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A super-efficiency model for ranking efficient units in data envelopment analysis
Applied Mathematics and Computation 184 (2007) 638–648 www.elsevier.com/locate/amc
A super-efficiency model for ranking efficient units in data envelopment analysis Shanling Li a, G.R. Jahanshahloo b, M. Khodabakhshi
c,*
a
b
Management Faculty of McGill University, Montreal, PQ, Canada H3A 1G5 Faculty of Mathematical Sciences and Computer Engineering, Teacher Training University, Tehran, Iran c Department of Mathematics, Faculty of Science, Lorestan University, Khorram Abad, Iran
Abstract Data envelopment analysis (DEA) is a body of research methodologies to evaluate overall efficiencies and identify the sources and estimate the amounts of inefficiencies in inputs and outputs. In DEA, the best performers are called DEA efficient and the efficiency score of a DEA efficient unit is denoted by an unity. In the last decade, ranking DEA efficient units has become the interests of many DEA researchers and a variety of models (called super-efficiency models) were developed to rank DEA efficient units. While the models developed in the past are interesting and meaningful, they have the disadvantages of being infeasible or instable occasionally. In this research, we develop a super-efficiency model to overcome some deficiencies in the earlier models. Both theoretical results and numerical examples are provided. 2006 Elsevier Inc. All rights reserved. Keywords: Data envelopment analysis; Super-efficiency; Ranking; Infeasibility; Instability
1. Introduction Data envelopment analysis was originated in 1978 by Charnes et al. [7] and the first DEA model was called CCR (Charnes, Cooper and Rhodes) model. The objective of DEA models is to evaluate overall efficiencies of decision making units (DMUs) that are responsible to convert a set of inputs into a set of outputs. Efficient DMUs are identified by an unity of 1.0 and inefficient DMUs have efficiency scores less than 1.0. When being evaluated, the efficiency score of a DMU is measured by the combination of a set of DEA efficient DMU(s), which form a part of the segments on the efficiency frontier. The efficient DMUs are not comparable among themselves in the CCR and other DEA models. In the last decade, some DEA researchers initiated a new area called super-efficiency to rank the DEA efficient DMUs and developed various models. Although the developed models are interesting and useful, in general, they have the drawbacks of lacking either stability or feasibility.
*
Corresponding author. E-mail address: [email protected] (M. Khodabakhshi).
0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.06.063
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In this paper, we propose a super-efficiency model to rank extreme DEA efficient DMUs that are identified by the CCR model. Our model is able to eliminate the drawbacks of earlier models such as the ones developed by Andersen and Petersen [2] and Mehrabian et al. [14]. The contributions of this research are several. First, we develop a super-efficiency model that is feasible and stable. Second, the proposed model is consistent with the models developed by Andersen and Petersen [2] in obtaining DEA efficient DMUs from the CCR model. Third, we compare the proposed model with Mehrabian et al.’s model [14] called MAJ model and show our model is superior to MAJ model. Forth, the model developed in this paper is simpler than the one recently developed by Tone [20] whose model needs to do extra steps to take care of zero inputs of the DMU being evaluating. Fifth, we point out that the SuperSBM (I) model of Tone [20] can also be infeasible. We present both theoretical and numerical results to illustrate the contributions of this paper. This paper is organized as follows. In Section 2, we provide a brief review of the literature in super-efficiency. In Section 3, we propose a super-efficiency model and present analytical results. The proposed model is compared with the models developed by Andersen and Petersen [2], Mehrabian et al. [14] and Tone [20]. In Section 4, we provide numerical examples to illustrate the differences. Section 5 discusses the future directions of this research. 2. Literature review As mentioned in the last section, the efficient DMUs obtained in most DEA models like CCR and BCC (Banker, Charnes and Cooper [3]) cannot be compared. In order to provide an overall assessment of the performances of all DMUs, ranking DMUs in DEA became an interesting topic in the last decade. According to Adler et al. [1], the research on ranking DMUs can be divided into 6 streams. We first briefly review the streams of research that are not directly related to this paper and then elaborate the stream that our research lies in. In the first stream, the research was pioneered by Sexton et al. [16]. In their research, the ranking of DMUs was based on a cross-efficiency ratio matrix. By comparing the efficiency scores of each DMU and its peers, a balanced ranking order of DMUs was developed. In the second stream, the ranking of DEA efficient DMUs is based on the benchmarking – an approach initially developed by Torgersen et al. [21]. They concluded that a DMU was highly ranked only if it was chosen as a reference by many other inefficient DMUs. In the third stream, researchers like Fridman and Sinuany-Stern [12], who initiated the research in this direction, used multi-variate statistical tools such as canonical correlation analysis and discriminant analysis to rank both efficient and inefficient DMUs. The research in the fourth stream focused on ranking inefficient DMUs. Bardhan et al. [4] developed an approach called measure of inefficiency dominance to rank the inefficient DMUs based on the average proportional inefficiency scores of all inputs and outputs. The research in the fifth stream combined DEA models with multi-criteria decision-making (MCDM) models to rank DMUs. The last yet most popular research stream in ranking DMUs is called super-efficiency. In this direction, researchers focused on ranking only DEA efficient DMUs based on the results obtained either from CCR or BCC models. The research in this area was first developed by Andesen and Petersen [2]. In their research, they ranked DEA efficient DMUs in such a way that superior DEA efficient DMUs may have efficiency scores greater than unity. Their approach became very popular and many research works extended their idea by addressing new issues such as outlier detection, sensitivity analysis and scale classification. Interesting research works in this area can be found in Charnes et al. [8], Thompson et al. [17], Durchholz [11], Bogetoft [6], Thrall [18], Zhu [23], Dula and Hickman [10], Sieford and Zhu [15], Mehrabian et al. [14], Xue and Harker [22] and Tone [20]. On the other hand, Thrall [18] pointed out that the model developed by Anderson and Petersen [2](called AP model) may result in infeasibility and instability when some inputs are close to zero. Similarly, Zhu [23] showed that when the constant-return-to-scale DEA models are used, the infeasibility could occur in the super-efficiency evaluation if and only if there is a zero in data. Recently, Mehrabian et al. [14] developed a super-efficiency model (called MAJ model) that does not have the drawbacks of infeasibility and instability as the AP model. However, the MAJ and the AP models used different ways to evaluate DEA efficiency scores. The DEA efficient DMUs in the AP model are obtained from the CCR model and that in the MAJ model are obtained through its own model. Another deficiency of the MAJ model is that at the optimal, all the inputs of
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the DMU being evaluated need to increase by the same variable. It is very difficult to explain the meaning of the variable. Xue and Harker [22] showed the necessary and sufficient conditions of infeasibility in super-efficiency evaluation when the BCC model with variable returns to scale is used. In ranking DEA efficiency DMUs, they identified four classes: (i) super-efficient, (ii) strongly efficient, (iii) efficient; and (iv) weakly efficient. Although their research results are rather insightful, Xue and Harker [22] did not provide new models to compute super-efficiency scores and rank DEA efficient DMUs. Recently, Tone [20] proposed a new super-efficiency model (called SuperSBM(I)) based on the measurement of slacks. Although Tone’s model did not have similar deficiencies as the AP and MAJ models but it can be difficult in ranking if any inputs of DMUs are zero. Adler et al. [1] provided the detailed discussion regarding the differences among the research works and the choices of the models in the above six areas. Overall, the research areas in super-efficiency and benchmarking are widely cited and have been applied in a wide range of settings such as financial institutions, industries, public regulations, education as well as health care. In this paper, we proposed a model that overcomes the disadvantages of the AP and MAJ models as discussed above. In addition, we provide a numerical example which shows the SuperSBM(I) model developed by Tone [20] can be infeasible. We also compare our model with SuperSBM(I), and show that our model is simpler than SuperSBM(I). However, our model allows zero inputs and does not need extra procedures to process zero inputs. 3. A super-efficiency model: LJK–CCR model In this section, we introduce a super-efficiency model called LJK (Li, Jahanshahloo and Khodabakhshi) model. Although the model is to rank extreme DEA efficient DMUs obtained by the CCR model, it can be used to evaluate efficient units directly. It means without solving the CCR model, one can rank efficient DMUs by solving just the super-efficiency model. Fore efficient DMUs have super-efficiency score greater than or equal to 1, while inefficient DMUs have super-efficiency score less than 1. We assume there are n homogeneous DMUs such that all the DMUs use m inputs xij (i = 1, . . . , m) to produce s outputs yrj (r = 1, . . . , s). We also assume that X j ¼ ðxij Þ 2 Rmn and Y j ¼ ðy rj Þ 2 Rsn are non-negative. The LJK model can be described as below: m 1 X sþ i2 Minimize 1 þ m i¼1 R i n X þ Subject to kj xij þ s i ¼ 1; . . . ; m; i1 si2 ¼ xi0 j¼1 j6¼0 n X
kj y rj sþ r ¼ y r0
r ¼ 1; . . . ; s;
j¼1 j6¼0 þ þ kj ; s i1 ; si2 ; sr P 0
i ¼ 1; . . . ; m; r ¼ 1; . . . ; s; j ¼ 1; . . . ; n;
where R i is maximum of all ith inputs including ith input of evaluating DMU i.e. Ri ¼ maxj ðxij Þ. Note that Ri is always positive because if Ri is zero, it means no DMU used the input i. For each DMU0 being evaluated, the objective of the LJK model is to minimize the unity plus the average ratio of the second input slacks over the maximum inputs among all DMUs. Given that the first and the second item are unitless, the objective function of the LJK model is unit invariant. The first constraint allows the input i of DMU0 to increase by sþ i2 or decrease by s i1 , see also [9,13]. One may suspect it is possible to use a free variable s instead of both slack þ variables, s i1 and si2 , in input constraint, while using free variable s lead to dual infeasibility of the super-efficiency model which causes problem to the model. Adding sþ i2 to input of DMU0 is also remove infeasibility problem in the super-efficiency model when some inputs of evaluating DMU are zero as will be discussed in numerical example of Section 4. Furthermore, one can interpret it as follows. ‘‘If we exclude (miss) DMU0, we have to use (pay) extra sþ i2 units of ith source such that a combination of the rest of DMUs can produce the output of excluding DMU that is yr0’’. The second constraint restricts that the output r of DMU0 can only increase by sþ r . Note that the non-negative linear combinations in both contraints 1 and 2 do not include DMU0.
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If the optimal objective value of LJK model is greater than 1, DMU0 that is DEA efficient in the CCR model is super-efficient in the LJK model. Otherwise, DMU0 is not super-efficient. Therefore, it is possible just solving super-efficiency model for ranking efficient units without solving the CCR model. The super-efficiency scores of the DMUs obtained by the LJK model can be ranked descendingly. Although the unity of 1.0 in the objective function is not necessary, we keep the unity for the sake of comparing the super-efficiency scores with those obtained by the AP, MAJ and SuperSBM(I) models. In what follows, we will show some important properties of the LJK model. 3.1. Analytical results of LJK model In this section, we provide some analytical results of the LJK model that will help understanding the model. þ Proposition 1. Between s i1 and si2 , only one of them can be positive in the optimal solution. þ Proof. It is easy to show that the vectors correspond to s i1 and si2 are linearly dependent. Thus by using simþ plex method, at most one of si1 and si2 is positive in the optimal solution. h þ þ þ Now denote s i1 and si2 ð8iÞ as the optimal amounts to si1 and si2 in the LJK model. Since si1 si2 ¼ 0, if þ is positive, DMU0 should increase its input i by si2 . Reversely, if si1 is positive, DMU0 should reduce its þ input i by s i1 . In the case of si2 ¼ 0 and si1 ¼ 0, the DMU being evaluated should not change its ith input amount. Note that at the optimal, it is possible that sþ i2 > 0 and sk1 > 0ði 6¼ kÞ. In what follows, we show that if there are two input–output combinations for DMU0, say (X0, Y0) and ðX 00 ; Y 0 Þ in which X 00 6 X 0 , then ðX 00 ; Y 0 Þ would not be ranked below (X0, Y0). In other words, if two DMUs have the same outputs, DMU with the lower inputs has better performance than the DMU with the higher inputs. This differentiation certainly implies that the super-efficiency model which is input oriented has real justification.
sþ i2
Proposition 2. Let X0 and X 00 be the two vectors to represent different input amounts of DMU0 in the LJK model with X 00 6 X 0 . If X 00 6 X 0 , the objective values of the LJK model corresponding to X 00 will be no less than that corresponding to X0. Proof. Consider following LJK model with X0 and X 00 as the two possible input vectors in the inequality constraints: m m X X sþ sþ i2 i2 1 Minimize 1 þ m1 Minimize 1 þ m R R i i i¼1 i¼1 n n X X 0 Subject to kj xij sþ i ¼ 1; . . . ; m; Subject to kj xij sþ i ¼ 1; . . . ; m; i2 6 xi0 ; i2 6 xi0 ; j¼1 j6¼0 n X
kj y rj P y r0 ;
r ¼ 1; . . . ; s;
j¼1 j6¼0 n X
: kj y rj P y r0 ;
j¼1 j6¼0
j¼1 j6¼0
kj ; s þ i2 P 0:
kj ; s þ i2 P 0
r ¼ 1; . . . ; s;
Let S and S 0 be the solution spaces of the above model corresponding to (X0, Y0) and ðX 00 ; Y 0 Þ respectively. We now show that S 0 S and the optimal value of objective function corresponding to S 0 will be no less than that 0 of S. Assume z ¼ ð k1 ; k2 ; . . . ; kn ; sþ sþ sþ 12 ; 22 ; . . . ; m2 Þ 2 S , then n X 0 kj xij sþ i ¼ 1; . . . ; m; i2 6 xi0 j¼1 j6¼0 n X
kj y rj P y r0
j¼1 j6¼0
kj ; sþ i2 P 0
r ¼ 1; . . . ; s;
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but x0i0 6 xi0 implies that n X 0 kj xij sþ i2 6 xi0 6 xi0 ; j¼1 j6¼0 n X
kj y rj P y r0 ;
i ¼ 1; . . . ; m;
r ¼ 1; . . . ; s;
j¼1 j6¼0
kj ; sþ i2 P 0; that is z 2 S therefore S 0 S and the proof is complete.
h
As it will be shown in next section, LJK model does not have the weaknesses of the AP model: (a) instability when some inputs are close to zero and (b) infeasibility (see Appendix A for the AP model). In addition, we will compare our model with the MAJ model in the next section. 3.2. Comparison with the MAJ model Mehrabian et al. [14] developed the MAJ model (see Appendix A for the model) to address the infeasibility and instability of the AP model. They assumed the data of both inputs and outputs should be non-negative. Their model first evaluates the DEA efficiencies and then ranks the DEA efficient DMUs according to the super-efficiency scores obtained by the model. The MAJ model requires a necessary and sufficient condition for the feasibility as below. Theorem. The MAJ model is feasible for the evaluation of DMU0 with output vector Y0 P 0 if and only if for each r, r = 1, . . . , s, either yr0 = 0 or there exists a DMUj, j 5 o such that yrj 5 0. Proof. Refer to Mehrabian et al. [14].
h
Proposition 3. The LJK model is also feasible under the same conditions that the MAJ model is feasible. Proof. For proving the feasibility of the LJK model we study its dual model, D-LJK, which follows: s m X X Maximize 1 þ lr y r0 vi xi0 r¼1
Subject to
s X
lr y rj
r¼1
i¼1 m X
vi xij 6 0;
j 6¼ 0; j ¼ 1; . . . ; n;
i¼1
1 ; mR i lr ; vi P 0;
vi 6
i ¼ 1; . . . ; m
ðD-LJKÞ;
i ¼ 1; . . . ; m; r ¼ 1; . . . ; s;
lr = 0("r), vi = 0("i) is a solution for the above model. We prove that the D-LJK model cannot be unbounded under the assumption of the previous Theorem, as a result the LJK model will be feasible under the same assumption. Note that we can exclude unity of 1 from the objective function of the LJK model. If the DLJK is unbounded, we must have: dv 9d ¼ “0 dl so that dl Y 0 dv X 0 > 0; dl Y j dv X j 6 0; dv 6 0; dl ; dv P 0:
j 6¼ 0;
j ¼ 1; . . . ; n;
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As a result it should be dv ¼ 0 and dl Y 0 > 0; dl Y j 6 0; j 6¼ 0; j ¼ 1; . . . ; n; dl P 0: Based on the assumption of the MAJ Theorem for all r, r = 1, . . ., s, either yr0 = 0, or there exist a (j 5 0) so that yrj > 0. Suppose that yr0 = 0, r = 1, . . . , k, and yr0 > 0, r = k + 1, . . . , s. Thus, for each r, r = k + 1, . . . , s there exist a jr 5 0 such that y rjr > 0. Therefore, dl Y jr ¼ 0 implies that (dl)r = 0, r = k + 1, . . ., s. Hence, dl Y 0 ¼
s X
ðdl Þr y r0 ¼
r¼1
k s X X ðdl Þr 0 þ 0 y r0 ¼ 0: r¼1
r¼kþ1
This contradiction shows that there is no such direction, see [5]. Therefore, the assumption of unboundedness for the D-LJK model is false. Consequently, the LJK envelopment model is feasible. On the other hand, if there exist an r such that yr0 > 0 and "j(j 5 o)yrj = 0, the rth output constraint will be inconsistent. Therefore, the LJK model is infeasible. h In comparison with the MAJ model, we now highlight some merits of the LJK model. Proposition 4. The LJK super-efficiency score is unit invariant. Proof. In the second item of the objective function of the LJK model, the numerators and denominators have identical units, the second item is unitless. Therefore, the objective function value is unit invariant. h To illustrate another drawback of the MAJ model, we now present their model as below. We note that each input constraint in the MAJ model has a w0 at the right side. This means, if w0 is positive at the optimal, each input should be increased by w0. For various units of inputs, the same w0 is added to each input, which makes the explanation of the physical meaning of w0 becomes difficult. For example, suppose one input represents the number of employees and the other represents capital. Then when w0 is added to each input, the unit of w0 has to change accordingly. We now present the LJK model together with the MAJ model to illustrate one more property. The LJK model
The MAJ model m X sþ i2 Minimize 1 þ w0 Minimize 1 þ R i i¼1 n n X X Subject to kj xij 6 xi0 þ sþ ; i ¼ 1; . . . ; m; Subject to kj xij 6 xi0 þ w0 ; i2 1 m
j¼1 j6¼0 n X
i ¼ 1; . . . ; m;
j¼1 j6¼0
kj y rj P y r0 ;
r ¼ 1; . . . ; s;
n X
kj y rj P y r0 ;
j¼1 j6¼0
j¼1 j6¼0
kj ; s þ i2 P 0:
kj P 0:
r ¼ 1; . . . ; s;
0 Þ be the optimal solutions to LJK and the MAJ modLet and ð k1 ; k2 ; . . . ; kn ; w els respectively. Comparing the right side constraints in both models implies following property. þ þ ðk1 ; k2 ; . . . ; kn ; sþ 12 ; s22 ; . . . ; sm2 Þ
0 , while in LJK Remark. In the MAJ model, all inputs must be changed by a fixed value of the variable of w þ þ model the inputs change by sþ ; s ; . . . ; s . 12 22 m2 3.3. LJK model and SuperSBM (I) model Tone [20] introduced two models called SuperSBM (I) and (O) that rank the DEA efficient DMUs obtained from the SBM model [19] or equivalently the CCR Model. One of the model SuperSBM (I), is input oriented
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(the objective function is composed by the inputs only.). Since the LJK model is also input oriented, we now compare the SuperSBM (I) model and the LJK model. The SuperSBM (I) model is presented as below: m xi0 1 X dI ¼ min m i¼1 xi0 n X Subject to X P kj X j ; j¼1 j6¼0
Y 6
n X
kj Y j ;
j¼1 j6¼0
X P X0
and
Y ¼ Y 0;
k P 0: In what follows, we show that the SuperSBM(I) can be infeasible. In fact, by the following illustrative example we show that the feasibility of the SuperSBM(I) is depended on the output of the evaluating DMU. 3.3.1. Illustrative example Consider DMUs A, B and C for which data are presented in Table 1. Each DMU uses one input to produce two outputs. We evaluate DMU A by the SuperSBM(I) model: dI ¼ min xA Subject to 2kB þ 3kC 6 xA ; 2kB þ 1kC P 1; 0kB þ 0kC P 1=2; xA P 1; kB ; kC P 0: Obviously, since the second output constraint is inconsistent, the above problem is infeasible. In what follows, we want to show that when there is no zero input in any DMU and define R i ¼ xi0 , the results from the LJK model and the SuperSBM (I) are identical, but the LJK model has some advantages over the SuperSBM (I) model in general. Proposition 5. The LJK and the SuperSBM (I) models provide the same results in ranking DEA efficient DMUs, if R i ¼ xi0 for which xi0 > 0 "i. Proof. If we replace X by X0 + S2 for which S2 is non-negative, the two models are equivalent. Therefore, in this case, the super-efficiency ratios and the ranking results for both the LJK and the SuperSBM (I) models are the same. Because in the SuperSBM(I), the objective function is as below: m m xi0 1 X 1 X sþ i2 ¼1þ ; dI ¼ min m i¼1 xi0 m i¼1 xi0 where xi0 instead of R i is used.
h
Table 1 Data for illustrative example DMUs
A
B
C
Input Output 1 Output 1
1 1 1/2
2 2 0
3 1 0
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Table 2 Data for comparison test DMUs
A1
A2
A3
B
C
D
E
Input 1 Input 2 Output 1 Output 2
2 8 1 2
0 8 1 2
0.1 8 1 2
5 5 1 1
10 4 2 1
10 6 2 1
2 12 1 2
Table 3 Computational results DMUs
CCR (%)
AP (%)
MAJ (%)
SuperSBM(I) (%)
LJK (%)
A1 A2 A3
100 100 100
147 Infeasible 2000
128 131 131
125 Indefinable 1075
117 127 126
Note that in the SuperSBM (I), if any of the inputs is zero, the objective function of the model cannot be defined since the denominator of the second term in the objective function will become zero. Although Tone [20], extended his model to consider the DMUs with zero inputs, the LJK model with inequality constraints is more direct and easier to use than the SuperSBM(I) model which requires additional work for zero inputs. 4. Numerical example In this section, we illustrate a numerical example by comparing the super-efficiency results obtained by AP, MAJ, SuperSBM(I) and the LJK models. The numerical example shows that while the results from the AP and SuperSBM(I) models can be infeasible or instable, the LJK model is feasible and stable. Table 2 provides the input and output data of 7 DMUs each of which uses two inputs and two outputs. A1, A2 and A3 are evaluated by the AP, MAJ, SuperSBM(I) and LJK models and their results are compared with each other. Every time one of the DMUs is compared with the other DMUs (B,C,D and E). The numerical example is from Meharabian et al. [14]. Since AP, SuperSBM(I) and LJK models are based on the results from the CCR model, the DMUs in Table 3 are DEA efficient in the CCR model. In Table 2, we notice that since input 1 of A2 is zero, it results in infeasibility in the AP model in Table 3. With the SuperSBM(I) model also without extra step it is impossible to obtain super-efficiency score of A2. In fact, since denominator in the objective function of the SuperSBM(I) is zero, the objective function cannot be defined in its current form. However, in Table 3 we also observe that the LJK model indicates that A2 is feasible with a super-efficiency score of 1.27. Therefore, one may find the LJK model simpler than the SuperSBM(I) model which need extra steps for evaluating DMUs with zero inputs. In addition, since Input 1 of A3 is close to zero (see Table 2), the super-efficiency scores of A3 in the AP and SuperSBM(I) models are as large as 20 and 10.75 respectively, while that of the LJK model is only 1.26. Note that with the MAJ model, A2 and A3 have super-efficiency scores of 1.31. However, the MAJ cannot differentiate A2 and A3, while A2 has better performance than A3. Because, although A2 and A3 produce the same outputs, A2 uses lower inputs than that of A3. Therefore, LJK model can be preferred to the MAJ model from accuracy point of view. See, also, Appendix B in which a real data set is used to compare the models. 5. Conclusion In this paper, we propose a super-efficiency model that does not have the weakness in earlier models. We first present several properties of our model and then compare our model with the models developed by Anderson and Peterson [2] and Mehrabian et al. [14]. We showed that our model is superior to these two models in removing the deficiencies. In addition, we also compare our model with SuperSBM(I) developed by Tone [20] and show that our model is simpler and easier to solve than SuperSBM(I). Moreover, we pointed out that
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the SuperSBM(I) can be infeasible. The research in this paper can be extended in different directions. First, we are in the process of developing a model to evaluate all DMUs instead of only DEA efficient DMUs. Second, we plan to apply our model to real data to explore the importance of our model. Acknowledgement We are also grateful to professor K. Tone, Dr. S. Mehrabian and Dr. F. H. Lotfi for their suggestions on the previous version of this paper. Appendix A The models related to the AP and MAJ approaches,which are defined on the non-negative inputs and outputs, are as follows: The AP model
The MAJ model
a0
j0 ¼ min
¼ min
Subject to
a0 X
kj X j 6 a0 X 0 ;
Subject to
j6¼0
X
w0 þ 1 X kj X j 6 X 0 þ w0 1; j6¼0
kj Y j P Y 0 ;
X
j6¼0
kj P 0;
kj Y j P Y 0 ;
j6¼0
j ¼ 1; . . . ; n:
kj P 0;
j ¼ 1; . . . ; n:
The AP model ranks the efficient DMUs under the CCR model while the MAJ model ranks the efficient DMUs under the MAJ including the evaluating DMU model.
Table 4 Data for 19 academic units of the UTE in the first semester, 1993–94 DMU
Department/Institute
I1
Faculty of Literature Persian Literature Theology and Islamic Culture History Geography Foreign Language Arabic Language and Literature Social Sciences
81 85 56.7 91 216 58 112.2
8 9 10 11 12 13 14
Faculty of Physical Education Men Physical Education Women Physical Education Mathematics Geology Biology Chemistry Physics
15 16 17 18 19
Faculty of Education foundations of Education Instructional Technology Psychology guidance and Counseling Institute of Mathematics
1 2 3 4 5 6 7
I2
O1
O2
87.6 12.8 55.2 78.8 72 25.6 8.8
5191 3629 3302 3379 5368 1674 2350
205 0 0 8 639 0 0
293.2 186 143.4 108.7 105.7 235 146.3
52 0 105.2 127 134.4 236.8 124
6315 2865 7689 2165 3963 6643 4611
414 0 66 266 315 236 128
57 118.7 58 146 0
203 48.2 47.4 50.8 91.3
4869 3313 1853 4578 0
540 16 230 217 508
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Table 5 Super-efficiency results for efficient units DMU 19 5 2 1 15 9
AP
MAJ
SuperSBM(I)
LJK
* 1.30 1.74 1.15 1.33 *
1.28 1.1 1.09 1.05 1.06 1.04
* 1.35 1.55 1.12 1.73 *
1.24 1.072 1.065 1.033 1.032 1.021
Appendix B. Case study Mehrabian et al. [14] evaluated teaching for 19 academic units in University for Teacher Education. Each unit has two inputs and two outputs which are presented in Table 4. Teaching inputs are expressed in teacher hours and classified in terms of two inputs, professional staff and instructors. Teaching outputs also are expressed in student hours and classified in terms of two outputs, course enrollments in undergraduate and graduate studies. There are six efficient departments, and ranking results are presented in Table 5. Based on the results, the AP model is infeasible for Department of Women’s Physical Education, DMU 9, and Institute of Mathematics, DMU 19, which have zero inputs. These DMUs are indicated by asterisks in Table 5. In addition, the SuperSBM(I) model is not definable for these two units in its current form. However, these DMUs have explicit super-efficiency scores and definite rank by the LJK model. Note that the ranking results for both LJK and MAJ models are somewhat similar. Based on the results of the LJK model (the last column in Table 5), Institute of Mathematics, DMU 19, has the first rank, while Women Physical Education, DMU 9, has the last rank. For the rest of DMUs the higher the super-efficiency score the better the DMU. References [1] N. Adler, L. Friedman, Z. Sinuany-Stern, Review of ranking methods in the data envelopment analysis context, European Journal of Operational Research 140 (2002) 249–265. [2] P. Andersen, N.C. Petersen, A procedure for ranking efficient units in data envelopment analysis, Management Science 39 (10) (1993) 1261–1264. [3] R.D. Banker, A. Charnes, W.W. Cooper, Some models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science 30 (1984) 1078–1092. [4] I. Bardhan, W.F. Bowlin, W.W. Cooper, T. Sueyoshi, Model for efficiency dominance in data envelopment analysis. Part I: Additive models and MED measures, Journal of the Operations Research Society of Japan 39 (1996) 322–332. [5] M.S. Bazaraa, J.J. Jarvis, H.D. Sherali, Linear Programming and Networks Flows, John Wiley & Sons, 1990. [6] P. Bogetoft, Incentive efficient production frontiers: an agency perspective on DEA, Management Science 40 (1994) 959–968. [7] A. Charnes, W.W. Cooper, E. Rhodes, Measuring the efficiencies of DMUs, European Journal of Operational Research 2 (6) (1978) 429–444. [8] A. Charnes, S. Haag, P. Jaska, J. Semple, Sensitivity of efficiency classifications in the additive model of data envelopment analysis, International Journal of System Science 23 (1992) 789–798. [9] G.R. Jahanshahloo, M. Khodabakhshi, Suitable combination of inputs for improving outputs in DEA with determining input congestion—Considering textile industry of China, Applied Mathematics and Computation 151 (1) (2004) 263–273. [10] J.H. Dula, B.L. Hickman, Effects of excluding the column being scored from the DEA envelopment LP technology matrix, The Journal of the Operational Research Society 48 (1997) 1001–1012. [11] M.L. Durchholz, Large-scale data envelopment analysis models and related applications, Ph.D. Thesis, Department of Computer Science and Engineering, Southern Methodist University, Dallas, TX 75275, 1994. [12] L. Fridman, Z. Sinuany-Stern, Scaling units via the canonical correlation analysis and the data envelopment analysis, European Journal of Operational Research 100 (3) (1997) 629–637. [13] G.R. Jahanshahloo, M. Khodabakhshi, Determining assurance interval for non-Archimedean element in the improving outputs model in DEA, Applied Mathematics and Computation 151 (2) (2004) 501–506. [14] S. Mehrabian, A. Alirezaee, G.R. Jahanshahloo, A complete efficiency ranking of decision making units in DEA, Computational Optimization and Applications (COAP) 14 (1999) 261–266. [15] L.M. Seiford, J. Zhu, Infeasibility of super-efficiency data envelopment analysis, INFOR 37 (2) (1999) 174–187. [16] T.R. Sexton, R.H. Silkman, A.J. Hogan, Data envelopment analysis: critique and extensions, in: R.H. Silkman (Ed.), Measuring Efficiency: An Assessment of Data Envelopment Analysis, Jossey-Bass, San Fransisco, CA, 1986, pp. 73–105.
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[17] R.G. Thompson, P.S. Darmapalah, R.M. Thrall, Importance for DEA of zeros in data, multipliers, and solutions, The Journal of Productivity Analysis 4 (1993) 379–390. [18] R.M. Thrall, Duality, classification and slacks in data envelopment analysis, The Annals of Operations Research 66 (1996) 109–138. [19] K. Tone, A slakes-based measure of efficiency in data envelopment analysis, European Journal of Operational Research 130 (2001) 498–509. [20] K. Tone, A slackes-based measure of super-efficiency in data envelopment analysis, European Journal of Operational Research, in press. [21] A.M. Torgersen, F.R. Forsund, S.A.C. Kittelsen, Slack-adjusted efficiency measures and ranking of efficient units, The Journal of Productivity Analysis 7 (1996) 379–398. [22] M. Xue, P.T. Harker, Note: Ranking DMUs with infeasible super-efficiency DEA models, Management Sciences 48 (5) (2002) 705– 710. [23] J. Zhu, Robustness of the efficient decision making units in data envelopment analysis, European Journal of Operational Research 90 (1996) 451–460.
A super-efficiency model for ranking efficient units in data envelopment analysis Shanling Li a, G.R. Jahanshahloo b, M. Khodabakhshi
c,*
a
b
Management Faculty of McGill University, Montreal, PQ, Canada H3A 1G5 Faculty of Mathematical Sciences and Computer Engineering, Teacher Training University, Tehran, Iran c Department of Mathematics, Faculty of Science, Lorestan University, Khorram Abad, Iran
Abstract Data envelopment analysis (DEA) is a body of research methodologies to evaluate overall efficiencies and identify the sources and estimate the amounts of inefficiencies in inputs and outputs. In DEA, the best performers are called DEA efficient and the efficiency score of a DEA efficient unit is denoted by an unity. In the last decade, ranking DEA efficient units has become the interests of many DEA researchers and a variety of models (called super-efficiency models) were developed to rank DEA efficient units. While the models developed in the past are interesting and meaningful, they have the disadvantages of being infeasible or instable occasionally. In this research, we develop a super-efficiency model to overcome some deficiencies in the earlier models. Both theoretical results and numerical examples are provided. 2006 Elsevier Inc. All rights reserved. Keywords: Data envelopment analysis; Super-efficiency; Ranking; Infeasibility; Instability
1. Introduction Data envelopment analysis was originated in 1978 by Charnes et al. [7] and the first DEA model was called CCR (Charnes, Cooper and Rhodes) model. The objective of DEA models is to evaluate overall efficiencies of decision making units (DMUs) that are responsible to convert a set of inputs into a set of outputs. Efficient DMUs are identified by an unity of 1.0 and inefficient DMUs have efficiency scores less than 1.0. When being evaluated, the efficiency score of a DMU is measured by the combination of a set of DEA efficient DMU(s), which form a part of the segments on the efficiency frontier. The efficient DMUs are not comparable among themselves in the CCR and other DEA models. In the last decade, some DEA researchers initiated a new area called super-efficiency to rank the DEA efficient DMUs and developed various models. Although the developed models are interesting and useful, in general, they have the drawbacks of lacking either stability or feasibility.
*
Corresponding author. E-mail address: [email protected] (M. Khodabakhshi).
0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.06.063
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In this paper, we propose a super-efficiency model to rank extreme DEA efficient DMUs that are identified by the CCR model. Our model is able to eliminate the drawbacks of earlier models such as the ones developed by Andersen and Petersen [2] and Mehrabian et al. [14]. The contributions of this research are several. First, we develop a super-efficiency model that is feasible and stable. Second, the proposed model is consistent with the models developed by Andersen and Petersen [2] in obtaining DEA efficient DMUs from the CCR model. Third, we compare the proposed model with Mehrabian et al.’s model [14] called MAJ model and show our model is superior to MAJ model. Forth, the model developed in this paper is simpler than the one recently developed by Tone [20] whose model needs to do extra steps to take care of zero inputs of the DMU being evaluating. Fifth, we point out that the SuperSBM (I) model of Tone [20] can also be infeasible. We present both theoretical and numerical results to illustrate the contributions of this paper. This paper is organized as follows. In Section 2, we provide a brief review of the literature in super-efficiency. In Section 3, we propose a super-efficiency model and present analytical results. The proposed model is compared with the models developed by Andersen and Petersen [2], Mehrabian et al. [14] and Tone [20]. In Section 4, we provide numerical examples to illustrate the differences. Section 5 discusses the future directions of this research. 2. Literature review As mentioned in the last section, the efficient DMUs obtained in most DEA models like CCR and BCC (Banker, Charnes and Cooper [3]) cannot be compared. In order to provide an overall assessment of the performances of all DMUs, ranking DMUs in DEA became an interesting topic in the last decade. According to Adler et al. [1], the research on ranking DMUs can be divided into 6 streams. We first briefly review the streams of research that are not directly related to this paper and then elaborate the stream that our research lies in. In the first stream, the research was pioneered by Sexton et al. [16]. In their research, the ranking of DMUs was based on a cross-efficiency ratio matrix. By comparing the efficiency scores of each DMU and its peers, a balanced ranking order of DMUs was developed. In the second stream, the ranking of DEA efficient DMUs is based on the benchmarking – an approach initially developed by Torgersen et al. [21]. They concluded that a DMU was highly ranked only if it was chosen as a reference by many other inefficient DMUs. In the third stream, researchers like Fridman and Sinuany-Stern [12], who initiated the research in this direction, used multi-variate statistical tools such as canonical correlation analysis and discriminant analysis to rank both efficient and inefficient DMUs. The research in the fourth stream focused on ranking inefficient DMUs. Bardhan et al. [4] developed an approach called measure of inefficiency dominance to rank the inefficient DMUs based on the average proportional inefficiency scores of all inputs and outputs. The research in the fifth stream combined DEA models with multi-criteria decision-making (MCDM) models to rank DMUs. The last yet most popular research stream in ranking DMUs is called super-efficiency. In this direction, researchers focused on ranking only DEA efficient DMUs based on the results obtained either from CCR or BCC models. The research in this area was first developed by Andesen and Petersen [2]. In their research, they ranked DEA efficient DMUs in such a way that superior DEA efficient DMUs may have efficiency scores greater than unity. Their approach became very popular and many research works extended their idea by addressing new issues such as outlier detection, sensitivity analysis and scale classification. Interesting research works in this area can be found in Charnes et al. [8], Thompson et al. [17], Durchholz [11], Bogetoft [6], Thrall [18], Zhu [23], Dula and Hickman [10], Sieford and Zhu [15], Mehrabian et al. [14], Xue and Harker [22] and Tone [20]. On the other hand, Thrall [18] pointed out that the model developed by Anderson and Petersen [2](called AP model) may result in infeasibility and instability when some inputs are close to zero. Similarly, Zhu [23] showed that when the constant-return-to-scale DEA models are used, the infeasibility could occur in the super-efficiency evaluation if and only if there is a zero in data. Recently, Mehrabian et al. [14] developed a super-efficiency model (called MAJ model) that does not have the drawbacks of infeasibility and instability as the AP model. However, the MAJ and the AP models used different ways to evaluate DEA efficiency scores. The DEA efficient DMUs in the AP model are obtained from the CCR model and that in the MAJ model are obtained through its own model. Another deficiency of the MAJ model is that at the optimal, all the inputs of
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the DMU being evaluated need to increase by the same variable. It is very difficult to explain the meaning of the variable. Xue and Harker [22] showed the necessary and sufficient conditions of infeasibility in super-efficiency evaluation when the BCC model with variable returns to scale is used. In ranking DEA efficiency DMUs, they identified four classes: (i) super-efficient, (ii) strongly efficient, (iii) efficient; and (iv) weakly efficient. Although their research results are rather insightful, Xue and Harker [22] did not provide new models to compute super-efficiency scores and rank DEA efficient DMUs. Recently, Tone [20] proposed a new super-efficiency model (called SuperSBM(I)) based on the measurement of slacks. Although Tone’s model did not have similar deficiencies as the AP and MAJ models but it can be difficult in ranking if any inputs of DMUs are zero. Adler et al. [1] provided the detailed discussion regarding the differences among the research works and the choices of the models in the above six areas. Overall, the research areas in super-efficiency and benchmarking are widely cited and have been applied in a wide range of settings such as financial institutions, industries, public regulations, education as well as health care. In this paper, we proposed a model that overcomes the disadvantages of the AP and MAJ models as discussed above. In addition, we provide a numerical example which shows the SuperSBM(I) model developed by Tone [20] can be infeasible. We also compare our model with SuperSBM(I), and show that our model is simpler than SuperSBM(I). However, our model allows zero inputs and does not need extra procedures to process zero inputs. 3. A super-efficiency model: LJK–CCR model In this section, we introduce a super-efficiency model called LJK (Li, Jahanshahloo and Khodabakhshi) model. Although the model is to rank extreme DEA efficient DMUs obtained by the CCR model, it can be used to evaluate efficient units directly. It means without solving the CCR model, one can rank efficient DMUs by solving just the super-efficiency model. Fore efficient DMUs have super-efficiency score greater than or equal to 1, while inefficient DMUs have super-efficiency score less than 1. We assume there are n homogeneous DMUs such that all the DMUs use m inputs xij (i = 1, . . . , m) to produce s outputs yrj (r = 1, . . . , s). We also assume that X j ¼ ðxij Þ 2 Rmn and Y j ¼ ðy rj Þ 2 Rsn are non-negative. The LJK model can be described as below: m 1 X sþ i2 Minimize 1 þ m i¼1 R i n X þ Subject to kj xij þ s i ¼ 1; . . . ; m; i1 si2 ¼ xi0 j¼1 j6¼0 n X
kj y rj sþ r ¼ y r0
r ¼ 1; . . . ; s;
j¼1 j6¼0 þ þ kj ; s i1 ; si2 ; sr P 0
i ¼ 1; . . . ; m; r ¼ 1; . . . ; s; j ¼ 1; . . . ; n;
where R i is maximum of all ith inputs including ith input of evaluating DMU i.e. Ri ¼ maxj ðxij Þ. Note that Ri is always positive because if Ri is zero, it means no DMU used the input i. For each DMU0 being evaluated, the objective of the LJK model is to minimize the unity plus the average ratio of the second input slacks over the maximum inputs among all DMUs. Given that the first and the second item are unitless, the objective function of the LJK model is unit invariant. The first constraint allows the input i of DMU0 to increase by sþ i2 or decrease by s i1 , see also [9,13]. One may suspect it is possible to use a free variable s instead of both slack þ variables, s i1 and si2 , in input constraint, while using free variable s lead to dual infeasibility of the super-efficiency model which causes problem to the model. Adding sþ i2 to input of DMU0 is also remove infeasibility problem in the super-efficiency model when some inputs of evaluating DMU are zero as will be discussed in numerical example of Section 4. Furthermore, one can interpret it as follows. ‘‘If we exclude (miss) DMU0, we have to use (pay) extra sþ i2 units of ith source such that a combination of the rest of DMUs can produce the output of excluding DMU that is yr0’’. The second constraint restricts that the output r of DMU0 can only increase by sþ r . Note that the non-negative linear combinations in both contraints 1 and 2 do not include DMU0.
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If the optimal objective value of LJK model is greater than 1, DMU0 that is DEA efficient in the CCR model is super-efficient in the LJK model. Otherwise, DMU0 is not super-efficient. Therefore, it is possible just solving super-efficiency model for ranking efficient units without solving the CCR model. The super-efficiency scores of the DMUs obtained by the LJK model can be ranked descendingly. Although the unity of 1.0 in the objective function is not necessary, we keep the unity for the sake of comparing the super-efficiency scores with those obtained by the AP, MAJ and SuperSBM(I) models. In what follows, we will show some important properties of the LJK model. 3.1. Analytical results of LJK model In this section, we provide some analytical results of the LJK model that will help understanding the model. þ Proposition 1. Between s i1 and si2 , only one of them can be positive in the optimal solution. þ Proof. It is easy to show that the vectors correspond to s i1 and si2 are linearly dependent. Thus by using simþ plex method, at most one of si1 and si2 is positive in the optimal solution. h þ þ þ Now denote s i1 and si2 ð8iÞ as the optimal amounts to si1 and si2 in the LJK model. Since si1 si2 ¼ 0, if þ is positive, DMU0 should increase its input i by si2 . Reversely, if si1 is positive, DMU0 should reduce its þ input i by s i1 . In the case of si2 ¼ 0 and si1 ¼ 0, the DMU being evaluated should not change its ith input amount. Note that at the optimal, it is possible that sþ i2 > 0 and sk1 > 0ði 6¼ kÞ. In what follows, we show that if there are two input–output combinations for DMU0, say (X0, Y0) and ðX 00 ; Y 0 Þ in which X 00 6 X 0 , then ðX 00 ; Y 0 Þ would not be ranked below (X0, Y0). In other words, if two DMUs have the same outputs, DMU with the lower inputs has better performance than the DMU with the higher inputs. This differentiation certainly implies that the super-efficiency model which is input oriented has real justification.
sþ i2
Proposition 2. Let X0 and X 00 be the two vectors to represent different input amounts of DMU0 in the LJK model with X 00 6 X 0 . If X 00 6 X 0 , the objective values of the LJK model corresponding to X 00 will be no less than that corresponding to X0. Proof. Consider following LJK model with X0 and X 00 as the two possible input vectors in the inequality constraints: m m X X sþ sþ i2 i2 1 Minimize 1 þ m1 Minimize 1 þ m R R i i i¼1 i¼1 n n X X 0 Subject to kj xij sþ i ¼ 1; . . . ; m; Subject to kj xij sþ i ¼ 1; . . . ; m; i2 6 xi0 ; i2 6 xi0 ; j¼1 j6¼0 n X
kj y rj P y r0 ;
r ¼ 1; . . . ; s;
j¼1 j6¼0 n X
: kj y rj P y r0 ;
j¼1 j6¼0
j¼1 j6¼0
kj ; s þ i2 P 0:
kj ; s þ i2 P 0
r ¼ 1; . . . ; s;
Let S and S 0 be the solution spaces of the above model corresponding to (X0, Y0) and ðX 00 ; Y 0 Þ respectively. We now show that S 0 S and the optimal value of objective function corresponding to S 0 will be no less than that 0 of S. Assume z ¼ ð k1 ; k2 ; . . . ; kn ; sþ sþ sþ 12 ; 22 ; . . . ; m2 Þ 2 S , then n X 0 kj xij sþ i ¼ 1; . . . ; m; i2 6 xi0 j¼1 j6¼0 n X
kj y rj P y r0
j¼1 j6¼0
kj ; sþ i2 P 0
r ¼ 1; . . . ; s;
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but x0i0 6 xi0 implies that n X 0 kj xij sþ i2 6 xi0 6 xi0 ; j¼1 j6¼0 n X
kj y rj P y r0 ;
i ¼ 1; . . . ; m;
r ¼ 1; . . . ; s;
j¼1 j6¼0
kj ; sþ i2 P 0; that is z 2 S therefore S 0 S and the proof is complete.
h
As it will be shown in next section, LJK model does not have the weaknesses of the AP model: (a) instability when some inputs are close to zero and (b) infeasibility (see Appendix A for the AP model). In addition, we will compare our model with the MAJ model in the next section. 3.2. Comparison with the MAJ model Mehrabian et al. [14] developed the MAJ model (see Appendix A for the model) to address the infeasibility and instability of the AP model. They assumed the data of both inputs and outputs should be non-negative. Their model first evaluates the DEA efficiencies and then ranks the DEA efficient DMUs according to the super-efficiency scores obtained by the model. The MAJ model requires a necessary and sufficient condition for the feasibility as below. Theorem. The MAJ model is feasible for the evaluation of DMU0 with output vector Y0 P 0 if and only if for each r, r = 1, . . . , s, either yr0 = 0 or there exists a DMUj, j 5 o such that yrj 5 0. Proof. Refer to Mehrabian et al. [14].
h
Proposition 3. The LJK model is also feasible under the same conditions that the MAJ model is feasible. Proof. For proving the feasibility of the LJK model we study its dual model, D-LJK, which follows: s m X X Maximize 1 þ lr y r0 vi xi0 r¼1
Subject to
s X
lr y rj
r¼1
i¼1 m X
vi xij 6 0;
j 6¼ 0; j ¼ 1; . . . ; n;
i¼1
1 ; mR i lr ; vi P 0;
vi 6
i ¼ 1; . . . ; m
ðD-LJKÞ;
i ¼ 1; . . . ; m; r ¼ 1; . . . ; s;
lr = 0("r), vi = 0("i) is a solution for the above model. We prove that the D-LJK model cannot be unbounded under the assumption of the previous Theorem, as a result the LJK model will be feasible under the same assumption. Note that we can exclude unity of 1 from the objective function of the LJK model. If the DLJK is unbounded, we must have: dv 9d ¼ “0 dl so that dl Y 0 dv X 0 > 0; dl Y j dv X j 6 0; dv 6 0; dl ; dv P 0:
j 6¼ 0;
j ¼ 1; . . . ; n;
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As a result it should be dv ¼ 0 and dl Y 0 > 0; dl Y j 6 0; j 6¼ 0; j ¼ 1; . . . ; n; dl P 0: Based on the assumption of the MAJ Theorem for all r, r = 1, . . ., s, either yr0 = 0, or there exist a (j 5 0) so that yrj > 0. Suppose that yr0 = 0, r = 1, . . . , k, and yr0 > 0, r = k + 1, . . . , s. Thus, for each r, r = k + 1, . . . , s there exist a jr 5 0 such that y rjr > 0. Therefore, dl Y jr ¼ 0 implies that (dl)r = 0, r = k + 1, . . ., s. Hence, dl Y 0 ¼
s X
ðdl Þr y r0 ¼
r¼1
k s X X ðdl Þr 0 þ 0 y r0 ¼ 0: r¼1
r¼kþ1
This contradiction shows that there is no such direction, see [5]. Therefore, the assumption of unboundedness for the D-LJK model is false. Consequently, the LJK envelopment model is feasible. On the other hand, if there exist an r such that yr0 > 0 and "j(j 5 o)yrj = 0, the rth output constraint will be inconsistent. Therefore, the LJK model is infeasible. h In comparison with the MAJ model, we now highlight some merits of the LJK model. Proposition 4. The LJK super-efficiency score is unit invariant. Proof. In the second item of the objective function of the LJK model, the numerators and denominators have identical units, the second item is unitless. Therefore, the objective function value is unit invariant. h To illustrate another drawback of the MAJ model, we now present their model as below. We note that each input constraint in the MAJ model has a w0 at the right side. This means, if w0 is positive at the optimal, each input should be increased by w0. For various units of inputs, the same w0 is added to each input, which makes the explanation of the physical meaning of w0 becomes difficult. For example, suppose one input represents the number of employees and the other represents capital. Then when w0 is added to each input, the unit of w0 has to change accordingly. We now present the LJK model together with the MAJ model to illustrate one more property. The LJK model
The MAJ model m X sþ i2 Minimize 1 þ w0 Minimize 1 þ R i i¼1 n n X X Subject to kj xij 6 xi0 þ sþ ; i ¼ 1; . . . ; m; Subject to kj xij 6 xi0 þ w0 ; i2 1 m
j¼1 j6¼0 n X
i ¼ 1; . . . ; m;
j¼1 j6¼0
kj y rj P y r0 ;
r ¼ 1; . . . ; s;
n X
kj y rj P y r0 ;
j¼1 j6¼0
j¼1 j6¼0
kj ; s þ i2 P 0:
kj P 0:
r ¼ 1; . . . ; s;
0 Þ be the optimal solutions to LJK and the MAJ modLet and ð k1 ; k2 ; . . . ; kn ; w els respectively. Comparing the right side constraints in both models implies following property. þ þ ðk1 ; k2 ; . . . ; kn ; sþ 12 ; s22 ; . . . ; sm2 Þ
0 , while in LJK Remark. In the MAJ model, all inputs must be changed by a fixed value of the variable of w þ þ model the inputs change by sþ ; s ; . . . ; s . 12 22 m2 3.3. LJK model and SuperSBM (I) model Tone [20] introduced two models called SuperSBM (I) and (O) that rank the DEA efficient DMUs obtained from the SBM model [19] or equivalently the CCR Model. One of the model SuperSBM (I), is input oriented
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(the objective function is composed by the inputs only.). Since the LJK model is also input oriented, we now compare the SuperSBM (I) model and the LJK model. The SuperSBM (I) model is presented as below: m xi0 1 X dI ¼ min m i¼1 xi0 n X Subject to X P kj X j ; j¼1 j6¼0
Y 6
n X
kj Y j ;
j¼1 j6¼0
X P X0
and
Y ¼ Y 0;
k P 0: In what follows, we show that the SuperSBM(I) can be infeasible. In fact, by the following illustrative example we show that the feasibility of the SuperSBM(I) is depended on the output of the evaluating DMU. 3.3.1. Illustrative example Consider DMUs A, B and C for which data are presented in Table 1. Each DMU uses one input to produce two outputs. We evaluate DMU A by the SuperSBM(I) model: dI ¼ min xA Subject to 2kB þ 3kC 6 xA ; 2kB þ 1kC P 1; 0kB þ 0kC P 1=2; xA P 1; kB ; kC P 0: Obviously, since the second output constraint is inconsistent, the above problem is infeasible. In what follows, we want to show that when there is no zero input in any DMU and define R i ¼ xi0 , the results from the LJK model and the SuperSBM (I) are identical, but the LJK model has some advantages over the SuperSBM (I) model in general. Proposition 5. The LJK and the SuperSBM (I) models provide the same results in ranking DEA efficient DMUs, if R i ¼ xi0 for which xi0 > 0 "i. Proof. If we replace X by X0 + S2 for which S2 is non-negative, the two models are equivalent. Therefore, in this case, the super-efficiency ratios and the ranking results for both the LJK and the SuperSBM (I) models are the same. Because in the SuperSBM(I), the objective function is as below: m m xi0 1 X 1 X sþ i2 ¼1þ ; dI ¼ min m i¼1 xi0 m i¼1 xi0 where xi0 instead of R i is used.
h
Table 1 Data for illustrative example DMUs
A
B
C
Input Output 1 Output 1
1 1 1/2
2 2 0
3 1 0
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Table 2 Data for comparison test DMUs
A1
A2
A3
B
C
D
E
Input 1 Input 2 Output 1 Output 2
2 8 1 2
0 8 1 2
0.1 8 1 2
5 5 1 1
10 4 2 1
10 6 2 1
2 12 1 2
Table 3 Computational results DMUs
CCR (%)
AP (%)
MAJ (%)
SuperSBM(I) (%)
LJK (%)
A1 A2 A3
100 100 100
147 Infeasible 2000
128 131 131
125 Indefinable 1075
117 127 126
Note that in the SuperSBM (I), if any of the inputs is zero, the objective function of the model cannot be defined since the denominator of the second term in the objective function will become zero. Although Tone [20], extended his model to consider the DMUs with zero inputs, the LJK model with inequality constraints is more direct and easier to use than the SuperSBM(I) model which requires additional work for zero inputs. 4. Numerical example In this section, we illustrate a numerical example by comparing the super-efficiency results obtained by AP, MAJ, SuperSBM(I) and the LJK models. The numerical example shows that while the results from the AP and SuperSBM(I) models can be infeasible or instable, the LJK model is feasible and stable. Table 2 provides the input and output data of 7 DMUs each of which uses two inputs and two outputs. A1, A2 and A3 are evaluated by the AP, MAJ, SuperSBM(I) and LJK models and their results are compared with each other. Every time one of the DMUs is compared with the other DMUs (B,C,D and E). The numerical example is from Meharabian et al. [14]. Since AP, SuperSBM(I) and LJK models are based on the results from the CCR model, the DMUs in Table 3 are DEA efficient in the CCR model. In Table 2, we notice that since input 1 of A2 is zero, it results in infeasibility in the AP model in Table 3. With the SuperSBM(I) model also without extra step it is impossible to obtain super-efficiency score of A2. In fact, since denominator in the objective function of the SuperSBM(I) is zero, the objective function cannot be defined in its current form. However, in Table 3 we also observe that the LJK model indicates that A2 is feasible with a super-efficiency score of 1.27. Therefore, one may find the LJK model simpler than the SuperSBM(I) model which need extra steps for evaluating DMUs with zero inputs. In addition, since Input 1 of A3 is close to zero (see Table 2), the super-efficiency scores of A3 in the AP and SuperSBM(I) models are as large as 20 and 10.75 respectively, while that of the LJK model is only 1.26. Note that with the MAJ model, A2 and A3 have super-efficiency scores of 1.31. However, the MAJ cannot differentiate A2 and A3, while A2 has better performance than A3. Because, although A2 and A3 produce the same outputs, A2 uses lower inputs than that of A3. Therefore, LJK model can be preferred to the MAJ model from accuracy point of view. See, also, Appendix B in which a real data set is used to compare the models. 5. Conclusion In this paper, we propose a super-efficiency model that does not have the weakness in earlier models. We first present several properties of our model and then compare our model with the models developed by Anderson and Peterson [2] and Mehrabian et al. [14]. We showed that our model is superior to these two models in removing the deficiencies. In addition, we also compare our model with SuperSBM(I) developed by Tone [20] and show that our model is simpler and easier to solve than SuperSBM(I). Moreover, we pointed out that
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the SuperSBM(I) can be infeasible. The research in this paper can be extended in different directions. First, we are in the process of developing a model to evaluate all DMUs instead of only DEA efficient DMUs. Second, we plan to apply our model to real data to explore the importance of our model. Acknowledgement We are also grateful to professor K. Tone, Dr. S. Mehrabian and Dr. F. H. Lotfi for their suggestions on the previous version of this paper. Appendix A The models related to the AP and MAJ approaches,which are defined on the non-negative inputs and outputs, are as follows: The AP model
The MAJ model
a0
j0 ¼ min
¼ min
Subject to
a0 X
kj X j 6 a0 X 0 ;
Subject to
j6¼0
X
w0 þ 1 X kj X j 6 X 0 þ w0 1; j6¼0
kj Y j P Y 0 ;
X
j6¼0
kj P 0;
kj Y j P Y 0 ;
j6¼0
j ¼ 1; . . . ; n:
kj P 0;
j ¼ 1; . . . ; n:
The AP model ranks the efficient DMUs under the CCR model while the MAJ model ranks the efficient DMUs under the MAJ including the evaluating DMU model.
Table 4 Data for 19 academic units of the UTE in the first semester, 1993–94 DMU
Department/Institute
I1
Faculty of Literature Persian Literature Theology and Islamic Culture History Geography Foreign Language Arabic Language and Literature Social Sciences
81 85 56.7 91 216 58 112.2
8 9 10 11 12 13 14
Faculty of Physical Education Men Physical Education Women Physical Education Mathematics Geology Biology Chemistry Physics
15 16 17 18 19
Faculty of Education foundations of Education Instructional Technology Psychology guidance and Counseling Institute of Mathematics
1 2 3 4 5 6 7
I2
O1
O2
87.6 12.8 55.2 78.8 72 25.6 8.8
5191 3629 3302 3379 5368 1674 2350
205 0 0 8 639 0 0
293.2 186 143.4 108.7 105.7 235 146.3
52 0 105.2 127 134.4 236.8 124
6315 2865 7689 2165 3963 6643 4611
414 0 66 266 315 236 128
57 118.7 58 146 0
203 48.2 47.4 50.8 91.3
4869 3313 1853 4578 0
540 16 230 217 508
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Table 5 Super-efficiency results for efficient units DMU 19 5 2 1 15 9
AP
MAJ
SuperSBM(I)
LJK
* 1.30 1.74 1.15 1.33 *
1.28 1.1 1.09 1.05 1.06 1.04
* 1.35 1.55 1.12 1.73 *
1.24 1.072 1.065 1.033 1.032 1.021
Appendix B. Case study Mehrabian et al. [14] evaluated teaching for 19 academic units in University for Teacher Education. Each unit has two inputs and two outputs which are presented in Table 4. Teaching inputs are expressed in teacher hours and classified in terms of two inputs, professional staff and instructors. Teaching outputs also are expressed in student hours and classified in terms of two outputs, course enrollments in undergraduate and graduate studies. There are six efficient departments, and ranking results are presented in Table 5. Based on the results, the AP model is infeasible for Department of Women’s Physical Education, DMU 9, and Institute of Mathematics, DMU 19, which have zero inputs. These DMUs are indicated by asterisks in Table 5. In addition, the SuperSBM(I) model is not definable for these two units in its current form. However, these DMUs have explicit super-efficiency scores and definite rank by the LJK model. Note that the ranking results for both LJK and MAJ models are somewhat similar. Based on the results of the LJK model (the last column in Table 5), Institute of Mathematics, DMU 19, has the first rank, while Women Physical Education, DMU 9, has the last rank. For the rest of DMUs the higher the super-efficiency score the better the DMU. References [1] N. Adler, L. Friedman, Z. Sinuany-Stern, Review of ranking methods in the data envelopment analysis context, European Journal of Operational Research 140 (2002) 249–265. [2] P. Andersen, N.C. Petersen, A procedure for ranking efficient units in data envelopment analysis, Management Science 39 (10) (1993) 1261–1264. [3] R.D. Banker, A. Charnes, W.W. Cooper, Some models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science 30 (1984) 1078–1092. [4] I. Bardhan, W.F. Bowlin, W.W. Cooper, T. Sueyoshi, Model for efficiency dominance in data envelopment analysis. Part I: Additive models and MED measures, Journal of the Operations Research Society of Japan 39 (1996) 322–332. [5] M.S. Bazaraa, J.J. Jarvis, H.D. Sherali, Linear Programming and Networks Flows, John Wiley & Sons, 1990. [6] P. Bogetoft, Incentive efficient production frontiers: an agency perspective on DEA, Management Science 40 (1994) 959–968. [7] A. Charnes, W.W. Cooper, E. Rhodes, Measuring the efficiencies of DMUs, European Journal of Operational Research 2 (6) (1978) 429–444. [8] A. Charnes, S. Haag, P. Jaska, J. Semple, Sensitivity of efficiency classifications in the additive model of data envelopment analysis, International Journal of System Science 23 (1992) 789–798. [9] G.R. Jahanshahloo, M. Khodabakhshi, Suitable combination of inputs for improving outputs in DEA with determining input congestion—Considering textile industry of China, Applied Mathematics and Computation 151 (1) (2004) 263–273. [10] J.H. Dula, B.L. Hickman, Effects of excluding the column being scored from the DEA envelopment LP technology matrix, The Journal of the Operational Research Society 48 (1997) 1001–1012. [11] M.L. Durchholz, Large-scale data envelopment analysis models and related applications, Ph.D. Thesis, Department of Computer Science and Engineering, Southern Methodist University, Dallas, TX 75275, 1994. [12] L. Fridman, Z. Sinuany-Stern, Scaling units via the canonical correlation analysis and the data envelopment analysis, European Journal of Operational Research 100 (3) (1997) 629–637. [13] G.R. Jahanshahloo, M. Khodabakhshi, Determining assurance interval for non-Archimedean element in the improving outputs model in DEA, Applied Mathematics and Computation 151 (2) (2004) 501–506. [14] S. Mehrabian, A. Alirezaee, G.R. Jahanshahloo, A complete efficiency ranking of decision making units in DEA, Computational Optimization and Applications (COAP) 14 (1999) 261–266. [15] L.M. Seiford, J. Zhu, Infeasibility of super-efficiency data envelopment analysis, INFOR 37 (2) (1999) 174–187. [16] T.R. Sexton, R.H. Silkman, A.J. Hogan, Data envelopment analysis: critique and extensions, in: R.H. Silkman (Ed.), Measuring Efficiency: An Assessment of Data Envelopment Analysis, Jossey-Bass, San Fransisco, CA, 1986, pp. 73–105.
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