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A note on a new method based on the dispersion of weights in data envelopment analysis
Computers & Industrial Engineering 56 (2009) 1703–1707
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Short Communication
A note on a new method based on the dispersion of weights in data envelopment analysis q Ying-Ming Wang a,b,*, Ying Luo c a
School of Economics & Management, Tongi University, Shanghai 200092, PR China School of Public Administration, Fuzhou University, Fuzhou 350002, PR China c School of Management, Xiamen University, Xiamen 361005, PR China b
a r t i c l e
i n f o
Article history: Received 10 January 2008 Received in revised form 26 April 2008 Accepted 31 August 2008 Available online 6 September 2008 Keywords: Data envelopment analysis Coefficient of variations Discrimination power Scale transformation Dimensions and measurement units
a b s t r a c t In a very recent paper by Bal et al. (Bal, H., Örkcü, H. H., & Çelebiog˘lu, S. (2008). A new method based on the dispersion of weights in data envelopment analysis. Computers & Industrial Engineering, 54(3), 502– 512), a data envelopment analysis (DEA) model which incorporates the coefficients of variations (CVs) of input–output weights was proposed to improve the discrimination power of DEA and balance input–output weights. This note points out that the input and output weights in DEA are of different dimensions and units. The weights with different dimensions and units cannot be simply added together and averaged. In other words, the DEA model with the inclusion of CVs of input–output weights, which was referred to as CVDEA model for short, makes no sense if input and output data are not normalized to eliminate their dimensions and units. This note also illustrates the facts that the CVDEA model can cause significant efficiency changes when a scale transformation is performed for an input or output and may produce multiple local optimal solutions due to its nonlinearity, leading to totally different assessment conclusions. These facts reveal that the CVDEA model suffers from serious drawbacks and its applications for efficiency assessment should be very cautious. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Data envelopment analysis (DEA), developed by Charnes, Cooper, and Rhodes (1978), is a powerful tool for efficiency assessment and has been widely applied and researched across the world. However, it also suffers from some drawbacks such as lack of discrimination power and zero weight problem. The former means efficient decision making units (DMUs) cannot be properly distinguished from one another by their efficiencies, whereas the latter refers to the problem that DEA may sometimes assign a zero weight to some of the inputs and outputs. To improve the discrimination power of DEA, super-efficiency model (Andersen & Petersen, 1993), cross-efficiency evaluation technique (Doyle & Green, 1994, 1995), and among others, have been suggested in the DEA literature and prove to be very effective in discriminating between DEA efficient units. To tackle the zero weight problem, weight restriction techniques such as assurance region (AR) approach (Thompson, Langemeier, Lee, & Thrall, 1990; Thompson, Singleton,
q The work described in this paper was supported by the National Natural Science Foundation of China (NSFC) under the Grant No. 70771027. * Corresponding author. Address: School of Public Administration, Fuzhou University, Fuzhou 350002, PR China. Tel.: +86 591 87893307; fax: +86 591 22866677. E-mail address: [email protected] (Y.-M. Wang).
0360-8352/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.08.014
Thrall, & Smith, 1986), cone ratio envelopment (Charnes, Cooper, Huang, & Sun, 1990) have been developed in the literature. Very recently, Bal, Örkcü, and Çelebiog˘lu (2008) proposed a CVDEA model which incorporates the coefficients of variations (CVs) of input–output weights for the improvement of the discrimination power of DEA and the tackling of the zero weight problem. It is claimed that the CVDEA model produces a more balanced dispersion of input–output weights with a reduced number of efficient DMUs. However, since the input and output weights in DEA are of different dimensions and units, they cannot be simply added together and averaged. On the other hand, the CVDEA model was found to cause significant efficiency changes of DMUs when a scale transformation was performed for an input or output. It was also found to produce multiple local optimal solutions due to its nonlinearity, arriving at totally different efficiency assessment conclusions. The purpose of this note is to make some comments on the CVDEA model and illustrate its significant drawbacks and incorrect calculations by Bal et al. The paper is organized as follows: Section 2 gives a brief description of the CVDEA model; Comments are made in Section 3 with the re-examination of two numerical examples; Section 4 concludes the paper. 2. The CVDEA model Suppose there are n DMUs to be evaluated in terms of m inputs and s outputs. Let xij (i = 1, . . ., m) and yrj (r = 1, . . ., s) be the input
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and output values of DMUj (j = 1, . . ., n). The efficiency of DMUj is defined as
Ps
hj ¼ Pr¼1 m
ur yrj
i¼1 vi xij
j ¼ 1; . . . ; n;
;
ð1Þ
where vi (i = 1, . . ., m) and ur (r = 1, . . ., s) are, respectively, the input and output weights assigned to the i-th input and the r-th output. To determine the input and output weights, Charnes et al. (1978) constructed the following well-known CCR model, which was named by their acronym:
Ps ur yr0 Maximize h0 ¼ Pr¼1 ; m i¼1 vi xi0 Ps ur yrj Subject to hj ¼ Pr¼1 6 1; m i¼1 vi xij ur P 0; vi P 0;
ð2Þ j ¼ 1; . . . ; n;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m;
where DMU0 refers to the DMU under evaluation. Through Charnes and Cooper transformation (Charnes & Cooper, 1962), the above fractional programming model is finally converted into the linear programming (LP) below for solution:
Maximize h0 ¼
s X
ur yr0 ;
ð3Þ
r¼1
Subject to
m X
vi xi0 ¼ 1;
i¼1 s X
ur yrj
m X
r¼1
vi xij 6 0;
j ¼ 1; . . . ; n;
i¼1
ur P 0;
r ¼ 1; . . . ; s;
vi P 0;
i ¼ 1; . . . ; m:
If there exists a set of input and output weights to make the efficiency of DMU0 be one, i.e. h0 ¼ 1, then DMU0 is referred to as DEA efficient or CCR efficient; otherwise, it is referred to as nonDEA efficient or non-CCR efficient. LP model (3) is solved n times, each time for one DMU only. Accordingly, at least one DMU is evaluated as DEA efficient. Meanwhile, a zero weight might be assigned to some of the inputs and outputs. To achieve a balanced input–output weights, Bal et al. (2008) suggest incorporating into the above LP model (3) the CVs of input–output weights, which are defined as
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 Xs Þ2 ; CV1 ¼ ður u ¼ r¼1 s1 u u rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rv 1 1 Xm Þ2 ; CV2 ¼ ðv v ¼ i¼1 i m1 v v
ru
ð4Þ ð5Þ
where rv and ru are the standard deviations of input and output P and u are their averages, namely, v ¼ m weights, and v i¼1 vi =m and Ps ¼ r¼1 ur =s. The DEA model that incorporates the CVs of input and u output weights is referred to as CVDEA model and is formulated as
Maximize
-0 ¼
s X
ur yr0 CV1 CV2 ;
ð6Þ
output weights with a reduced number of efficient DMUs than the traditional DEA CCR model. In the next section, we make some comments on the CVDEA model to show its drawbacks and wrong calculations by Bal et al. 3. Problems with the CVDEA model As is known to all, the inputs and outputs in DEA are usually of different dimensions and units, but DEA does not require them to eliminate their dimensions and unify their scales due to its good property of unit-invariance, which means that scale transformation has no effect on DEA efficiency. In order that the inputs and outputs with different dimensions and units can be added together, their weights have to be of different dimensions. However, the weights with different dimensions and scales cannot be averaged or measured by their variance. We therefore have the following comment on the CVDEA model. Comment 1. The CVDEA model makes no sense by measuring the coefficients of variations of the input and output weights which are of different dimensions and scales. Consider the following numerical example examined by Bal et al. Example 1 . Efficiency evaluation of six nursing homes ( Sexton, Silkman, & Hogan,1986). Six nursing homes are evaluated in terms of two inputs: staff hours per day (x1), including nurses, physicians etc., and suppliers per day (x2), measured in thousands of dollars, and two outputs: total medicare-plus medicaid-reimbursed patient days (y1) and total privately paid patient days (y2). The input and output data for the six nursing homes are shown in Table 1. For this example, the two inputs x1 and x2 have different dimensions which are day and dollars, respectively. In order for them to be added together, their weights must have reciprocal dimensions, i.e., 1/day and 1/dollars. Similarly, the weights for the two outputs y1 and y2 also have dimensions, but they are the same in this specific numerical example, which are both 1/days. Obviously, the weights for the two inputs cannot be averaged or measured by their variance or coefficient of variations. Their average and variance make no sense because of their differences in dimensions and scales. To make them meaningful, the input and output data should first be normalized to eliminate their dimensions and units. For example, the input and output data can be normalized by dividing them by their maximums or by their sums. Due to the unit-invariance property of DEA models, such normalizations have no impact on DEA efficiency. Comment 2. Scale transformation of inputs and outputs can cause significant changes of CVDEA efficiency. As mentioned before, DEA models (2) and (3) have a good property of unit-invariance, which means that scale transformation has no effect on DEA efficiency. It is found that, however, the CVDEA model (6) has no such a good property. In the numerical example
r¼1
Subject to
m X
vi xi0 ¼ 1;
i¼1 s X
ur yrj
r¼1
m X
Table 1 Data for six nursing homes
vi xij 6 0;
j ¼ 1; . . . ; n;
i¼1
ur P 0;
r ¼ 1; . . . ; s;
vi P 0;
i ¼ 1; . . . ; m;
which is a nonlinear optimization model. It is claimed that the CVDEA model can produce a more balanced dispersion of input–
DMU
y1
y2
x1
x2
A B C D E F
1.40 1.40 4.20 2.80 1.90 1.40
0.35 2.10 1.05 4.20 2.50 1.50
1.50 4.00 3.20 5.20 3.50 3.20
0.2 0.7 1.2 2.0 1.2 0.7
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Y.-M. Wang, Y. Luo / Computers & Industrial Engineering 56 (2009) 1703–1707 Table 2 Effects of the scale transformation of input x2 on input–output weights and CVDEA efficiencies by CVDEA model DMU
Before scale transformation
A B C D E F
After scale transformation
u1
u2
v1
v2
CVDEA efficiency
u1
u2
v1
v2
CVDEA efficiency
0.493 0.178 0.190 0.116 0.178 0.215
0.493 0.178 0.190 0.116 0.178 0.215
0.588 0.213 0.227 0.139 0.213 0.256
0.588 0.213 0.227 0.139 0.213 0.256
0.8627 0.6241 1 0.8148 0.7846 0.6232
0.571 0.164 0.096 0.057 0.096 0.164
0.571 0.164 0.096 0.057 0.096 0.164
0.0050 0.0014 0.0008 0.0005 0.0008 0.0014
0.0050 0.0014 0.0008 0.0005 0.0008 0.0014
1 0.5724 0.5024 0.4020 0.4210 0.4748
Table 3 CVDEA efficiencies after scale transformation without consideration of input x1 DMU
A B C D E F
Outputs and input
Output and input weights
y1
y2
x2
u1
u2
v2
1.40 1.40 4.20 2.80 1.90 1.40
0.35 2.10 1.05 4.20 2.50 1.50
200 700 1200 2000 1200 700
0.571 0.163 0.095 0.057 0.095 0.163
0.571 0.163 0.095 0.057 0.095 0.163
0.005 0.00143 0.00083 0.0005 0.00083 0.00143
CVDEA efficiency
1.0000 0.5714 0.5000 0.4000 0.4190 0.4735
above, x2 is measured in thousands of dollars. We now change its values to dollars rather than thousands of dollars. In other words, the input values for x2 in Table 1 are all multiplied by 1000. Such a scale transformation has certainly no impact on DEA efficiency. Table 2 shows the input and output weights and the CVDEA efficiencies of the six nursing homes before and after the scale transformation of input 2. It is observed that the CVDEA efficiencies before and after the scale transformation are significantly different from each other. Before the scale transformation, nursing home C is CVDEA efficient. However, after the scale transformation, nursing home C is no longer CVDEA efficient while nursing home A appears as a CVDEA efficient unit. Such an efficiency change is obviously illogic and unacceptable. It is also observed from Table 2 that the input weights v1 and v2 become much far smaller than they are before the scale transformation. Since input x1 is not rescaled, a very small input
Table 4 Data for seven departments in a university DMU
1 2 3 4 5 6 7
Outputs
Inputs
y1
y2
y3
x1
x2
x3
60 139 225 90 253 132 305
35 41 68 12 145 45 159
17 40 75 17 130 45 97
12 19 42 15 45 19 41
400 750 1500 600 2000 730 2350
20 70 70 100 250 50 600
weight makes its contribution to CVDEA efficiency very trivial. Table 3 shows the CVDEA efficiencies of the six nursing homes without input x1 being taken into consideration. It is seen that the CVDEA efficiencies in the last column of Table 3 are very close to those in the last column of Table 2. This shows that input x1 makes very little contribution to CVDEA efficiency when input x2 is rescaled and measured in dollars rather than thousand of dollars. This again calls for a normalization to be performed to unify the scales of inputs and outputs before CVDEA model (6) is applied. Comment 3. The CVDEA model may have multiple local optimal solutions due to its nonlinearity, leading to totally different efficiency assessment conclusions. Consider the following numerical example examined by Bal et al. Example 2 . Efficiency evaluation of seven departments in a university (Wong & Beasley, 1990). Seven departments (DMUs) in a university are evaluated in terms of three inputs and three outputs given below and their related input and output data are provided in Table 4. x1: Number of academic staff. x2: Academic staff salaries in thousands of pounds. x3: Support staff salaries in thousands of pounds. y1: Number of undergraduate students. y2: Number of postgraduate students. y3: Number of research papers. For the data in Table 4, the CVDEA model produces multiple local optimal solutions for input and output weights. Tables 5 and 6 show two sets of local optimal solutions derived by the CVDEA model for each department. It is easy to verify that the local optimal solution in Table 6 is also the global optimal solution and cannot be improved any further. Apparently, multiple local optimal solutions complicate efficiency assessment and comparison because people have to be very cautious about the solutions produced by the CVDEA model. Any efficiency assessment and comparison based upon local rather than global optimal solutions are certainly not reliable. It is, however, absolutely not easy to
Table 5 Local optimal solution for input and output weights derived by CVDEA model for Example 2 DMU
1 2 3 4 5 6 7
Output weights
Input weights
u1
u2
u3
v1
v2
v3
0.0089 0.0044 0.0021 0.0056 0.0018 0.0045 0.0018
0.0089 0.0044 0.0021 0.0056 0.0018 0.0045 0.0018
0.0089 0.0044 0.0021 0.0056 0.0018 0.0045 0.0018
0.0115 0.0463 0.0232 0.0429 0.0205 0.0507 0.0196
0.0014 0.0001 0.0000 0.0006 0.0000 0.0000 0.0000
0.0141 0.0006 0.0003 0.0000 0.0003 0.0008 0.0003
CVDEA efficiency
-0
1 0.9775 0.7603 0.6694 0.9640 1 1
0.2589 0.7171 0.9339 1.0272 0.7303 0.6943 0.6944
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Y.-M. Wang, Y. Luo / Computers & Industrial Engineering 56 (2009) 1703–1707
Table 6 Global optimal solution for input and output weights derived by CVDEA model for Example 2 DMU
Output weights
Input weights
u1
u2
u3
v1
v2
v3
1 2 3 4 5 6 7
0.0083 0.0043 0.0022 0.0050 0.0016 0.0045 0.0012
0.0083 0.0043 0.0022 0.0050 0.0016 0.0045 0.0012
0.0083 0.0043 0.0022 0.0050 0.0016 0.0045 0.0012
0.0023 0.0012 0.0006 0.0014 0.0004 0.0013 0.0003
0.0023 0.0012 0.0006 0.0014 0.0004 0.0013 0.0003
0.0023 0.0012 0.0006 0.0014 0.0004 0.0013 0.0003
u1
u2
v1
v2
CVDEA efficiency
-0
-0
A B C D E F
0.571 0.176 0.189 0.103 0.158 0.190
0.571 0.293 0.189 0.165 0.259 0.312
0.517 0.181 0.227 0.138 0.212 0.256
1.120 0.392 0.227 0.138 0.212 0.256
1.000 0.863 0.993 0.988 0.951 0.733
0.4788 0.0105 0.9934 0.6606 0.6086 0.3915
0.8627 0.6241 1.0000 0.8148 0.7846 0.6232
-0
0.9331 0.9437 0.8216 0.5990 0.8280 1.0000 0.6751
0.9331 0.9437 0.8216 0.5990 0.8280 1.0000 0.6751
It has been observed that the CVDEA model tends to derive an equal weight for all inputs and also an equal weight for all outputs. This can be verified from the input and output weights in Tables 2, 3 and 6. Although it is not clear if the CVDEA model always produces equal weights for inputs and outputs, when this is true, the CVDEA efficiency can be simplified as
Table 7 Incorrect results provided by Bal et al. for Example 1 DMU
CVDEA efficiency
h0
Pm Ps xi0 r¼1 yr0 Ps i¼1 Pm : ¼ max y r¼1 rj i¼1 xij
ð7Þ
j¼1;...;n
Tables 9 and 10 show the results for Examples 1 and 2 obtained by the above equation. These results are exactly the same as those in the left part of Table 2 and those in Table 6. The problem here is why the inputs and outputs should be aggregated equally. It seems that there is no evidence to support such an equal aggregation. To avoid aggregating inputs and outputs equally, more weight should P be placed on the efficiency sr¼1 ur yr0 in the objective function of model (6). For example, the CVDEA model could be reformulated as
judge if a solution provided by the CVDEA model is globally optimal or not. Comment 4 . The computational results provided by Bal et al. (2008) are not globally optimal and therefore not correct. Tables 7 and 8 show the input–output weights, CVDEA efficiencies and objective function values -0 provided by Bal et al. (2008) for Examples 1 and 2, respectively. It is seen that these results are significantly different from those in the left part of Table 2 and those in Table 6. It is not clear how Bal et al. obtained these results. It is very likely that these results are only local optimal solutions to model (6), but they are certainly not the global optimal solutions. For contrast, we present in the last columns of Tables 7 and 8 the optimal objective function values -0 at global optimality. It is easy to see that the objective function values -0 provided by Bal et al. are much less than the true optimal objective function values -0 except for DMU4 in Table 8. This reveals that the results provided by Bal et al. cannot be correct. To find out the reason why -0 is bigger than -0 for DMU4 in Table 8, we checked its input and output weights u1 = 0.812, u2 = 0.420, u3 = 0.350, v1 = 1.107, v2 = 0.109, v3 = 0.179 provided by Bal et al. and found that the efficiencies of the seven DMUs under this set of weights are 1.147, 1.250, 1.067, 0.842, 0.998, 1.295, 0.852 for DMU1 to DMU7, respectively, four of which violate the constraints of model (6). It is obvious that the results for this DMU4 provided by Bal et al. cannot be correct either.
Maximize
-0 ¼ M
s X
ur yr0 CV1 CV2 ;
ð8Þ
r¼1
Subject to
m X i¼1 s X
vi xi0 ¼ 1; ur yrj
r¼1
m X
vi xij 6 0;
j ¼ 1; . . . ; n;
i¼1
ur P 0;
r ¼ 1; . . . ; s;
vi P 0;
i ¼ 1; . . . ; m;
P where M > 1 is a big weight assigned to the efficiency sr¼1 ur yr0 . However, its setting is not easy and seems to be a tough problem.
Table 9 Simplified computations for CVDEA efficiencies of six nursing homes in Example 1 P P P P DMU y1 y2 x1 x2 Normalization yr xi yr = xi A B C D E F
Comment 5 . The CVDEA model tends to aggregate inputs with equal weights and outputs equally, but there is no evidence to support such an aggregation.
1.40 1.40 4.20 2.80 1.90 1.40
0.35 2.10 1.05 4.20 2.50 1.50
1.50 4.00 3.20 5.20 3.50 3.20
0.2 0.7 1.2 2.0 1.2 0.7
1.75 3.50 5.25 7.00 4.40 2.90
1.7 4.7 4.4 7.2 4.7 3.9
1.0294 0.7447 1.1932 0.9722 0.9362 0.7436
0.8627 0.6241 1.0000 0.8148 0.7846 0.6232
Table 8 Incorrect results provided by Bal et al. for Example 2 DMU
u1
u2
u3
v1
v2
v3
CVDEA efficiency
-0
-0
1 2 3 4 5 6 7
0.847 0.462 0.293 0.812 0.038 0.440 0.178
0.971 0.403 0.162 0.420 0.293 0.440 0.178
0.893 0.438 0.293 0.350 0.366 0.440 0.178
0.870 0.106 0.756 1.107 0.101 0.133 0.513
0.193 0.124 0.021 0.109 0.032 0.133 0.025
0.618 0.063 0.514 0.179 0.125 0 0.033
1 0.988 0.997 0.841 0.999 0.981 1
0.3198 0.5991 0.1770 0.8287 0.3038 0.1145 0.4655
0.9331 0.9437 0.8216 0.5990 0.8280 1.0000 0.6751
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Y.-M. Wang, Y. Luo / Computers & Industrial Engineering 56 (2009) 1703–1707 Table 10 Simplified computations for CVDEA efficiencies of seven departments in Example 2 DMU
y1
y2
y3
x1
x2
x3
P
1 2 3 4 5 6 7
60 139 225 90 253 132 305
35 41 68 12 145 45 159
17 40 75 17 130 45 97
12 19 42 15 45 19 41
400 750 1500 600 2000 730 2350
20 70 70 100 250 50 600
112 220 368 119 528 222 561
If M is set too big, CV1 and CV2 may play little or no role in the model. If M is set too small, CV1 and CV2 may dominate the efficiency P measure sr¼1 ur yr0 and lead to an equal weight for inputs and an equal weight for outputs being derived. Form the above comments and analyses, it can be concluded that the CVDEA method suffers from serious problems. Its use should be very cautious. The following is a brief summary of its significant drawbacks: The input and output weights with different dimensions and units are added together and averaged, which makes no sense unless their dimensions and units are eliminated by normalizing input and output data in advance. Scale transformation can cause significant changes in CVDEA efficiency, which makes no sense either. This also calls for normalization of input and output data. Multiple local optimal solutions produced by the CVDEA model complicate efficiency assessment and comparison of DMUs. People have to be very careful about the solutions produced by the CVDEA model. Any efficiency assessment and comparison based upon local rather than global optimal solutions are certainly not reliable. The aggregation of inputs and outputs with equal weights lacks of theoretical evidence. To avoid this from happening, the CVDEA model needs to be revised and more weight should be placed on the efficiency measure in its objective function.
4. Conclusions In this paper we have briefly reviewed the CVDEA model proposed by Bal et al. (2008) and made some comments on the model to show its serious problems, including averaging the input and output weights with different dimensions and scales and measuring them with coefficients of variations, significant impacts of scale transformation on input and output weights as well as efficiencies,
yr
P
xi
432 839 1612 715 2295 799 2991
P
yr =
P
0.2593 0.2622 0.2283 0.1664 0.2301 0.2778 0.1876
xi
Normalization 0.9331 0.9437 0.8216 0.5990 0.8280 1.0000 0.6751
lack of theoretical evidence to support the aggregation of inputs and outputs with equal weights, and complication of efficiency assessment and comparisons by multiple local optimal solutions. We point out these problems to avoid any possible misapplications in the future. Acknowledgement The authors would like to thank two anonymous reviewers for their helpful comments, which are very helpful to improve the paper. References Andersen, P., & Petersen, N. C. (1993). A procedure for ranking efficient units in data envelopment analysis. Management Science, 39, 1261–1264. Bal, H., Örkcü, H. H., & Çelebiog˘lu, S. (2008). A new method based on the dispersion of weights in data envelopment analysis. Computers & Industrial Engineering, 54(3), 502–512. Charnes, A., & Cooper, W. W. (1962). Programming with linear fractional functional. Naval Research Logistics Quarterly, 9, 181–185. Charnes, A., Cooper, W. W., Huang, Z. M., & Sun, D. B. (1990). Polyhedral cone-ratio models with an illustrative application to large commercial banks. Journal of Econometrics, 46, 73–91. Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2, 429–444. Doyle, J. R., & Green, R. H. (1994). Efficiency and cross-efficiency in DEA: Derivations, meanings and uses. Journal of the Operational Research Society, 45, 567–578. Doyle, J. R., & Green, R. H. (1995). Cross-evaluation in DEA: Improving discrimination among DMUs. INFOR, 33, 205–222. Sexton, T. R., Silkman, R. H., & Hogan, A. J. (1986). The methodology of data envelopment analysis. In R. H. Silkman (Ed.), Measuring efficiency: An assessment of data envelopment analysis (pp. 7–29). San Fransisco: Jossey-Bass. Thompson, R. G., Langemeier, L. N., Lee, C. T., & Thrall, R. M. (1990). The role of multiplier bounds in efficiency analysis with application to Kansas farming. Journal of Econometrics, 46, 93–108. Thompson, R. G., Singleton, F. G., Thrall, R. M., & Smith, B. A. (1986). Comparative site evaluations for locating a high-energy physics lab in Texas. Interfaces, 16, 35–49. Wong, Y. H. B., & Beasley, J. E. (1990). Restricting weight flexibility in data envelopment analysis. Journal of the Operational Research Society, 41(9), 829–835.
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Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Short Communication
A note on a new method based on the dispersion of weights in data envelopment analysis q Ying-Ming Wang a,b,*, Ying Luo c a
School of Economics & Management, Tongi University, Shanghai 200092, PR China School of Public Administration, Fuzhou University, Fuzhou 350002, PR China c School of Management, Xiamen University, Xiamen 361005, PR China b
a r t i c l e
i n f o
Article history: Received 10 January 2008 Received in revised form 26 April 2008 Accepted 31 August 2008 Available online 6 September 2008 Keywords: Data envelopment analysis Coefficient of variations Discrimination power Scale transformation Dimensions and measurement units
a b s t r a c t In a very recent paper by Bal et al. (Bal, H., Örkcü, H. H., & Çelebiog˘lu, S. (2008). A new method based on the dispersion of weights in data envelopment analysis. Computers & Industrial Engineering, 54(3), 502– 512), a data envelopment analysis (DEA) model which incorporates the coefficients of variations (CVs) of input–output weights was proposed to improve the discrimination power of DEA and balance input–output weights. This note points out that the input and output weights in DEA are of different dimensions and units. The weights with different dimensions and units cannot be simply added together and averaged. In other words, the DEA model with the inclusion of CVs of input–output weights, which was referred to as CVDEA model for short, makes no sense if input and output data are not normalized to eliminate their dimensions and units. This note also illustrates the facts that the CVDEA model can cause significant efficiency changes when a scale transformation is performed for an input or output and may produce multiple local optimal solutions due to its nonlinearity, leading to totally different assessment conclusions. These facts reveal that the CVDEA model suffers from serious drawbacks and its applications for efficiency assessment should be very cautious. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Data envelopment analysis (DEA), developed by Charnes, Cooper, and Rhodes (1978), is a powerful tool for efficiency assessment and has been widely applied and researched across the world. However, it also suffers from some drawbacks such as lack of discrimination power and zero weight problem. The former means efficient decision making units (DMUs) cannot be properly distinguished from one another by their efficiencies, whereas the latter refers to the problem that DEA may sometimes assign a zero weight to some of the inputs and outputs. To improve the discrimination power of DEA, super-efficiency model (Andersen & Petersen, 1993), cross-efficiency evaluation technique (Doyle & Green, 1994, 1995), and among others, have been suggested in the DEA literature and prove to be very effective in discriminating between DEA efficient units. To tackle the zero weight problem, weight restriction techniques such as assurance region (AR) approach (Thompson, Langemeier, Lee, & Thrall, 1990; Thompson, Singleton,
q The work described in this paper was supported by the National Natural Science Foundation of China (NSFC) under the Grant No. 70771027. * Corresponding author. Address: School of Public Administration, Fuzhou University, Fuzhou 350002, PR China. Tel.: +86 591 87893307; fax: +86 591 22866677. E-mail address: [email protected] (Y.-M. Wang).
0360-8352/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.08.014
Thrall, & Smith, 1986), cone ratio envelopment (Charnes, Cooper, Huang, & Sun, 1990) have been developed in the literature. Very recently, Bal, Örkcü, and Çelebiog˘lu (2008) proposed a CVDEA model which incorporates the coefficients of variations (CVs) of input–output weights for the improvement of the discrimination power of DEA and the tackling of the zero weight problem. It is claimed that the CVDEA model produces a more balanced dispersion of input–output weights with a reduced number of efficient DMUs. However, since the input and output weights in DEA are of different dimensions and units, they cannot be simply added together and averaged. On the other hand, the CVDEA model was found to cause significant efficiency changes of DMUs when a scale transformation was performed for an input or output. It was also found to produce multiple local optimal solutions due to its nonlinearity, arriving at totally different efficiency assessment conclusions. The purpose of this note is to make some comments on the CVDEA model and illustrate its significant drawbacks and incorrect calculations by Bal et al. The paper is organized as follows: Section 2 gives a brief description of the CVDEA model; Comments are made in Section 3 with the re-examination of two numerical examples; Section 4 concludes the paper. 2. The CVDEA model Suppose there are n DMUs to be evaluated in terms of m inputs and s outputs. Let xij (i = 1, . . ., m) and yrj (r = 1, . . ., s) be the input
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and output values of DMUj (j = 1, . . ., n). The efficiency of DMUj is defined as
Ps
hj ¼ Pr¼1 m
ur yrj
i¼1 vi xij
j ¼ 1; . . . ; n;
;
ð1Þ
where vi (i = 1, . . ., m) and ur (r = 1, . . ., s) are, respectively, the input and output weights assigned to the i-th input and the r-th output. To determine the input and output weights, Charnes et al. (1978) constructed the following well-known CCR model, which was named by their acronym:
Ps ur yr0 Maximize h0 ¼ Pr¼1 ; m i¼1 vi xi0 Ps ur yrj Subject to hj ¼ Pr¼1 6 1; m i¼1 vi xij ur P 0; vi P 0;
ð2Þ j ¼ 1; . . . ; n;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m;
where DMU0 refers to the DMU under evaluation. Through Charnes and Cooper transformation (Charnes & Cooper, 1962), the above fractional programming model is finally converted into the linear programming (LP) below for solution:
Maximize h0 ¼
s X
ur yr0 ;
ð3Þ
r¼1
Subject to
m X
vi xi0 ¼ 1;
i¼1 s X
ur yrj
m X
r¼1
vi xij 6 0;
j ¼ 1; . . . ; n;
i¼1
ur P 0;
r ¼ 1; . . . ; s;
vi P 0;
i ¼ 1; . . . ; m:
If there exists a set of input and output weights to make the efficiency of DMU0 be one, i.e. h0 ¼ 1, then DMU0 is referred to as DEA efficient or CCR efficient; otherwise, it is referred to as nonDEA efficient or non-CCR efficient. LP model (3) is solved n times, each time for one DMU only. Accordingly, at least one DMU is evaluated as DEA efficient. Meanwhile, a zero weight might be assigned to some of the inputs and outputs. To achieve a balanced input–output weights, Bal et al. (2008) suggest incorporating into the above LP model (3) the CVs of input–output weights, which are defined as
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 Xs Þ2 ; CV1 ¼ ður u ¼ r¼1 s1 u u rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rv 1 1 Xm Þ2 ; CV2 ¼ ðv v ¼ i¼1 i m1 v v
ru
ð4Þ ð5Þ
where rv and ru are the standard deviations of input and output P and u are their averages, namely, v ¼ m weights, and v i¼1 vi =m and Ps ¼ r¼1 ur =s. The DEA model that incorporates the CVs of input and u output weights is referred to as CVDEA model and is formulated as
Maximize
-0 ¼
s X
ur yr0 CV1 CV2 ;
ð6Þ
output weights with a reduced number of efficient DMUs than the traditional DEA CCR model. In the next section, we make some comments on the CVDEA model to show its drawbacks and wrong calculations by Bal et al. 3. Problems with the CVDEA model As is known to all, the inputs and outputs in DEA are usually of different dimensions and units, but DEA does not require them to eliminate their dimensions and unify their scales due to its good property of unit-invariance, which means that scale transformation has no effect on DEA efficiency. In order that the inputs and outputs with different dimensions and units can be added together, their weights have to be of different dimensions. However, the weights with different dimensions and scales cannot be averaged or measured by their variance. We therefore have the following comment on the CVDEA model. Comment 1. The CVDEA model makes no sense by measuring the coefficients of variations of the input and output weights which are of different dimensions and scales. Consider the following numerical example examined by Bal et al. Example 1 . Efficiency evaluation of six nursing homes ( Sexton, Silkman, & Hogan,1986). Six nursing homes are evaluated in terms of two inputs: staff hours per day (x1), including nurses, physicians etc., and suppliers per day (x2), measured in thousands of dollars, and two outputs: total medicare-plus medicaid-reimbursed patient days (y1) and total privately paid patient days (y2). The input and output data for the six nursing homes are shown in Table 1. For this example, the two inputs x1 and x2 have different dimensions which are day and dollars, respectively. In order for them to be added together, their weights must have reciprocal dimensions, i.e., 1/day and 1/dollars. Similarly, the weights for the two outputs y1 and y2 also have dimensions, but they are the same in this specific numerical example, which are both 1/days. Obviously, the weights for the two inputs cannot be averaged or measured by their variance or coefficient of variations. Their average and variance make no sense because of their differences in dimensions and scales. To make them meaningful, the input and output data should first be normalized to eliminate their dimensions and units. For example, the input and output data can be normalized by dividing them by their maximums or by their sums. Due to the unit-invariance property of DEA models, such normalizations have no impact on DEA efficiency. Comment 2. Scale transformation of inputs and outputs can cause significant changes of CVDEA efficiency. As mentioned before, DEA models (2) and (3) have a good property of unit-invariance, which means that scale transformation has no effect on DEA efficiency. It is found that, however, the CVDEA model (6) has no such a good property. In the numerical example
r¼1
Subject to
m X
vi xi0 ¼ 1;
i¼1 s X
ur yrj
r¼1
m X
Table 1 Data for six nursing homes
vi xij 6 0;
j ¼ 1; . . . ; n;
i¼1
ur P 0;
r ¼ 1; . . . ; s;
vi P 0;
i ¼ 1; . . . ; m;
which is a nonlinear optimization model. It is claimed that the CVDEA model can produce a more balanced dispersion of input–
DMU
y1
y2
x1
x2
A B C D E F
1.40 1.40 4.20 2.80 1.90 1.40
0.35 2.10 1.05 4.20 2.50 1.50
1.50 4.00 3.20 5.20 3.50 3.20
0.2 0.7 1.2 2.0 1.2 0.7
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Y.-M. Wang, Y. Luo / Computers & Industrial Engineering 56 (2009) 1703–1707 Table 2 Effects of the scale transformation of input x2 on input–output weights and CVDEA efficiencies by CVDEA model DMU
Before scale transformation
A B C D E F
After scale transformation
u1
u2
v1
v2
CVDEA efficiency
u1
u2
v1
v2
CVDEA efficiency
0.493 0.178 0.190 0.116 0.178 0.215
0.493 0.178 0.190 0.116 0.178 0.215
0.588 0.213 0.227 0.139 0.213 0.256
0.588 0.213 0.227 0.139 0.213 0.256
0.8627 0.6241 1 0.8148 0.7846 0.6232
0.571 0.164 0.096 0.057 0.096 0.164
0.571 0.164 0.096 0.057 0.096 0.164
0.0050 0.0014 0.0008 0.0005 0.0008 0.0014
0.0050 0.0014 0.0008 0.0005 0.0008 0.0014
1 0.5724 0.5024 0.4020 0.4210 0.4748
Table 3 CVDEA efficiencies after scale transformation without consideration of input x1 DMU
A B C D E F
Outputs and input
Output and input weights
y1
y2
x2
u1
u2
v2
1.40 1.40 4.20 2.80 1.90 1.40
0.35 2.10 1.05 4.20 2.50 1.50
200 700 1200 2000 1200 700
0.571 0.163 0.095 0.057 0.095 0.163
0.571 0.163 0.095 0.057 0.095 0.163
0.005 0.00143 0.00083 0.0005 0.00083 0.00143
CVDEA efficiency
1.0000 0.5714 0.5000 0.4000 0.4190 0.4735
above, x2 is measured in thousands of dollars. We now change its values to dollars rather than thousands of dollars. In other words, the input values for x2 in Table 1 are all multiplied by 1000. Such a scale transformation has certainly no impact on DEA efficiency. Table 2 shows the input and output weights and the CVDEA efficiencies of the six nursing homes before and after the scale transformation of input 2. It is observed that the CVDEA efficiencies before and after the scale transformation are significantly different from each other. Before the scale transformation, nursing home C is CVDEA efficient. However, after the scale transformation, nursing home C is no longer CVDEA efficient while nursing home A appears as a CVDEA efficient unit. Such an efficiency change is obviously illogic and unacceptable. It is also observed from Table 2 that the input weights v1 and v2 become much far smaller than they are before the scale transformation. Since input x1 is not rescaled, a very small input
Table 4 Data for seven departments in a university DMU
1 2 3 4 5 6 7
Outputs
Inputs
y1
y2
y3
x1
x2
x3
60 139 225 90 253 132 305
35 41 68 12 145 45 159
17 40 75 17 130 45 97
12 19 42 15 45 19 41
400 750 1500 600 2000 730 2350
20 70 70 100 250 50 600
weight makes its contribution to CVDEA efficiency very trivial. Table 3 shows the CVDEA efficiencies of the six nursing homes without input x1 being taken into consideration. It is seen that the CVDEA efficiencies in the last column of Table 3 are very close to those in the last column of Table 2. This shows that input x1 makes very little contribution to CVDEA efficiency when input x2 is rescaled and measured in dollars rather than thousand of dollars. This again calls for a normalization to be performed to unify the scales of inputs and outputs before CVDEA model (6) is applied. Comment 3. The CVDEA model may have multiple local optimal solutions due to its nonlinearity, leading to totally different efficiency assessment conclusions. Consider the following numerical example examined by Bal et al. Example 2 . Efficiency evaluation of seven departments in a university (Wong & Beasley, 1990). Seven departments (DMUs) in a university are evaluated in terms of three inputs and three outputs given below and their related input and output data are provided in Table 4. x1: Number of academic staff. x2: Academic staff salaries in thousands of pounds. x3: Support staff salaries in thousands of pounds. y1: Number of undergraduate students. y2: Number of postgraduate students. y3: Number of research papers. For the data in Table 4, the CVDEA model produces multiple local optimal solutions for input and output weights. Tables 5 and 6 show two sets of local optimal solutions derived by the CVDEA model for each department. It is easy to verify that the local optimal solution in Table 6 is also the global optimal solution and cannot be improved any further. Apparently, multiple local optimal solutions complicate efficiency assessment and comparison because people have to be very cautious about the solutions produced by the CVDEA model. Any efficiency assessment and comparison based upon local rather than global optimal solutions are certainly not reliable. It is, however, absolutely not easy to
Table 5 Local optimal solution for input and output weights derived by CVDEA model for Example 2 DMU
1 2 3 4 5 6 7
Output weights
Input weights
u1
u2
u3
v1
v2
v3
0.0089 0.0044 0.0021 0.0056 0.0018 0.0045 0.0018
0.0089 0.0044 0.0021 0.0056 0.0018 0.0045 0.0018
0.0089 0.0044 0.0021 0.0056 0.0018 0.0045 0.0018
0.0115 0.0463 0.0232 0.0429 0.0205 0.0507 0.0196
0.0014 0.0001 0.0000 0.0006 0.0000 0.0000 0.0000
0.0141 0.0006 0.0003 0.0000 0.0003 0.0008 0.0003
CVDEA efficiency
-0
1 0.9775 0.7603 0.6694 0.9640 1 1
0.2589 0.7171 0.9339 1.0272 0.7303 0.6943 0.6944
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Table 6 Global optimal solution for input and output weights derived by CVDEA model for Example 2 DMU
Output weights
Input weights
u1
u2
u3
v1
v2
v3
1 2 3 4 5 6 7
0.0083 0.0043 0.0022 0.0050 0.0016 0.0045 0.0012
0.0083 0.0043 0.0022 0.0050 0.0016 0.0045 0.0012
0.0083 0.0043 0.0022 0.0050 0.0016 0.0045 0.0012
0.0023 0.0012 0.0006 0.0014 0.0004 0.0013 0.0003
0.0023 0.0012 0.0006 0.0014 0.0004 0.0013 0.0003
0.0023 0.0012 0.0006 0.0014 0.0004 0.0013 0.0003
u1
u2
v1
v2
CVDEA efficiency
-0
-0
A B C D E F
0.571 0.176 0.189 0.103 0.158 0.190
0.571 0.293 0.189 0.165 0.259 0.312
0.517 0.181 0.227 0.138 0.212 0.256
1.120 0.392 0.227 0.138 0.212 0.256
1.000 0.863 0.993 0.988 0.951 0.733
0.4788 0.0105 0.9934 0.6606 0.6086 0.3915
0.8627 0.6241 1.0000 0.8148 0.7846 0.6232
-0
0.9331 0.9437 0.8216 0.5990 0.8280 1.0000 0.6751
0.9331 0.9437 0.8216 0.5990 0.8280 1.0000 0.6751
It has been observed that the CVDEA model tends to derive an equal weight for all inputs and also an equal weight for all outputs. This can be verified from the input and output weights in Tables 2, 3 and 6. Although it is not clear if the CVDEA model always produces equal weights for inputs and outputs, when this is true, the CVDEA efficiency can be simplified as
Table 7 Incorrect results provided by Bal et al. for Example 1 DMU
CVDEA efficiency
h0
Pm Ps xi0 r¼1 yr0 Ps i¼1 Pm : ¼ max y r¼1 rj i¼1 xij
ð7Þ
j¼1;...;n
Tables 9 and 10 show the results for Examples 1 and 2 obtained by the above equation. These results are exactly the same as those in the left part of Table 2 and those in Table 6. The problem here is why the inputs and outputs should be aggregated equally. It seems that there is no evidence to support such an equal aggregation. To avoid aggregating inputs and outputs equally, more weight should P be placed on the efficiency sr¼1 ur yr0 in the objective function of model (6). For example, the CVDEA model could be reformulated as
judge if a solution provided by the CVDEA model is globally optimal or not. Comment 4 . The computational results provided by Bal et al. (2008) are not globally optimal and therefore not correct. Tables 7 and 8 show the input–output weights, CVDEA efficiencies and objective function values -0 provided by Bal et al. (2008) for Examples 1 and 2, respectively. It is seen that these results are significantly different from those in the left part of Table 2 and those in Table 6. It is not clear how Bal et al. obtained these results. It is very likely that these results are only local optimal solutions to model (6), but they are certainly not the global optimal solutions. For contrast, we present in the last columns of Tables 7 and 8 the optimal objective function values -0 at global optimality. It is easy to see that the objective function values -0 provided by Bal et al. are much less than the true optimal objective function values -0 except for DMU4 in Table 8. This reveals that the results provided by Bal et al. cannot be correct. To find out the reason why -0 is bigger than -0 for DMU4 in Table 8, we checked its input and output weights u1 = 0.812, u2 = 0.420, u3 = 0.350, v1 = 1.107, v2 = 0.109, v3 = 0.179 provided by Bal et al. and found that the efficiencies of the seven DMUs under this set of weights are 1.147, 1.250, 1.067, 0.842, 0.998, 1.295, 0.852 for DMU1 to DMU7, respectively, four of which violate the constraints of model (6). It is obvious that the results for this DMU4 provided by Bal et al. cannot be correct either.
Maximize
-0 ¼ M
s X
ur yr0 CV1 CV2 ;
ð8Þ
r¼1
Subject to
m X i¼1 s X
vi xi0 ¼ 1; ur yrj
r¼1
m X
vi xij 6 0;
j ¼ 1; . . . ; n;
i¼1
ur P 0;
r ¼ 1; . . . ; s;
vi P 0;
i ¼ 1; . . . ; m;
P where M > 1 is a big weight assigned to the efficiency sr¼1 ur yr0 . However, its setting is not easy and seems to be a tough problem.
Table 9 Simplified computations for CVDEA efficiencies of six nursing homes in Example 1 P P P P DMU y1 y2 x1 x2 Normalization yr xi yr = xi A B C D E F
Comment 5 . The CVDEA model tends to aggregate inputs with equal weights and outputs equally, but there is no evidence to support such an aggregation.
1.40 1.40 4.20 2.80 1.90 1.40
0.35 2.10 1.05 4.20 2.50 1.50
1.50 4.00 3.20 5.20 3.50 3.20
0.2 0.7 1.2 2.0 1.2 0.7
1.75 3.50 5.25 7.00 4.40 2.90
1.7 4.7 4.4 7.2 4.7 3.9
1.0294 0.7447 1.1932 0.9722 0.9362 0.7436
0.8627 0.6241 1.0000 0.8148 0.7846 0.6232
Table 8 Incorrect results provided by Bal et al. for Example 2 DMU
u1
u2
u3
v1
v2
v3
CVDEA efficiency
-0
-0
1 2 3 4 5 6 7
0.847 0.462 0.293 0.812 0.038 0.440 0.178
0.971 0.403 0.162 0.420 0.293 0.440 0.178
0.893 0.438 0.293 0.350 0.366 0.440 0.178
0.870 0.106 0.756 1.107 0.101 0.133 0.513
0.193 0.124 0.021 0.109 0.032 0.133 0.025
0.618 0.063 0.514 0.179 0.125 0 0.033
1 0.988 0.997 0.841 0.999 0.981 1
0.3198 0.5991 0.1770 0.8287 0.3038 0.1145 0.4655
0.9331 0.9437 0.8216 0.5990 0.8280 1.0000 0.6751
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Y.-M. Wang, Y. Luo / Computers & Industrial Engineering 56 (2009) 1703–1707 Table 10 Simplified computations for CVDEA efficiencies of seven departments in Example 2 DMU
y1
y2
y3
x1
x2
x3
P
1 2 3 4 5 6 7
60 139 225 90 253 132 305
35 41 68 12 145 45 159
17 40 75 17 130 45 97
12 19 42 15 45 19 41
400 750 1500 600 2000 730 2350
20 70 70 100 250 50 600
112 220 368 119 528 222 561
If M is set too big, CV1 and CV2 may play little or no role in the model. If M is set too small, CV1 and CV2 may dominate the efficiency P measure sr¼1 ur yr0 and lead to an equal weight for inputs and an equal weight for outputs being derived. Form the above comments and analyses, it can be concluded that the CVDEA method suffers from serious problems. Its use should be very cautious. The following is a brief summary of its significant drawbacks: The input and output weights with different dimensions and units are added together and averaged, which makes no sense unless their dimensions and units are eliminated by normalizing input and output data in advance. Scale transformation can cause significant changes in CVDEA efficiency, which makes no sense either. This also calls for normalization of input and output data. Multiple local optimal solutions produced by the CVDEA model complicate efficiency assessment and comparison of DMUs. People have to be very careful about the solutions produced by the CVDEA model. Any efficiency assessment and comparison based upon local rather than global optimal solutions are certainly not reliable. The aggregation of inputs and outputs with equal weights lacks of theoretical evidence. To avoid this from happening, the CVDEA model needs to be revised and more weight should be placed on the efficiency measure in its objective function.
4. Conclusions In this paper we have briefly reviewed the CVDEA model proposed by Bal et al. (2008) and made some comments on the model to show its serious problems, including averaging the input and output weights with different dimensions and scales and measuring them with coefficients of variations, significant impacts of scale transformation on input and output weights as well as efficiencies,
yr
P
xi
432 839 1612 715 2295 799 2991
P
yr =
P
0.2593 0.2622 0.2283 0.1664 0.2301 0.2778 0.1876
xi
Normalization 0.9331 0.9437 0.8216 0.5990 0.8280 1.0000 0.6751
lack of theoretical evidence to support the aggregation of inputs and outputs with equal weights, and complication of efficiency assessment and comparisons by multiple local optimal solutions. We point out these problems to avoid any possible misapplications in the future. Acknowledgement The authors would like to thank two anonymous reviewers for their helpful comments, which are very helpful to improve the paper. References Andersen, P., & Petersen, N. C. (1993). A procedure for ranking efficient units in data envelopment analysis. Management Science, 39, 1261–1264. Bal, H., Örkcü, H. H., & Çelebiog˘lu, S. (2008). A new method based on the dispersion of weights in data envelopment analysis. Computers & Industrial Engineering, 54(3), 502–512. Charnes, A., & Cooper, W. W. (1962). Programming with linear fractional functional. Naval Research Logistics Quarterly, 9, 181–185. Charnes, A., Cooper, W. W., Huang, Z. M., & Sun, D. B. (1990). Polyhedral cone-ratio models with an illustrative application to large commercial banks. Journal of Econometrics, 46, 73–91. Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2, 429–444. Doyle, J. R., & Green, R. H. (1994). Efficiency and cross-efficiency in DEA: Derivations, meanings and uses. Journal of the Operational Research Society, 45, 567–578. Doyle, J. R., & Green, R. H. (1995). Cross-evaluation in DEA: Improving discrimination among DMUs. INFOR, 33, 205–222. Sexton, T. R., Silkman, R. H., & Hogan, A. J. (1986). The methodology of data envelopment analysis. In R. H. Silkman (Ed.), Measuring efficiency: An assessment of data envelopment analysis (pp. 7–29). San Fransisco: Jossey-Bass. Thompson, R. G., Langemeier, L. N., Lee, C. T., & Thrall, R. M. (1990). The role of multiplier bounds in efficiency analysis with application to Kansas farming. Journal of Econometrics, 46, 93–108. Thompson, R. G., Singleton, F. G., Thrall, R. M., & Smith, B. A. (1986). Comparative site evaluations for locating a high-energy physics lab in Texas. Interfaces, 16, 35–49. Wong, Y. H. B., & Beasley, J. E. (1990). Restricting weight flexibility in data envelopment analysis. Journal of the Operational Research Society, 41(9), 829–835.