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A new method based on the dispersion of weights in data envelopment analysis
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Computers & Industrial Engineering 54 (2008) 502–512 www.elsevier.com/locate/dsw
A new method based on the dispersion of weights in data envelopment analysis ¨ rkcu¨, Salih C Hasan Bal *, H. Hasan O ¸ elebiog˘lu Department of Statistics, Gazi University, 06500 Teknikokullar, Ankara, Turkey Received 15 April 2006; received in revised form 20 August 2007; accepted 4 September 2007 Available online 11 September 2007
Abstract One of the drawbacks of the data envelopment analysis (DEA) is the problem of lack of discrimination among efficient decision making units (DMUs) and hence yielding many numbers of DMUs as efficient. The main purpose of this study is to overcome this inability. In the case in which the minimization of the coefficient of variation (CV) for input–output weights is added to the DEA model, more reasonable and more homogeneous input–output weights are obtained. For this new proposed model based on the CV it is observed that the number of efficient DMUs is reduced, improving the discrimination power. When this new approach is applied to two well-known examples in the literature, and a real-world data of OECD countries, it has been seen that the new model yielded a more balanced dispersion of input–output weights and reduced the number of efficient DMUs. In addition, the applicability of the new model is tested by a simulation study. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Data envelopment analysis; Weight dispersion; Weight restriction; Coefficient of variation; Discrimination power
1. Introduction Data envelopment analysis (DEA) is a fractional mathematical programming technique that has been developed by Charnes, Cooper, and Rhodes (1978). It is used to measure the productive efficiency of decision making units (DMUs) and evaluate their relative efficiency. This analysis determines the productivities of DMUs, specified as the ratio of the weighted sum of outputs to the weighted sum of inputs, compares them to each other and determines the most efficient DMU(s). DEA obtains the optimal weights for all inputs and outputs of each unit without imposing any constraints on these weights. In DEA we sometimes encounter extreme values or zeroes in input and/or output weights for examined DMUs. In some cases we meet the unfitness of weights, i.e., a solution giving a big weight to variables with less importance or giving a small or zero weight to important variables. Especially in the zero cases, weights of input and/or output do not contribute to interpret the results of analysis. In the literature, various efforts have been done to overcome this problem. The problem of unrealistic weights in DEA has been tackled mainly by *
Corresponding author. E-mail address: [email protected] (H. Bal).
0360-8352/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2007.09.001
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the techniques of weight restriction. Thompson, Singleton, Thrall, and Smith (1986, 1990) developed the assurance region approach to help choosing a best site for the location of high energy physics laboratory in Texas. The name of assurance region comes from the constraint which limits the region of weights to some special area. Charnes, Cooper, Huang, and Sun (1990) developed an approach that they called cone ratio envelopment in order to evaluate bank performances when unknown allowances for risk and similar factors needed to be taken into account. Wong and Beasley (1990) provide a weight restriction method by setting bounds on the proportions of individual inputs (or outputs) to total input (or output). The value efficiency developed by Halme, Joro, Korhonen, Salo, and Wallenius (1999) is an efficiency concept, which takes into consideration the decision maker’s preferences. Li and Reeves (1999) have been developed multi-criteria data envelopment analysis (MCDEA) with the aid of one criterion efficiency evaluation methods. In a multi-criteria problem, it is generally impossible to find a solution that optimizes all criteria simultaneously. The Li and Reeves approach gives non-dominant (non-optimal) solutions. These non-dominant solutions can also be different to the preferences of decision maker. Recently, using the goal programming and DEA, Jahanshahloo, Memariani, Hosseinzadeh, and Shoja (2005) have been proposed a feasible interval of weights. This method uses the bounds on weights considered by decision maker. Because all of these methods incorporate additional constraints to the model they make harder to solve the problem and may cause to infeasibility. In this study, a new method is developed for the betterment of dispersion of input–output weights based on the minimization of coefficient of variation (CV), and hence improving the discrimination power of the DEA method. For our new approach it is not necessary any a priori information involving human value judgement and it does not need any additional constraints on weights, as well. Moreover, it seems there is no unfeasibility problem for the solutions to our approach. The paper is organized as follows. In Section 2, the basic DEA model and related concepts are given. In Section 3, the data envelopment analysis based on CV model (CVDEA) is presented and its formulation is explained. In Section 4, both the DEA and the CVDEA models are applied to two examples and their solutions are compared in respects of the number of efficient DMUs and the dispersion of input–output weights. In Section 5, the performances of the DEA and the CVDEA models are compared by a detailed simulation data. In Section 6, the DEA and the CVDEA models are applied to a real data set related to the OECD countries. Lastly, in Section 7, a summary of the research and its results are provided.
2. Data envelopment analysis DEA evaluates the relative efficiency of homogeneous units by considering multiple inputs and outputs. Inputs can be resources used by a DMU and outputs can be products produced and/or performance measures of the DMU. The efficiency is defined as a ratio of the weighted sum of outputs to the weighted sum of inputs. DEA has been extensively used to compare the efficiencies of non-profit and profit organizations such as schools, hospitals, shops, bank branches and other environments in which there are relatively homogeneous DMUs (Baker and Talluri, 1997; Cooper et al., 2000). Assuming that there are n DMUs each with m inputs and s outputs, the relative efficiency of a particular DMUo is obtained by solving the following fractional programming problem: s P
wo ¼ max r¼1 m P i¼1 s P
s:t:
r¼1 m P
ur y ro vi xio ur y rj
ð1Þ 6 1;
j ¼ 1; 2; . . . ; n;
vi xij
i¼1
ur P 0;
r ¼ 1; 2; . . . ; s;
vi P 0;
i ¼ 1; 2; . . . ; m;
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where j is the DMU index, j = 1, . . . , n; r the output index, r = 1, . . . , s; i the input index, i = 1, . . . ,m; yrj the value of the rth output for the jth DMU, xij the value of the ith input for the jth DMU, ur the weight given to the rth output; vi the weight given to the ith input, and wo is the relative efficiency of DMUo, the DMU under evaluation or the target DMU. In this model, DMUo is efficient if and only if wo = 1. In DEA, a DMU is considered individually in determining its relative efficiency. This target DMU effectively selects weights that maximize its output to input ratio, subject to the constraints that the output to input ratios of all the n DMUs with these weights are 61. A relative efficiency score of 1 indicates that the DMU under consideration is efficient whereas a score less than 1 implies that it is inefficient. This fractional program can be converted into a linear programming problem where the optimal value of the objective function indicates the relative efficiency of DMUo. The reformulated linear programming problem, also known as the CCR model, is as follows: s X wo ¼ max ur y ro r¼1
s:t:
m X i¼1 s X
vi xio ¼ 1; ur y rj
r¼1
m X
ð2Þ vi xij 6 0;
j ¼ 1; 2; . . . ; n
i¼1
ur P 0; vi P 0;
r ¼ 1; 2; . . . ; s; i ¼ 1; 2; . . . ; m:
In model (2), i.e., in the classical DEA, the weighted sum of the inputs for the target DMU is forced to 1, thus allowing for the conversion of the fractional programming problem into a linear programming problem that can be solved by using any linear programming software. The solution to model (2) assigns the value 1 to all efficient DMUs. The super efficiency concept is proposed to differentiate completely among all efficient DMUs when there are more than one efficient DMUs. One of the super efficiency models for ranking efficient DMUs in DEA was introduced by Andersen and Petersen (1993). This method enables an extreme efficient unit o to achieve an efficiency score greater than one by removing the oth constraint in the envelopment LP formulation, as shown in model (3) (Adler et al., 2002). s X wo ¼ max ur y ro r¼1
s:t:
s X r¼1 m X
ur y rj
m X
vi xij 6 0;
j ¼ 1; 2; . . . ; n;
j 6¼ o
i¼1
ð3Þ
vi xio ¼ 1;
i¼1
ur P 0;
r ¼ 1; 2; . . . ; s;
vi P 0;
i ¼ 1; 2; . . . ; m:
3. The inclusion of coefficient of variation for input–output weights in the minimization process The coefficient of variation (CV), the ratio of sample standard deviation to the sample mean, measures the variability of the weights relative to their mean (or average). It compares the relative dispersion in one type of data with the relative dispersion in another type of data. Let ur(r = 1, 2, . . . , s) be the weight on output r and let u be the mean of the ur(r = 1,2, . . . , s). We define the CV for the weights ur as follows: ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s P ður uÞ2 =ðs 1Þ : CV ¼ r¼1 u
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Similarly, we can calculate the CV for the weights vi (i = 1, 2, . . . , m) as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m P ðvi vÞ2 =ðm 1Þ CV ¼
i¼1
: v Incorporating the coefficient of variation for input–output weights to the model 2 in the minimization process subject to the same constraints (i.e., min CV = max(CV)), our suggested model, we call it as CVDEA, becomes sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m P s P 2 ðvi vÞ2 =ðm 1Þ ðu u Þ =ðs 1Þ r s X i¼1 wo ¼ max ur y ro r¼1 v u r¼1 s m X X s:t: ur y rj vi xij 6 0; j ¼ 1; 2; . . . ; n; ð4Þ r¼1 i¼1 m X vi xio ¼ 1; i¼1
ur P 0;
r ¼ 1; 2; . . . ; s;
vi P 0;
i ¼ 1; 2; . . . ; m:
This nonlinear optimization model, based on the CCR model, can be easily solved with the Kuhn–Tucker algorithm. When there are more than one efficient DMUs, the CV is incorporated into the model 3 instead of model 2 in the minimization process and then all efficient DMUs are ranked over again. This model serves the same purposes of the Li and Reeves (1999) model MCDEA, i.e., can be used to improve discriminating power of the DEA and also effectively yields more reasonable/homogeneous input and output weights. On the other hand, our proposed model has only one objective function, namely simple, and is easily solved without any preference information of decision maker in contrary to the Li–Reeves model. 4. A comparison of the models by means of examples In order to exhibit the improvement of the dispersion of input–output weights and the increasing discrimination power for our suggested model, CVDEA, we have used the data sets of two examples from previous studies (Li & Reeves, 1999; Sexton, 1986; Wong & Beasley, 1990). Example 1 (Efficiency evaluation of six nursing homes). Two inputs and two output variables for six nursing homes are staff hours per day, including nurses, physicians etc. (x1); supplies per day, measured in thousands of dollars (x2); total medicare-plus medicaid-reimbursed patient days (y1); and total privately paid patient days (y2), respectively, and the related data are given in Table 1. For detailed descriptions of the data see Sexton (1986). When our suggested CVDEA and the DEA models are applied to the data, the results in Tables 2 and 3 are obtained. In Table 2, it is seen that the first output variable for DMUs A and B, and the first input variable for DMUs B and D, and the second input variable for DMUs A and C are neglected but the second output variable is Table 1 Data of six nursing homes 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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Table 2 Results of the DEA model DMU
Efficiency
Super efficiency
u1
u2
v1
v2
A B C D E F
1 1 1 1 0.977 0.867
2 1.395 1.412 1.131 0.977 0.867
0.714 0 0.238 0 0.115 0.162
0 0.476 0 0.238 0.304 0.427
0 0 0.172 0.069 0.110 0.155
5.000 1.429 0.374 0.321 0.513 0.722
Table 3 Results of the CVDEA model DMU
Efficiency
u1
u2
v1
v2
A B C D E F
1 0.863 0.991 0.983 0.948 0.735
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
considered as extremely important in the DEA for DMUs A and B, respectively. In Table 3, the input–output weights for which are obtained by the CVDEA model are dispersed more homogeneous than that of the DEA. On the other hand, the CVDEA identifies only DMU A as efficient while the DEA identifies DMUs A, B, C and D as efficient. Hence we see that the CVDEA model reduces the number of efficient DMUs for the data in Example 1. In Table 4, the ranks of DMUs obtained by the DEA under the super efficiency and the CVDEA model, and a comparison of variances for input–output weights are given. The comparison of ranks for DMUs in both model is performed by Spearman’s and Kendall’s rank correlation coefficients, and the comparison of variances for input–output weights assigned by the models is conducted by the Mann–Whitney test. Kendall’s tau value s = 0.733 (or the Spearman’s value rs = 0.828) shows that under the significance level a = 0.05 there is a powerful correlation in the same direction between the efficiency ranking values of the DMUs obtained by the two models. The Mann–Whitney test inferred that the variance of weights for the CVDEA model is less than that of the DEA (p value = 0.023). Example 2 (Efficiency evaluation of seven departments in a university). The input–output variables for seven departments in a university are defined as follows and the related data are given in Table 5: y1 number of undergraduate students y2 number of postgraduate students y3 number of research papers x1 number of academic staff Table 4 Ranks of the DMUs and test of homogeneity of weights for the models DMU
Rank of DMUs
Weight variances
DEA
CVDEA
DEA
CVDEA
A B C D E F
1 3 2 4 5 6
1 5 2 3 4 6
5.7824 0.4538 0.0241 0.0220 0.0365 0.0722
0.0800 0.0017 0.0005 0.0007 0.0017 0.0025
Test
Spearman’s rs = 0.829, Kendall’s s = 0.733
p value = 0.023
H. Bal et al. / Computers & Industrial Engineering 54 (2008) 502–512
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Table 5 Data of seven departments in a university DMU
y1
y2
y3
x1
x2
x3
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
x2 x3
academic staff salaries in thousands of pounds support staff salaries in thousands of pounds
Again, when our suggested DEA and CVDEA models are applied to this data, the results in Tables 6 and 7 are obtained. In Example 2, it is obvious that, the CVDEA model results a more better dispersion of weights than that of the DEA when Tables 6 and 7 are examined. In the CVDEA model, almost the same efficiency scores of the DEA are assigned to the DMUs and almost all of the input–output variables are consumed while in the DEA for most of DMUs the variables corresponding to the second output and the first and third input are neglected. On the other hand, the CVDEA identifies DMU 1, DMU 5 and DMU 7 as efficient while the DEA rates all of the DMUs as efficient except DMU 4. Hence we see that the CVDEA model reduces the number of efficient DMUs in Example 2. In Table 8, the ranks of DMUs obtained by the DEA under super efficiency and the CVDEA model, and a comparison of variances for input–output weights are given. The Kendall’s tau value s = 0.809 is also an indicator that under the significance level a = 0.05 there is a powerful correlation in the same direction between the efficiency ranking values of the DMUs obtained by the two models. As well, the variances of the input–output weights of each DMU for the two models are given. The hypothesis related to the variances is tested with Mann–Whitney. Hence it is statistically Table 6 Results of the DEA model DMU
Efficiency
Super efficiency
u1
u2
u3
v1
v2
v3
1 2 3 4 5 6 7
1 1 1 0.820 1 1 1
1.829 1.048 1.198 0.819 1.220 1.190 1.266
0.983 0.719 0 0.911 0 0.639 0.121
1.172 0 0 0 0.432 0 0.334
0 0 1.333 0 0.288 0.347 0.105
0 0 0 6.415 0 0 0.732
0.250 0.133 0.033 0.006 0.05 0.137 0.030
0 0 0.711 0 0 0 0
Table 7 Results of the CVDEA model DMU
Efficiency
Super efficiency
u1
u2
u3
v1
v2
v3
1 2 3 4 5 6 7
1 0.983 0.990 0.820 1 0.980 1
1.368 0.983 0.990 0.820 1.311 0.980 1.253
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
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H. Bal et al. / Computers & Industrial Engineering 54 (2008) 502–512
Table 8 Ranks of the DMUs and testing the homogeneity of weights for models DMU
Rank of DMUs
Weight variances
DEA
CVDEA
DEA
CVDEA
1 2 3 4 5 6 7
1 6 4 7 3 5 2
1 5 4 7 2 6 3
0.2877 0.0827 0.2406 6.6045 0.0357 0.0675 0.0765
0.0838 0.0347 0.0682 0.1501 0.0192 0.0394 0.0313
Test
Spearman’s rs = 0.928, Kendall’s s = 0.809
p value = 0.042
concluded that the variance of weights for the CVDEA model is less than that of the DEA (p value = 0.042). However we have seen the CVDEA improves the dispersion of input–output weights and increases the discrimination power of efficient DMUs in the foregoing two examples, it will be presented for supporting these achievements by a simulation study reflecting the diversity of input–output and DMU numbers. 5. Simulation experiment In the simulation study, the efficiency scores and the weights by the two models are evaluated according to the formed possible cases for the different number of DMUs (n = 10, 15, 20, 25, 30, 35, 40, 45, 50, 100), and of input variables (i = 1, 2, . . . , 7), and of output variables (o = 1, 2, . . . , 7). Each case is repeated for 100 times. The input and output variables are drawn randomly from Uniform(10, 500) distribution. For each case, using the following hypotheses it is tested if the two approaches give the similar rankings to DMUs. H0: The DMU efficiency scores for the CVDEA are uncorrelated to the DMU efficiency scores for the DEA. H1: The DMU efficiency scores for the CVDEA are correlated in the same direction to the DMU efficiency scores for the DEA. For testing the hypotheses, the Spearman’s rho and the Kendall’s tau correlations are used. In Table 9, in the column of testing the ranks of methods, the number of cases agreed with the hypothesis H1 per 100 repeats and the average value for the Spearman’s and Kendall’s rank correlations are given. Later, in order to compare the weight variances corresponding to the CVDEA and the DEA models the following hypotheses are tested. H0: There are no difference between the weight variances corresponding to the CVDEA and the DEA models. H1: The weight variances for the CVDEA are less than that of the DEA. For testing the hypotheses, the Mann–Whitney statistic is used. In Table 9, in the column of testing the homogeneity, the number of cases agreed with the hypothesis H1 per 100 repeats and the average value for p values are given. All of the results given in Table 9 are obtained by using the programs MATLAB 7.0 and LINGO 9.0 (Pentium(R) 4 CPU 2.60 GHz., 248 MB RAM). Table 9 shows that there is a high correlation in the same direction between the efficiency scores of DMUs assigned by our suggested approach and the DEA in more than 94% of the cases and reaches 100% of the case for n = 10, i = 2, and s = 1. On the other hand, the weight variance for the CVDEA is less than that of the DEA in more than 95% of the cases and reaches 100% of the case for n = 10, i = 2, and s = 1. In other words, all results of the simulation show that there is statistically a significant correlation and hence a similarity between the ranks obtained by the DEA and the CVDEA models for each case. In addition, in almost all cases it is observed that the weight dispersion for the input and output variables is more homogeneous in the CVDEA. The simulation data results also confirmed that the CVDEA approach realizes a more convenient dispersion of weights for all variables.
H. Bal et al. / Computers & Industrial Engineering 54 (2008) 502–512
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Table 9 Results of the hypotheses testing n
i
o
Testing the ranks of methods
Testing the homogeneity
Number of accepting H1
rs
s
Number of accepting H1
p value
10
2 1 2 1
1 2 2 3
100 98 99 98
0.901 0.867 0.899 0.851
0.828 0.801 0.812 0.795
100 100 100 99
0.0009 0.0012 0.0013 0.0028
15
2 2 2 3
1 2 3 2
100 98 99 99
0.853 0.808 0.831 0.839
0.761 0.723 0.746 0.729
99 100 100 99
0.0021 0.0008 0.0011 0.0038
20
3 2 4 2
1 3 1 4
99 98 100 97
0.761 0.719 0.772 0.705
0.701 0.682 0.716 0.673
100 100 100 98
0.0018 0.0020 0.0016 0.0061
25
2 2 5 3
2 4 2 4
100 98 97 99
0.749 0.700 0.682 0.745
0.649 0.624 0.609 0.632
100 97 99 96
0.0008 0.0057 0.0029 0.0069
30
3 2 4 3
2 5 3 5
100 97 98 97
0.709 0.635 0.659 0.640
0.627 0.553 0.599 0.537
100 98 96 97
0.0026 0.0049 0.0083 0.0077
35
2 3 5 5
3 6 4 5
99 96 96 98
0.648 0.579 0.599 0.621
0.608 0.528 0.564 0.588
100 96 96 97
0.0015 0.0078 0.0075 0.0059
40
4 2 6 4
3 5 3 4
99 99 96 100
0.641 0.645 0.598 0.649
0.589 0.591 0.500 0.602
99 97 97 99
0.0038 0.0071 0.0078 0.0031
45
3 4 5 5
4 6 3 5
99 97 98 99
0.538 0.478 0.512 0.547
0.500 0.451 0.467 0.486
99 97 95 98
0.0033 0.0064 0.0091 0.0047
50
5 4 6 5
2 6 3 6
97 98 96 98
0.495 0.513 0.476 0.521
0.477 0.500 0.451 0.511
98 96 95 97
0.0050 0.0081 0.0090 0.0065
100
3 4 6 7
5 7 4 7
99 96 96 94
0.459 0.424 0.437 0.399
0.388 0.361 0.346 0.311
98 95 96 95
0.0021 0.0097 0.0088 0.0095
6. A real-world application In order to compare the two methods with a new real-world data, we make use of the variables for 30 OECD countries consisting of three input and five output variables (The data of the year 2005 are abridged from the site OECD home page http://www.oecd.org/countrieslist see Appendix A):
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Table 10 Efficiency and super efficiency scores, ranking and weight variances of the DEA and the CVDEA Countries
Efficiency
Australia Austria Belgium Canada Chec. Rep. Denmark Finland France Germany Greece Hungary Iceland Ireland Italy Japan S. Korea Luxembourg Mexico Netherland N. Zealand Norway Poland Portugal Slovak. Rep. Spain Sweden Switzerland Turkey England USA
y1 y2 y3 y4 y5 x1 x2 x3
Super efficiency
Rank of the countries
Weight variances
Maverick index
DEA
CVDEA
DEA
CVDEA
DEA
CVDEA
DEA
CVDEA
DEA
CVDEA
0.500 0.657 0.573 0.593 0.600 0.750 0.750 0.957 1 0.582 0.405 0.940 0.528 0.593 1 0.643 1 1 0.767 0.500 0.870 0.330 0.515 0.370 0.645 1 1 0.254 0.655 1
0.466 0.624 0.547 0.631 0.553 0.708 0.724 0.941 1 0.546 0.352 0.972 0.455 0.615 1 0.548 1 0.833 0.731 0.490 0.836 0.368 0.489 0.323 0.684 0.989 1 0.358 0.763 1
0.500 0.657 0.573 0.593 0.600 0.750 0.750 0.957 1.133 0.582 0.405 0.94 0.528 0.593 7.637 0.643 2.437 1.058 0.767 0.500 0.870 0.330 0.515 0.370 0.645 1.025 1.679 0.254 0.655 3.940
0.466 0.624 0.547 0.631 0.553 0.708 0.724 0.941 1.084 0.546 0.352 0.972 0.455 0.615 7.811 0.548 1.325 0.833 0.731 0.490 0.836 0.368 0.489 0.323 0.684 0.989 1.061 0.358 0.763 2.295
25 14 22 19 18 12 13 8 5 21 27 9 23 20 1 17 3 6 11 26 10 29 24 28 16 7 4 30 15 2
25 17 21 16 19 14 13 8 4 22 29 7 26 18 1 20 3 10 12 23 9 27 24 30 15 6 5 28 11 2
0.003012 0.002831 0.002938 0.002938 0.004341 0.006783 0.006783 0.006616 0.004379 0.004341 0.001029 0.005464 0.002356 0.002938 3.783673 0.004234 0.014809 0.021114 0.002122 0.003012 0.004628 0.001340 0.001949 0.001696 0.004283 0.012057 0.070788 0.002005 0.002938 0.005258
0.001352 0.003043 0.002501 0.000888 0.003386 0.002513 0.004100 0.003455 0.001069 0.000516 0.000526 0.001886 0.000809 0.000482 0.002456 0.001008 0.003482 0.000189 0.001400 0.000899 0.003081 0.000577 0.001038 0.00059 0.001009 0.001417 0.005364 0.001503 0.001059 0.001217
0.195 0.218 0.378 0.281 0.377 0.258 0.436 0.571 0.657 0.435 0.315 0.419 0.227 0.433 0.228 0.299 0.465 3.973 0.326 0.255 0.254 0.251 0.311 0.455 0.545 0.396 0.092 1.999 0.296 0.386
0.232 0.245 0.308 0.232 0.352 0.181 0.695 0.128 0.137 0.151 0.364 0.231 0.213 0.124 0.109 0.349 0.223 3.431 0.366 0.327 0.321 0.243 0.523 0.523 0.531 0.244 0.113 3.419 0.149 0.236
health expenditure per capita national income per capita ratio by literacy portion in world import portion in world export baby death rate unemployment ratio rate of inflation
When the DEA and our suggested CVDEA models are applied to this data the efficiency and the super efficiency scores, the ranks, the weight variances and the maverick index1 of DMUs (countries) are given in Table 10, respectively. In order to save space the input–output weights for the DMUs obtained for the DEA and the CVDEA models are not given in the tables. In Table 10, the DEA determines seven countries, USA, Germany, Japan, Luxembourg, Mexico, Sweden and Switzerland, as efficient while the CVDEA finds five efficient countries, USA, Germany, Japan, Luxembourg, and Switzerland. According to this analysis results, we see that the CVDEA model reduces the number of efficient DMUs for this real data. Here Mexico found as an efficient country by the DEA while it is identified as an inefficient country by the CVDEA. It is also known that Mexico is not a country having a powerful 1
A high maverick index, also called as the false positive index (FPI) of DMUs, is a measure showing that a DMU obtains an efficiency score just by using inappropriate weights. For details and comments, see Doyle and Green (1994) and Baker and Talluri (1997).
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economic structure and a high level of development such as the abovementioned efficient countries in reality. In addition, the maverick indices for Mexico are also evaluated very high for both methods, the DEA and the CVDEA. These results point out that our suggested CVDEA method is more consistent than the DEA. In other words the dispersion of weights is more reasonable in the CVDEA approach than that of the DEA. As seen from Table 10, it is statistically found that there is a significant correlation and hence a similarity between the ranks resulted by the DEA and the CVDEA, i.e., with Spearman‘s rho, rs = 0.974. In addition, it is observed that the weight dispersion for the input and output variables is more better in the CVDEA (p value = 0.0000507). In summary, the CVDEA approach attaches importance to all variables in general while the DEA ignores some variables. These two examples, the simulated data and the real data results also confirmed that the CVDEA approach realizes a more convenient dispersion of weights for all variables, refines the results of DEA model and eliminates apparently efficient DMUs and reduces the number of efficient DMUs. 7. Conclusions In generally, DEA models yield in many instances non-homogeneous weight dispersion of input and output parameters. Indeed, they yield several input–output weights that are zero or that have extreme values which imply that the corresponding parameters are not taken into account to interpret the efficiency of the DMUs. This paper overcomes this problem by using of the CVDEA model, i.e., incorporating the coefficient of variation for input–output weights to the DEA model in the minimization process. When the new CVDEA model is applied to the two instances in the literature, a real-world data of OECD countries, and simulation data it has been seen that the new model yielded a more balanced dispersion of input–output weights and reduced the number of efficient DMUs. Thus, we can objectively make more correct decisions by using CVDEA method in determining the efficiency of DMUs. Moreover, it is observed that there is a high correlation in the same direction between the efficiency scores of DMUs assigned by our suggested approach and the DEA. The CVDEA does not also need any constraints on weights. For a further study, the performance of the CVDEA method for the other forms of DEA models, such as the BCC, additive etc., may be investigated. Acknowledgements The authors are indebted to the anonymous referees for valuable comments and suggestions which have improved the earlier version of this paper. Appendix A Data of 30 OECD countries Countries
y1
y2
y3
Australia Austria Belgium Canada Chec. Rep. Denmark Finland France Germany Greece Hungary Iceland
2036 1968 2081 2312 930 2133 1502 2055 2424 1167 705 2103
25693 26765 27178 27840 13991 27627 24996 24223 25103 16501 12416 29581
100 100 100 100 100 100 100 100 100 97 99 100
y4 1.108 1.096 2.989 3.47 0.652 0.727 0.513 4.676 7.526 0.475 0.576 0.034
y5 1.016 1.134 3.332 3.939 0.598 0.87 0.696 4.82 8.562 0.161 0.538 0.034
x1
x2
6 7 5 4.3 6 11.7 6 7.6 5 8.8 4 5.6 4 10.2 4 11.3 5 8.7 5 10.2 8 7 4 2.8 (continued on next
x3 3 1.8 1.6 2.2 1.8 2.4 1.7 1.9 2.3 4.6 5.3 5.2 page)
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Appendix A (continued) Countries
y1
y2
y3
y4
y5
x1
x2
x3
Ireland Italy Japan S. Korea Luxembourg Mexico Netherland N. Zealand Norway Poland Portugal Slovak. Rep. Spain Sweden Switzerland Turkey England USA
1436 1783 1822 730 2215 356 2070 1424 2330 496 1237 930 1218 1746 2794 255 1461 4178
29886 23626 26755 17380 50061 9023 25657 20070 29918 9051 17290 11243 19472 24277 28769 6974 23509 34142
100 98 100 98 100 91 100 100 100 100 92 100 98 100 100 85 100 100
0.785 3.725 5.144 2.322 0.177 2.694 2.953 0.23 0.532 0.84 0.584 0.266 2.494 1.008 1.207 0.781 5.111 18.344
1.364 3.952 6.504 2.536 0.133 2.508 3.469 0.224 0.931 0.624 0.398 0.225 1.927 1.266 1.309 0.558 4.312 10.83
6 6 4 5 5 25 5 6 4 9 6 8 5 3 3 38 6 7
5.8 11.5 4.7 6.3 2.1 2.1 4.3 6.8 3.2 10.5 5 16.4 15.9 7.2 3 7.3 6 4.2
4.7 2.5 0.15 2.8 1.1 5 3.5 2.7 1.3 1.9 3.5 3.3 3.1 2.2 0.6 21.7 1.6 1.6
References Adler, N., Friedman, L., & Sinuany-Stern, Z. (2002). Review of ranking methods in the data envelopment analysis context. European Journal of Operational Research, 140, 249–265. Andersen, P., & Petersen, N. (1993). A procedure for ranking efficient units in data envelopment analysis. Management Science, 39(10), 1261–1264. Baker, R. C., & Talluri, S. (1997). A closer look at the use of data envelopment analysis for technology selection. Computers and Industrial Engineering, 32(1), 101–108. 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. Cooper, W. W., Seiford, L. M., & Tone, K. (2000). Data envelopment analysis. Boston, USA: Kluwer Academic Publishers. Doyle, J. R., & Green, R. (1994). Efficiency and cross-efficiency in data envelopment analysis: Derivatives, meanings and uses. Journal of Operational Research Society, 45(5), 567–578. Halme, M., Joro, T., Korhonen, P., Salo, S., & Wallenius, J. (1999). A value efficiency approach to incorporating preference information in data envelopment analysis. Management Science, 45, 103–115. Jahanshahloo, G. R., Memariani, A., Hosseinzadeh, F., & Shoja, N. (2005). A feasible interval for weights in data envelopment analysis. Applied Mathematics and Computation, 160, 155–168. Li, X. B., & Reeves, G. R. (1999). A multiple criteria approach to data envelopment analysis. European Journal of Operational Research, 115, 507–517. Sexton, T. R. (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.
Computers & Industrial Engineering 54 (2008) 502–512 www.elsevier.com/locate/dsw
A new method based on the dispersion of weights in data envelopment analysis ¨ rkcu¨, Salih C Hasan Bal *, H. Hasan O ¸ elebiog˘lu Department of Statistics, Gazi University, 06500 Teknikokullar, Ankara, Turkey Received 15 April 2006; received in revised form 20 August 2007; accepted 4 September 2007 Available online 11 September 2007
Abstract One of the drawbacks of the data envelopment analysis (DEA) is the problem of lack of discrimination among efficient decision making units (DMUs) and hence yielding many numbers of DMUs as efficient. The main purpose of this study is to overcome this inability. In the case in which the minimization of the coefficient of variation (CV) for input–output weights is added to the DEA model, more reasonable and more homogeneous input–output weights are obtained. For this new proposed model based on the CV it is observed that the number of efficient DMUs is reduced, improving the discrimination power. When this new approach is applied to two well-known examples in the literature, and a real-world data of OECD countries, it has been seen that the new model yielded a more balanced dispersion of input–output weights and reduced the number of efficient DMUs. In addition, the applicability of the new model is tested by a simulation study. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Data envelopment analysis; Weight dispersion; Weight restriction; Coefficient of variation; Discrimination power
1. Introduction Data envelopment analysis (DEA) is a fractional mathematical programming technique that has been developed by Charnes, Cooper, and Rhodes (1978). It is used to measure the productive efficiency of decision making units (DMUs) and evaluate their relative efficiency. This analysis determines the productivities of DMUs, specified as the ratio of the weighted sum of outputs to the weighted sum of inputs, compares them to each other and determines the most efficient DMU(s). DEA obtains the optimal weights for all inputs and outputs of each unit without imposing any constraints on these weights. In DEA we sometimes encounter extreme values or zeroes in input and/or output weights for examined DMUs. In some cases we meet the unfitness of weights, i.e., a solution giving a big weight to variables with less importance or giving a small or zero weight to important variables. Especially in the zero cases, weights of input and/or output do not contribute to interpret the results of analysis. In the literature, various efforts have been done to overcome this problem. The problem of unrealistic weights in DEA has been tackled mainly by *
Corresponding author. E-mail address: [email protected] (H. Bal).
0360-8352/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2007.09.001
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the techniques of weight restriction. Thompson, Singleton, Thrall, and Smith (1986, 1990) developed the assurance region approach to help choosing a best site for the location of high energy physics laboratory in Texas. The name of assurance region comes from the constraint which limits the region of weights to some special area. Charnes, Cooper, Huang, and Sun (1990) developed an approach that they called cone ratio envelopment in order to evaluate bank performances when unknown allowances for risk and similar factors needed to be taken into account. Wong and Beasley (1990) provide a weight restriction method by setting bounds on the proportions of individual inputs (or outputs) to total input (or output). The value efficiency developed by Halme, Joro, Korhonen, Salo, and Wallenius (1999) is an efficiency concept, which takes into consideration the decision maker’s preferences. Li and Reeves (1999) have been developed multi-criteria data envelopment analysis (MCDEA) with the aid of one criterion efficiency evaluation methods. In a multi-criteria problem, it is generally impossible to find a solution that optimizes all criteria simultaneously. The Li and Reeves approach gives non-dominant (non-optimal) solutions. These non-dominant solutions can also be different to the preferences of decision maker. Recently, using the goal programming and DEA, Jahanshahloo, Memariani, Hosseinzadeh, and Shoja (2005) have been proposed a feasible interval of weights. This method uses the bounds on weights considered by decision maker. Because all of these methods incorporate additional constraints to the model they make harder to solve the problem and may cause to infeasibility. In this study, a new method is developed for the betterment of dispersion of input–output weights based on the minimization of coefficient of variation (CV), and hence improving the discrimination power of the DEA method. For our new approach it is not necessary any a priori information involving human value judgement and it does not need any additional constraints on weights, as well. Moreover, it seems there is no unfeasibility problem for the solutions to our approach. The paper is organized as follows. In Section 2, the basic DEA model and related concepts are given. In Section 3, the data envelopment analysis based on CV model (CVDEA) is presented and its formulation is explained. In Section 4, both the DEA and the CVDEA models are applied to two examples and their solutions are compared in respects of the number of efficient DMUs and the dispersion of input–output weights. In Section 5, the performances of the DEA and the CVDEA models are compared by a detailed simulation data. In Section 6, the DEA and the CVDEA models are applied to a real data set related to the OECD countries. Lastly, in Section 7, a summary of the research and its results are provided.
2. Data envelopment analysis DEA evaluates the relative efficiency of homogeneous units by considering multiple inputs and outputs. Inputs can be resources used by a DMU and outputs can be products produced and/or performance measures of the DMU. The efficiency is defined as a ratio of the weighted sum of outputs to the weighted sum of inputs. DEA has been extensively used to compare the efficiencies of non-profit and profit organizations such as schools, hospitals, shops, bank branches and other environments in which there are relatively homogeneous DMUs (Baker and Talluri, 1997; Cooper et al., 2000). Assuming that there are n DMUs each with m inputs and s outputs, the relative efficiency of a particular DMUo is obtained by solving the following fractional programming problem: s P
wo ¼ max r¼1 m P i¼1 s P
s:t:
r¼1 m P
ur y ro vi xio ur y rj
ð1Þ 6 1;
j ¼ 1; 2; . . . ; n;
vi xij
i¼1
ur P 0;
r ¼ 1; 2; . . . ; s;
vi P 0;
i ¼ 1; 2; . . . ; m;
504
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where j is the DMU index, j = 1, . . . , n; r the output index, r = 1, . . . , s; i the input index, i = 1, . . . ,m; yrj the value of the rth output for the jth DMU, xij the value of the ith input for the jth DMU, ur the weight given to the rth output; vi the weight given to the ith input, and wo is the relative efficiency of DMUo, the DMU under evaluation or the target DMU. In this model, DMUo is efficient if and only if wo = 1. In DEA, a DMU is considered individually in determining its relative efficiency. This target DMU effectively selects weights that maximize its output to input ratio, subject to the constraints that the output to input ratios of all the n DMUs with these weights are 61. A relative efficiency score of 1 indicates that the DMU under consideration is efficient whereas a score less than 1 implies that it is inefficient. This fractional program can be converted into a linear programming problem where the optimal value of the objective function indicates the relative efficiency of DMUo. The reformulated linear programming problem, also known as the CCR model, is as follows: s X wo ¼ max ur y ro r¼1
s:t:
m X i¼1 s X
vi xio ¼ 1; ur y rj
r¼1
m X
ð2Þ vi xij 6 0;
j ¼ 1; 2; . . . ; n
i¼1
ur P 0; vi P 0;
r ¼ 1; 2; . . . ; s; i ¼ 1; 2; . . . ; m:
In model (2), i.e., in the classical DEA, the weighted sum of the inputs for the target DMU is forced to 1, thus allowing for the conversion of the fractional programming problem into a linear programming problem that can be solved by using any linear programming software. The solution to model (2) assigns the value 1 to all efficient DMUs. The super efficiency concept is proposed to differentiate completely among all efficient DMUs when there are more than one efficient DMUs. One of the super efficiency models for ranking efficient DMUs in DEA was introduced by Andersen and Petersen (1993). This method enables an extreme efficient unit o to achieve an efficiency score greater than one by removing the oth constraint in the envelopment LP formulation, as shown in model (3) (Adler et al., 2002). s X wo ¼ max ur y ro r¼1
s:t:
s X r¼1 m X
ur y rj
m X
vi xij 6 0;
j ¼ 1; 2; . . . ; n;
j 6¼ o
i¼1
ð3Þ
vi xio ¼ 1;
i¼1
ur P 0;
r ¼ 1; 2; . . . ; s;
vi P 0;
i ¼ 1; 2; . . . ; m:
3. The inclusion of coefficient of variation for input–output weights in the minimization process The coefficient of variation (CV), the ratio of sample standard deviation to the sample mean, measures the variability of the weights relative to their mean (or average). It compares the relative dispersion in one type of data with the relative dispersion in another type of data. Let ur(r = 1, 2, . . . , s) be the weight on output r and let u be the mean of the ur(r = 1,2, . . . , s). We define the CV for the weights ur as follows: ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s P ður uÞ2 =ðs 1Þ : CV ¼ r¼1 u
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Similarly, we can calculate the CV for the weights vi (i = 1, 2, . . . , m) as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m P ðvi vÞ2 =ðm 1Þ CV ¼
i¼1
: v Incorporating the coefficient of variation for input–output weights to the model 2 in the minimization process subject to the same constraints (i.e., min CV = max(CV)), our suggested model, we call it as CVDEA, becomes sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m P s P 2 ðvi vÞ2 =ðm 1Þ ðu u Þ =ðs 1Þ r s X i¼1 wo ¼ max ur y ro r¼1 v u r¼1 s m X X s:t: ur y rj vi xij 6 0; j ¼ 1; 2; . . . ; n; ð4Þ r¼1 i¼1 m X vi xio ¼ 1; i¼1
ur P 0;
r ¼ 1; 2; . . . ; s;
vi P 0;
i ¼ 1; 2; . . . ; m:
This nonlinear optimization model, based on the CCR model, can be easily solved with the Kuhn–Tucker algorithm. When there are more than one efficient DMUs, the CV is incorporated into the model 3 instead of model 2 in the minimization process and then all efficient DMUs are ranked over again. This model serves the same purposes of the Li and Reeves (1999) model MCDEA, i.e., can be used to improve discriminating power of the DEA and also effectively yields more reasonable/homogeneous input and output weights. On the other hand, our proposed model has only one objective function, namely simple, and is easily solved without any preference information of decision maker in contrary to the Li–Reeves model. 4. A comparison of the models by means of examples In order to exhibit the improvement of the dispersion of input–output weights and the increasing discrimination power for our suggested model, CVDEA, we have used the data sets of two examples from previous studies (Li & Reeves, 1999; Sexton, 1986; Wong & Beasley, 1990). Example 1 (Efficiency evaluation of six nursing homes). Two inputs and two output variables for six nursing homes are staff hours per day, including nurses, physicians etc. (x1); supplies per day, measured in thousands of dollars (x2); total medicare-plus medicaid-reimbursed patient days (y1); and total privately paid patient days (y2), respectively, and the related data are given in Table 1. For detailed descriptions of the data see Sexton (1986). When our suggested CVDEA and the DEA models are applied to the data, the results in Tables 2 and 3 are obtained. In Table 2, it is seen that the first output variable for DMUs A and B, and the first input variable for DMUs B and D, and the second input variable for DMUs A and C are neglected but the second output variable is Table 1 Data of six nursing homes 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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Table 2 Results of the DEA model DMU
Efficiency
Super efficiency
u1
u2
v1
v2
A B C D E F
1 1 1 1 0.977 0.867
2 1.395 1.412 1.131 0.977 0.867
0.714 0 0.238 0 0.115 0.162
0 0.476 0 0.238 0.304 0.427
0 0 0.172 0.069 0.110 0.155
5.000 1.429 0.374 0.321 0.513 0.722
Table 3 Results of the CVDEA model DMU
Efficiency
u1
u2
v1
v2
A B C D E F
1 0.863 0.991 0.983 0.948 0.735
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
considered as extremely important in the DEA for DMUs A and B, respectively. In Table 3, the input–output weights for which are obtained by the CVDEA model are dispersed more homogeneous than that of the DEA. On the other hand, the CVDEA identifies only DMU A as efficient while the DEA identifies DMUs A, B, C and D as efficient. Hence we see that the CVDEA model reduces the number of efficient DMUs for the data in Example 1. In Table 4, the ranks of DMUs obtained by the DEA under the super efficiency and the CVDEA model, and a comparison of variances for input–output weights are given. The comparison of ranks for DMUs in both model is performed by Spearman’s and Kendall’s rank correlation coefficients, and the comparison of variances for input–output weights assigned by the models is conducted by the Mann–Whitney test. Kendall’s tau value s = 0.733 (or the Spearman’s value rs = 0.828) shows that under the significance level a = 0.05 there is a powerful correlation in the same direction between the efficiency ranking values of the DMUs obtained by the two models. The Mann–Whitney test inferred that the variance of weights for the CVDEA model is less than that of the DEA (p value = 0.023). Example 2 (Efficiency evaluation of seven departments in a university). The input–output variables for seven departments in a university are defined as follows and the related data are given in Table 5: y1 number of undergraduate students y2 number of postgraduate students y3 number of research papers x1 number of academic staff Table 4 Ranks of the DMUs and test of homogeneity of weights for the models DMU
Rank of DMUs
Weight variances
DEA
CVDEA
DEA
CVDEA
A B C D E F
1 3 2 4 5 6
1 5 2 3 4 6
5.7824 0.4538 0.0241 0.0220 0.0365 0.0722
0.0800 0.0017 0.0005 0.0007 0.0017 0.0025
Test
Spearman’s rs = 0.829, Kendall’s s = 0.733
p value = 0.023
H. Bal et al. / Computers & Industrial Engineering 54 (2008) 502–512
507
Table 5 Data of seven departments in a university DMU
y1
y2
y3
x1
x2
x3
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
x2 x3
academic staff salaries in thousands of pounds support staff salaries in thousands of pounds
Again, when our suggested DEA and CVDEA models are applied to this data, the results in Tables 6 and 7 are obtained. In Example 2, it is obvious that, the CVDEA model results a more better dispersion of weights than that of the DEA when Tables 6 and 7 are examined. In the CVDEA model, almost the same efficiency scores of the DEA are assigned to the DMUs and almost all of the input–output variables are consumed while in the DEA for most of DMUs the variables corresponding to the second output and the first and third input are neglected. On the other hand, the CVDEA identifies DMU 1, DMU 5 and DMU 7 as efficient while the DEA rates all of the DMUs as efficient except DMU 4. Hence we see that the CVDEA model reduces the number of efficient DMUs in Example 2. In Table 8, the ranks of DMUs obtained by the DEA under super efficiency and the CVDEA model, and a comparison of variances for input–output weights are given. The Kendall’s tau value s = 0.809 is also an indicator that under the significance level a = 0.05 there is a powerful correlation in the same direction between the efficiency ranking values of the DMUs obtained by the two models. As well, the variances of the input–output weights of each DMU for the two models are given. The hypothesis related to the variances is tested with Mann–Whitney. Hence it is statistically Table 6 Results of the DEA model DMU
Efficiency
Super efficiency
u1
u2
u3
v1
v2
v3
1 2 3 4 5 6 7
1 1 1 0.820 1 1 1
1.829 1.048 1.198 0.819 1.220 1.190 1.266
0.983 0.719 0 0.911 0 0.639 0.121
1.172 0 0 0 0.432 0 0.334
0 0 1.333 0 0.288 0.347 0.105
0 0 0 6.415 0 0 0.732
0.250 0.133 0.033 0.006 0.05 0.137 0.030
0 0 0.711 0 0 0 0
Table 7 Results of the CVDEA model DMU
Efficiency
Super efficiency
u1
u2
u3
v1
v2
v3
1 2 3 4 5 6 7
1 0.983 0.990 0.820 1 0.980 1
1.368 0.983 0.990 0.820 1.311 0.980 1.253
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
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Table 8 Ranks of the DMUs and testing the homogeneity of weights for models DMU
Rank of DMUs
Weight variances
DEA
CVDEA
DEA
CVDEA
1 2 3 4 5 6 7
1 6 4 7 3 5 2
1 5 4 7 2 6 3
0.2877 0.0827 0.2406 6.6045 0.0357 0.0675 0.0765
0.0838 0.0347 0.0682 0.1501 0.0192 0.0394 0.0313
Test
Spearman’s rs = 0.928, Kendall’s s = 0.809
p value = 0.042
concluded that the variance of weights for the CVDEA model is less than that of the DEA (p value = 0.042). However we have seen the CVDEA improves the dispersion of input–output weights and increases the discrimination power of efficient DMUs in the foregoing two examples, it will be presented for supporting these achievements by a simulation study reflecting the diversity of input–output and DMU numbers. 5. Simulation experiment In the simulation study, the efficiency scores and the weights by the two models are evaluated according to the formed possible cases for the different number of DMUs (n = 10, 15, 20, 25, 30, 35, 40, 45, 50, 100), and of input variables (i = 1, 2, . . . , 7), and of output variables (o = 1, 2, . . . , 7). Each case is repeated for 100 times. The input and output variables are drawn randomly from Uniform(10, 500) distribution. For each case, using the following hypotheses it is tested if the two approaches give the similar rankings to DMUs. H0: The DMU efficiency scores for the CVDEA are uncorrelated to the DMU efficiency scores for the DEA. H1: The DMU efficiency scores for the CVDEA are correlated in the same direction to the DMU efficiency scores for the DEA. For testing the hypotheses, the Spearman’s rho and the Kendall’s tau correlations are used. In Table 9, in the column of testing the ranks of methods, the number of cases agreed with the hypothesis H1 per 100 repeats and the average value for the Spearman’s and Kendall’s rank correlations are given. Later, in order to compare the weight variances corresponding to the CVDEA and the DEA models the following hypotheses are tested. H0: There are no difference between the weight variances corresponding to the CVDEA and the DEA models. H1: The weight variances for the CVDEA are less than that of the DEA. For testing the hypotheses, the Mann–Whitney statistic is used. In Table 9, in the column of testing the homogeneity, the number of cases agreed with the hypothesis H1 per 100 repeats and the average value for p values are given. All of the results given in Table 9 are obtained by using the programs MATLAB 7.0 and LINGO 9.0 (Pentium(R) 4 CPU 2.60 GHz., 248 MB RAM). Table 9 shows that there is a high correlation in the same direction between the efficiency scores of DMUs assigned by our suggested approach and the DEA in more than 94% of the cases and reaches 100% of the case for n = 10, i = 2, and s = 1. On the other hand, the weight variance for the CVDEA is less than that of the DEA in more than 95% of the cases and reaches 100% of the case for n = 10, i = 2, and s = 1. In other words, all results of the simulation show that there is statistically a significant correlation and hence a similarity between the ranks obtained by the DEA and the CVDEA models for each case. In addition, in almost all cases it is observed that the weight dispersion for the input and output variables is more homogeneous in the CVDEA. The simulation data results also confirmed that the CVDEA approach realizes a more convenient dispersion of weights for all variables.
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509
Table 9 Results of the hypotheses testing n
i
o
Testing the ranks of methods
Testing the homogeneity
Number of accepting H1
rs
s
Number of accepting H1
p value
10
2 1 2 1
1 2 2 3
100 98 99 98
0.901 0.867 0.899 0.851
0.828 0.801 0.812 0.795
100 100 100 99
0.0009 0.0012 0.0013 0.0028
15
2 2 2 3
1 2 3 2
100 98 99 99
0.853 0.808 0.831 0.839
0.761 0.723 0.746 0.729
99 100 100 99
0.0021 0.0008 0.0011 0.0038
20
3 2 4 2
1 3 1 4
99 98 100 97
0.761 0.719 0.772 0.705
0.701 0.682 0.716 0.673
100 100 100 98
0.0018 0.0020 0.0016 0.0061
25
2 2 5 3
2 4 2 4
100 98 97 99
0.749 0.700 0.682 0.745
0.649 0.624 0.609 0.632
100 97 99 96
0.0008 0.0057 0.0029 0.0069
30
3 2 4 3
2 5 3 5
100 97 98 97
0.709 0.635 0.659 0.640
0.627 0.553 0.599 0.537
100 98 96 97
0.0026 0.0049 0.0083 0.0077
35
2 3 5 5
3 6 4 5
99 96 96 98
0.648 0.579 0.599 0.621
0.608 0.528 0.564 0.588
100 96 96 97
0.0015 0.0078 0.0075 0.0059
40
4 2 6 4
3 5 3 4
99 99 96 100
0.641 0.645 0.598 0.649
0.589 0.591 0.500 0.602
99 97 97 99
0.0038 0.0071 0.0078 0.0031
45
3 4 5 5
4 6 3 5
99 97 98 99
0.538 0.478 0.512 0.547
0.500 0.451 0.467 0.486
99 97 95 98
0.0033 0.0064 0.0091 0.0047
50
5 4 6 5
2 6 3 6
97 98 96 98
0.495 0.513 0.476 0.521
0.477 0.500 0.451 0.511
98 96 95 97
0.0050 0.0081 0.0090 0.0065
100
3 4 6 7
5 7 4 7
99 96 96 94
0.459 0.424 0.437 0.399
0.388 0.361 0.346 0.311
98 95 96 95
0.0021 0.0097 0.0088 0.0095
6. A real-world application In order to compare the two methods with a new real-world data, we make use of the variables for 30 OECD countries consisting of three input and five output variables (The data of the year 2005 are abridged from the site OECD home page http://www.oecd.org/countrieslist see Appendix A):
510
H. Bal et al. / Computers & Industrial Engineering 54 (2008) 502–512
Table 10 Efficiency and super efficiency scores, ranking and weight variances of the DEA and the CVDEA Countries
Efficiency
Australia Austria Belgium Canada Chec. Rep. Denmark Finland France Germany Greece Hungary Iceland Ireland Italy Japan S. Korea Luxembourg Mexico Netherland N. Zealand Norway Poland Portugal Slovak. Rep. Spain Sweden Switzerland Turkey England USA
y1 y2 y3 y4 y5 x1 x2 x3
Super efficiency
Rank of the countries
Weight variances
Maverick index
DEA
CVDEA
DEA
CVDEA
DEA
CVDEA
DEA
CVDEA
DEA
CVDEA
0.500 0.657 0.573 0.593 0.600 0.750 0.750 0.957 1 0.582 0.405 0.940 0.528 0.593 1 0.643 1 1 0.767 0.500 0.870 0.330 0.515 0.370 0.645 1 1 0.254 0.655 1
0.466 0.624 0.547 0.631 0.553 0.708 0.724 0.941 1 0.546 0.352 0.972 0.455 0.615 1 0.548 1 0.833 0.731 0.490 0.836 0.368 0.489 0.323 0.684 0.989 1 0.358 0.763 1
0.500 0.657 0.573 0.593 0.600 0.750 0.750 0.957 1.133 0.582 0.405 0.94 0.528 0.593 7.637 0.643 2.437 1.058 0.767 0.500 0.870 0.330 0.515 0.370 0.645 1.025 1.679 0.254 0.655 3.940
0.466 0.624 0.547 0.631 0.553 0.708 0.724 0.941 1.084 0.546 0.352 0.972 0.455 0.615 7.811 0.548 1.325 0.833 0.731 0.490 0.836 0.368 0.489 0.323 0.684 0.989 1.061 0.358 0.763 2.295
25 14 22 19 18 12 13 8 5 21 27 9 23 20 1 17 3 6 11 26 10 29 24 28 16 7 4 30 15 2
25 17 21 16 19 14 13 8 4 22 29 7 26 18 1 20 3 10 12 23 9 27 24 30 15 6 5 28 11 2
0.003012 0.002831 0.002938 0.002938 0.004341 0.006783 0.006783 0.006616 0.004379 0.004341 0.001029 0.005464 0.002356 0.002938 3.783673 0.004234 0.014809 0.021114 0.002122 0.003012 0.004628 0.001340 0.001949 0.001696 0.004283 0.012057 0.070788 0.002005 0.002938 0.005258
0.001352 0.003043 0.002501 0.000888 0.003386 0.002513 0.004100 0.003455 0.001069 0.000516 0.000526 0.001886 0.000809 0.000482 0.002456 0.001008 0.003482 0.000189 0.001400 0.000899 0.003081 0.000577 0.001038 0.00059 0.001009 0.001417 0.005364 0.001503 0.001059 0.001217
0.195 0.218 0.378 0.281 0.377 0.258 0.436 0.571 0.657 0.435 0.315 0.419 0.227 0.433 0.228 0.299 0.465 3.973 0.326 0.255 0.254 0.251 0.311 0.455 0.545 0.396 0.092 1.999 0.296 0.386
0.232 0.245 0.308 0.232 0.352 0.181 0.695 0.128 0.137 0.151 0.364 0.231 0.213 0.124 0.109 0.349 0.223 3.431 0.366 0.327 0.321 0.243 0.523 0.523 0.531 0.244 0.113 3.419 0.149 0.236
health expenditure per capita national income per capita ratio by literacy portion in world import portion in world export baby death rate unemployment ratio rate of inflation
When the DEA and our suggested CVDEA models are applied to this data the efficiency and the super efficiency scores, the ranks, the weight variances and the maverick index1 of DMUs (countries) are given in Table 10, respectively. In order to save space the input–output weights for the DMUs obtained for the DEA and the CVDEA models are not given in the tables. In Table 10, the DEA determines seven countries, USA, Germany, Japan, Luxembourg, Mexico, Sweden and Switzerland, as efficient while the CVDEA finds five efficient countries, USA, Germany, Japan, Luxembourg, and Switzerland. According to this analysis results, we see that the CVDEA model reduces the number of efficient DMUs for this real data. Here Mexico found as an efficient country by the DEA while it is identified as an inefficient country by the CVDEA. It is also known that Mexico is not a country having a powerful 1
A high maverick index, also called as the false positive index (FPI) of DMUs, is a measure showing that a DMU obtains an efficiency score just by using inappropriate weights. For details and comments, see Doyle and Green (1994) and Baker and Talluri (1997).
H. Bal et al. / Computers & Industrial Engineering 54 (2008) 502–512
511
economic structure and a high level of development such as the abovementioned efficient countries in reality. In addition, the maverick indices for Mexico are also evaluated very high for both methods, the DEA and the CVDEA. These results point out that our suggested CVDEA method is more consistent than the DEA. In other words the dispersion of weights is more reasonable in the CVDEA approach than that of the DEA. As seen from Table 10, it is statistically found that there is a significant correlation and hence a similarity between the ranks resulted by the DEA and the CVDEA, i.e., with Spearman‘s rho, rs = 0.974. In addition, it is observed that the weight dispersion for the input and output variables is more better in the CVDEA (p value = 0.0000507). In summary, the CVDEA approach attaches importance to all variables in general while the DEA ignores some variables. These two examples, the simulated data and the real data results also confirmed that the CVDEA approach realizes a more convenient dispersion of weights for all variables, refines the results of DEA model and eliminates apparently efficient DMUs and reduces the number of efficient DMUs. 7. Conclusions In generally, DEA models yield in many instances non-homogeneous weight dispersion of input and output parameters. Indeed, they yield several input–output weights that are zero or that have extreme values which imply that the corresponding parameters are not taken into account to interpret the efficiency of the DMUs. This paper overcomes this problem by using of the CVDEA model, i.e., incorporating the coefficient of variation for input–output weights to the DEA model in the minimization process. When the new CVDEA model is applied to the two instances in the literature, a real-world data of OECD countries, and simulation data it has been seen that the new model yielded a more balanced dispersion of input–output weights and reduced the number of efficient DMUs. Thus, we can objectively make more correct decisions by using CVDEA method in determining the efficiency of DMUs. Moreover, it is observed that there is a high correlation in the same direction between the efficiency scores of DMUs assigned by our suggested approach and the DEA. The CVDEA does not also need any constraints on weights. For a further study, the performance of the CVDEA method for the other forms of DEA models, such as the BCC, additive etc., may be investigated. Acknowledgements The authors are indebted to the anonymous referees for valuable comments and suggestions which have improved the earlier version of this paper. Appendix A Data of 30 OECD countries Countries
y1
y2
y3
Australia Austria Belgium Canada Chec. Rep. Denmark Finland France Germany Greece Hungary Iceland
2036 1968 2081 2312 930 2133 1502 2055 2424 1167 705 2103
25693 26765 27178 27840 13991 27627 24996 24223 25103 16501 12416 29581
100 100 100 100 100 100 100 100 100 97 99 100
y4 1.108 1.096 2.989 3.47 0.652 0.727 0.513 4.676 7.526 0.475 0.576 0.034
y5 1.016 1.134 3.332 3.939 0.598 0.87 0.696 4.82 8.562 0.161 0.538 0.034
x1
x2
6 7 5 4.3 6 11.7 6 7.6 5 8.8 4 5.6 4 10.2 4 11.3 5 8.7 5 10.2 8 7 4 2.8 (continued on next
x3 3 1.8 1.6 2.2 1.8 2.4 1.7 1.9 2.3 4.6 5.3 5.2 page)
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H. Bal et al. / Computers & Industrial Engineering 54 (2008) 502–512
Appendix A (continued) Countries
y1
y2
y3
y4
y5
x1
x2
x3
Ireland Italy Japan S. Korea Luxembourg Mexico Netherland N. Zealand Norway Poland Portugal Slovak. Rep. Spain Sweden Switzerland Turkey England USA
1436 1783 1822 730 2215 356 2070 1424 2330 496 1237 930 1218 1746 2794 255 1461 4178
29886 23626 26755 17380 50061 9023 25657 20070 29918 9051 17290 11243 19472 24277 28769 6974 23509 34142
100 98 100 98 100 91 100 100 100 100 92 100 98 100 100 85 100 100
0.785 3.725 5.144 2.322 0.177 2.694 2.953 0.23 0.532 0.84 0.584 0.266 2.494 1.008 1.207 0.781 5.111 18.344
1.364 3.952 6.504 2.536 0.133 2.508 3.469 0.224 0.931 0.624 0.398 0.225 1.927 1.266 1.309 0.558 4.312 10.83
6 6 4 5 5 25 5 6 4 9 6 8 5 3 3 38 6 7
5.8 11.5 4.7 6.3 2.1 2.1 4.3 6.8 3.2 10.5 5 16.4 15.9 7.2 3 7.3 6 4.2
4.7 2.5 0.15 2.8 1.1 5 3.5 2.7 1.3 1.9 3.5 3.3 3.1 2.2 0.6 21.7 1.6 1.6
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