 =
0.6127.Looking at the average values in Table 2, for example, we
can see that the average global profit Malmquist index ofall of the DMUs from the first period to the second period(1.0298) is higher than their average global profit Malmquist
Journal of Applied Mathematics 7
Table 3: The results of the RDM within-period-efficiency and allocative efficiency change.
index from the second period to the third period (1.0037).The average of columns 2, 3, and 4 in Table 2 shows that theaverage growth of all of the DMUs from the first period to thethird period is less than their growth from period 1 to period2 and that from period 2 to period 3. A similar discussion canbe made for the other columns of Table 2.
Tables 3 and 4 illustrate the results from the RDMwithin-period-efficiency change, allocative efficiency change, thefrontier shift between the VRS frontiers, and residual pricechange from different periods to the periods 2 and 3, and theaverages are shown in the last row of these tables.
For example, the frontier shift between the VRS frontiersalong the ray (𝑥2
1, 𝑦2
1) from period 1 to period 2 is 0.6064; that
is, there is a decrease in technical progress for DMU1 fromperiod 1 to period 2.
Using the average values in Table 4, for example, we cansee that the average frontier shifts between the VRS frontiersalong the rays (𝑥2
1, 𝑦2
1), (𝑥22, 𝑦2
2), (𝑥23, 𝑦2
3), and (𝑥2
4, 𝑦2
4) of all of
the DMUs from the first period to the second period (0.862)are less than their average frontier shifts between the VRSfrontiers along the rays (𝑥3
1, 𝑦3
1), (𝑥32, 𝑦3
2), (𝑥33, 𝑦3
3), and (𝑥3
4, 𝑦3
4)
from the second period to the third period (1.1366).The valuePE𝐺1,2= 0.8749 < 1, for DMU2 as an example, for the PE𝐺
component, indicates that the effect of input cost changesfrom the first period to the second period is detrimental toproductivity.
8. Conclusions
Under the VRS assumption allocative and profit efficiencieshave been defined for the case that the dataset containsthe inputs and/or outputs with the negative values in dataenvelopment analysis. The definitions could be generalizedwhen the unit cost of the ith input (𝑖 = 1, . . . , 𝑚) orthe unit price of the 𝑟th output (𝑟 = 1, . . . , 𝑠) of all ofDMUs is different. The proposed PM𝐺 index decomposed
into some components in two stages. The RDM and thedefined efficiencies were used to compute the proposed indexand its components. The interpretations of the componentswere presented. A numerical example has been presented forthree successive periods of time to illustrate the propertiesof PM𝐺 and its components in the presence of the negativeinputs and outputs.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgment
Financial support of this research, in a project entitled AnOverall Index to Determine the Progress or Regress of theProductivity of DMUs and Its Global Form in the Presence ofthe Negative Data, Islamic Azad University, Central TehranBranch, is acknowledged.
References
[1] D. W. Caves, L. R. Christensen, and W. E. Diewert, “Theeconomic theory of index numbers and the measurement ofinput, output and productivity,” Econometrica, vol. 50, pp. 1393–1414, 1982.
[2] R. Fare, S. Grosskopf, M. Norris, and Z. Zhang, “Productivitygrowth, technical progress, and efficiency change in industrial-ized countries,” American Economic Review, vol. 84, no. 1, pp.66–83, 1994.
[3] R. Fare, S. Grosskopf, B. Lindgren, and P. Roos, “Productivitydevelopments in Swedish hospitals: a Malmquist output indexapproach,” in Data Envelopment Analysis: Theory, Methodologyand Applications, A. Charnes, W. W. Cooper, A. Lewin, and L.Seiford, Eds., pp. 253–272, Kluwer Academic Publishers, 1994.
8 Journal of Applied Mathematics
[4] A. Q. Nyrud and S. Baardsen, “Production efficiency andproductivity growth in Norwegian sawmilling,” Forest Science,vol. 49, no. 1, pp. 89–97, 2003.
[5] J. T. Pastor andC.A.K. Lovell, “A globalMalmquist productivityindex,” Economics Letters, vol. 88, no. 2, pp. 266–271, 2005.
[6] M. Portela and E. Thanassoulis, “A circular Malmquist-typeindex for measuring productivity,” AstonWorking Paper RP08-02, Aston University, Birmingham, UK, 2008.
[7] F. Hosseinzadeh Lotfi, G. Jahanshahloo, M. Vaez-Ghasemi,and Z. Moghaddas, “Modified Malmquist productivity indexbased on present time value of money,” Journal of AppliedMathematics, vol. 2013, Article ID 607190, 8 pages, 2013.
[8] F. H. Lotfi, G. R. Jahanshahloo, M. Vaez-Ghasemi, and Z.Moghaddas, “Evaluation progress and regress of balancedscorecards by multi-stage malmquist productivity index,” Jour-nal of Industrial and Production Engineering, vol. 30, no. 5, 2013.
[9] N. Maniadakis and E.Thanassoulis, “A cost Malmquist produc-tivity index,” European Journal of Operational Research, vol. 154,no. 2, pp. 396–409, 2004.
[10] G. Tohidi, S. Razavyan, and S. Tohidnia, “A profit Malmquistproductivity index,” Journal of Industrial Engineering Interna-tional, vol. 6, pp. 23–30, 2008.
[11] G. Tohidi and S. Razavyan, “A circular global profit Malmquistproductivity index in data envelopment analysis,”AppliedMath-ematical Modelling, vol. 37, no. 1-2, pp. 216–227, 2013.
[12] G. Tohidi, S. Razavyan, and S. Tohidnia, “A global costMalmquist productivity index using data envelopment analy-sis,” Journal of the Operational Research Society, vol. 63, no. 1,pp. 72–78, 2012.
[13] W. W. Cooper, L. M. Seiford, and K. Tone, Data EnvelopmentAnalysis: A Comprehensive Text with Models, Applications,References and DEA-Solver Software, Springer, New York, NY,USA, 2nd edition, 2007.
[14] J. T. Pastor and J. L. Ruiz, “Variables with negative values inDEA,” inModeling Data Irregularities and Structural Complexi-ties in Data Envelopment Analysis, J. Zhu andW. D. Cook, Eds.,Springer, Berlin, Germany, 2007.
[15] M. C. A. S. Portela, E. Thanassoulis, and G. Simpson, “Negativedata in DEA: a directional distance approach applied to bankbranches,” Journal of the Operational Research Society, vol. 55,no. 10, pp. 1111–1121, 2004.
[16] M. C. A. S. Portela and E. Thanassoulis, “Malmquist-typeindices in the presence of negative data: an application to bankbranches,” Journal of Banking and Finance, vol. 34, no. 7, pp.1472–1483, 2010.
[17] A. Emrouznejad, A. L. Anouze, and E. Thanassoulis, “A semi-oriented radial measure formeasuring the efficiency of decisionmaking units with negative data, using DEA,” European Journalof Operational Research, vol. 200, no. 1, pp. 297–304, 2010.
[18] J. A. Sharp, W. Meng, and W. Liu, “A modified slacks-basedmeasure model for data envelopment analysis with “natural”negative outputs and inputs,” Journal of theOperational ResearchSociety, vol. 58, no. 12, pp. 1672–1677, 2007.
