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An output oriented super-efficiency measure in stochastic data envelopment analysis: Considering Iranian electricity distribution companies
Computers & Industrial Engineering 58 (2010) 663–671
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An output oriented super-efficiency measure in stochastic data envelopment analysis: Considering Iranian electricity distribution companies q Mohammad Khodabakhshi * Department of Mathematics, Faculty of Science, Lorestan University, Khorram Abad, Iran
a r t i c l e
i n f o
Article history: Received 16 March 2009 Received in revised form 13 January 2010 Accepted 14 January 2010 Available online 20 January 2010 Keywords: Chance constrained programming Electricity distribution units Super-efficiency Sensitivity analysis
a b s t r a c t In this paper, the concept of chance constrained programming approaches is used to develop output oriented super-efficiency model in stochastic data envelopment analysis. Output oriented super-efficiency model is one of the classic models in data envelopment analysis widely used by DEA people and practitioners. However, in many real applications, data is often imprecise. A successful method to address uncertainty in data is replacing deterministic data by random variables, leading to stochastic DEA. Therefore, in this paper, output oriented super-efficiency model is developed in stochastic data envelopment analysis, and its deterministic equivalent which is a nonlinear program is derived. Moreover, it is shown that the deterministic equivalent of the stochastic super-efficiency model can be converted to a quadratic program. Furthermore, sensitivity analysis of the proposed super-efficiency model is also discussed with respect to changes on parameter variables. Finally, data related to seventeen Iranian electricity distribution companies is used to illustrate the methods developed in this article. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Data envelopment analysis (DEA) was initiated in 1978 by Charnes, Cooper, and Rhodes (1978) and the first DEA model was called CCR model. Later, Banker, Charnes, and Cooper (1984) extended DEA to obtain a variable returns to scale version of the CCR model called BCC model. Since 1978 there has been a surge of research on DEA and many further models were introduced in the literature. Cooper, Seiford, and Tone (2000) reviewed most of these developments and extensions in DEA up to 2000. The objective of these models, DEA models, is evaluating overall efficiencies of decision making units (DMUs) that are responsible to convert a set of inputs into a set of outputs. The results of DEA models divide the DMUs into two sets, efficient and inefficient DMUs. While decision makers are interested in a complete ranking of all DMUs, efficient units obtained by original models cannot be differentiated among themselves. In other words, original models such as CCR and BCC models cannot rank efficient units. Therefore, in the last two decades, some DEA researchers initiated a new area called super-efficiency to rank the DEA efficient DMUs. The research in this area was first developed by Andersen and Petersen (1993). Mehrabian, Alirezaee, and Jahanshahloo (1999), also, developed a super-efficiency model to rank efficient units obtained by their own model. Martic and Savic (2001) applied super-efficiency model developed by Andersen and Petersen (1993) to rank regions in Serbia with regards to q
This manuscript was processed by Area Editor Imed Kacem. * Tel.: +98 912 6278846; fax: +98 661 2201333. E-mail addresses: [email protected], [email protected].
0360-8352/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2010.01.009
social-economic development. Moreover, Tone (2002) developed a super-efficiency measure based on the slack based measure introduced in the literature. The research on super-efficiency has widely been cited and became very popular in DEA. Ranking papers using super-efficiency models up to 2002 have been reviewed in Adler, Friedman, and Sinuany-Stern (2002). Furthermore, Khodabakhshi (2007) developed a super-efficiency measure based on, improved outputs, input relaxation model introduced in Jahanshahloo and Khodabakhshi (2004). Li, Jahanshahloo, and Khodabakhshi (2007), also, provided a super-efficiency model to rank efficient units obtained by the CCR model. Several papers have been published on DEA in Journal Computers & Industrial Engineering. In what follows, some of these applications are reported. Emrouznejad and Shale (2009) proposed a neural network background data envelopment analysis to evaluate the efficiency of large scale datasets. In their own approach requirement for computer memory and CPU time are far less than that needed by conventional DEA methods. Liu and Chen (2009), for the purpose of identifying bad performers such as bankrupt firms in the worst-case scenario proposed worst-practice frontier DEA model. While the original DEA model construct an efficient best-practice frontiers, the proposed model by Liu and Chen (2009) construct a worstpractice frontier. Liu (2008), also, developed a fuzzy DEA/AR method that is able to evaluate the performance flexible Manufacturing System (FMS) alternatives when the input -output are represented as crisp and fuzzy data. The lower and upper bound of alternatives is calculated by a pair of two level mathematical programs which can be transformed into a pair of conventional one-level DEA/Ar method.
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Baker and Talluri (1997) proposed an alternative methodology for technology selection using DEA. They addressed some of shortcomings in the methodology suggested by Khouja (1995), and provided a more robust analysis based on cross efficiencies in DEA. They, also, discussed applicability of the methodology from buyer and manufacturer view points. It is noticeable that Khouja (1995) proposed a decision model for technology selection problems using a two-phase procedure. In first phase, DEA is used to identify technologies that provide the best combinations of vender specifications on the performance parameters of the technologies. In the second phase, a multi attribute decision making model is used to select a technology from these identification the first phase. Papers mentioned above are quite interesting, but none of them considers stochastic variations in input–output data, or imprecise data, to rank efficient units. However, this study extends output oriented super-efficiency in stochastic DEA to consider stochastic variations in input–output data. Note that we know data envelopment analysis methodology has many advantages, such as no requirement for a priori weights or explicit specification of functional relations among the multiple inputs and outputs. However, there is a weakness in conventional DEA models, in fact, original DEA models do not allow stochastic variations in input and output such as data entry errors. As a result, DEA efficiency measurement may be sensitive to such variations. A DMU which is measured as efficient relative to other DMUs, may turn inefficient if such random variations are considered. Stochastic input and output variations into DEA have been studied by, for example, Cooper, Deng, Huang, and Li (2002, 2004), Land, Lovell, and Thore (1988), and Olesen and Petersen (1995), Morita and Seiford (1999), Khodabakhshi and Asgharian (2009), Khodabakhshi (2009, 2010). Although original DEA models such as CCR or BCC models have been extended in stochastic data envelopment analysis, the research on output oriented super-efficiency models solely have been done in deterministic DEA. To close this gap, in this paper, output oriented version of the popular super-efficiency measure introduced in Andersen and Petersen (1993) is extended in stochastic data envelopment analysis which allows stochastic variations in input–output data. The concept of chance constrained programming approach introduced by Cooper et al. (2004) and Khodabakhshi and Asgharian (2009), Khodabakhshi (2009, 2010) is used to develop output oriented stochastic super-efficiency model. Moreover, deterministic equivalent of the stochastic super-efficiency model is obtained to solve the stochastic model. Sensitivity of the stochastic model and its deterministic equivalent with respect to changes in the parameter values is done, too. Furthermore, as an empirical example, the proposed approach is applied on data of Iranian electricity distribution units. The rest of the paper is organized as follows: The output oriented version of the super-efficiency model introduced in Andersen and Petersen (1993) is described in Section 2. In Section 3, stochastic version of the output oriented super-efficiency model is developed, and its deterministic equivalent is also obtained. In addition, it is shown that the deterministic equivalent of the stochastic output oriented super-efficiency model can be converted to a quadratic program. Section 4 discusses sensitivity analysis. An illustrative example is used to describe some theoretical results, too. As an empirical example, the model is applied to data of Iranian electricity distribution companies in Section 5. Section 6 concludes the paper. 2. Preliminaries Suppose that all input and output are non-negative deterministic elements. Let DMUj ; ðj ¼ 1; 2; ; nÞ be n decision making units (DMU) that convert m inputs xij ði ¼ 1; . . . ; mÞ into s outputs yrj ðr ¼ 1; . . . ; sÞ. One of the basic models used to evaluate DMUs
efficiency is output oriented model introduced in Banker et al. (1984) called output oriented BCC (Banker, Charnes, Cooper) model. This model which evaluates DMUo is as follows:
Maximize
m X
/o þ e
si
subject to xio ¼
!
sþr
r¼1
i¼1 n X
þ
s X
kj xij þ si ;
i ¼ 1; . . . ; m
j¼1
/o yro ¼
n X
kj yrj sþr ;
r ¼ 1; . . . ; s
ð1Þ
j¼1
1¼
n X
kj
j¼1
kj ; si ; sþr P 0 where /o is maximum possible proportional outputs amount that DMUo can produce with assisting its available resources in comparison to other DMUs. Variable s i represents input decrement for evaluating DMU, and sþ r shows the rth output amount that can be further produced by evaluating DMU. kj is, also, a non-negative variable to construct a convex combination of other DMUs to compare evaluating DMU, and e is a non-Archimedean element in order to þ optimize s i and sr and ensure that these variables are considered in solution. Definition 1. (Efficiency): DMU0 is efficient when in optimal solution
(i) /0 ¼ 1 (ii) All slack variables are zero. Excluding the column vector correspond to DMUo from the LP coefficients matrix of model (1), super-efficiency model is defined as follows.
Maximize
/sup o
subject to xio ¼
n X
kj xij þ si ;
i ¼ 1; . . . ; m
j¼1
0¼
j–o n X
þ kj yrj /sup o yro sr ;
j¼1
1¼
j–o n X
r ¼ 1; . . . ; s ð2Þ
kj
j¼1 j–o si ; kj ; sþr
P0
This is an output oriented version of Andersen, Petersen’s model under variable returns to scale for model (1). Inefficient DMUs are assigned an index of efficiency greater than 1 that could be interpreted as the minimum increase in output vector that is required to make a DMU efficient. Efficient DMUs have an index equal to or less than 1. It represents the maximum possible proportional decrease in an output vector retaining DMU efficiency. It is worth emphasizing that we can solve the proposed super-efficiency model to find efficient DMUs as well as ranking them. This derives from this fact that efficient DMUs have efficiency score less than or equal to 1. Therefore, DMUs with super-efficiency score not greater than 1 are efficient with model (1). For inefficient ones super-efficiency scores which are equal to efficiency scores in the model (1) are greater than 1.
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3. Stochastic output oriented super-efficiency model In this section, stochastic super-efficiency model is developed which permits the possible presence of stochastic variability in the data. As we know, the conventional DEA models do not allow stochastic variations in input and output, therefore, DEA efficiency measurement may be sensitive to such variations. For example, a DMU which is measured as efficient relative to other DMUs, may turn inefficient if such random variations are considered. To remove this weakness in the conventional DEA models, some authors incorporated stochastic input and output variations into the DEA. See, for example, Huang and Li (1996) and Cooper et al. (2004), Khodabakhshi and Asgharian (2009), Khodabakhshi (2009), and Khodabakhshi et al. (2009a, 2009b) among others. In what follows, stochastic version of the output oriented super-efficiency model is introduced which allows for the possibility of stochastic variations in input–output data. Following Cooper et al. (2004), let ~j ¼ ðy ~1j ; . . . ; y ~sj Þt be random input and output ~ x1j ; . . . ; ~ xmj Þt ; y xj ¼ ð~ related to DMUj ðj ¼ 1; . . . ; n). Let, also, xj ¼ ðx1j ; . . . ; xmj Þt ; yj ¼ ðy1j ; . . . ; ysj Þt show the corresponding vectors of expected values of inputs and outputs for DMUj . Suppose that all input and output components are jointly Normally distributed. Therefore, from Cooper et al. (2004) we know that stochastic version of the model (1) with inequality constraints, where slack variables are all excluded from the objective function is as follows:
Maximize
/o ( ) n X subject to P kj ~xij 6 ~xio P 1 a; j¼1
P
( n X
i ¼ 1; . . . ; m
) ~rj P /o y ~ro kj y
P 1 a;
r ¼ 1; . . . ; s
ð3Þ
j¼1
1¼
n X
kj
~rj P /sup ~ P 1 a; kj y o yro > > > > > > j ¼ 1 > > > > ; : j–o n X 1¼ kj j¼1
P
j–o
3.2. Deterministic equivalent for the super-efficiency model In what follows, Normality assumption is exploited to introduce a deterministic equivalent to model (4). For the stochastic efficiency model, model (3), deterministic equivalent quadratic program have been obtained by Cooper et al. (2004). We first need to recall a well-known fact about normally distributed random lk1 ; Rkk Þ, where vectors that is used below. Suppose that ~ X k Nð~ lk1 and Rkk are, respectively, the mean value vector and the variance–covariance matrix. Then for any matrix Amk we have ! l; ARkk AT Þ, where AT is the transpose of A. Using this reA X NðA~ sult, one can obtain the following deterministic equivalent to the stochastic model, model (3), see Cooper et al. (2004).
subject to
Based on the previous assumptions the stochastic version of the proposed super-efficiency model can be defined as below:
/sup o 9 8 > > > > > > > > > > n = < X subject to P kj ~xij 6 ~xio P 1 a; > > > > > > j¼1 > > > > ; : j–o
si þ
n X
s X
! sþr
r¼1
kj xij þ si U1 ðaÞrIi ðkÞ ¼ xio ; i ¼ 1;...;m
j¼1
/o yro
n X
kj yrj þ sþr U1 ðaÞror ð/o ;kÞ ¼ 0; r ¼ 1;...;s
ð5Þ
j¼1 n X
(i) /o ¼ 1 (ii) Slack variables are zero in all alternative optimal solutions.
3.1. Stochastic output oriented super-efficiency model
m X i¼1
Definition 2. (Stochastic efficiency according to model (3)) DMUo is stochastically efficient if and only if the following conditions are satisfied:
DMUo is called stochastically inefficient if it does not fulfill the conditions of Definition 2. In other words, if for an optimal solution /o > 1, or some of slacks are non zero, then DMUo is stochastically inefficient. In fact, if /o > 1, then all outputs for evaluating DMUo can be increased to /o yro ; ðr ¼ 1; . . . ; sÞ by using a convex combination of the other DMUs at the significant level a.
ð4Þ
where a is a predetermined value between 0 and 1 which specifies the significance level, and P represents the probability. DMUo is stochastically super efficient at significance a if the optimal value of the objective function is less than 1. Therefore, if /sup < 1 it means that DMUo can reduce its output to /sup percent of its current output and still remain efficient, hence the lower the /sup , the better the DMU. In the next subsection, the deterministic equivalent of the above stochastic super-efficiency model is obtained.
Maximize /o þ e
where P represents probability, and a is a predetermined value between 0 and 1 which specifies the significant level. Stochastic efficiency with the model (3) can therefore be defined as below:
r ¼ 1; . . . ; s
kj P 0
j¼1
kj P 0;
9 > > > > > =
8 > > > > > < X n
kj ¼ 1
j¼1
si ;kj ;sþr P 0; where U is the cumulative distribution function (cdf) of a standard Normal random variable and U1 is its inverse. It is assumed that xij and yrj are the means of the input and output variables, which are estimated in application by the observed values of the inputs and outputs. To see how above equations have been obtained, or how rIi ðkÞ and ror ð/o ; kÞ have been defined, let fr > 0 be an external slack for converting rth output chance constrained to following equality form:
P
( n X
) ~rj /o y ~ro P 0 kj y
¼ ð1 aÞ þ fr
j¼1
Maximize
Then sþ r > 0 will exist such that
i ¼ 1; . . . ; m P
( n X j¼1
) ~rj /o y ~ro P kj y
sþr
¼ ð1 aÞ
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M. Khodabakhshi / Computers & Industrial Engineering 58 (2010) 663–671
Therefore,
8 ! !9 > > > > n n n P P P > > > þ ~rj /o y ~ro E ~rj /o y ~ro ~rj /o y ~ro > > > k k s E k jy jy jy > > r = > > > n n u u > > > tv ar P kj y tv ar P kj y > > ~rj /o y ~ro ~rj /o y ~ro > > > ; : j¼1
j¼1
m X
Maximize /o þ e
si þ
subject to
! sþr
r¼1
i¼1 n X
s X
kj xij þ si U1 ðaÞwIi ¼ xio ; i ¼ 1;. . .; m
j¼1
/o yro
n X
kj yrj þ sþr U1 ðaÞwor ¼ 0; r ¼ 1;. .. ;s
j¼1
¼a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nP o n ~ ~ For the sake of simplicity we denote v ar by j¼1 kj yrj /o yro ror ð/o ; kÞ. Hence,
P
8n n P P > > ~ ~ > < kj yrj /o yro kj yrj þ /o yro j¼1
j¼1
ror ð/o ; kÞ
> > > :
6
9 n P > sþr kj yrj þ /o yro > > = j¼1
ror ð/o ; kÞ
> > > ;
n X
kj ¼ 1
j¼1
ðwIi Þ2 ¼
XX
kj kk Cov ð~xij ; ~xik Þ þ 2ðko 1Þ
j–o k–o
¼a
X
kj Cov ð~xij ; ~xio Þ
j–o
þ ðko 1Þ2 Varð~xio Þ XX X ~rk ; y ~rj Þ þ 2ðko /o Þ ~rk ; y ~ro Þ ðwor Þ2 ¼ kk kj Cov ðy kk Cov ðy k–o j–o
k–o 2
~ro Þ þ ðko /o Þ Varðy In other words,
( P Z6
sþr
Pn
j¼1 kj yrj þ o ð/ ; kÞ o r
/o yro
si ; kj ;sþr ;wIi ;wor P 0:
)
r
ð6Þ
¼a
where Z is a normal standard variable, and we can have
9 8 n P > > þ > > s k y þ / y > j o ro > rj = < r j¼1
U
ror ð/o ; kÞ
> > > :
> > > ;
¼a
Maximize /sup o n X subject to kj xij þ si U1 ðaÞrIi ðkÞ ¼ xio ; i ¼ 1;. .. ;m j¼1 j–o
or
/sup o yro
/o yro
n X
ror ð/o ; kÞ 2 ¼ Var
( n X
¼ Var
8 > > > : j¼1 j–o 0
j–o
~rj /o y ~ro kj y
kj yrj þ ðko /o Þyro
si ; kj ;sþr P 0;
9 > > =
ð7Þ
> > ;
1
n BX C ¼ Var B kj yrj C @ A þ Varððko /o Þyro Þ j¼1 j–o
0
kj ¼ 1
j¼1
)
j¼1
1
n BX C kj yrj ; ðko /o Þyro C þ 2Cov B @ A j¼1 j–o
Finally, the following deterministic equivalent to our stochastic super-efficiency model is obtained. þ One can, therefore, obtain the optimal values /sup s o i and sr by solving the quadratic program.
Maximize /sup o n X subject to kj xij þ si U1 ðaÞwIi ¼ xio ; i ¼ 1; . . . ; m j¼1 j–o
/sup o yro
ror ð/o ; kÞ 2 ¼
n X
kj yrj þ sþr U1 ðaÞwor ¼ 0; r ¼ 1; . . . ; s
j¼1 j–o
Therefore,
kj yrj þ sþr U1 ðaÞror /sup o ;k ¼ 0; r ¼ 1;. .. ;s
j–o n X
ror ð/o ; kÞ note that
To derive equations for
n X
j¼1
kj yrj þ sþr U1 ðaÞror ð/o ; kÞ ¼ 0
j¼1
In a similar fashion, one can obtain the following deterministic equivalent to the stochastic super-efficiency model, model (4).
XX
~rk ; y ~rj Þ þ 2ðko /o Þ kk kj Cov ðy
k–o j–o
X
~rk ; y ~ro Þ þ ðko /o Þ2 Varðy ~ro Þ kk Cov ðy
k–o
n X
kj ¼ 1
j¼1 j–o
wIi
2
¼
XX
kj kk Cov ð~xij ; ~xik Þ 2
j–o k–o
Using the aforementioned property of Normal distribution, and replacing rIi ðkÞ and ror ð/o ; kÞ, respectively, by non-negative variables wIi and wor , and adding the following quadratic equality constraints
wIi
2
2
rIi ðkÞ 2 o 2 o wr ¼ rr ð/o ; kÞ ¼
model (5) is transformed to a quadratic programming problem.
2 wor
X
kj Cov ð~xij ; ~xio Þ
j–o
þ Varð~xio Þ X XX ~rk ; y ~rj Þ 2/sup ~rk ; y ~ro Þ ¼ kk kj Cov ðy kk Cov ðy o k–o j–o
k–o
2 ~ þ ð/sup o Þ Varðyro Þ
si ; kj ; sþr ; wIi ; wor P 0: ð8Þ
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M. Khodabakhshi / Computers & Industrial Engineering 58 (2010) 663–671
Proof.
4. Sensitivity analysis In our sensitivity analysis discussion, following Cooper et al. (2002), allowable limits of data variations for only one DMU at a time is permitted. There are other approaches to sensitivity analysis in DEA; for example, Charnes and Neralic´ (1990) and Seiford and Zhu (1998) that allow all data for all DMUs to be varied simultaneously until at least one DMU changes its status from efficient to inefficient, or vice versa. See, also, Charnes and Zlobec (1989) which studied stability of efficiency tests used in DEA. We are going to simplify matter in the previous part by assuming that only DMUo has random variations in its input and outputs, i.e. rIio –0; roro –0; rIij ¼ 0, and rOrj ¼ 0ðj–oÞ for all i and r. In this case, the model (7) can be written:
~ sup ¼ Maximize / o
/sup o
(i) Note that since 0:5 < a < 1; U1 ðaÞ > 0. Therefore, yro > y0ro and xio < x0io . Thus, if /sup > 1, then there exists a solution o ~ sup ¼ /sup > 1 for the model (9), when evaluating with / o o ~ sup 6 1. Therefore, this DMUo , which is in contrast with / o 6 1. contradiction shows that we must have /sup o (ii) This follows directly from (i). h
Theorem 4. For 0:5 < a < 1Suppose that for DMUo /sup 6 1 in ~ sup 6 1 in model (9), if model (2), then / m X
I io
r þ
n X
kj x0ij þ si ;
i ¼ 1; . . . ; m
where
0¼
kj y0rj
0 /sup o yro
sþr ;
r ¼ 1; . . . ; s
j¼1 j–o
1¼
n X
Maximize ð9Þ subject to
kj
x0ij
þ r¼1 hr
m X i¼1 n X
hi þ
s X
!, hþ r
U1 ðaÞ
r¼1
is the optimal value of
hþr
r¼1
kj xij 6 xio þ hi ;
i ¼ 1; . . . ; m
kj yrj P yro hþr ;
r ¼ 1; . . . ; s
ð12Þ
j¼1
1¼
n X
kj
j¼1
r ¼ 1; . . . ; s
j–o; r ¼ 1; . . . ; s
¼ xij ; j–o;
Ps
þ
n X
where
x0io ¼ xio þ rIio U1 ðaÞ;
þ
s X
j¼1
si ; kj ; sþr P 0;
y0ro ¼ yro roro U1 ðaÞ;
h i
i¼1
i¼1 hi
j¼1 j–o
y0rj ¼ yrj ;
r 6
Pm
j¼1 j–o n X
m X
o ro
r¼1
i¼1
subject to x0io ¼
s X
hi ; hþr ; kj P 0
ð10Þ
i ¼ 1; . . . ; m
i ¼ 1; . . . ; m
ð11Þ
Therefore, the model (9) is the deterministic equivalent of stochastic super-efficiency model (4) under the above assumptions. Theorem 1. If a ¼ 0:5, then results of the output oriented superefficiency model (2) and model (9) are the same.
~ sup > 1 in the model (9). This Proof. Suppose on the contrary that / P kj P 0ðj–0Þ and 1 ¼ nj¼1 kj implies that there is a k with ko ¼ 0; such that
X
kj y0 > y0 ; rj ro
r ¼ 1; . . . ; s
kj x0 6 x0 ; ij io
i ¼ 1; . . . ; m
j–o
X j–o
Proof. Since U1 ð0:5Þ ¼ 0, it is obvious. h
by the definition of y0o and x0o
Theorem 2. For 0 < a < 0:5, DMUo /sup o
n X
~ sup / o
6 1 in model (2), then 6 1 in (i) Suppose that for model (9). ~ sup > 1 in model (9), then /sup > 1 in (ii) Suppose that for DMUo / o o the model (2). Proof. (i) Note that since 0 < a < 0:5; U1 ðaÞ < 0. Therefore, y0ro > yro ~ sup and x0io < xio . Thus, if / > 1, then there exists a solution o ~ sup ¼ / > 1 for the super-efficiency model (2), with /sup o o 6 1. when evaluating DMUo , which is in contrast with /sup o Therefore, this contradiction shows that we must have ~ sup / 6 1. o (ii) This follows directly from (i). h
Theorem 3. For 0:5 < a < 1, ~ sup 6 1 in model (9), then /sup 6 1 in (i) Suppose that for DMUo / o o the model (2). ~ sup > 1 in model (2), then / > 1 in (ii) Suppose that for DMUo /sup o o model (9).
kj y ¼ rj
kj y0 > y U1 ðaÞro ro rj ro
j–o
j¼1 n X
X
kj xij ¼
j¼1
X
kj x0 6 xio þ U1 ðaÞrI ij io
j–o
letting
hþ ¼ U1 ðaÞro ; r ro
r ¼ 1; . . . ; s
and
h ¼ U1 ðaÞrI ; i io
i ¼ 1; . . . ; m
we find that ð hþ ; h ; kÞ satisfies model (12) for which Pm Ps þ Pm Ps þ i¼1 hi þ r¼1 hr < i¼1 hi þ r¼1 hr , a contradiction to assump ~ sup 6 1 in modtion that hþ r ; hi are optimal for model (12). Hence, / el (9). h ~ sup 6 1 in Theorem 5. For 0 < a < 0:5Suppose that for DMUo / Pm I P sup 6 1 in model (2), if rio þ sr¼1 roro < model (9), then / i¼1P Pm Ps þ P m s þ 1 ðaÞÞ where is the =ðU i¼1 hi þ r¼1 hr i¼1 hi þ r¼1 hr optimal value of
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M. Khodabakhshi / Computers & Industrial Engineering 58 (2010) 663–671 m X
Maximize subject to
i¼1 n X
hi þ
s X
hþr
r¼1
kj xij 6 x0io þ hi ;
i ¼ 1; . . . ; m
kj yrj P y0ro hþr ;
r ¼ 1; . . . ; s
j¼1 n X
ð13Þ
j¼1
1¼
n X
kj
j¼1
Fig. 1. Numerical example.
hi ; hþr ; kj P 0 Proof. Suppose on the contrary that /sup > 1 in model (2). ThereP kj P 0ðj–0Þ and 1 ¼ nj¼1 kj such fore, there exists k with ko ¼ 0; that
X
kj y > y rj ro
j–o
X
kj xij 6 xio
Table 1 Data for illustrative example. DMU
x1
y1
/o
/sup o
A B C D E
1 2 3 5 4
0.5 2 2 1 1
1 1 1 2 2
0 0.625 1 2 2
j–o
by definition of x0o and y0o we then have n X
kj y ¼ rj
X
j¼1
j–o
n X
X
j¼1
kj xij ¼
kj y > y ¼ y0 þ U1 ðaÞro rj ro ro ro kj xij 6 x0 U1 ðaÞrI io io
j–o
1 o letting hþ U1 ðaÞrIio , we find r ¼ U ðaÞrro ; r ¼ 1; . . . ; s and hi ¼ P Ps þ m h ; kÞ is a solution of (13) for which that ð hþ ; i¼1 hi þ r¼1 hr < Pm Ps þ þ h þ h , a contradiction to assumption that h , hi are r i¼1 i r¼1 r optimal for model (13). Therefore, we must have /sup 6 1 in the model (2). h
Table 2 ~ sup Computational results of model (9), / . o Possibility level, a
DMUC
DMUE
0.2 0.8
0.8264 1.266
1.4084 3.448
C0 : x0 ¼ 3 þ ð0:42Þ ¼ 3:42 and y0 ¼ 2 þ ð0:42Þ ¼ 1:58: ~ sup / is 1.266, therefore, C0 is inefficient, see Fig. 4 in Appendix A. C0 Furthermore, adjusted input–output for DMU E is E0 (4.42,0.58) with ~ sup / ¼ 3:448. Therefore, E0 is inefficient, see Fig. 5 in Appendix A. E0
4.1. Numerical example 5. Application Data and numerical results for five DMUs depicted in Fig. 1 are presented in Table 1. Columns 4 and 5 of Table 1 show the optimal value of the objective function of the BCC and output oriented super-efficiency models, respectively. In order to illustrate some of our theoretical results, only two cases with a ¼ 0:2 and a ¼ 0:8 are considered. It is also assumed that, rI ¼ ro ¼ 0:5. Computational results are presented in Table 2. Case 1. a ¼ 0:2. From a cumulative normal distribution table, we have U1 ¼ 0:84 and therefore rI U1 ðaÞ ¼ ro U1 ðaÞ ¼ 0:42 for use in the following example. Assume that only point C has random variations in its input and output. Based on (10) and (11), the adjusted input and output for point C with a ¼ 0:2 is
C0 : x0 ¼ 3 þ ð0:42Þ ¼ 2:58 and y0 ¼ 2 ð0:42Þ ¼ 2:42: ~ sup We have / ¼ 0:8264, therefore, C0 is super efficient. The condiC0 6 1. This is tion of Theorem 5 is satisfied, and from Table 1, /sup C consistent with Theorem 5, see Fig. 2 in Appendix A. Adjusted input–output for point E is also E0 (3.58,1.42). For E0 we have ~ sup / ¼ 1:4084. Therefore, E0 is inefficient, see Fig. 3 in Appendix E0 > 1 in Table 1. This is consistent with TheA. Note that for E, /sup E orem 2(ii). Case 2. a ¼ 0:8. In this case, we again have U1 ðaÞ ¼ 0:84 and therefore rI U1 ðaÞ ¼ ro U1 ðaÞ ¼ 0:42. Assume that only point C has random variations in its input and output. Based on (10) and (11), the adjusted input and output for point C with a ¼ 0:8 is
As an empirical example, the proposed method is applied using some actual data of year 2000 to Iranian electricity distribution units. The Iranian electricity distribution units established in 1992, are public and act under the supervision of TAVANIR company (Iran power, Generation, Transmission and Distribution Management company). According to the extensive review in Jamasb and Pollitt (2001), the most frequently used inputs are operating costs, number of employees transformer capacity and network length. The most widely useful outputs are also units of energy delivered, number of customers and size service area. The cost data usually is not available. This study used network length, transformer capacity, and employee variables as inputs and units delivery, and service area variables as outputs. The size of service area is considered as an environmental variable in our study. The service area is out of units control by nothing that the electricity distribution units are public and act in provinces in Iran. So, the units can-
Fig. 2. a ¼ 0:2, C has random variations in its input and output.
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M. Khodabakhshi / Computers & Industrial Engineering 58 (2010) 663–671
Fig. 3. a ¼ 0:2, E has random variations in its input and output.
1.582*(1806, 24,641). Therefore, it is inefficient in its current status. After Kermanshah, DMU 11, Lorestan, with score 1.423, DMU 7, Kurdistan, 1.376, DMU 1, Azarbaijan Sharghi, with score 1.361, DMU 15, Gilan, with score 1.327 are the worst companies in terms of efficiency. Inefficient companies have scores ranking from 1.143, Markazi, to 1.582, Kermanshah. Among inefficient companies DMU 10, 1.143 is the best one, but this DMU can still produce 1.143 times of its current outputs. To compute results for stochastic data, a ¼ 0:4 and a ¼ 0:6 have been chosen for which U1 ðaÞ 0:25 and U1 ðaÞ 0:25, respectively. These rather large values of a are deliberately chosen to illustrate differences between the results based on model (5) and model (2). It is worth noting that model (5) and model (2) produce similar results when a is small. The computational results of the equivalent deterministic problem are presented in columns 5 and 6 of Table 4. It is assumed that all DMUs have the same variance, but they can have different means. The variances for the outputs and the inputs can therefore be estimated by:
~r Þ ¼ Varðy Fig. 4. a ¼ 0:8, C has random variations in its input and output.
not change the value of this variable. The measurement units for the network length, transformers capacity and total electricity sales are Kilometer(KM), MVA and MWh, respectively. Table 3 presents data of seventeen companies used in Azadeh, Ghaderi, and Omrani (2008). Computational results of the deterministic output oriented super-efficiency model, model (2) and deterministic equivalent of the stochastic super-efficiency model, model (8), are presented in Table 4. Columns 3–4 and 5–6 of the Table represent super-efficiency score and rank of units for both deterministic and stochastic models, respectively. Note that a super-efficiency score less than 1 implies that the DMU is super efficient, scores equal to 1 imply they are just efficient; scores greater than 1 imply that the DMUs are inefficient. Therefore, the lower the super-efficiency score the better the company. Based on the numerical results presented in columns 3 and 4 of Table 4, DMU 3, Esfahan, DMU 5, Khozestan, DMU 8, Fars, DMU 9, Ardabil, DMU 12, Ghazvin, DMU 13, Semnan, DMU 17, Yazd for which /sup is less than one are super efficient with the model (2). The rest of companies for which /sup is greater than 1 are inefficient. DMU 13, Semnan, with super-efficiency score 0.133 is ranked the first. In fact, even if this unit produce 0.133 percent of its current outputs, i.e., 0.133*(1418, 96,816) in which the first component is units delivery and the second one is size service area, it still remains efficient with the deterministic model (1). Next top units are DMU 12, Ghazvin and DMU 5, Khozestan, with super-efficiency scores 0.528 and 0.568, respectively. The worst company is DMU 14, Kermanshah with score 1.582. This company can produce 1.582 times of its current outputs, i.e.,
Fig. 5. a ¼ 0:8, E has random variations in its input and output.
17 1 X ðy yr Þ2 16 j¼1 rj
and Varðx~i Þ ¼
17 1 X ðxij xi Þ2 16 j¼1
where
yr ¼
17 1 X y 17 j¼1 rj
and xi ¼
17 1 X xij 17 j¼1
and xij and yrj are the observed values of inputs and outputs for DMUj which are used as an estimate for the expected values of the stochastic inputs and outputs. It is also assumed that outputs and inputs for different DMUs are independent. This independence ~rk ; y ~rj Þ ¼ 0 and also assumption, then, implies that Cov ðy xik Þ ¼ 0. The stochastic super-efficiency scores obtained Cov ðx~ij ; ~ from GAMS software are presented in Table 4. Again, the best company is DMU 13, Semnan, with stochastic score 0.146(0.148) correspond to a ¼ 0:4ða ¼ 0:6Þ. The worst company is, again, DMU 14, Kermanshah with score 1.564(1.601). Numerical results of the stochastic super-efficiency model correspond to a ¼ 0:4 and a ¼ 0:6 which are presented in columns 4–5 and 6–7 show that both cases are quite similar. For example, DMU 3, Esfahan with a score 0.802(0.804) correspond to a ¼ 0:4 ða ¼ 0:6Þ, DMU 5, Khozestan with a score of 0.567(0.570), DMU 8, Fars, with a score of 0.912(0.926), DMU 9, Ardabil, with a score of 0.850(0.976), DMU 12, Ghazvin, with a score of 0.529(0.550), DMU 13, Semnan, with a score of 0.146(0.148)and DMU 17, Yazd with a score of 0.911(0.938) are stochastically efficient in both cases. These DMUs also were efficient in deterministic sense. However, the super-efficiency scores are not the same. Comparing numerical results presented in columns 3–4 and 5– 6(7–8) of Table 4, DMU 14, Kermanshah, and DMU 15, Gilan, DMU 10, Markazi, DMU 16, Hormozgan, DMU 4, Hamedan, DMU 6, Zanjan, DMU 2, Azarbaijan Gharbi, DMU 1, Azarbaijan Sharghi, DMU 7, Kurdistan, and DMU 11, Lorestan, are inefficient with both deterministic and stochastic models. Therefore, the computational results are quite identical. From computational results presented in columns 5–6 and 7–8 of Table 4, Semnan, DMU 13, with stochastic super-efficiency score 0.146 and 0.148 correspond to a ¼ 0:4 an a ¼ 0:6, respectively is ranked 1st by the stochastic super-efficiency model. Ghazvin, DMU 12, with a score of 0.529(0.550) correspond to a ¼ 0:4 ða ¼ 0:6Þ, is ranked the second. Another two top companies are Khozestan, DMU 5, with a score of 0.567(0.570), Esfahan, DMU 3, with a score of 0.802(0.804). At the bottom of ranking, the companies are Kermanshah, DMU 14, with a score of 1.564(1.601) and Lorestan, DMU 11 with a score of 1.402(1.445).
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Table 3 Data for electricity distribution units. DMU
Company
Net. len. (km)
Tr. cap. (MVA)
Empl.
Un. del. (MW h)
S. area (km2)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Azarbaijan Sharghi Azarbaijan Gharbi Esfahan Hamedan Khozestan Zanjan Kurdistan Fars Ardabil Markazi Lorestan Ghazvin Semnan Kermanshah Gilan Hormozgan Yazd
15,151 19,610 25,566 12,340 19,380 10,347 11,697 24,634 10,129 14,505 12,078 8766 8063 12,795 21,187 16,185 11,990
990 1306 2713 1101 3932 788 725 1617 507 1489 1006 946 730 1147 1534 1786 963
867 1047 1072 595 1471 351 426 872 441 600 594 302 374 538 938 938 576
1825 2597 5835 2369 8048 1526 1174 4015 936 3063 1784 2414 1418 1806 2848 3411 2400
40,968 37,463 97,923 19,574 53,442 21,841 28,817 83,575 17,881 29,406 28,392 15,491 96,816 24,641 13,952 71,193 73,467
Table 4 Deterministic and stochastic super-efficiency scores for electricity distribution units. DMU
Company
Determ. score
Rank
Stocha. score, a ¼ 0:4
Rank
Stocha. score, a ¼ 0:6
Rank
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Azarbaijan Sharghi Azarbaijan Gharbi Esfahan Hamedan Khozestan Zanjan Kurdistan Fars Ardabil Markazi Lorestan Ghazvin Semnan Kermanshah Gilan Hormozgan Yazd
1.361 1.249 0.802 1.154 0.568 1.213 1.377 0.919 0.711 1.143 1.423 0.528 0.133 1.582 1.327 1.147 0.924
14 12 5 10 3 11 15 6 4 8 16 2 1 17 13 9 7
1.341 1.246 0.802 1.159 0.567 1.284 1.340 0.912 0.850 1.142 1.402 0.529 0.146 1.564 1.316 1.141 0.911
15 11 4 10 3 12 14 7 5 9 16 2 1 17 13 8 6
1.382 1.274 0.804 1.188 0.570 1.332 1.414 0.926 0.976 1.163 1.445 0.550 0.148 1.601 1.339 1.154 0.938
14 11 4 10 3 12 15 5 7 9 16 2 1 17 13 8 6
6. Conclusion
Acknowledgement
In this paper, super-efficiency issue is discussed in stochastic data envelopment analysis. We described its characterizations theoretically and empirically. In addition to developing stochastic version of the output oriented super-efficiency model, we obtained the deterministic equivalent of the stochastic version which can be converted to a quadratic problem. As an empirical example, the proposed approach was also applied to data of Iranian electricity distribution companies. Computational results of both deterministic and stochastic super-efficiency models show that DMU 13, Semnan is the best company in terms of technical efficiency. DMU 14, Kermanshah is, also, the worst company among seventeen companies studied in this paper. Although ranking results of both models for the evaluating companies are similar, the superefficiency scores are different. The stochastic model allows the data errors, therefore, if the data is rather inaccurate, and a rough estimate is required, the stochastic model might be preferred. However, if the data is accurate, the deterministic model will be preferred. Sensitivity analysis of the proposed super-efficiency model w. r. t. changes on parameter a which specifies significance level, also, has been discussed. Finally, developing the proposed model in fuzzy data envelopment analysis can be suggested for further research.
The author would like to thank three anonymous referees for their constructive comments. Appendix A Figs. 1–5 are related to numerical example of Section 4. When
a ¼ 0:2, Fig. 2 is related to the case C has random variations in its input–output, and adjusted input–output of point C which is named C0 is used to depict production possibility set of the given DMUs. From the Fig. 2, it is clear that DMU C0 is efficient. Fig. 3, also, shows that DMU E0 is inefficient. Furthermore, from Figs. 4 and 5 it is obvious that DMUs C0 and E0 are inefficient. References Adler, N., Friedman, L., & Sinuany-Stern, Z. (2002). Review of ranking methods in the data envelopment analysis context. European Journal of Operational Research, 140(2), 249–265. Andersen, P., & Petersen, N. C. (1993). A procedure for ranking efficient units in data envelopment analysis. Management Science, 39(10), 1261–1264. Azadeh, A., Ghaderi, S. F., & Omrani, H. (2008). A deterministic approach for performance assesment and optimization of power distribution units in Iran. Energy Policy. doi:10.1016/j.enpol.2008.08.027.
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Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
An output oriented super-efficiency measure in stochastic data envelopment analysis: Considering Iranian electricity distribution companies q Mohammad Khodabakhshi * Department of Mathematics, Faculty of Science, Lorestan University, Khorram Abad, Iran
a r t i c l e
i n f o
Article history: Received 16 March 2009 Received in revised form 13 January 2010 Accepted 14 January 2010 Available online 20 January 2010 Keywords: Chance constrained programming Electricity distribution units Super-efficiency Sensitivity analysis
a b s t r a c t In this paper, the concept of chance constrained programming approaches is used to develop output oriented super-efficiency model in stochastic data envelopment analysis. Output oriented super-efficiency model is one of the classic models in data envelopment analysis widely used by DEA people and practitioners. However, in many real applications, data is often imprecise. A successful method to address uncertainty in data is replacing deterministic data by random variables, leading to stochastic DEA. Therefore, in this paper, output oriented super-efficiency model is developed in stochastic data envelopment analysis, and its deterministic equivalent which is a nonlinear program is derived. Moreover, it is shown that the deterministic equivalent of the stochastic super-efficiency model can be converted to a quadratic program. Furthermore, sensitivity analysis of the proposed super-efficiency model is also discussed with respect to changes on parameter variables. Finally, data related to seventeen Iranian electricity distribution companies is used to illustrate the methods developed in this article. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Data envelopment analysis (DEA) was initiated in 1978 by Charnes, Cooper, and Rhodes (1978) and the first DEA model was called CCR model. Later, Banker, Charnes, and Cooper (1984) extended DEA to obtain a variable returns to scale version of the CCR model called BCC model. Since 1978 there has been a surge of research on DEA and many further models were introduced in the literature. Cooper, Seiford, and Tone (2000) reviewed most of these developments and extensions in DEA up to 2000. The objective of these models, DEA models, is evaluating overall efficiencies of decision making units (DMUs) that are responsible to convert a set of inputs into a set of outputs. The results of DEA models divide the DMUs into two sets, efficient and inefficient DMUs. While decision makers are interested in a complete ranking of all DMUs, efficient units obtained by original models cannot be differentiated among themselves. In other words, original models such as CCR and BCC models cannot rank efficient units. Therefore, in the last two decades, some DEA researchers initiated a new area called super-efficiency to rank the DEA efficient DMUs. The research in this area was first developed by Andersen and Petersen (1993). Mehrabian, Alirezaee, and Jahanshahloo (1999), also, developed a super-efficiency model to rank efficient units obtained by their own model. Martic and Savic (2001) applied super-efficiency model developed by Andersen and Petersen (1993) to rank regions in Serbia with regards to q
This manuscript was processed by Area Editor Imed Kacem. * Tel.: +98 912 6278846; fax: +98 661 2201333. E-mail addresses: [email protected], [email protected].
0360-8352/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2010.01.009
social-economic development. Moreover, Tone (2002) developed a super-efficiency measure based on the slack based measure introduced in the literature. The research on super-efficiency has widely been cited and became very popular in DEA. Ranking papers using super-efficiency models up to 2002 have been reviewed in Adler, Friedman, and Sinuany-Stern (2002). Furthermore, Khodabakhshi (2007) developed a super-efficiency measure based on, improved outputs, input relaxation model introduced in Jahanshahloo and Khodabakhshi (2004). Li, Jahanshahloo, and Khodabakhshi (2007), also, provided a super-efficiency model to rank efficient units obtained by the CCR model. Several papers have been published on DEA in Journal Computers & Industrial Engineering. In what follows, some of these applications are reported. Emrouznejad and Shale (2009) proposed a neural network background data envelopment analysis to evaluate the efficiency of large scale datasets. In their own approach requirement for computer memory and CPU time are far less than that needed by conventional DEA methods. Liu and Chen (2009), for the purpose of identifying bad performers such as bankrupt firms in the worst-case scenario proposed worst-practice frontier DEA model. While the original DEA model construct an efficient best-practice frontiers, the proposed model by Liu and Chen (2009) construct a worstpractice frontier. Liu (2008), also, developed a fuzzy DEA/AR method that is able to evaluate the performance flexible Manufacturing System (FMS) alternatives when the input -output are represented as crisp and fuzzy data. The lower and upper bound of alternatives is calculated by a pair of two level mathematical programs which can be transformed into a pair of conventional one-level DEA/Ar method.
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Baker and Talluri (1997) proposed an alternative methodology for technology selection using DEA. They addressed some of shortcomings in the methodology suggested by Khouja (1995), and provided a more robust analysis based on cross efficiencies in DEA. They, also, discussed applicability of the methodology from buyer and manufacturer view points. It is noticeable that Khouja (1995) proposed a decision model for technology selection problems using a two-phase procedure. In first phase, DEA is used to identify technologies that provide the best combinations of vender specifications on the performance parameters of the technologies. In the second phase, a multi attribute decision making model is used to select a technology from these identification the first phase. Papers mentioned above are quite interesting, but none of them considers stochastic variations in input–output data, or imprecise data, to rank efficient units. However, this study extends output oriented super-efficiency in stochastic DEA to consider stochastic variations in input–output data. Note that we know data envelopment analysis methodology has many advantages, such as no requirement for a priori weights or explicit specification of functional relations among the multiple inputs and outputs. However, there is a weakness in conventional DEA models, in fact, original DEA models do not allow stochastic variations in input and output such as data entry errors. As a result, DEA efficiency measurement may be sensitive to such variations. A DMU which is measured as efficient relative to other DMUs, may turn inefficient if such random variations are considered. Stochastic input and output variations into DEA have been studied by, for example, Cooper, Deng, Huang, and Li (2002, 2004), Land, Lovell, and Thore (1988), and Olesen and Petersen (1995), Morita and Seiford (1999), Khodabakhshi and Asgharian (2009), Khodabakhshi (2009, 2010). Although original DEA models such as CCR or BCC models have been extended in stochastic data envelopment analysis, the research on output oriented super-efficiency models solely have been done in deterministic DEA. To close this gap, in this paper, output oriented version of the popular super-efficiency measure introduced in Andersen and Petersen (1993) is extended in stochastic data envelopment analysis which allows stochastic variations in input–output data. The concept of chance constrained programming approach introduced by Cooper et al. (2004) and Khodabakhshi and Asgharian (2009), Khodabakhshi (2009, 2010) is used to develop output oriented stochastic super-efficiency model. Moreover, deterministic equivalent of the stochastic super-efficiency model is obtained to solve the stochastic model. Sensitivity of the stochastic model and its deterministic equivalent with respect to changes in the parameter values is done, too. Furthermore, as an empirical example, the proposed approach is applied on data of Iranian electricity distribution units. The rest of the paper is organized as follows: The output oriented version of the super-efficiency model introduced in Andersen and Petersen (1993) is described in Section 2. In Section 3, stochastic version of the output oriented super-efficiency model is developed, and its deterministic equivalent is also obtained. In addition, it is shown that the deterministic equivalent of the stochastic output oriented super-efficiency model can be converted to a quadratic program. Section 4 discusses sensitivity analysis. An illustrative example is used to describe some theoretical results, too. As an empirical example, the model is applied to data of Iranian electricity distribution companies in Section 5. Section 6 concludes the paper. 2. Preliminaries Suppose that all input and output are non-negative deterministic elements. Let DMUj ; ðj ¼ 1; 2; ; nÞ be n decision making units (DMU) that convert m inputs xij ði ¼ 1; . . . ; mÞ into s outputs yrj ðr ¼ 1; . . . ; sÞ. One of the basic models used to evaluate DMUs
efficiency is output oriented model introduced in Banker et al. (1984) called output oriented BCC (Banker, Charnes, Cooper) model. This model which evaluates DMUo is as follows:
Maximize
m X
/o þ e
si
subject to xio ¼
!
sþr
r¼1
i¼1 n X
þ
s X
kj xij þ si ;
i ¼ 1; . . . ; m
j¼1
/o yro ¼
n X
kj yrj sþr ;
r ¼ 1; . . . ; s
ð1Þ
j¼1
1¼
n X
kj
j¼1
kj ; si ; sþr P 0 where /o is maximum possible proportional outputs amount that DMUo can produce with assisting its available resources in comparison to other DMUs. Variable s i represents input decrement for evaluating DMU, and sþ r shows the rth output amount that can be further produced by evaluating DMU. kj is, also, a non-negative variable to construct a convex combination of other DMUs to compare evaluating DMU, and e is a non-Archimedean element in order to þ optimize s i and sr and ensure that these variables are considered in solution. Definition 1. (Efficiency): DMU0 is efficient when in optimal solution
(i) /0 ¼ 1 (ii) All slack variables are zero. Excluding the column vector correspond to DMUo from the LP coefficients matrix of model (1), super-efficiency model is defined as follows.
Maximize
/sup o
subject to xio ¼
n X
kj xij þ si ;
i ¼ 1; . . . ; m
j¼1
0¼
j–o n X
þ kj yrj /sup o yro sr ;
j¼1
1¼
j–o n X
r ¼ 1; . . . ; s ð2Þ
kj
j¼1 j–o si ; kj ; sþr
P0
This is an output oriented version of Andersen, Petersen’s model under variable returns to scale for model (1). Inefficient DMUs are assigned an index of efficiency greater than 1 that could be interpreted as the minimum increase in output vector that is required to make a DMU efficient. Efficient DMUs have an index equal to or less than 1. It represents the maximum possible proportional decrease in an output vector retaining DMU efficiency. It is worth emphasizing that we can solve the proposed super-efficiency model to find efficient DMUs as well as ranking them. This derives from this fact that efficient DMUs have efficiency score less than or equal to 1. Therefore, DMUs with super-efficiency score not greater than 1 are efficient with model (1). For inefficient ones super-efficiency scores which are equal to efficiency scores in the model (1) are greater than 1.
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3. Stochastic output oriented super-efficiency model In this section, stochastic super-efficiency model is developed which permits the possible presence of stochastic variability in the data. As we know, the conventional DEA models do not allow stochastic variations in input and output, therefore, DEA efficiency measurement may be sensitive to such variations. For example, a DMU which is measured as efficient relative to other DMUs, may turn inefficient if such random variations are considered. To remove this weakness in the conventional DEA models, some authors incorporated stochastic input and output variations into the DEA. See, for example, Huang and Li (1996) and Cooper et al. (2004), Khodabakhshi and Asgharian (2009), Khodabakhshi (2009), and Khodabakhshi et al. (2009a, 2009b) among others. In what follows, stochastic version of the output oriented super-efficiency model is introduced which allows for the possibility of stochastic variations in input–output data. Following Cooper et al. (2004), let ~j ¼ ðy ~1j ; . . . ; y ~sj Þt be random input and output ~ x1j ; . . . ; ~ xmj Þt ; y xj ¼ ð~ related to DMUj ðj ¼ 1; . . . ; n). Let, also, xj ¼ ðx1j ; . . . ; xmj Þt ; yj ¼ ðy1j ; . . . ; ysj Þt show the corresponding vectors of expected values of inputs and outputs for DMUj . Suppose that all input and output components are jointly Normally distributed. Therefore, from Cooper et al. (2004) we know that stochastic version of the model (1) with inequality constraints, where slack variables are all excluded from the objective function is as follows:
Maximize
/o ( ) n X subject to P kj ~xij 6 ~xio P 1 a; j¼1
P
( n X
i ¼ 1; . . . ; m
) ~rj P /o y ~ro kj y
P 1 a;
r ¼ 1; . . . ; s
ð3Þ
j¼1
1¼
n X
kj
~rj P /sup ~ P 1 a; kj y o yro > > > > > > j ¼ 1 > > > > ; : j–o n X 1¼ kj j¼1
P
j–o
3.2. Deterministic equivalent for the super-efficiency model In what follows, Normality assumption is exploited to introduce a deterministic equivalent to model (4). For the stochastic efficiency model, model (3), deterministic equivalent quadratic program have been obtained by Cooper et al. (2004). We first need to recall a well-known fact about normally distributed random lk1 ; Rkk Þ, where vectors that is used below. Suppose that ~ X k Nð~ lk1 and Rkk are, respectively, the mean value vector and the variance–covariance matrix. Then for any matrix Amk we have ! l; ARkk AT Þ, where AT is the transpose of A. Using this reA X NðA~ sult, one can obtain the following deterministic equivalent to the stochastic model, model (3), see Cooper et al. (2004).
subject to
Based on the previous assumptions the stochastic version of the proposed super-efficiency model can be defined as below:
/sup o 9 8 > > > > > > > > > > n = < X subject to P kj ~xij 6 ~xio P 1 a; > > > > > > j¼1 > > > > ; : j–o
si þ
n X
s X
! sþr
r¼1
kj xij þ si U1 ðaÞrIi ðkÞ ¼ xio ; i ¼ 1;...;m
j¼1
/o yro
n X
kj yrj þ sþr U1 ðaÞror ð/o ;kÞ ¼ 0; r ¼ 1;...;s
ð5Þ
j¼1 n X
(i) /o ¼ 1 (ii) Slack variables are zero in all alternative optimal solutions.
3.1. Stochastic output oriented super-efficiency model
m X i¼1
Definition 2. (Stochastic efficiency according to model (3)) DMUo is stochastically efficient if and only if the following conditions are satisfied:
DMUo is called stochastically inefficient if it does not fulfill the conditions of Definition 2. In other words, if for an optimal solution /o > 1, or some of slacks are non zero, then DMUo is stochastically inefficient. In fact, if /o > 1, then all outputs for evaluating DMUo can be increased to /o yro ; ðr ¼ 1; . . . ; sÞ by using a convex combination of the other DMUs at the significant level a.
ð4Þ
where a is a predetermined value between 0 and 1 which specifies the significance level, and P represents the probability. DMUo is stochastically super efficient at significance a if the optimal value of the objective function is less than 1. Therefore, if /sup < 1 it means that DMUo can reduce its output to /sup percent of its current output and still remain efficient, hence the lower the /sup , the better the DMU. In the next subsection, the deterministic equivalent of the above stochastic super-efficiency model is obtained.
Maximize /o þ e
where P represents probability, and a is a predetermined value between 0 and 1 which specifies the significant level. Stochastic efficiency with the model (3) can therefore be defined as below:
r ¼ 1; . . . ; s
kj P 0
j¼1
kj P 0;
9 > > > > > =
8 > > > > > < X n
kj ¼ 1
j¼1
si ;kj ;sþr P 0; where U is the cumulative distribution function (cdf) of a standard Normal random variable and U1 is its inverse. It is assumed that xij and yrj are the means of the input and output variables, which are estimated in application by the observed values of the inputs and outputs. To see how above equations have been obtained, or how rIi ðkÞ and ror ð/o ; kÞ have been defined, let fr > 0 be an external slack for converting rth output chance constrained to following equality form:
P
( n X
) ~rj /o y ~ro P 0 kj y
¼ ð1 aÞ þ fr
j¼1
Maximize
Then sþ r > 0 will exist such that
i ¼ 1; . . . ; m P
( n X j¼1
) ~rj /o y ~ro P kj y
sþr
¼ ð1 aÞ
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M. Khodabakhshi / Computers & Industrial Engineering 58 (2010) 663–671
Therefore,
8 ! !9 > > > > n n n P P P > > > þ ~rj /o y ~ro E ~rj /o y ~ro ~rj /o y ~ro > > > k k s E k jy jy jy > > r =
j¼1
m X
Maximize /o þ e
si þ
subject to
! sþr
r¼1
i¼1 n X
s X
kj xij þ si U1 ðaÞwIi ¼ xio ; i ¼ 1;. . .; m
j¼1
/o yro
n X
kj yrj þ sþr U1 ðaÞwor ¼ 0; r ¼ 1;. .. ;s
j¼1
¼a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nP o n ~ ~ For the sake of simplicity we denote v ar by j¼1 kj yrj /o yro ror ð/o ; kÞ. Hence,
P
8n n P P > > ~ ~ > < kj yrj /o yro kj yrj þ /o yro j¼1
j¼1
ror ð/o ; kÞ
> > > :
6
9 n P > sþr kj yrj þ /o yro > > = j¼1
ror ð/o ; kÞ
> > > ;
n X
kj ¼ 1
j¼1
ðwIi Þ2 ¼
XX
kj kk Cov ð~xij ; ~xik Þ þ 2ðko 1Þ
j–o k–o
¼a
X
kj Cov ð~xij ; ~xio Þ
j–o
þ ðko 1Þ2 Varð~xio Þ XX X ~rk ; y ~rj Þ þ 2ðko /o Þ ~rk ; y ~ro Þ ðwor Þ2 ¼ kk kj Cov ðy kk Cov ðy k–o j–o
k–o 2
~ro Þ þ ðko /o Þ Varðy In other words,
( P Z6
sþr
Pn
j¼1 kj yrj þ o ð/ ; kÞ o r
/o yro
si ; kj ;sþr ;wIi ;wor P 0:
)
r
ð6Þ
¼a
where Z is a normal standard variable, and we can have
9 8 n P > > þ > > s k y þ / y > j o ro > rj = < r j¼1
U
ror ð/o ; kÞ
> > > :
> > > ;
¼a
Maximize /sup o n X subject to kj xij þ si U1 ðaÞrIi ðkÞ ¼ xio ; i ¼ 1;. .. ;m j¼1 j–o
or
/sup o yro
/o yro
n X
ror ð/o ; kÞ 2 ¼ Var
( n X
¼ Var
8 > >
j–o
~rj /o y ~ro kj y
kj yrj þ ðko /o Þyro
si ; kj ;sþr P 0;
9 > > =
ð7Þ
> > ;
1
n BX C ¼ Var B kj yrj C @ A þ Varððko /o Þyro Þ j¼1 j–o
0
kj ¼ 1
j¼1
)
j¼1
1
n BX C kj yrj ; ðko /o Þyro C þ 2Cov B @ A j¼1 j–o
Finally, the following deterministic equivalent to our stochastic super-efficiency model is obtained. þ One can, therefore, obtain the optimal values /sup s o i and sr by solving the quadratic program.
Maximize /sup o n X subject to kj xij þ si U1 ðaÞwIi ¼ xio ; i ¼ 1; . . . ; m j¼1 j–o
/sup o yro
ror ð/o ; kÞ 2 ¼
n X
kj yrj þ sþr U1 ðaÞwor ¼ 0; r ¼ 1; . . . ; s
j¼1 j–o
Therefore,
kj yrj þ sþr U1 ðaÞror /sup o ;k ¼ 0; r ¼ 1;. .. ;s
j–o n X
ror ð/o ; kÞ note that
To derive equations for
n X
j¼1
kj yrj þ sþr U1 ðaÞror ð/o ; kÞ ¼ 0
j¼1
In a similar fashion, one can obtain the following deterministic equivalent to the stochastic super-efficiency model, model (4).
XX
~rk ; y ~rj Þ þ 2ðko /o Þ kk kj Cov ðy
k–o j–o
X
~rk ; y ~ro Þ þ ðko /o Þ2 Varðy ~ro Þ kk Cov ðy
k–o
n X
kj ¼ 1
j¼1 j–o
wIi
2
¼
XX
kj kk Cov ð~xij ; ~xik Þ 2
j–o k–o
Using the aforementioned property of Normal distribution, and replacing rIi ðkÞ and ror ð/o ; kÞ, respectively, by non-negative variables wIi and wor , and adding the following quadratic equality constraints
wIi
2
2
rIi ðkÞ 2 o 2 o wr ¼ rr ð/o ; kÞ ¼
model (5) is transformed to a quadratic programming problem.
2 wor
X
kj Cov ð~xij ; ~xio Þ
j–o
þ Varð~xio Þ X XX ~rk ; y ~rj Þ 2/sup ~rk ; y ~ro Þ ¼ kk kj Cov ðy kk Cov ðy o k–o j–o
k–o
2 ~ þ ð/sup o Þ Varðyro Þ
si ; kj ; sþr ; wIi ; wor P 0: ð8Þ
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M. Khodabakhshi / Computers & Industrial Engineering 58 (2010) 663–671
Proof.
4. Sensitivity analysis In our sensitivity analysis discussion, following Cooper et al. (2002), allowable limits of data variations for only one DMU at a time is permitted. There are other approaches to sensitivity analysis in DEA; for example, Charnes and Neralic´ (1990) and Seiford and Zhu (1998) that allow all data for all DMUs to be varied simultaneously until at least one DMU changes its status from efficient to inefficient, or vice versa. See, also, Charnes and Zlobec (1989) which studied stability of efficiency tests used in DEA. We are going to simplify matter in the previous part by assuming that only DMUo has random variations in its input and outputs, i.e. rIio –0; roro –0; rIij ¼ 0, and rOrj ¼ 0ðj–oÞ for all i and r. In this case, the model (7) can be written:
~ sup ¼ Maximize / o
/sup o
(i) Note that since 0:5 < a < 1; U1 ðaÞ > 0. Therefore, yro > y0ro and xio < x0io . Thus, if /sup > 1, then there exists a solution o ~ sup ¼ /sup > 1 for the model (9), when evaluating with / o o ~ sup 6 1. Therefore, this DMUo , which is in contrast with / o 6 1. contradiction shows that we must have /sup o (ii) This follows directly from (i). h
Theorem 4. For 0:5 < a < 1Suppose that for DMUo /sup 6 1 in ~ sup 6 1 in model (9), if model (2), then / m X
I io
r þ
n X
kj x0ij þ si ;
i ¼ 1; . . . ; m
where
0¼
kj y0rj
0 /sup o yro
sþr ;
r ¼ 1; . . . ; s
j¼1 j–o
1¼
n X
Maximize ð9Þ subject to
kj
x0ij
þ r¼1 hr
m X i¼1 n X
hi þ
s X
!, hþ r
U1 ðaÞ
r¼1
is the optimal value of
hþr
r¼1
kj xij 6 xio þ hi ;
i ¼ 1; . . . ; m
kj yrj P yro hþr ;
r ¼ 1; . . . ; s
ð12Þ
j¼1
1¼
n X
kj
j¼1
r ¼ 1; . . . ; s
j–o; r ¼ 1; . . . ; s
¼ xij ; j–o;
Ps
þ
n X
where
x0io ¼ xio þ rIio U1 ðaÞ;
þ
s X
j¼1
si ; kj ; sþr P 0;
y0ro ¼ yro roro U1 ðaÞ;
h i
i¼1
i¼1 hi
j¼1 j–o
y0rj ¼ yrj ;
r 6
Pm
j¼1 j–o n X
m X
o ro
r¼1
i¼1
subject to x0io ¼
s X
hi ; hþr ; kj P 0
ð10Þ
i ¼ 1; . . . ; m
i ¼ 1; . . . ; m
ð11Þ
Therefore, the model (9) is the deterministic equivalent of stochastic super-efficiency model (4) under the above assumptions. Theorem 1. If a ¼ 0:5, then results of the output oriented superefficiency model (2) and model (9) are the same.
~ sup > 1 in the model (9). This Proof. Suppose on the contrary that / P kj P 0ðj–0Þ and 1 ¼ nj¼1 kj implies that there is a k with ko ¼ 0; such that
X
kj y0 > y0 ; rj ro
r ¼ 1; . . . ; s
kj x0 6 x0 ; ij io
i ¼ 1; . . . ; m
j–o
X j–o
Proof. Since U1 ð0:5Þ ¼ 0, it is obvious. h
by the definition of y0o and x0o
Theorem 2. For 0 < a < 0:5, DMUo /sup o
n X
~ sup / o
6 1 in model (2), then 6 1 in (i) Suppose that for model (9). ~ sup > 1 in model (9), then /sup > 1 in (ii) Suppose that for DMUo / o o the model (2). Proof. (i) Note that since 0 < a < 0:5; U1 ðaÞ < 0. Therefore, y0ro > yro ~ sup and x0io < xio . Thus, if / > 1, then there exists a solution o ~ sup ¼ / > 1 for the super-efficiency model (2), with /sup o o 6 1. when evaluating DMUo , which is in contrast with /sup o Therefore, this contradiction shows that we must have ~ sup / 6 1. o (ii) This follows directly from (i). h
Theorem 3. For 0:5 < a < 1, ~ sup 6 1 in model (9), then /sup 6 1 in (i) Suppose that for DMUo / o o the model (2). ~ sup > 1 in model (2), then / > 1 in (ii) Suppose that for DMUo /sup o o model (9).
kj y ¼ rj
kj y0 > y U1 ðaÞro ro rj ro
j–o
j¼1 n X
X
kj xij ¼
j¼1
X
kj x0 6 xio þ U1 ðaÞrI ij io
j–o
letting
hþ ¼ U1 ðaÞro ; r ro
r ¼ 1; . . . ; s
and
h ¼ U1 ðaÞrI ; i io
i ¼ 1; . . . ; m
we find that ð hþ ; h ; kÞ satisfies model (12) for which Pm Ps þ Pm Ps þ i¼1 hi þ r¼1 hr < i¼1 hi þ r¼1 hr , a contradiction to assump ~ sup 6 1 in modtion that hþ r ; hi are optimal for model (12). Hence, / el (9). h ~ sup 6 1 in Theorem 5. For 0 < a < 0:5Suppose that for DMUo / Pm I P sup 6 1 in model (2), if rio þ sr¼1 roro < model (9), then / i¼1P Pm Ps þ P m s þ 1 ðaÞÞ where is the =ðU i¼1 hi þ r¼1 hr i¼1 hi þ r¼1 hr optimal value of
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M. Khodabakhshi / Computers & Industrial Engineering 58 (2010) 663–671 m X
Maximize subject to
i¼1 n X
hi þ
s X
hþr
r¼1
kj xij 6 x0io þ hi ;
i ¼ 1; . . . ; m
kj yrj P y0ro hþr ;
r ¼ 1; . . . ; s
j¼1 n X
ð13Þ
j¼1
1¼
n X
kj
j¼1
Fig. 1. Numerical example.
hi ; hþr ; kj P 0 Proof. Suppose on the contrary that /sup > 1 in model (2). ThereP kj P 0ðj–0Þ and 1 ¼ nj¼1 kj such fore, there exists k with ko ¼ 0; that
X
kj y > y rj ro
j–o
X
kj xij 6 xio
Table 1 Data for illustrative example. DMU
x1
y1
/o
/sup o
A B C D E
1 2 3 5 4
0.5 2 2 1 1
1 1 1 2 2
0 0.625 1 2 2
j–o
by definition of x0o and y0o we then have n X
kj y ¼ rj
X
j¼1
j–o
n X
X
j¼1
kj xij ¼
kj y > y ¼ y0 þ U1 ðaÞro rj ro ro ro kj xij 6 x0 U1 ðaÞrI io io
j–o
1 o letting hþ U1 ðaÞrIio , we find r ¼ U ðaÞrro ; r ¼ 1; . . . ; s and hi ¼ P Ps þ m h ; kÞ is a solution of (13) for which that ð hþ ; i¼1 hi þ r¼1 hr < Pm Ps þ þ h þ h , a contradiction to assumption that h , hi are r i¼1 i r¼1 r optimal for model (13). Therefore, we must have /sup 6 1 in the model (2). h
Table 2 ~ sup Computational results of model (9), / . o Possibility level, a
DMUC
DMUE
0.2 0.8
0.8264 1.266
1.4084 3.448
C0 : x0 ¼ 3 þ ð0:42Þ ¼ 3:42 and y0 ¼ 2 þ ð0:42Þ ¼ 1:58: ~ sup / is 1.266, therefore, C0 is inefficient, see Fig. 4 in Appendix A. C0 Furthermore, adjusted input–output for DMU E is E0 (4.42,0.58) with ~ sup / ¼ 3:448. Therefore, E0 is inefficient, see Fig. 5 in Appendix A. E0
4.1. Numerical example 5. Application Data and numerical results for five DMUs depicted in Fig. 1 are presented in Table 1. Columns 4 and 5 of Table 1 show the optimal value of the objective function of the BCC and output oriented super-efficiency models, respectively. In order to illustrate some of our theoretical results, only two cases with a ¼ 0:2 and a ¼ 0:8 are considered. It is also assumed that, rI ¼ ro ¼ 0:5. Computational results are presented in Table 2. Case 1. a ¼ 0:2. From a cumulative normal distribution table, we have U1 ¼ 0:84 and therefore rI U1 ðaÞ ¼ ro U1 ðaÞ ¼ 0:42 for use in the following example. Assume that only point C has random variations in its input and output. Based on (10) and (11), the adjusted input and output for point C with a ¼ 0:2 is
C0 : x0 ¼ 3 þ ð0:42Þ ¼ 2:58 and y0 ¼ 2 ð0:42Þ ¼ 2:42: ~ sup We have / ¼ 0:8264, therefore, C0 is super efficient. The condiC0 6 1. This is tion of Theorem 5 is satisfied, and from Table 1, /sup C consistent with Theorem 5, see Fig. 2 in Appendix A. Adjusted input–output for point E is also E0 (3.58,1.42). For E0 we have ~ sup / ¼ 1:4084. Therefore, E0 is inefficient, see Fig. 3 in Appendix E0 > 1 in Table 1. This is consistent with TheA. Note that for E, /sup E orem 2(ii). Case 2. a ¼ 0:8. In this case, we again have U1 ðaÞ ¼ 0:84 and therefore rI U1 ðaÞ ¼ ro U1 ðaÞ ¼ 0:42. Assume that only point C has random variations in its input and output. Based on (10) and (11), the adjusted input and output for point C with a ¼ 0:8 is
As an empirical example, the proposed method is applied using some actual data of year 2000 to Iranian electricity distribution units. The Iranian electricity distribution units established in 1992, are public and act under the supervision of TAVANIR company (Iran power, Generation, Transmission and Distribution Management company). According to the extensive review in Jamasb and Pollitt (2001), the most frequently used inputs are operating costs, number of employees transformer capacity and network length. The most widely useful outputs are also units of energy delivered, number of customers and size service area. The cost data usually is not available. This study used network length, transformer capacity, and employee variables as inputs and units delivery, and service area variables as outputs. The size of service area is considered as an environmental variable in our study. The service area is out of units control by nothing that the electricity distribution units are public and act in provinces in Iran. So, the units can-
Fig. 2. a ¼ 0:2, C has random variations in its input and output.
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M. Khodabakhshi / Computers & Industrial Engineering 58 (2010) 663–671
Fig. 3. a ¼ 0:2, E has random variations in its input and output.
1.582*(1806, 24,641). Therefore, it is inefficient in its current status. After Kermanshah, DMU 11, Lorestan, with score 1.423, DMU 7, Kurdistan, 1.376, DMU 1, Azarbaijan Sharghi, with score 1.361, DMU 15, Gilan, with score 1.327 are the worst companies in terms of efficiency. Inefficient companies have scores ranking from 1.143, Markazi, to 1.582, Kermanshah. Among inefficient companies DMU 10, 1.143 is the best one, but this DMU can still produce 1.143 times of its current outputs. To compute results for stochastic data, a ¼ 0:4 and a ¼ 0:6 have been chosen for which U1 ðaÞ 0:25 and U1 ðaÞ 0:25, respectively. These rather large values of a are deliberately chosen to illustrate differences between the results based on model (5) and model (2). It is worth noting that model (5) and model (2) produce similar results when a is small. The computational results of the equivalent deterministic problem are presented in columns 5 and 6 of Table 4. It is assumed that all DMUs have the same variance, but they can have different means. The variances for the outputs and the inputs can therefore be estimated by:
~r Þ ¼ Varðy Fig. 4. a ¼ 0:8, C has random variations in its input and output.
not change the value of this variable. The measurement units for the network length, transformers capacity and total electricity sales are Kilometer(KM), MVA and MWh, respectively. Table 3 presents data of seventeen companies used in Azadeh, Ghaderi, and Omrani (2008). Computational results of the deterministic output oriented super-efficiency model, model (2) and deterministic equivalent of the stochastic super-efficiency model, model (8), are presented in Table 4. Columns 3–4 and 5–6 of the Table represent super-efficiency score and rank of units for both deterministic and stochastic models, respectively. Note that a super-efficiency score less than 1 implies that the DMU is super efficient, scores equal to 1 imply they are just efficient; scores greater than 1 imply that the DMUs are inefficient. Therefore, the lower the super-efficiency score the better the company. Based on the numerical results presented in columns 3 and 4 of Table 4, DMU 3, Esfahan, DMU 5, Khozestan, DMU 8, Fars, DMU 9, Ardabil, DMU 12, Ghazvin, DMU 13, Semnan, DMU 17, Yazd for which /sup is less than one are super efficient with the model (2). The rest of companies for which /sup is greater than 1 are inefficient. DMU 13, Semnan, with super-efficiency score 0.133 is ranked the first. In fact, even if this unit produce 0.133 percent of its current outputs, i.e., 0.133*(1418, 96,816) in which the first component is units delivery and the second one is size service area, it still remains efficient with the deterministic model (1). Next top units are DMU 12, Ghazvin and DMU 5, Khozestan, with super-efficiency scores 0.528 and 0.568, respectively. The worst company is DMU 14, Kermanshah with score 1.582. This company can produce 1.582 times of its current outputs, i.e.,
Fig. 5. a ¼ 0:8, E has random variations in its input and output.
17 1 X ðy yr Þ2 16 j¼1 rj
and Varðx~i Þ ¼
17 1 X ðxij xi Þ2 16 j¼1
where
yr ¼
17 1 X y 17 j¼1 rj
and xi ¼
17 1 X xij 17 j¼1
and xij and yrj are the observed values of inputs and outputs for DMUj which are used as an estimate for the expected values of the stochastic inputs and outputs. It is also assumed that outputs and inputs for different DMUs are independent. This independence ~rk ; y ~rj Þ ¼ 0 and also assumption, then, implies that Cov ðy xik Þ ¼ 0. The stochastic super-efficiency scores obtained Cov ðx~ij ; ~ from GAMS software are presented in Table 4. Again, the best company is DMU 13, Semnan, with stochastic score 0.146(0.148) correspond to a ¼ 0:4ða ¼ 0:6Þ. The worst company is, again, DMU 14, Kermanshah with score 1.564(1.601). Numerical results of the stochastic super-efficiency model correspond to a ¼ 0:4 and a ¼ 0:6 which are presented in columns 4–5 and 6–7 show that both cases are quite similar. For example, DMU 3, Esfahan with a score 0.802(0.804) correspond to a ¼ 0:4 ða ¼ 0:6Þ, DMU 5, Khozestan with a score of 0.567(0.570), DMU 8, Fars, with a score of 0.912(0.926), DMU 9, Ardabil, with a score of 0.850(0.976), DMU 12, Ghazvin, with a score of 0.529(0.550), DMU 13, Semnan, with a score of 0.146(0.148)and DMU 17, Yazd with a score of 0.911(0.938) are stochastically efficient in both cases. These DMUs also were efficient in deterministic sense. However, the super-efficiency scores are not the same. Comparing numerical results presented in columns 3–4 and 5– 6(7–8) of Table 4, DMU 14, Kermanshah, and DMU 15, Gilan, DMU 10, Markazi, DMU 16, Hormozgan, DMU 4, Hamedan, DMU 6, Zanjan, DMU 2, Azarbaijan Gharbi, DMU 1, Azarbaijan Sharghi, DMU 7, Kurdistan, and DMU 11, Lorestan, are inefficient with both deterministic and stochastic models. Therefore, the computational results are quite identical. From computational results presented in columns 5–6 and 7–8 of Table 4, Semnan, DMU 13, with stochastic super-efficiency score 0.146 and 0.148 correspond to a ¼ 0:4 an a ¼ 0:6, respectively is ranked 1st by the stochastic super-efficiency model. Ghazvin, DMU 12, with a score of 0.529(0.550) correspond to a ¼ 0:4 ða ¼ 0:6Þ, is ranked the second. Another two top companies are Khozestan, DMU 5, with a score of 0.567(0.570), Esfahan, DMU 3, with a score of 0.802(0.804). At the bottom of ranking, the companies are Kermanshah, DMU 14, with a score of 1.564(1.601) and Lorestan, DMU 11 with a score of 1.402(1.445).
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M. Khodabakhshi / Computers & Industrial Engineering 58 (2010) 663–671
Table 3 Data for electricity distribution units. DMU
Company
Net. len. (km)
Tr. cap. (MVA)
Empl.
Un. del. (MW h)
S. area (km2)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Azarbaijan Sharghi Azarbaijan Gharbi Esfahan Hamedan Khozestan Zanjan Kurdistan Fars Ardabil Markazi Lorestan Ghazvin Semnan Kermanshah Gilan Hormozgan Yazd
15,151 19,610 25,566 12,340 19,380 10,347 11,697 24,634 10,129 14,505 12,078 8766 8063 12,795 21,187 16,185 11,990
990 1306 2713 1101 3932 788 725 1617 507 1489 1006 946 730 1147 1534 1786 963
867 1047 1072 595 1471 351 426 872 441 600 594 302 374 538 938 938 576
1825 2597 5835 2369 8048 1526 1174 4015 936 3063 1784 2414 1418 1806 2848 3411 2400
40,968 37,463 97,923 19,574 53,442 21,841 28,817 83,575 17,881 29,406 28,392 15,491 96,816 24,641 13,952 71,193 73,467
Table 4 Deterministic and stochastic super-efficiency scores for electricity distribution units. DMU
Company
Determ. score
Rank
Stocha. score, a ¼ 0:4
Rank
Stocha. score, a ¼ 0:6
Rank
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Azarbaijan Sharghi Azarbaijan Gharbi Esfahan Hamedan Khozestan Zanjan Kurdistan Fars Ardabil Markazi Lorestan Ghazvin Semnan Kermanshah Gilan Hormozgan Yazd
1.361 1.249 0.802 1.154 0.568 1.213 1.377 0.919 0.711 1.143 1.423 0.528 0.133 1.582 1.327 1.147 0.924
14 12 5 10 3 11 15 6 4 8 16 2 1 17 13 9 7
1.341 1.246 0.802 1.159 0.567 1.284 1.340 0.912 0.850 1.142 1.402 0.529 0.146 1.564 1.316 1.141 0.911
15 11 4 10 3 12 14 7 5 9 16 2 1 17 13 8 6
1.382 1.274 0.804 1.188 0.570 1.332 1.414 0.926 0.976 1.163 1.445 0.550 0.148 1.601 1.339 1.154 0.938
14 11 4 10 3 12 15 5 7 9 16 2 1 17 13 8 6
6. Conclusion
Acknowledgement
In this paper, super-efficiency issue is discussed in stochastic data envelopment analysis. We described its characterizations theoretically and empirically. In addition to developing stochastic version of the output oriented super-efficiency model, we obtained the deterministic equivalent of the stochastic version which can be converted to a quadratic problem. As an empirical example, the proposed approach was also applied to data of Iranian electricity distribution companies. Computational results of both deterministic and stochastic super-efficiency models show that DMU 13, Semnan is the best company in terms of technical efficiency. DMU 14, Kermanshah is, also, the worst company among seventeen companies studied in this paper. Although ranking results of both models for the evaluating companies are similar, the superefficiency scores are different. The stochastic model allows the data errors, therefore, if the data is rather inaccurate, and a rough estimate is required, the stochastic model might be preferred. However, if the data is accurate, the deterministic model will be preferred. Sensitivity analysis of the proposed super-efficiency model w. r. t. changes on parameter a which specifies significance level, also, has been discussed. Finally, developing the proposed model in fuzzy data envelopment analysis can be suggested for further research.
The author would like to thank three anonymous referees for their constructive comments. Appendix A Figs. 1–5 are related to numerical example of Section 4. When
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