An Approach to Judge Homogeneity of Decision Making Units

Transcription

1 An Approach to Judge Homogeneity of Decision Making Units Zhongsheng HUA * Ping HE School of Management University of Science & Technology of China Hefei, Anhui People s Republic of China [ ] Submitted to POMS International Conference - Shanghai 2006 Corresponding author. address: zshua@ustc.edu.cn Phone: Fax:

2 An Approach to Judge Homogeneity of Decision Making Units Abstract: It is a general precondition of Data Envelopment Analysis (DEA) that all Decision Making Units (DMUs) to be evaluated are homogeneous. However, it is hard to find a method in the literature to udge whether DMUs are homogeneous or not. This paper firstly analyzes the impacts of DMUs non-homogeneity caused by technology difference on the efficiency estimates. It is showed that the DMU under evaluated should be compared to only those DMUs utilizing similar technologies. Then a hypothesis test-based method is proposed to udge DMUs homogeneity in technology. Under the assumption that observed input/output data of DMUs are generated from Cobb-Douglas production functions, we illustrate and verify our approach through simulations. Key words: Data Envelopment Analysis; Homogenous; Technology Difference; Hypothesis Test 2

3 1. Introduction Data envelopment analysis (DEA) was originally developed to measure the relative efficiency of peer decision making units (DMUs) in multiple input-multiple output settings (Charnes et al. 1978). Many theoretical extensions and successful applications in various fields of DEA have been reported ever since (Anderson et al. 2002; Haas and Murphy 2003; Cook and Green 2005). A general precondition of DEA is that all DMUs to be evaluated are homogeneous (Charnes et al. 1978), i.e. the units are assumed to be undertaking similar activities, utilizing a similar range of resources and common technologies, and operating in similar environments (Dyson et al. 2001; Haas and Murphy 2003). Some authors have done some research on non-homogeneity in DEA. In a study of 144 bank branches, Soteriou and Zenios (1999) dealt with non-homogeneous DMU groups by location category and size. They compared efficiency within the groups and also compare the efficiency of the groups. Cook et al. (1998) introduced the concept of hierarchical DEA, where efficiency can be viewed at various levels. They provided a means for adusting the ratings of DMUs at one level to account for the ratings received by the groups (into which these DMUs fall) at a higher level. These researches separate non-homogeneous DMUs into different groups and evaluate them in different models. When non-homogeneity is caused by environment factors, some researchers developed models to include non-homogeneous DMUs in a single model. They took environment factors as non-discretionary inputs, and discriminate DMUs in different environment by defining different reference sets (Banker and Morey 1986; Ruggiero 1996; Hua et al. 2005). 3

4 However, when non-homogeneity is caused by technology, we have not found any research results on how DEA analysis can be performed. Actually, it is hard even to find an exact definition of homogeneity in technology from current literature. By surveying the literature (e.g. Dyson et al. 2001), we suggest that, two DMUs are homogeneous in technology should satisfy the following three conditions: (1) undertaking similar activities; (2) utilizing a similar range of resources; and (3) using common technologies, i.e., a similar increase in an input of each DMU will lead to a similar increase in output. The first two conditions can be easily verified by directly observing the data or applying a cluster analysis. To verify the third condition of homogeneous in technology, this paper proposes a method for homogeneity udgment through hypotheses test (Anderson et al. 2002). The remainder of the paper is organized as follows: Section 2 discusses the DEA technique; Section 3 analyzes the impacts of DMUs non-homogeneity caused by technology difference on the efficiency estimates; Section 4 provides the hypothesis test-based method for udging DMUs homogeneity in technology; Section 5 describes the simulation data and result for illustrating and verifying our approach; Section 6 summarizes the result and concludes the paper. 2. The DEA technique DEA is a non-parametric technique that measures the relative efficiency of peer DMUs in multiple input-multiple output settings. In a DEA framework, performance is evaluated with respect to an efficient frontier. The efficient frontier is constructed by examining linear combinations of DMUs and determining the minimum input usage necessary to achieve a given output level. 4

5 Suppose we have a set of n peer DMUs, which produce multiple outputs ( 1,2,..., n) =, by utilizing multiple inputs x ( 1, 2,..., n) y =. During a production process, it is expected that minimum inputs be used and maximum output be produced. The Production Possibility Set (PPS) could be one of the following two types. n n T = {( x, y) λ x x, λ y y, λ 0, = 1, 2,..., } C n = 1 = 1, (1) n n n T = {( x, y) η x x, λ y y, λ = 1, λ 0, = 1, 2,..., n} V. (2) = 1 = 1 = 1 T implies that the activities of all DMUs are under constant returns to scale. implies C T V that the activities of all DMUs are under variable returns to scale. The CCR model (Charnes et al. 1978) is a typical CRS DEA model, and the BCC model (Banker et al. 1984) is a typical VRS DEA model. When a DMU 0 is under evaluation by the CCR model, we have: Min θ subect to n = 1 n = 1 λ 0, = 1, 2,, n. λ x λ y θ x y 0 0 (3) The optimal valueθ of above model is the overall technical efficiency of DMU 0, which signifies the extent to which the inputs need to be reduced to bring DMU 0 onto the best practice frontier without worsening outputs under constant returns to scale. When a DMU 0 is under evaluation by the BCC model, we have: 5

6 Min φ subect to n = 1 n = 1 n = 1 λ =1 0 0 λ 0, =1,2,, n λ x λ y φx y (4) The optimal value φ of above model is the pure technical efficiency of DMU 0, which signifies the extent to which the inputs need to be reduced to bring DMU 0 onto the best practice frontier without worsening outputs under variable returns to scale. A precondition of applying the above two models is that all the DMUs to be evaluated are homogeneous. 3. Impact of non-homogeneity caused by technology As mentioned before, all the DMUs to be evaluated should take common technology, i.e., a similar increase in an input of each DMU will lead to a similar increase in output. The difference of technologies can be expressed by discrepancy of production functions. To illustrate more clearly the impact of pooling DMUs with different production functions on efficiencies evaluating, we construct some DMUs which produce with two different production functions, each transform two inputs into one single output. Designate y α ( α = 1, 2 ) as the maximal output of DMU which is possible for each of different α α α production functions, and similarly index the input vectors x ( x1, x2 ) = in order to evaluate the technical efficiency under each of these two different sets of production possibilities. Hence we have 6

7 1 1 1 y f1( x1, x2 ) = (5) y f2( x1, x2 ) = (6) as the maximal output obtainable from these input values under each production function. Notice, for instance, that one production function may admit greater output values than another over some ranges of input values but the reverse may be true over other input ranges. The situation is, of course, even more complicated when more than two functions are involved. The following two-input dimensional figure will help analysis. DMU A, B, C, D, etc, designated as asterisks, are producing with technology (5). DMU E, F, G, H etc, designated as dots, are producing with technology (6). Figure 1 portrays the distribution of these DMUs under the same amount of output. As we can see from this figure, the DMUs in the first category are enveloped by the solid piece-wise line A B C D, and the DMUs in the second category are enveloped by the dotted piece-wise line E F G H. x 2 A E F B Q C G D O x 1 Legend: = DMU using the first technology = DMU using the second technology FIGURE 1. DMUs Using Different Technologies 7

8 As showed by figure 1, DMU F is on the frontier of the production possibility set composed of DMUs under the second technology. Therefore, DMU F should be technical efficient, since it cannot improve its input without worsening other inputs or output, given the technology unchanged, which is one of the homogeneity assumption. However, if DMU F is compared with all the DMUs under two technologies, it is technical inefficient. In fact, keeping the level of output remain the same, DMU F can possibly decrease its inputs x 1 and x 2 proportionally to DMU Q ---the intersection of line OF and BC. Consequently, the technical efficiency of DMU F is underestimated. Similar impact can be seen for DMU E, C, D, etc. From above analysis we can conclude that, pooling DMUs utilizing different technologies together as reference set when evaluating efficiencies will underestimate the efficiencies of part of DMUs. To get the true technical efficiencies of non-homogeneous DMUs utilizing different technologies, it is appropriate to discriminate them and evaluate each cluster of homogeneous DMUs separately, which implies that we should udge the homogeneity of several categories of DMUs before we evaluate the efficiencies. 4. Hypotheses test If two groups of DMUs are homogeneous, they can be viewed as two samples of the same population. From statistical knowledge (Anderson et al. 2002), we know that these two groups of DMUs should have the same sample distribution of efficiency. This implies that two groups of DMUs can be udged as non-homogeneous if their sample distributions of efficiency are significantly different. If two sample distributions of efficiency are not significantly different, we can regard them homogeneous. We adopt Wilcoxon signed-rank 8

9 test to check if two groups of DMUs are non-homogeneous. The Wilcoxon signed-rank method tests the null hypothesis that two related medians of variables are the same. Denote the efficiencies of two groups of DMUs evaluated in the same DEA model as two random variables and. We assume that and Y have probability distributions Y1 Y2 Y1 2 that are continuous and symmetric, with density functions f1 ( y) and f2 ( ) y, respectively. The null and alternative hypotheses are as follows: H0 : μ1 = μ 2 H : μ μ where μ 1 and 2 μ are the median of f ( y) and f ( ) 1 2 y, respectively. In the Wilcoxon test, ranks are based on the absolute value of the difference between the two test variables. The sign of the difference is used to classify cases into one of three groups: differences below 0 (negative ranks), above 0 (positive rank), or equal to 0 (ties) (see Kani 1994). Let y1i and y2i be the i th member of variables Y1 and 2, therefore, Y y11 y21 y y 12 22,, y y denote the set of absolute values of the deviations of 1n 2n y 1i from y 2i. These absolute deviations can be sorted in ascending order. Define Ri, i = 1, 2,..., n to be the rank of y1 y2 in the ordered set. Define the sign indicator Z, i = 1,2,..., nas i i i Z i 1 if y y > 0 = 1 0 1i 2i if y1 i y2i < The Wilcoxon test statistic is calculated as: W n = Z R (7) i= 1 i i by: If the null hypothesis is true, the mean and variance of the Wilcoxon test statistic are given 9

10 ( ) EW ( + 1) n n =, (8) 4 ( ) Var W n n = ( + 1)( 2n+ 1) 24. (9) For n >12, the Z score of W has an approximately standard normal distribution, i.e. Z W W n( n+ ) ( + )( n+ ) 1/4 = N n n /24 ( 0,1 ) as n (10) Therefore, if is large, will be reected in favor of H if n H0 1 ZW < or Z Z α /2 W > Z1 α /2 The Wilcoxon signed-rank test considers information about both the sign of the differences and the magnitude of the differences between pairs. If the two variables are similarly distributed, the number of positive and negative differences will not differ significantly. If H 0 is reected, that means the two variables are significantly different, i.e. the two groups of DMUs are non-homogeneous. If H 0 is accepted, we regard the two groups of DMUs homogeneous. 5. Simulation analysis To verify our hypotheses test-based approach for homogeneity udgment, we have done many simulation experiments. In all of our experiments, input/output data of DMUs are generated from Cobb-Douglas production functions. Our experiments are composed of two parts, depending on the PPS, one of which is CRS and the other VRS. 5.1 Constant returns to scale When PPS is of CRS, input/output data of DMUs in two groups are generated according to the following functions, respectively: y p 1 ( 1 ) p = r x1 x 2, (11) 10

11 q 1 ( 1 r) x x y =. (12) q 1 2 Where, r refers to inefficiency, x 1 and x 2 refer to two inputs of each DMU. In our experiments, these two inputs are randomly generated from a random uniform distribution for 200 observations (100 observations in each group) within the following intervals: x 1 : (10, 60) x 2 : (5, 20) The true level of technical inefficiencies r is generated from a normal distribution with zero mean, i.e. r N(0,0.2036). The value of r is further restricted by setting r equal to zero if the value of r generated is greater than We test our approach by varying p and q values of the production functions, which are listed in Table 1. For specific values of p and q, outputs of DMUs in a group are computed based on Eqs. (11) or (12). TABLE 1 List of p and q Values in Production Functions (CRS) p q q-p

12 Based on the input/output data generated as the way described above, we evaluate efficiency scores of 200 DMUs in the two groups by applying a single CCR model. We then use Wilcoxon signed-rank non-parametric test to check whether the sample distributions of efficiency of DMUs in different groups are significantly different or not (we set significant level of the test as 0.05). Given specific values of p and q, we repeat the experiments 30 times and report the average results about the 30 experiments in Table 2. In Table 2, Significant percent describes the percentage that the null hypothesis is reected in the 30 experiments. Average of Z value is the average of Z scores over the 30 experiments. Average difference between mean efficiencies represents the average difference between the mean efficiencies of the two groups of DMUs over the 30 experiments. Based on the approach described in the previous section, it can be observed from Table 2 that, when q p is greater than 0.04, the two groups of DMUs are non-homogeneous in technology. As it is elaborated in Section 4, for each experiment, Z W satisfies a normal distribution of N(0,1). Because each of the 30 experiments is independent of others, the average Z value over 30 experiments also satisfies the normal distribution of N(0,1). When q-p equals 0.04, the average Z value ( ) over 30 experiments is less than Z α /2 (-1.960), thus the two groups of DMUs are also deemed as non-homogeneous. 12

13 TABLE 2 Results of Wilcoxon Signed-Rank Non-Parametric Test (CRS) q-p Significant Average of Average of mean Homogeneity percent Z value efficiency difference udgment % Non-homogeneous % Non-homogeneous % Non-homogeneous % Non-homogeneous % Non-homogeneous % Non-homogeneous % Non-homogeneous % Non-homogeneous % Non-homogeneous % Non-homogeneous % Non-homogeneous % Non-homogeneous % Non-homogeneous % Non-homogeneous % Non-homogeneous % homogeneous % homogeneous When q p is less than 0.04, null hypothesis H is accepted, we regard the two groups 0 of DMUs homogeneous. 5.2 Variable returns to scale When PPS is of VRS, input/output data of DMUs in two groups are generated according to the following functions, respectively: p/2 p/2 ( 1 ) y = r x x, (13) 1 2 q/2 q/2 ( 1 ) y = r x x. (14) 1 2 We test our approach by varying p and q values, which are listed in Table 3. For specific values of p and q, outputs of DMUs in a group are computed based on Eqs. (13) or (14). 13

14 TABLE 3 List of p and q Values In Production Functions (VRS) p q q-p Based on the input/output data generated, we evaluate efficiency scores of 200 DMUs in the two groups by applying a single BCC model. Wilcoxon signed-rank test is used again to check whether the sample distributions of efficiency of DMUs in different groups are significantly different or not (again, we set significant level of the test as 0.05). Given specific values of p and q, we repeat the experiments 30 times and report the average results about the 30 experiments in Table 4. Analogously, it can be observed from Table 4 that, when q p is greater than 0.01, the two groups of DMUs are non-homogeneous in technology. 0 When q p is less than or equal to 0.01, null hypothesis H is accepted, we regard the two groups of DMUs homogeneous. Both results of CRS and VRS show that there are monotone relations between the difference in technologies and the Z-value of Wilcoxon signed-rank non-parametric test. 14

15 TABLE 4 Wilcoxon Signed-Rank NonParametric Test (VRS) q-p Significant Average of Average of mean Homogeneity percent Z value efficiency difference udgment % Non-homogeneous % Non-homogeneous % Non-homogeneous % Non-homogeneous % Non-homogeneous % Non-homogeneous % homogeneous % homogeneous % homogeneous % homogeneous % homogeneous % homogeneous 6. Summary & conclusion A general precondition of DEA analysis is that all DMUs to be evaluated are homogeneous. However, it is hard even to find an exact definition of homogeneity in current literature. This paper firstly analyzes the impacts of DMUs non-homogeneity caused by technology difference on the efficiency estimates. It is showed that the DMU under evaluated should be compared to only those DMUs utilizing similar technologies. Based on our definition of homogeneity, this paper proposes a hypothesis test-based approach to udge DMUs homogeneity in technology. When input/output data of DMUs are generated from Cobb-Douglas production functions, numerical experiments confirm the efficacy of our approach. Many extensions to this research are necessary and important. It is an interesting problem to udge homogeneity when input/output data of DMUs are generated from other way than Cobb-Douglas production functions. Further research effort can also be invested on the 15

16 reverse problem of this paper, i.e. find appropriate methods to separate non-homogeneous DMUs into different groups. References Anderson R.I., R.Fok, T.Springer, J.webb Technical efficiency and economies of scale: A non-parametric analysis of REIT operating efficiency. European Journal of Operational Research, 139, Banker, R.D., H.H.Chang A simulation study of hypothesis tests for differences in efficiencies. International Journal of Production Economics, 39, Banker, R.D., A.Charnes, W.W. Cooper Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science, 30, Banker, R.D., R.Morey Efficiency analysis for exogenously fixed inputs and outputs. Operations Research, 34, Charnes, A., W.W. Cooper., E. Rhodes Measuring the efficiency of decision making units. European Journal of Operational Research, 2, Charnes, A., W.W. Cooper., E. Rhodes Evaluating program and managerial efficiency: an application of data envelopment analysis to program follow through. Management Science, 16

17 27, Cook, W. D., D.Chai, J. Doyle, R. Green Hierarchies and Groups in DEA. Journal of Productivity Analysis, 10, Cook, W. D., R. Green Evaluating power plant efficiency: a hierarchical model. Computers & Operations Research, 32, Dyson, R.G., R.Allen, A.S.Camanho, V.V.Podinovski, C.S.Sarrico, E.A.Shale Pitfalls and protocols in DEA. European Journal of Operational Research, 132, Haas, D.A., F.H. Murphy Compensating for non-homogeneity in decision-making units in data envelopment analysis. European Journal of Operational Research, 144, Hua, Z.S, Y.W. Bian., L. Liang Eco-efficiency analysis of paper mills along the Huai River: An extended DEA approach. Omega, International Journal of Management Science. In press, available online at sciencedirect.com. Kani.G Statistical Tests, CA: SAGE Publications Kao, C., H.T. Hung Efficiency Analysis of University Departments: An Empirical Study. National Cheng Kung University, Tainan, Taiwan, working paper 17

18 Ray, SC Resource use efficiency in public schools: A study of Connecticut data. Management Science, 37, Ruggiero, J On the measurement of technical efficiency in the public sector. European Journal of Operational Research, 90, Sexton, T.R., S. Sleeper., R.E. Taggart Improving pupil transportation in North Carolina. Interfaces, 24, Soteriou, A., S.A. Zenios Operations, quality, and profitability in the provision of banking services. Management Science, 45, Zhang, Y., R. Bartels The Effect of Sample Size on the Mean Efficiency in DEA with an Application to Electricity Distribution in Australia, Sweden and New Zealand. Journal of Productivity Analysis, 9,