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Minimizing deviations of input and output weights from their means in data envelopment analysis
Computers & Industrial Engineering 60 (2011) 527–533
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Minimizing deviations of input and output weights from their means in data envelopment analysis q Kim Fung Lam ⇑, Feng Bai Department of Management Sciences, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 12 April 2010 Received in revised form 14 September 2010 Accepted 11 December 2010 Available online 21 December 2010 Keywords: Data envelopment analysis Cross-efficiency matrix Linear programming Goal programming Weights
In this paper, we propose a model that minimizes deviations of input and output weights from their means for efficient decision-making units in data envelopment analysis. The mean of an input or output weight is defined as the average of the maximum and the minimum attainable values of the weight when the efficient decision making unit under evaluation remains efficient. Alternate optimal weights usually exist in the linear programming solutions of efficient decision-making units and the optimal weights obtained from most of the linear programming software are somewhat arbitrary. Our proposed model can yield more rational weights without a priori information about the weights. Input and output weights can be used to compute cross-efficiencies of decision-making units in peer evaluations or group decisionmaking units, which have similar production processes via cluster analysis. If decision makers want to avoid using weights with extreme or zero values to access performance of decision-making units, then choosing weights that are close to their means, may be a rational choice. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Data Envelopment Analysis (DEA) measures the relative performance of a group of decision-making units (DMUs) which consume multiple inputs and produce a number of outputs. Let n be the number of decision-making units, DMUj (j = 1, 2, . . . , n), and each of which has m inputs xij (i = 1, . . . , m) and s outputs yrj (r = 1, . . . , s). The relative efficiency of a DMU (DMU0) can be obtained from the following linear programming (LP) model (Charnes, Cooper, & Rhodes, 1978), usually known as the CCR model:
Max s:t:
s X r¼1 s X r¼1 m X
ur yr0 ur yrj
m X
v i xij 0;
j ¼ 1; . . . ; n;
i¼1
ð1Þ
v i xi0 ¼ 1;
i¼1
ur ; v i 0;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m;
where ur and vi are the weights given to output r and input i, respectively. A frequently discussed shortcoming of DEA in the literature is the lack of differential capabilities in DEA applications. The need to discriminate between efficient DMUs becomes more significant when the number of DMUs is small relative to the total number of q
This manuscript was handled by area editor Imed Kacem.
⇑ Corresponding author. Tel.: +852 34428582; fax: +852 34420189. E-mail address: [email protected] (K.F. Lam). 0360-8352/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2010.12.007
inputs and outputs. Some studies (Andersen & Petersen, 1993; Azizi & Ajirlu, 2010; Bal, Orkcu, & Celebioglu, 2008; Doyle & Green, 1994; Lam, 2010; Li & Reeves, 1999; Sexton, Silkman, & Hogan, 1986; Tofallis, 1997) propose different approaches to enhance discrimination in DEA. Among those studies, Sexton et al. (1986) and Doyle and Green (1994) suggest using the cross-efficiency matrix to measure efficiencies of DMUs. This approach of cross-efficiency matrix applies weights of other DMUs to rate the efficiency of a DMU. If its own weights are also applied, then the DMU is both self and peer evaluated. It is common for efficient DMUs to have alternate optimal weights in the LP solutions in DEA. However, most LP software programs, regardless of the existence of alternate optimal solutions, will stop their iteration processes whenever they hit an optimal solution. Therefore, the obtained optimal weights are to some extent arbitrary. Sexton et al. (1986) and Doyle and Green (1994, 1995) propose using secondary objectives: aggressive or benevolent, to explore more suitable weights to compute crossefficiencies. Li and Reeves (1999) propose an interesting multiple criteria approach in order to enhance the discriminating power and solve the problem of having too many zero weights exist in some DEA studies. Cooper, Ruiz, and Sirvent (2007) point out that the existences of alternate optimal weights and zero weights hinder the applications of substitution and transformation analyses in DEA. Since the weights given to inputs and outputs may provide additional insights to some efficiency analyses in DEA, they propose a model that selects weights that have the maximum support from the production possibility set and contains no zero weight. They further apply their model to evaluate effectiveness of basketball players (Cooper, Ruiz, & Sirvent, 2009). Liu and Chen
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(2009) identify bad performers in the most unfavorable scenario using a radial worst-practice frontier DEA model. Wu, Liang, and Chen (2009) introduce a new DEA model to evaluate the performance of nations in the Summer Olympic Game. More recently, Mohajeri and Amin (2010) apply analytical hierarchy process (AHP) to obtain local weights and priorities of railway-station candidates and then use a DEA model to determine the optimum site for a railway station in a railway station site selection problem. In this paper, we introduce a model to determine weight sets that minimize deviations of input and output weights from their means of efficient DMUs in DEA. We define the mean of a weight as the average of the maximum and the minimum attainable values of the weight when the efficient DMU under evaluation remains efficient. The proposed model chooses weights within the alternate optimal solutions of the CCR model. Among the alternate optimal solutions, extreme weights may exist. As pointed out by Cooper et al. (2007), some optimal weight sets may have very different values even though they are all within the same alternate optimal solutions. These weight sets may lead to very different conclusions in an efficiency analysis. Intuitively, if decision makers want to avoid using weights with extreme or zero values to assess performance of DMUs, then choosing weights that are close to their means is expected to be a rational choice. Another advantage of our model is the fact that no a priori information about the weights is required. Similar to the model suggested by Cooper et al. (2007), our model only applies to efficient DMUs, as inefficient DMUs usually have a unique optimal solution. Furthermore, Cooper et al. (2007, p. 445) also stated that, ‘‘. . .we restrict attention to extreme efficient points. As already noted, these points play a crucial role in DEA, since they are generally used to evaluate the performances associated with all the other points. . . .’’. It should be pointed out that our proposed model is different from the methods with weight restrictions discussed in the DEA literature (Allen, Athanassopolus, & Dyson, 1997; Podinovski & Athanassopoulos, 1998). Some methods, which apply weight restrictions, may cause infeasibility to the LP model. Sometimes, restrictions in weights may limit the DMU under evaluation to achieve its maximum efficiency. Our model only selects solutions among the alternate optimal solutions of the CCR model. As a result, our model is always feasible and retains original efficiencies of all efficient DMUs. Therefore, our model is different from those methods, which apply weight restrictions to DEA. Our model is also different from those methods, which find the non-Archimedean epsilon in DEA models (Amin & Toloo, 2004; Mehrabian, Jahanshahloo, Mohammad, & Amin, 2000). The nonArchimedean epsilon provides a lower bound for weights to keep them away from zero. Our proposed model, instead of providing a lower bound for weights, actually attempts to pick up weights close to their means in the alternate optimal solutions of the efficient DMU under evaluation. In the following sections, we discuss the proposed model with applications to two data sets used in the literature. Finally, we will give a conclusion. 2. The proposed model In this paper, we propose a model that minimizes deviations of weights from their means in DEA. Our model can be implemented in two steps as follows: Step 1. From the optimal solutions of the CCR model, for each efficient DMU, the maximum and the minimum attainable values of each input weight and output weight are determined via the following LP model. For example, the maximum attainable value of u1 for an efficient DMU, DMU0, can be obtained by solving the following LP model:
Max u1 s m X X s:t: ur yrj v i xij 0; r¼1 m X i¼1 s X
j ¼ 1; . . . ; n; and j–o;
i¼1
v i xio ¼ 1;
ð2Þ
ur yro ¼ 1;
r¼1
ur ; v i 0;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m:
While preserving efficiency of DMU0, the maximum attainable value of u1 is obtained in (2). To obtain the minimum attainable value of u1, one simply changes the objective function to minimizing u1. Similarly, using the same procedures, the maximum and the minimum values of all other weights can be obtained. The maximum and the minimum attainable values of output weights and input weights are represented by, umax and umin (where r r min r = 1, . . . , s) and v max and v (where i = 1, . . . , m), respectively. i i Then the mean of a weight is equal to the average of the maximum and the minimum attainable values of the weight. Means of output weights and means of input weights are represented by r (where r = 1, . . . , s) and v i (where i = 1, . . . , m), respectively. u Step 2. In this step, we propose a goal programming (GP) model to determine a weight set, which is feasible in the CCR model and is also close to the means of the weights. The primary goal of the proposed model is to minimize the maximum percentage deviation of weights from their means and the secondary goal is to minimize the sum of percentage deviations. Consider the situation when umax ¼ umin , then no r r deviation of ur is allowed if the DMU under evaluation is kept as efficient. Therefore, we define X = {r if umax –umin , for r r min r = 1, . . . , s} to exclude those outputs if umax ¼ u . Similarly, r r we define U ¼ fi if v max – v min ; for i ¼ 1; . . . ; mg to exclude i i max min those inputs if v i ¼ v i . We call the proposed GP model, Percentage Deviation from Mean (PDM). The formulation of PDM of an efficient DMU, DMU0, is stated as follows:
Min Mz þ
X
þ
dr
dr
0:5 umax umin r r ! X eþi ei þ þ 0:5 v max v min 0:5 v max v min i i i i i2U r2X
s:t:
s X r¼1 m X i¼1 s X
ur yrj
0:5 umax umin r r
þ
m X
v i xij 0;
!
j ¼ 1; . . . ; n; and j–o;
ð3:1Þ ð3:2Þ
i¼1
v i xio ¼ 1;
ð3:3Þ
ur yro ¼ 1;
ð3:4Þ
r¼1
þ
r ; r 2 X ur þ dr dr ¼ u v i þ ei eþi ¼ v i ; i 2 U dr 0 z max 0:5 ur umin r þ dr 0 z 0:5 umax umin r r e maxi 0 z 0:5 i min i eþ maxi 0 z 0:5 i min i þ z; dr ; dr ; ei ; eþi ; ur ; i 0;
v
v
v
v
v
ð3:5Þ ð3:6Þ r2X
ð3:7Þ
r2X
ð3:8Þ
i2U
ð3:9Þ
i2U
ð3:10Þ
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where M is a large positive number. Values of umax ; umin ; v max ; r r i min v i ; ur and v i are obtained from Step 1. The objective function is bi-criteria. PDM is solved as a preemptive goal programming problem, where the primary goal is to minimize the value of z P d r and the secondary goal is to minimize r2X 0:5ðumax umin Þ þ r r P þ þ ei ei dr þ . Constraints in þ i2U 0:5ðv max v min Þ 0:5ðumax umin 0:5ðv max v min Þ Þ r r i i i i (3.2) ensure that no efficiency ratio is greater than unity. Constraints (3.3) and (3.4) enforce that the DMU under evaluation is always efficient. Constraints (3.5) and (3.6) measure deviations of weights from their means using deviational variables,
3. Examples We apply the proposed model to two data sets, which have been studied by other researchers. All linear programming problems in this paper are solved using LINDO (Industrial 6.1, 2003).
þ
dr þ dr ; dr ; e i ; ei . The ratio, 0:5ðumax umin Þ, in constraint (3.7) represents r r r when ur < u r . Similarly, the percentage deviations of ur from u
the ratio,
max min smaller and all e i would be smaller than max thanmin0:5 ur ur 0:5 v i v i , and this implies z < 1. Therefore, this solution is always better than any solution contains one or more zero weights, since z is equal to 1 when the solution contains one or more zero weights.
dþ r , 0:5ðumax umin Þ r r
in constraint (3.8) represents the percent-
r when ur > u r . Similar arguments apage deviations of ur from u ply to constraints (3.9) and (3.10). Since z must be greater than or equal to all those deviations in constraints (3.7)–(3.10), minimizing the value of z is the same as minimizing the maximum percentage deviation. It can be shown that all the original optimal weight sets of DMU0 from the CCR model are also feasible in the PDM model. Since constraints (3.2) and (3.3) are from the CCR model and constraint (3.4) designates that DMU0 remains efficient in the PDM model, therefore, all the optimal weight sets that are originally feasible from the CCR model are also feasible in constraints 3.2, 3.3, and 3.4. In addition, the same weight sets must also be feasible þ þ in constraints (3.5)–(3.10) for z, dr ; dr ; e i ; ei 0, where r 2 X and i 2 U. Therefore, all the optimal weight sets, which are originally feasible from the CCR model, are also feasible in the PDM model. Furthermore, the proposed model has an advantage that it attempts to avoid taking weights with extreme or zero values in DEA. DEA allows flexibility in determining values of weights. Consequently, many different forms of weight sets may be found within the same alternate optimal solutions in DEA. Although some studies (Cooper et al., 2007; Doyle & Green, 1994; Li & Reeves, 1999; Sexton et al., 1986) have suggested different methods in picking weight sets among the alternate optimal solutions, our proposed model is unique in the sense that it is the first method, which considers the feasible span of the weights, i.e., the maximum and the minimum attainable values in determining weights. In our model, PDM minimizes weights’ deviations from their means. Alternatively, when considering the feasible spread of weights, other interesting approaches can be developed. For example, if constraints (3.8) and (3.10) are removed from the PDM model, then only the maximum of the negative percentage deviations is penalized in the primary goal. Subsequently, if there exists at least one non-zero weight set among the original alternate optimal solutions, then it can be shown that the revised PDM model will guarantee the weights selected to be strictly positive. Given that the primary goal of the revised PDM model is to minimize the value of z, and if all ur and vi are strictly positive, then all dr would be
3.1. Example 1 Example 1 uses a data set that is also studied in Wong and Beasley (1990) and Li and Reeves (1999). The performance of seven departments in a university is studied. Each department contains scores of three inputs and three outputs. The three inputs are number of academic staff (x1), academic staff salaries (x2) and support staff salaries (x3). The three outputs are number of undergraduate students (y1), number of postgraduate students (y2) and number of research papers (y3). Wong and Beasley (1990) point out that the classical DEA results of Example 1 contain too many zero weights. We apply the CCR model to Example 1 and the DEA results are reported in Table 1. In Table 1, among the 42 weights from the seven departments, 22 of them are zero weights. Li and Reeves (1999) also apply their model, Minsum to Example 1. Seven zero weights are obtained by Minsum (Li & Reeves, 1999, p. 514). However, according to the results of Minsum, all the seven departments have zero weights for input 3. This result may require explanation of why support staff salaries are not included in measuring efficiencies among all the departments. Li and Reeves (1999, p. 513) point out the problem of zero weights for input 3 as follows: Notice that in all these results, the third input is avoided by almost all DMUs. This may indicate that the third input is the most expensive one for most DMUs. If this input cannot be ignored, a lower bound needs to be specified. Bal et al. (2008) propose a super efficiency model that minimizes the coefficient of variation for input and output weights. They also apply their model to Example 1. Only one zero weight is obtained for input 3 in Department 6. Non-zero weights are obtained for other inputs and outputs in the other departments. The results of the PDM model are reported in Table 2. Since our model applies only to efficient DMUs, we apply PDM to the six efficient departments. All weights obtained are strictly positive, including weights of input 3. Therefore, in this study, PDM has successfully excluded zero weights, especially for input 3, which has caused some problems to some previous studies. Moreover, all the six departments remain efficient with their new weight sets. In addition, the maximum percentage deviation (z) for each efficient department is also reported in Table 2. The z values range from 0.1853 (Department 5) to 0.3796 (Department 2).
Table 1 CCR DEA results of Example 1. Dept.
Efficiency
v1
v2
v3
u1
u2
u3
1 2 3 4 5 6 7
1 1 1 0.819679 1 1 1
0 0 0 0.06415 0 0 0.007319
0.0025 0.001333 0.000335 0.000063 0.0005 0.00137 0.000298
0 0 0.007113 0 0 0 0
0 0.007194 0 0.009108 0 0.001162 0.001207
0.028579 0 0 0 0 0 0.003337
0 0 0.013333 0 0.007692 0.018813 0.001045
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Table 2 PDM DEA results of Example 1.
a
Dept.
za
Efficiency
v1
v2
v3
u1
u2
u3
1 2 3 4 5 6 7
0.3232 0.3796 0.3031 N/A 0.1853 0.2608 0.2793
1 1 1 0.819679 1 1 1
0.026934 0.033413 0.004043 0.06415 0.009323 0.019299 0.015450
0.000846 0.000414 0.000130 0.000063 0.000204 0.000506 0.000107
0.016920 0.000781 0.009067 0 0.000692 0.005274 0.000190
0.005275 0.006020 0.002222 0.009108 0.000664 0.002800 0.001406
0.011129 0.001926 0.001508 0 0.002929 0.005496 0.002402
0.017294 0.002107 0.005300 0 0.003133 0.008513 0.001950
The maximum percentage deviation of weights from their means.
Table 3 illustrates relationships between the maximum attainable value, the minimum attainable value and the percentage deviation of a weight, using results of Department 1. As previously discussed, the ratio,
d r , 0:5ðumax umin Þ r r
represents the percentage devia-
r when ur < u r and the ratio, tions of ur from u
dþ r , 0:5ðumax umin Þ r r
repre-
r when ur > u r . In sents the percentage deviations of ur from u Table 3, the percentage deviation of a weight from its mean can be represented by the deviation of the weight from its mean divided by half of the difference between its maximum attainable value and its minimum attainable value. The z value of Department 1 is 0.3232. Notice that Table 3 is for illustration purpose, where all the values of the weights and z are obtained directly from the PDM model.
Table 4 shows the cross-efficiency scores of departments using weights obtained from the CCR model (in Table 1). Each row in Table 4 shows the self and peer evaluated scores of a department. Table 5 shows the cross-efficiency scores of departments using weights obtained from the PDM model (in Table 2). After a comparison of results in Table 4 and Table 5, it is evident that Department 2 becomes more efficient than Departments 1 and 3 becomes more efficient than Department 5 in Table 5. Departments 6 and 4 remain as the most efficient department and the least efficient department respectively in both tables. The average cross-efficiency score of Department 6 increases from 0.932 (in Table 4) to 0.987 (in Table 5). Moreover, Department 6 is efficient when peer evaluated by four other departments in Table 5. These results further suggest that Department 6 is possibly the most efficient department in this case.
Table 3 An illustration of the results of Department 1 in Example 1.
Maximum weight Mean of weight Minimum weight Weights from PDM
v1
v2
v3
u1
u2
u3
0.079592 0.039796 0 0.026934
0.002500 0.001250 0 0.000846
0.050000 0.025000 0 0.016920
0.015588 0.007794 0 0.005275
0.028579 0.015214 0.001849 0.011129
0.039706 0.019853 0 0.017294
(0.039796 0.026934)/ (0.001250 0.000846)/ (0.025000 0.016920)/ (0.007794 0.005275)/ (0.015214 0.011129)/ (0.019853 0.017294)/ Percentage 0.039796 = 0.3232 0.001250 = 0.3232 0.025000 = 0.3232 0.007794 = 0.3232 0.013365a = 0.3056 deviation 0.019853 = 0.1289 from mean a ¼ 0:5ð0:028579 0:001849Þ ¼ 0:013365. 0:5 umax umin 2 2
Table 4 Cross-efficiency scores of departments using weights obtained from CCR in Example 1. Dept.
1
2
3
4
5
6
7
Average cross-efficiency score
Rank
1 2 3 4 5 6 7
1.000 0.625 0.518 0.229 0.829 0.704 0.773
0.811 1.000 0.811 0.811 0.684 0.978 0.702
0.826 0.716 1.000 0.249 0.711 1.000 0.256
0.689 1.000 0.736 0.821 0.767 0.952 1.000
1.000 0.780 0.680 0.336 1.000 0.888 0.822
0.807 1.000 0.826 0.771 0.755 1.000 0.710
1.000 0.953 0.763 0.576 1.000 1.000 1.000
0.876 0.868 0.762 0.542 0.821 0.932 0.752
2 3 5 7 4 1 6
Table 5 Cross-efficiency scores of departments using weights obtained from PDM in Example 1. Dept.
1
2
3
4
5
6
7
Average cross-efficiency score
Rank
1 2 3 4 5 6 7
1.000 0.807 0.904 0.347 0.728 1.000 0.382
0.798 1.000 0.791 0.726 0.822 1.000 0.835
0.980 0.720 1.000 0.295 0.543 0.959 0.242
0.687 1.000 0.735 0.820 0.765 0.951 1.000
0.944 0.892 0.782 0.447 1.000 1.000 0.761
0.936 0.856 0.847 0.413 0.817 1.000 0.496
0.869 0.961 0.761 0.599 1.000 1.000 1.000
0.888 0.891 0.831 0.521 0.811 0.987 0.674
3 2 4 7 5 1 6
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3.2. Example 2 Example 2 uses a data set that is studied in Shang and Sueyoshi (1995) and Li and Reeves (1999). The problem is to select the best flexible manufacturing system among a group of 12 similar systems. Each system contains scores of two inputs and four outputs. The two inputs are capital and operating cost (x1) and floor space requirements (x2). The four outputs are qualitative benefits (y1), work in process (y2), average number of tardy jobs (y3) and average yield (y4). We apply the CCR model to the data set and the DEA results of Example 2 are reported in Table 6. Table 6 shows the DEA results of the CCR model contain many zero weights and that seven systems are efficient. Therefore, it is difficult to determine the best system using results from the CCR model and other methods must be employed to find the best system. Solving model (2) gives the maximum value and the minimum value to each weight (see Table 7). Many minimum values are equal to zero. For efficient systems, many of the weights obtained from the CCR model (in Table 6) reach either their maximum or minimum values in Table 7. For example, two weights of System 1 reach their maximum values (v1 = 0.058754, u1 = 0.023810) and the other four arrive at their minimum values (v2 = u2 = u3 = u4 = 0). Systems 5 and 9 have similar results. As previously discussed, most LP software programs, regardless of the existence of alternate optimal solutions, will stop their iteration processes whenever they hit an optimal solution. Usually, they will first hit
an extreme point of the alternate optimal solutions. In Example 2, an extreme point generally contains many zero weights. We apply our proposed model PDM to the seven efficient systems, and the results are reported in Table 8. The z values range from 0.3309 (System 1) to 0.8370 (System 7). Table 8 shows for all the seven efficient systems, no zero weight is obtained. All the seven systems remain efficient with their new weights. Table 9 shows the cross-efficiency scores of systems using weights obtained from the CCR model (in Table 6), whereas Table 10 shows the cross-efficiency scores of systems using weights obtained from the PDM model (in Table 8). After a comparison of results in Tables 9 and 10, it is evident that some systems have similar rankings in both tables. For example, System 9 is the least efficient system in both tables. However, it is also one of the efficient systems in the CCR model (see Table 6). Thus, System 9 may require more assessments on its actual efficiency. Despite its poor peer-evaluated performance, System 9 has strictly positive weights in the results of the PDM model (see Table 8). However, much heavier virtual weights are placed on input 1 and output 4 than input 2 and the other outputs. For instance, based on the weights of System 9 in Table 8, 69.2% of the total virtual input is accounted for by the value of capital and operating cost (x1), while 75.7% of the total virtual output is accounted for by the value of the average yield (y4). Furthermore, if the weights of System 9 are based on Table 6, then 100% of the total virtual input is accounted for by the value of x1, and 100% of the total virtual output is accounted for by the value of y4. Another example of systems having
Table 6 CCR DEA results of Example 2. System
Efficiency
v1
v2
u1
u2
u3
u4
1 2 3 4 5 6 7 8 9 10 11 12
1 1 0.982369 1 1 1 1 0.961428 1 0.953505 0.983144 0.801173
0.058754 0.049889 0.085034 0.084674 0.105263 0.208768 0.161031 0.075790 0.272480 0.087680 0.056370 0.059349
0 0.039738 0 0.027306 0 0 0 0.026203 0 0.061002 0 0.019139
0.023810 0.014474 0.028327 0.023820 0 0 0 0.021082 0 0 0.020600 0.016696
0 0 0 0 0 0 0.018763 0.003977 0 0.027800 0 0
0 0 0.016164 0.007040 0.083333 0.061430 0.031183 0 0 0 0 0.004935
0 0.014614 0.000931 0.015856 0 0.041991 0.014441 0.011806 0.055249 0 0.003129 0.011114
Table 7 Maximum and minimum values of weights in Example 2. Sa
a b
Efficiency
1
1
2
1
3 4
0.982369b 1
5
1
6
1
7
1
8 9
0.961428 1
10 11 12
0.953505 0.983144 0.801173
Max Min Max Min Max Min Max Min Max Min Max Min Max Min
v1
v2
u1
u2
u3
u4
0.058754 0 0.049889 0 0.085034 0.084674 0.000085 0.105263 0 0.208768 0.111077 0.161031 0.100813 0.075790 0.272480 0.102303 0.087680 0.056370 0.059349
0.2 0 0.222222 0.039738 0 0.249776 0.027306 0.263158 0 0.086656 0 0.060315 0 0.026203 0.078068 0 0.061002 0 0.019139
0.023810 0 0.025641 0 0.028327 0.023820 0 0.035466 0 0.049748 0 0.058849 0 0.021082 0.061945 0 0 0.020600 0.016696
0.022075 0 0.021067 0 0 0.027568 0 0.029240 0 0.024326 0 0.037736 0 0.003977 0.057471 0 0.027800 0 0
0.040092 0 0.061197 0 0.016164 0.046479 0 0.083333 0 0.099497 0 0.104027 0 0 0.080177 0 0 0 0.004935
0.013947 0 0.033557 0 0.000931 0.04 0.000302 0.027244 0 0.060606 0.018722 0.036880 0 0.011806 0.055249 0 0 0.003129 0.011114
S = system. For all inefficient systems, maximum value = minimum value.
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K.F. Lam, F. Bai / Computers & Industrial Engineering 60 (2011) 527–533
Table 8 PDM DEA results of Example 2.
a b
Sa
z
b
1 2 3 4 5 6 7 8 9 10 11 12
0.3309 0.4506 N/A 0.3433 0.3940 0.6295 0.8370 N/A 0.6135 N/A N/A N/A
Efficiency
v1
v2
u1
u2
u3
u4
1 1 0.982369 1 1 1 1 0.961428 1 0.953505 0.983144 0.801173
0.024458 0.024737 0.085034 0.042337 0.052632 0.135131 0.105718 0.075790 0.188439 0.087680 0.056370 0.059349
0.116747 0.131739 0 0.138654 0.131579 0.065379 0.055401 0.026203 0.038554 0.061002 0 0.019139
0.007965 0.007044 0.028327 0.007821 0.010745 0.009217 0.004795 0.021082 0.011972 0 0.020600 0.016696
0.007385 0.005787 0 0.009052 0.008859 0.004884 0.010100 0.003977 0.011108 0.027800 0 0
0.013412 0.016810 0.016164 0.015261 0.025248 0.018433 0.008476 0 0.015496 0 0 0.004935
0.004666 0.009218 0.000931 0.013185 0.008254 0.043485 0.030756 0.011806 0.041839 0 0.003129 0.011114
S = system. The maximum percentage deviation of weights from their means.
Table 9 Cross-efficiency scores of systems using weights obtained from CCR in Example 2.
a
System
1
2
3
4
5
6
7
8
9
10
11
12
ACEa
Rank
1 2 3 4 5 6 7 8 9 10 11 12
1.00 0.96 0.90 0.85 0.90 0.85 0.91 0.91 0.44 0.73 0.98 0.74
1.00 1.00 0.89 1.00 0.96 0.85 0.88 0.91 0.64 0.74 0.92 0.77
1.00 0.96 0.98 0.93 1.00 0.93 1.00 0.93 0.42 0.76 0.98 0.80
1.00 0.98 0.95 1.00 1.00 0.97 1.00 0.95 0.72 0.80 0.95 0.80
0.66 0.63 0.93 0.85 1.00 0.83 0.89 0.64 0.02 0.58 0.62 0.74
0.60 0.60 0.76 0.79 0.80 1.00 0.97 0.68 1.00 0.68 0.58 0.62
0.63 0.60 0.81 0.82 0.86 1.00 1.00 0.73 1.00 0.80 0.61 0.64
1.00 0.98 0.95 1.00 1.00 0.96 1.00 0.96 0.75 0.83 0.95 0.79
0.36 0.37 0.42 0.48 0.44 0.70 0.64 0.45 1.00 0.47 0.36 0.35
0.70 0.65 0.79 0.86 0.89 0.75 0.80 0.74 0.60 0.79 0.64 0.64
1.00 0.97 0.92 0.90 0.93 0.95 1.00 0.94 0.67 0.78 0.98 0.76
1.00 0.98 0.95 1.00 1.00 0.97 1.00 0.95 0.72 0.80 0.95 0.80
0.829 0.806 0.855 0.873 0.898 0.896 0.925 0.817 0.667 0.728 0.794 0.704
6 8 5 4 2 3 1 7 12 10 9 11
ACE: average cross-efficiency score.
Table 10 Cross-efficiency scores of systems using weights obtained from PDM in Example 2.
a
System
1
2
3
4
5
6
7
8
9
10
11
12
ACEa
Rank
1 2 3 4 5 6 7 8 9 10 11 12
1.00 0.99 0.81 0.98 1.00 0.50 0.56 0.72 0.24 0.54 0.80 0.74
0.99 1.00 0.80 1.00 1.00 0.51 0.57 0.72 0.26 0.53 0.79 0.74
1.00 0.96 0.98 0.93 1.00 0.93 1.00 0.93 0.42 0.76 0.98 0.80
0.96 0.95 0.82 1.00 1.00 0.58 0.64 0.75 0.34 0.60 0.79 0.74
0.94 0.92 0.84 0.97 1.00 0.57 0.63 0.74 0.28 0.58 0.79 0.75
0.83 0.83 0.88 0.99 0.96 1.00 1.00 0.87 0.89 0.80 0.78 0.74
0.82 0.81 0.89 1.00 0.97 0.99 1.00 0.87 0.91 0.83 0.77 0.74
1.00 0.98 0.95 1.00 1.00 0.96 1.00 0.96 0.75 0.83 0.95 0.79
0.73 0.72 0.81 0.88 0.86 1.00 0.99 0.80 1.00 0.79 0.70 0.67
0.70 0.65 0.79 0.86 0.89 0.75 0.80 0.74 0.60 0.79 0.64 0.64
1.00 0.97 0.92 0.90 0.93 0.95 1.00 0.94 0.67 0.78 0.98 0.76
1.00 0.98 0.95 1.00 1.00 0.97 1.00 0.95 0.72 0.80 0.95 0.80
0.914 0.898 0.871 0.958 0.967 0.809 0.849 0.834 0.591 0.720 0.827 0.744
3 4 5 2 1 9 6 7 12 11 8 10
ACE: average cross-efficiency score.
similar rankings in both tables is System 3. The rank of System 3 is 5 in both tables. This ranking is higher than those of three other efficient systems. However, System 3 is also one of the inefficient systems in the CCR model. Therefore, System 3 may as well require more assessments. While some systems have similar rankings in Tables 9 and 10, other systems have very different rankings. For example, the ranks of Systems 6 and 7 are 3 and 1, respectively in Table 9. Nevertheless, the ranks of Systems 6 and 7 are 9 and 6, respectively in Table 10. In fact, substantial differences exist in the rankings of the two systems in the two tables. Intuitively, one may prefer to use weights obtained from the PDM model since they contain less zero weights and are more homogeneous. After a comparison of the average cross-efficiency scores in Table 10, it is evident that System 5 is the most efficient system. Furthermore, System 5 is also efficient when peer evaluated by six other systems in Table 10. Therefore, we choose System 5 as the best system. This
result is also consistent with the results of some previous studies (Li & Reeves, 1999; Shang & Sueyoshi, 1995). If we used the weights obtained from the CCR model to compute cross-efficiency scores, then we would choose System 7 as the best model. The average cross-efficiency scores with their efficiency rankings of the PDM model, alongside with those results in the previous studies (Li & Reeves, 1999; Shang & Sueyoshi, 1995) are reported in Table 11. Li and Reeves (1999) use two multiple criteria approaches Minimax and Min M 0.2do, and Shang and Sueyoshi (1995) use AHP and DEA (AHP–DEA) to solve this problem. Results of Minimax and that of AHP–DEA indicated that there are two systems tied at the highest efficiency score; therefore, these two methods cannot be used to determine the best system. Both PDM and Min M 0.2do choose System 5 as the best system. To compare the results of our proposed model with the other existing methods, we further compute the rankings of the average effi-
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K.F. Lam, F. Bai / Computers & Industrial Engineering 60 (2011) 527–533 Table 11 Efficiency scores and rankings of different methods of Example 2.
a b c d
System
Minimaxa
Min M 0.2dob
AHP–DEAc
Averaged of the three methods
Ranks of average of the three methods
Average crossefficiency (PDM)
Ranks of average cross- efficiency (PDM)
1 2 3 4 5 6 7 8 9 10 11 12
1 0.959 0.953 0.927 1 0.847 0.891 0.950 0.604 0.728 0.977 0.801
0.837 0.804 0.927 0.922 1 0.812 0.891 0.904 0.604 0.728 0.813 0.801
0.986 0.980 0.941 0.989 1 0.961 1 0.895 0.115 0.786 0.882 0.788
0.941 0.914 0.940 0.946 1 0.873 0.927 0.916 0.441 0.747 0.891 0.797
3 7 4 2 1 9 5 6 12 11 8 10
0.914 0.897 0.871 0.958 0.967 0.809 0.849 0.834 0.591 0.720 0.827 0.744
3 4 5 2 1 9 6 7 12 11 8 10
Efficiency scores of Minimax (Li & Reeves, 1999, Table 14, p. 515). Efficiency scores of Min M 0.2do (Li & Reeves, 1999, Table 14, p. 515). Efficiency scores of AHP–DEA (Shang & Sueyoshi, 1995, Table 6, p. 311). Average efficiency scores of the three methods: Minimax, Min M 0.2do and AHP.
ciency scores of the three approaches, namely, Minimax, Min M 0.2do and AHP–DEA and compare them with the rankings of the cross-efficiency scores of our proposed model. The Spearman’s rho of 0.958 between the two rankings represents that our model is also comparable to some good existing methods. Moreover, our model can choose weights with fewer zero values and are close to their means. 4. Conclusion This paper proposes a model to obtain weight sets, which minimize the percentage deviations of weights from their means in DEA. The proposed model can be used to yield more reasonable weights without a priori information about the weights; therefore, it can be applied to any DEA studies. Furthermore, the optimal weight set, which obtained from our proposed model, is also feasible in the CCR model. We have demonstrated that the proposed model stays within the framework of classical DEA, and attempts to find weight sets only from the alternate optimal solutions of the CCR model. Another advantage of the proposed model is that it attempts to select weights that are close to their mean values. This objective, I believe, is a general agreeable objective among most decision makers. References Allen, R., Athanassopolus, A., & Dyson, R. G. (1997). Weights restrictions and value judgments in data envelopment analysis: Evolution, development and future directions. Annals of Operations Research, 73, 13–34. Amin, G. R., & Toloo, M. (2004). A polynomial-time algorithm for finding e in DEA models. Computers & Operations Research, 31, 803–805. Andersen, P., & Petersen, N. C. (1993). A procedure for ranking efficient units in data envelopment analysis. Management Science, 39, 1261–1294. Azizi, H., & Ajirlu, S. F. (2010). Measurement of overall performances of decisionmaking units using ideal and anti-ideal decision-making units. Computers & Industrial Engineering, 59, 411–418.
Bal, H., Orkcu, H. H., & Celebioglu, S. (2008). A new method based on the dispersion of weights in data envelopment analysis. Computers & Industrial Engineering, 54, 502–512. Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of the decision making units. European Journal of Operational Research, 2, 429–444. Cooper, W. W., Ruiz, J. L., & Sirvent, I. (2007). Choosing weights from alternative optimal solutions of dual multiplier models in DEA. European Journal of Operational Research, 180, 443–458. Cooper, W. W., Ruiz, J. L., & Sirvent, I. (2009). Selecting non-zero weights to evaluate effectiveness of basketball players with DEA. European Journal of Operational Research, 195, 563–574. Doyle, J., & Green, R. (1994). Efficiency and cross-efficiency in DEA: Derivations, meanings and uses. Journal of Operational Research Society, 45, 567–578. Doyle, J., & Green, R. (1995). Cross-evaluation in DEA: Improving discrimination among DMUs. INFOR, 33, 205–222. Lam, K. F. (2010). In the determination of weight sets to compute cross-efficiency ratios in DEA. Journal of Operational Research Society, 61, 134–143. Li, X. B., & Reeves, G. R. (1999). A multiple criteria approach to data envelopment analysis. European Journal of Operational Research, 115, 507–517. Liu, F. F., & Chen, C. L. (2009). The worst-practice DEA model with slack-based measurement. Computers & Industrial Engineering, 57, 496–505. Mehrabian, S., Jahanshahloo, G. R., Mohammad, R. A., & Amin, G. R. (2000). An assurance interval for the non-archimedean epsilon in DEA models. Operations Research, 48, 344–347. Mohajeri, N., & Amin, G. R. (2010). Railway station site selection using analytical hierarchy process and data envelopment analysis. Computers & Industrial Engineering, 59, 107–114. Podinovski, V. V., & Athanassopoulos, A. (1998). Assessing the relative efficiency of decision making units using DEA models with weight restrictions. Journal of Operational Research Society, 49, 500–508. Sexton, T. R., Silkman, R. H., & Hogan, A. J. (1986). Data envelopment analysis: Critique and extensions. In R. H. Silkman (Ed.), Measuring efficiency: An assessment of data envelopment analysis (pp. 73–105). San Francisco: JosseyBass. Shang, J., & Sueyoshi, T. (1995). A unified framework for the selection of a flexible manufacturing systems. European Journal of Operational Research, 85, 297–315. Tofallis, C. (1997). Input efficiency profiling: An application to airlines. Computers & Operations Research, 24, 253–258. Wong, Y. H. B., & Beasley, J. E. (1990). Restricting weight flexibility in data envelopment analysis. Journal of Operational Research Society, 41, 829–835. Wu, J., Liang, L., & Chen, Y. (2009). DEA game cross-efficiency approach to Olympic rankings. Omega, 37, 909–918.
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Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Minimizing deviations of input and output weights from their means in data envelopment analysis q Kim Fung Lam ⇑, Feng Bai Department of Management Sciences, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 12 April 2010 Received in revised form 14 September 2010 Accepted 11 December 2010 Available online 21 December 2010 Keywords: Data envelopment analysis Cross-efficiency matrix Linear programming Goal programming Weights
In this paper, we propose a model that minimizes deviations of input and output weights from their means for efficient decision-making units in data envelopment analysis. The mean of an input or output weight is defined as the average of the maximum and the minimum attainable values of the weight when the efficient decision making unit under evaluation remains efficient. Alternate optimal weights usually exist in the linear programming solutions of efficient decision-making units and the optimal weights obtained from most of the linear programming software are somewhat arbitrary. Our proposed model can yield more rational weights without a priori information about the weights. Input and output weights can be used to compute cross-efficiencies of decision-making units in peer evaluations or group decisionmaking units, which have similar production processes via cluster analysis. If decision makers want to avoid using weights with extreme or zero values to access performance of decision-making units, then choosing weights that are close to their means, may be a rational choice. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Data Envelopment Analysis (DEA) measures the relative performance of a group of decision-making units (DMUs) which consume multiple inputs and produce a number of outputs. Let n be the number of decision-making units, DMUj (j = 1, 2, . . . , n), and each of which has m inputs xij (i = 1, . . . , m) and s outputs yrj (r = 1, . . . , s). The relative efficiency of a DMU (DMU0) can be obtained from the following linear programming (LP) model (Charnes, Cooper, & Rhodes, 1978), usually known as the CCR model:
Max s:t:
s X r¼1 s X r¼1 m X
ur yr0 ur yrj
m X
v i xij 0;
j ¼ 1; . . . ; n;
i¼1
ð1Þ
v i xi0 ¼ 1;
i¼1
ur ; v i 0;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m;
where ur and vi are the weights given to output r and input i, respectively. A frequently discussed shortcoming of DEA in the literature is the lack of differential capabilities in DEA applications. The need to discriminate between efficient DMUs becomes more significant when the number of DMUs is small relative to the total number of q
This manuscript was handled by area editor Imed Kacem.
⇑ Corresponding author. Tel.: +852 34428582; fax: +852 34420189. E-mail address: [email protected] (K.F. Lam). 0360-8352/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2010.12.007
inputs and outputs. Some studies (Andersen & Petersen, 1993; Azizi & Ajirlu, 2010; Bal, Orkcu, & Celebioglu, 2008; Doyle & Green, 1994; Lam, 2010; Li & Reeves, 1999; Sexton, Silkman, & Hogan, 1986; Tofallis, 1997) propose different approaches to enhance discrimination in DEA. Among those studies, Sexton et al. (1986) and Doyle and Green (1994) suggest using the cross-efficiency matrix to measure efficiencies of DMUs. This approach of cross-efficiency matrix applies weights of other DMUs to rate the efficiency of a DMU. If its own weights are also applied, then the DMU is both self and peer evaluated. It is common for efficient DMUs to have alternate optimal weights in the LP solutions in DEA. However, most LP software programs, regardless of the existence of alternate optimal solutions, will stop their iteration processes whenever they hit an optimal solution. Therefore, the obtained optimal weights are to some extent arbitrary. Sexton et al. (1986) and Doyle and Green (1994, 1995) propose using secondary objectives: aggressive or benevolent, to explore more suitable weights to compute crossefficiencies. Li and Reeves (1999) propose an interesting multiple criteria approach in order to enhance the discriminating power and solve the problem of having too many zero weights exist in some DEA studies. Cooper, Ruiz, and Sirvent (2007) point out that the existences of alternate optimal weights and zero weights hinder the applications of substitution and transformation analyses in DEA. Since the weights given to inputs and outputs may provide additional insights to some efficiency analyses in DEA, they propose a model that selects weights that have the maximum support from the production possibility set and contains no zero weight. They further apply their model to evaluate effectiveness of basketball players (Cooper, Ruiz, & Sirvent, 2009). Liu and Chen
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(2009) identify bad performers in the most unfavorable scenario using a radial worst-practice frontier DEA model. Wu, Liang, and Chen (2009) introduce a new DEA model to evaluate the performance of nations in the Summer Olympic Game. More recently, Mohajeri and Amin (2010) apply analytical hierarchy process (AHP) to obtain local weights and priorities of railway-station candidates and then use a DEA model to determine the optimum site for a railway station in a railway station site selection problem. In this paper, we introduce a model to determine weight sets that minimize deviations of input and output weights from their means of efficient DMUs in DEA. We define the mean of a weight as the average of the maximum and the minimum attainable values of the weight when the efficient DMU under evaluation remains efficient. The proposed model chooses weights within the alternate optimal solutions of the CCR model. Among the alternate optimal solutions, extreme weights may exist. As pointed out by Cooper et al. (2007), some optimal weight sets may have very different values even though they are all within the same alternate optimal solutions. These weight sets may lead to very different conclusions in an efficiency analysis. Intuitively, if decision makers want to avoid using weights with extreme or zero values to assess performance of DMUs, then choosing weights that are close to their means is expected to be a rational choice. Another advantage of our model is the fact that no a priori information about the weights is required. Similar to the model suggested by Cooper et al. (2007), our model only applies to efficient DMUs, as inefficient DMUs usually have a unique optimal solution. Furthermore, Cooper et al. (2007, p. 445) also stated that, ‘‘. . .we restrict attention to extreme efficient points. As already noted, these points play a crucial role in DEA, since they are generally used to evaluate the performances associated with all the other points. . . .’’. It should be pointed out that our proposed model is different from the methods with weight restrictions discussed in the DEA literature (Allen, Athanassopolus, & Dyson, 1997; Podinovski & Athanassopoulos, 1998). Some methods, which apply weight restrictions, may cause infeasibility to the LP model. Sometimes, restrictions in weights may limit the DMU under evaluation to achieve its maximum efficiency. Our model only selects solutions among the alternate optimal solutions of the CCR model. As a result, our model is always feasible and retains original efficiencies of all efficient DMUs. Therefore, our model is different from those methods, which apply weight restrictions to DEA. Our model is also different from those methods, which find the non-Archimedean epsilon in DEA models (Amin & Toloo, 2004; Mehrabian, Jahanshahloo, Mohammad, & Amin, 2000). The nonArchimedean epsilon provides a lower bound for weights to keep them away from zero. Our proposed model, instead of providing a lower bound for weights, actually attempts to pick up weights close to their means in the alternate optimal solutions of the efficient DMU under evaluation. In the following sections, we discuss the proposed model with applications to two data sets used in the literature. Finally, we will give a conclusion. 2. The proposed model In this paper, we propose a model that minimizes deviations of weights from their means in DEA. Our model can be implemented in two steps as follows: Step 1. From the optimal solutions of the CCR model, for each efficient DMU, the maximum and the minimum attainable values of each input weight and output weight are determined via the following LP model. For example, the maximum attainable value of u1 for an efficient DMU, DMU0, can be obtained by solving the following LP model:
Max u1 s m X X s:t: ur yrj v i xij 0; r¼1 m X i¼1 s X
j ¼ 1; . . . ; n; and j–o;
i¼1
v i xio ¼ 1;
ð2Þ
ur yro ¼ 1;
r¼1
ur ; v i 0;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m:
While preserving efficiency of DMU0, the maximum attainable value of u1 is obtained in (2). To obtain the minimum attainable value of u1, one simply changes the objective function to minimizing u1. Similarly, using the same procedures, the maximum and the minimum values of all other weights can be obtained. The maximum and the minimum attainable values of output weights and input weights are represented by, umax and umin (where r r min r = 1, . . . , s) and v max and v (where i = 1, . . . , m), respectively. i i Then the mean of a weight is equal to the average of the maximum and the minimum attainable values of the weight. Means of output weights and means of input weights are represented by r (where r = 1, . . . , s) and v i (where i = 1, . . . , m), respectively. u Step 2. In this step, we propose a goal programming (GP) model to determine a weight set, which is feasible in the CCR model and is also close to the means of the weights. The primary goal of the proposed model is to minimize the maximum percentage deviation of weights from their means and the secondary goal is to minimize the sum of percentage deviations. Consider the situation when umax ¼ umin , then no r r deviation of ur is allowed if the DMU under evaluation is kept as efficient. Therefore, we define X = {r if umax –umin , for r r min r = 1, . . . , s} to exclude those outputs if umax ¼ u . Similarly, r r we define U ¼ fi if v max – v min ; for i ¼ 1; . . . ; mg to exclude i i max min those inputs if v i ¼ v i . We call the proposed GP model, Percentage Deviation from Mean (PDM). The formulation of PDM of an efficient DMU, DMU0, is stated as follows:
Min Mz þ
X
þ
dr
dr
0:5 umax umin r r ! X eþi ei þ þ 0:5 v max v min 0:5 v max v min i i i i i2U r2X
s:t:
s X r¼1 m X i¼1 s X
ur yrj
0:5 umax umin r r
þ
m X
v i xij 0;
!
j ¼ 1; . . . ; n; and j–o;
ð3:1Þ ð3:2Þ
i¼1
v i xio ¼ 1;
ð3:3Þ
ur yro ¼ 1;
ð3:4Þ
r¼1
þ
r ; r 2 X ur þ dr dr ¼ u v i þ ei eþi ¼ v i ; i 2 U dr 0 z max 0:5 ur umin r þ dr 0 z 0:5 umax umin r r e maxi 0 z 0:5 i min i eþ maxi 0 z 0:5 i min i þ z; dr ; dr ; ei ; eþi ; ur ; i 0;
v
v
v
v
v
ð3:5Þ ð3:6Þ r2X
ð3:7Þ
r2X
ð3:8Þ
i2U
ð3:9Þ
i2U
ð3:10Þ
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K.F. Lam, F. Bai / Computers & Industrial Engineering 60 (2011) 527–533
where M is a large positive number. Values of umax ; umin ; v max ; r r i min v i ; ur and v i are obtained from Step 1. The objective function is bi-criteria. PDM is solved as a preemptive goal programming problem, where the primary goal is to minimize the value of z P d r and the secondary goal is to minimize r2X 0:5ðumax umin Þ þ r r P þ þ ei ei dr þ . Constraints in þ i2U 0:5ðv max v min Þ 0:5ðumax umin 0:5ðv max v min Þ Þ r r i i i i (3.2) ensure that no efficiency ratio is greater than unity. Constraints (3.3) and (3.4) enforce that the DMU under evaluation is always efficient. Constraints (3.5) and (3.6) measure deviations of weights from their means using deviational variables,
3. Examples We apply the proposed model to two data sets, which have been studied by other researchers. All linear programming problems in this paper are solved using LINDO (Industrial 6.1, 2003).
þ
dr þ dr ; dr ; e i ; ei . The ratio, 0:5ðumax umin Þ, in constraint (3.7) represents r r r when ur < u r . Similarly, the percentage deviations of ur from u
the ratio,
max min smaller and all e i would be smaller than max thanmin0:5 ur ur 0:5 v i v i , and this implies z < 1. Therefore, this solution is always better than any solution contains one or more zero weights, since z is equal to 1 when the solution contains one or more zero weights.
dþ r , 0:5ðumax umin Þ r r
in constraint (3.8) represents the percent-
r when ur > u r . Similar arguments apage deviations of ur from u ply to constraints (3.9) and (3.10). Since z must be greater than or equal to all those deviations in constraints (3.7)–(3.10), minimizing the value of z is the same as minimizing the maximum percentage deviation. It can be shown that all the original optimal weight sets of DMU0 from the CCR model are also feasible in the PDM model. Since constraints (3.2) and (3.3) are from the CCR model and constraint (3.4) designates that DMU0 remains efficient in the PDM model, therefore, all the optimal weight sets that are originally feasible from the CCR model are also feasible in constraints 3.2, 3.3, and 3.4. In addition, the same weight sets must also be feasible þ þ in constraints (3.5)–(3.10) for z, dr ; dr ; e i ; ei 0, where r 2 X and i 2 U. Therefore, all the optimal weight sets, which are originally feasible from the CCR model, are also feasible in the PDM model. Furthermore, the proposed model has an advantage that it attempts to avoid taking weights with extreme or zero values in DEA. DEA allows flexibility in determining values of weights. Consequently, many different forms of weight sets may be found within the same alternate optimal solutions in DEA. Although some studies (Cooper et al., 2007; Doyle & Green, 1994; Li & Reeves, 1999; Sexton et al., 1986) have suggested different methods in picking weight sets among the alternate optimal solutions, our proposed model is unique in the sense that it is the first method, which considers the feasible span of the weights, i.e., the maximum and the minimum attainable values in determining weights. In our model, PDM minimizes weights’ deviations from their means. Alternatively, when considering the feasible spread of weights, other interesting approaches can be developed. For example, if constraints (3.8) and (3.10) are removed from the PDM model, then only the maximum of the negative percentage deviations is penalized in the primary goal. Subsequently, if there exists at least one non-zero weight set among the original alternate optimal solutions, then it can be shown that the revised PDM model will guarantee the weights selected to be strictly positive. Given that the primary goal of the revised PDM model is to minimize the value of z, and if all ur and vi are strictly positive, then all dr would be
3.1. Example 1 Example 1 uses a data set that is also studied in Wong and Beasley (1990) and Li and Reeves (1999). The performance of seven departments in a university is studied. Each department contains scores of three inputs and three outputs. The three inputs are number of academic staff (x1), academic staff salaries (x2) and support staff salaries (x3). The three outputs are number of undergraduate students (y1), number of postgraduate students (y2) and number of research papers (y3). Wong and Beasley (1990) point out that the classical DEA results of Example 1 contain too many zero weights. We apply the CCR model to Example 1 and the DEA results are reported in Table 1. In Table 1, among the 42 weights from the seven departments, 22 of them are zero weights. Li and Reeves (1999) also apply their model, Minsum to Example 1. Seven zero weights are obtained by Minsum (Li & Reeves, 1999, p. 514). However, according to the results of Minsum, all the seven departments have zero weights for input 3. This result may require explanation of why support staff salaries are not included in measuring efficiencies among all the departments. Li and Reeves (1999, p. 513) point out the problem of zero weights for input 3 as follows: Notice that in all these results, the third input is avoided by almost all DMUs. This may indicate that the third input is the most expensive one for most DMUs. If this input cannot be ignored, a lower bound needs to be specified. Bal et al. (2008) propose a super efficiency model that minimizes the coefficient of variation for input and output weights. They also apply their model to Example 1. Only one zero weight is obtained for input 3 in Department 6. Non-zero weights are obtained for other inputs and outputs in the other departments. The results of the PDM model are reported in Table 2. Since our model applies only to efficient DMUs, we apply PDM to the six efficient departments. All weights obtained are strictly positive, including weights of input 3. Therefore, in this study, PDM has successfully excluded zero weights, especially for input 3, which has caused some problems to some previous studies. Moreover, all the six departments remain efficient with their new weight sets. In addition, the maximum percentage deviation (z) for each efficient department is also reported in Table 2. The z values range from 0.1853 (Department 5) to 0.3796 (Department 2).
Table 1 CCR DEA results of Example 1. Dept.
Efficiency
v1
v2
v3
u1
u2
u3
1 2 3 4 5 6 7
1 1 1 0.819679 1 1 1
0 0 0 0.06415 0 0 0.007319
0.0025 0.001333 0.000335 0.000063 0.0005 0.00137 0.000298
0 0 0.007113 0 0 0 0
0 0.007194 0 0.009108 0 0.001162 0.001207
0.028579 0 0 0 0 0 0.003337
0 0 0.013333 0 0.007692 0.018813 0.001045
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Table 2 PDM DEA results of Example 1.
a
Dept.
za
Efficiency
v1
v2
v3
u1
u2
u3
1 2 3 4 5 6 7
0.3232 0.3796 0.3031 N/A 0.1853 0.2608 0.2793
1 1 1 0.819679 1 1 1
0.026934 0.033413 0.004043 0.06415 0.009323 0.019299 0.015450
0.000846 0.000414 0.000130 0.000063 0.000204 0.000506 0.000107
0.016920 0.000781 0.009067 0 0.000692 0.005274 0.000190
0.005275 0.006020 0.002222 0.009108 0.000664 0.002800 0.001406
0.011129 0.001926 0.001508 0 0.002929 0.005496 0.002402
0.017294 0.002107 0.005300 0 0.003133 0.008513 0.001950
The maximum percentage deviation of weights from their means.
Table 3 illustrates relationships between the maximum attainable value, the minimum attainable value and the percentage deviation of a weight, using results of Department 1. As previously discussed, the ratio,
d r , 0:5ðumax umin Þ r r
represents the percentage devia-
r when ur < u r and the ratio, tions of ur from u
dþ r , 0:5ðumax umin Þ r r
repre-
r when ur > u r . In sents the percentage deviations of ur from u Table 3, the percentage deviation of a weight from its mean can be represented by the deviation of the weight from its mean divided by half of the difference between its maximum attainable value and its minimum attainable value. The z value of Department 1 is 0.3232. Notice that Table 3 is for illustration purpose, where all the values of the weights and z are obtained directly from the PDM model.
Table 4 shows the cross-efficiency scores of departments using weights obtained from the CCR model (in Table 1). Each row in Table 4 shows the self and peer evaluated scores of a department. Table 5 shows the cross-efficiency scores of departments using weights obtained from the PDM model (in Table 2). After a comparison of results in Table 4 and Table 5, it is evident that Department 2 becomes more efficient than Departments 1 and 3 becomes more efficient than Department 5 in Table 5. Departments 6 and 4 remain as the most efficient department and the least efficient department respectively in both tables. The average cross-efficiency score of Department 6 increases from 0.932 (in Table 4) to 0.987 (in Table 5). Moreover, Department 6 is efficient when peer evaluated by four other departments in Table 5. These results further suggest that Department 6 is possibly the most efficient department in this case.
Table 3 An illustration of the results of Department 1 in Example 1.
Maximum weight Mean of weight Minimum weight Weights from PDM
v1
v2
v3
u1
u2
u3
0.079592 0.039796 0 0.026934
0.002500 0.001250 0 0.000846
0.050000 0.025000 0 0.016920
0.015588 0.007794 0 0.005275
0.028579 0.015214 0.001849 0.011129
0.039706 0.019853 0 0.017294
(0.039796 0.026934)/ (0.001250 0.000846)/ (0.025000 0.016920)/ (0.007794 0.005275)/ (0.015214 0.011129)/ (0.019853 0.017294)/ Percentage 0.039796 = 0.3232 0.001250 = 0.3232 0.025000 = 0.3232 0.007794 = 0.3232 0.013365a = 0.3056 deviation 0.019853 = 0.1289 from mean a ¼ 0:5ð0:028579 0:001849Þ ¼ 0:013365. 0:5 umax umin 2 2
Table 4 Cross-efficiency scores of departments using weights obtained from CCR in Example 1. Dept.
1
2
3
4
5
6
7
Average cross-efficiency score
Rank
1 2 3 4 5 6 7
1.000 0.625 0.518 0.229 0.829 0.704 0.773
0.811 1.000 0.811 0.811 0.684 0.978 0.702
0.826 0.716 1.000 0.249 0.711 1.000 0.256
0.689 1.000 0.736 0.821 0.767 0.952 1.000
1.000 0.780 0.680 0.336 1.000 0.888 0.822
0.807 1.000 0.826 0.771 0.755 1.000 0.710
1.000 0.953 0.763 0.576 1.000 1.000 1.000
0.876 0.868 0.762 0.542 0.821 0.932 0.752
2 3 5 7 4 1 6
Table 5 Cross-efficiency scores of departments using weights obtained from PDM in Example 1. Dept.
1
2
3
4
5
6
7
Average cross-efficiency score
Rank
1 2 3 4 5 6 7
1.000 0.807 0.904 0.347 0.728 1.000 0.382
0.798 1.000 0.791 0.726 0.822 1.000 0.835
0.980 0.720 1.000 0.295 0.543 0.959 0.242
0.687 1.000 0.735 0.820 0.765 0.951 1.000
0.944 0.892 0.782 0.447 1.000 1.000 0.761
0.936 0.856 0.847 0.413 0.817 1.000 0.496
0.869 0.961 0.761 0.599 1.000 1.000 1.000
0.888 0.891 0.831 0.521 0.811 0.987 0.674
3 2 4 7 5 1 6
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3.2. Example 2 Example 2 uses a data set that is studied in Shang and Sueyoshi (1995) and Li and Reeves (1999). The problem is to select the best flexible manufacturing system among a group of 12 similar systems. Each system contains scores of two inputs and four outputs. The two inputs are capital and operating cost (x1) and floor space requirements (x2). The four outputs are qualitative benefits (y1), work in process (y2), average number of tardy jobs (y3) and average yield (y4). We apply the CCR model to the data set and the DEA results of Example 2 are reported in Table 6. Table 6 shows the DEA results of the CCR model contain many zero weights and that seven systems are efficient. Therefore, it is difficult to determine the best system using results from the CCR model and other methods must be employed to find the best system. Solving model (2) gives the maximum value and the minimum value to each weight (see Table 7). Many minimum values are equal to zero. For efficient systems, many of the weights obtained from the CCR model (in Table 6) reach either their maximum or minimum values in Table 7. For example, two weights of System 1 reach their maximum values (v1 = 0.058754, u1 = 0.023810) and the other four arrive at their minimum values (v2 = u2 = u3 = u4 = 0). Systems 5 and 9 have similar results. As previously discussed, most LP software programs, regardless of the existence of alternate optimal solutions, will stop their iteration processes whenever they hit an optimal solution. Usually, they will first hit
an extreme point of the alternate optimal solutions. In Example 2, an extreme point generally contains many zero weights. We apply our proposed model PDM to the seven efficient systems, and the results are reported in Table 8. The z values range from 0.3309 (System 1) to 0.8370 (System 7). Table 8 shows for all the seven efficient systems, no zero weight is obtained. All the seven systems remain efficient with their new weights. Table 9 shows the cross-efficiency scores of systems using weights obtained from the CCR model (in Table 6), whereas Table 10 shows the cross-efficiency scores of systems using weights obtained from the PDM model (in Table 8). After a comparison of results in Tables 9 and 10, it is evident that some systems have similar rankings in both tables. For example, System 9 is the least efficient system in both tables. However, it is also one of the efficient systems in the CCR model (see Table 6). Thus, System 9 may require more assessments on its actual efficiency. Despite its poor peer-evaluated performance, System 9 has strictly positive weights in the results of the PDM model (see Table 8). However, much heavier virtual weights are placed on input 1 and output 4 than input 2 and the other outputs. For instance, based on the weights of System 9 in Table 8, 69.2% of the total virtual input is accounted for by the value of capital and operating cost (x1), while 75.7% of the total virtual output is accounted for by the value of the average yield (y4). Furthermore, if the weights of System 9 are based on Table 6, then 100% of the total virtual input is accounted for by the value of x1, and 100% of the total virtual output is accounted for by the value of y4. Another example of systems having
Table 6 CCR DEA results of Example 2. System
Efficiency
v1
v2
u1
u2
u3
u4
1 2 3 4 5 6 7 8 9 10 11 12
1 1 0.982369 1 1 1 1 0.961428 1 0.953505 0.983144 0.801173
0.058754 0.049889 0.085034 0.084674 0.105263 0.208768 0.161031 0.075790 0.272480 0.087680 0.056370 0.059349
0 0.039738 0 0.027306 0 0 0 0.026203 0 0.061002 0 0.019139
0.023810 0.014474 0.028327 0.023820 0 0 0 0.021082 0 0 0.020600 0.016696
0 0 0 0 0 0 0.018763 0.003977 0 0.027800 0 0
0 0 0.016164 0.007040 0.083333 0.061430 0.031183 0 0 0 0 0.004935
0 0.014614 0.000931 0.015856 0 0.041991 0.014441 0.011806 0.055249 0 0.003129 0.011114
Table 7 Maximum and minimum values of weights in Example 2. Sa
a b
Efficiency
1
1
2
1
3 4
0.982369b 1
5
1
6
1
7
1
8 9
0.961428 1
10 11 12
0.953505 0.983144 0.801173
Max Min Max Min Max Min Max Min Max Min Max Min Max Min
v1
v2
u1
u2
u3
u4
0.058754 0 0.049889 0 0.085034 0.084674 0.000085 0.105263 0 0.208768 0.111077 0.161031 0.100813 0.075790 0.272480 0.102303 0.087680 0.056370 0.059349
0.2 0 0.222222 0.039738 0 0.249776 0.027306 0.263158 0 0.086656 0 0.060315 0 0.026203 0.078068 0 0.061002 0 0.019139
0.023810 0 0.025641 0 0.028327 0.023820 0 0.035466 0 0.049748 0 0.058849 0 0.021082 0.061945 0 0 0.020600 0.016696
0.022075 0 0.021067 0 0 0.027568 0 0.029240 0 0.024326 0 0.037736 0 0.003977 0.057471 0 0.027800 0 0
0.040092 0 0.061197 0 0.016164 0.046479 0 0.083333 0 0.099497 0 0.104027 0 0 0.080177 0 0 0 0.004935
0.013947 0 0.033557 0 0.000931 0.04 0.000302 0.027244 0 0.060606 0.018722 0.036880 0 0.011806 0.055249 0 0 0.003129 0.011114
S = system. For all inefficient systems, maximum value = minimum value.
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K.F. Lam, F. Bai / Computers & Industrial Engineering 60 (2011) 527–533
Table 8 PDM DEA results of Example 2.
a b
Sa
z
b
1 2 3 4 5 6 7 8 9 10 11 12
0.3309 0.4506 N/A 0.3433 0.3940 0.6295 0.8370 N/A 0.6135 N/A N/A N/A
Efficiency
v1
v2
u1
u2
u3
u4
1 1 0.982369 1 1 1 1 0.961428 1 0.953505 0.983144 0.801173
0.024458 0.024737 0.085034 0.042337 0.052632 0.135131 0.105718 0.075790 0.188439 0.087680 0.056370 0.059349
0.116747 0.131739 0 0.138654 0.131579 0.065379 0.055401 0.026203 0.038554 0.061002 0 0.019139
0.007965 0.007044 0.028327 0.007821 0.010745 0.009217 0.004795 0.021082 0.011972 0 0.020600 0.016696
0.007385 0.005787 0 0.009052 0.008859 0.004884 0.010100 0.003977 0.011108 0.027800 0 0
0.013412 0.016810 0.016164 0.015261 0.025248 0.018433 0.008476 0 0.015496 0 0 0.004935
0.004666 0.009218 0.000931 0.013185 0.008254 0.043485 0.030756 0.011806 0.041839 0 0.003129 0.011114
S = system. The maximum percentage deviation of weights from their means.
Table 9 Cross-efficiency scores of systems using weights obtained from CCR in Example 2.
a
System
1
2
3
4
5
6
7
8
9
10
11
12
ACEa
Rank
1 2 3 4 5 6 7 8 9 10 11 12
1.00 0.96 0.90 0.85 0.90 0.85 0.91 0.91 0.44 0.73 0.98 0.74
1.00 1.00 0.89 1.00 0.96 0.85 0.88 0.91 0.64 0.74 0.92 0.77
1.00 0.96 0.98 0.93 1.00 0.93 1.00 0.93 0.42 0.76 0.98 0.80
1.00 0.98 0.95 1.00 1.00 0.97 1.00 0.95 0.72 0.80 0.95 0.80
0.66 0.63 0.93 0.85 1.00 0.83 0.89 0.64 0.02 0.58 0.62 0.74
0.60 0.60 0.76 0.79 0.80 1.00 0.97 0.68 1.00 0.68 0.58 0.62
0.63 0.60 0.81 0.82 0.86 1.00 1.00 0.73 1.00 0.80 0.61 0.64
1.00 0.98 0.95 1.00 1.00 0.96 1.00 0.96 0.75 0.83 0.95 0.79
0.36 0.37 0.42 0.48 0.44 0.70 0.64 0.45 1.00 0.47 0.36 0.35
0.70 0.65 0.79 0.86 0.89 0.75 0.80 0.74 0.60 0.79 0.64 0.64
1.00 0.97 0.92 0.90 0.93 0.95 1.00 0.94 0.67 0.78 0.98 0.76
1.00 0.98 0.95 1.00 1.00 0.97 1.00 0.95 0.72 0.80 0.95 0.80
0.829 0.806 0.855 0.873 0.898 0.896 0.925 0.817 0.667 0.728 0.794 0.704
6 8 5 4 2 3 1 7 12 10 9 11
ACE: average cross-efficiency score.
Table 10 Cross-efficiency scores of systems using weights obtained from PDM in Example 2.
a
System
1
2
3
4
5
6
7
8
9
10
11
12
ACEa
Rank
1 2 3 4 5 6 7 8 9 10 11 12
1.00 0.99 0.81 0.98 1.00 0.50 0.56 0.72 0.24 0.54 0.80 0.74
0.99 1.00 0.80 1.00 1.00 0.51 0.57 0.72 0.26 0.53 0.79 0.74
1.00 0.96 0.98 0.93 1.00 0.93 1.00 0.93 0.42 0.76 0.98 0.80
0.96 0.95 0.82 1.00 1.00 0.58 0.64 0.75 0.34 0.60 0.79 0.74
0.94 0.92 0.84 0.97 1.00 0.57 0.63 0.74 0.28 0.58 0.79 0.75
0.83 0.83 0.88 0.99 0.96 1.00 1.00 0.87 0.89 0.80 0.78 0.74
0.82 0.81 0.89 1.00 0.97 0.99 1.00 0.87 0.91 0.83 0.77 0.74
1.00 0.98 0.95 1.00 1.00 0.96 1.00 0.96 0.75 0.83 0.95 0.79
0.73 0.72 0.81 0.88 0.86 1.00 0.99 0.80 1.00 0.79 0.70 0.67
0.70 0.65 0.79 0.86 0.89 0.75 0.80 0.74 0.60 0.79 0.64 0.64
1.00 0.97 0.92 0.90 0.93 0.95 1.00 0.94 0.67 0.78 0.98 0.76
1.00 0.98 0.95 1.00 1.00 0.97 1.00 0.95 0.72 0.80 0.95 0.80
0.914 0.898 0.871 0.958 0.967 0.809 0.849 0.834 0.591 0.720 0.827 0.744
3 4 5 2 1 9 6 7 12 11 8 10
ACE: average cross-efficiency score.
similar rankings in both tables is System 3. The rank of System 3 is 5 in both tables. This ranking is higher than those of three other efficient systems. However, System 3 is also one of the inefficient systems in the CCR model. Therefore, System 3 may as well require more assessments. While some systems have similar rankings in Tables 9 and 10, other systems have very different rankings. For example, the ranks of Systems 6 and 7 are 3 and 1, respectively in Table 9. Nevertheless, the ranks of Systems 6 and 7 are 9 and 6, respectively in Table 10. In fact, substantial differences exist in the rankings of the two systems in the two tables. Intuitively, one may prefer to use weights obtained from the PDM model since they contain less zero weights and are more homogeneous. After a comparison of the average cross-efficiency scores in Table 10, it is evident that System 5 is the most efficient system. Furthermore, System 5 is also efficient when peer evaluated by six other systems in Table 10. Therefore, we choose System 5 as the best system. This
result is also consistent with the results of some previous studies (Li & Reeves, 1999; Shang & Sueyoshi, 1995). If we used the weights obtained from the CCR model to compute cross-efficiency scores, then we would choose System 7 as the best model. The average cross-efficiency scores with their efficiency rankings of the PDM model, alongside with those results in the previous studies (Li & Reeves, 1999; Shang & Sueyoshi, 1995) are reported in Table 11. Li and Reeves (1999) use two multiple criteria approaches Minimax and Min M 0.2do, and Shang and Sueyoshi (1995) use AHP and DEA (AHP–DEA) to solve this problem. Results of Minimax and that of AHP–DEA indicated that there are two systems tied at the highest efficiency score; therefore, these two methods cannot be used to determine the best system. Both PDM and Min M 0.2do choose System 5 as the best system. To compare the results of our proposed model with the other existing methods, we further compute the rankings of the average effi-
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K.F. Lam, F. Bai / Computers & Industrial Engineering 60 (2011) 527–533 Table 11 Efficiency scores and rankings of different methods of Example 2.
a b c d
System
Minimaxa
Min M 0.2dob
AHP–DEAc
Averaged of the three methods
Ranks of average of the three methods
Average crossefficiency (PDM)
Ranks of average cross- efficiency (PDM)
1 2 3 4 5 6 7 8 9 10 11 12
1 0.959 0.953 0.927 1 0.847 0.891 0.950 0.604 0.728 0.977 0.801
0.837 0.804 0.927 0.922 1 0.812 0.891 0.904 0.604 0.728 0.813 0.801
0.986 0.980 0.941 0.989 1 0.961 1 0.895 0.115 0.786 0.882 0.788
0.941 0.914 0.940 0.946 1 0.873 0.927 0.916 0.441 0.747 0.891 0.797
3 7 4 2 1 9 5 6 12 11 8 10
0.914 0.897 0.871 0.958 0.967 0.809 0.849 0.834 0.591 0.720 0.827 0.744
3 4 5 2 1 9 6 7 12 11 8 10
Efficiency scores of Minimax (Li & Reeves, 1999, Table 14, p. 515). Efficiency scores of Min M 0.2do (Li & Reeves, 1999, Table 14, p. 515). Efficiency scores of AHP–DEA (Shang & Sueyoshi, 1995, Table 6, p. 311). Average efficiency scores of the three methods: Minimax, Min M 0.2do and AHP.
ciency scores of the three approaches, namely, Minimax, Min M 0.2do and AHP–DEA and compare them with the rankings of the cross-efficiency scores of our proposed model. The Spearman’s rho of 0.958 between the two rankings represents that our model is also comparable to some good existing methods. Moreover, our model can choose weights with fewer zero values and are close to their means. 4. Conclusion This paper proposes a model to obtain weight sets, which minimize the percentage deviations of weights from their means in DEA. The proposed model can be used to yield more reasonable weights without a priori information about the weights; therefore, it can be applied to any DEA studies. Furthermore, the optimal weight set, which obtained from our proposed model, is also feasible in the CCR model. We have demonstrated that the proposed model stays within the framework of classical DEA, and attempts to find weight sets only from the alternate optimal solutions of the CCR model. Another advantage of the proposed model is that it attempts to select weights that are close to their mean values. This objective, I believe, is a general agreeable objective among most decision makers. References Allen, R., Athanassopolus, A., & Dyson, R. G. (1997). Weights restrictions and value judgments in data envelopment analysis: Evolution, development and future directions. Annals of Operations Research, 73, 13–34. Amin, G. R., & Toloo, M. (2004). A polynomial-time algorithm for finding e in DEA models. Computers & Operations Research, 31, 803–805. Andersen, P., & Petersen, N. C. (1993). A procedure for ranking efficient units in data envelopment analysis. Management Science, 39, 1261–1294. Azizi, H., & Ajirlu, S. F. (2010). Measurement of overall performances of decisionmaking units using ideal and anti-ideal decision-making units. Computers & Industrial Engineering, 59, 411–418.
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