An Approach to Discriminate Non-Homogeneous DMUs

Transcription

1 An Approach to Discriminate Non-Homogeneous DMUs Zhongsheng HUA * Ping HE School of Management University of Science & Technology of China Hefei, Anhui 3006 People s Republic of China [ ] Submitted to POMS International Conference - Shanghai 006 Corresponding author. address: zshua@ustc.edu.cn Phone: Fax:

2 An Approach to Discriminate Non-Homogeneous DMUs Abstract: DMUs under DEA evaluation are assumed to be homogeneous. That means the units should undertake similar activities, use common technologies, and operate in similar environments. If one of these assumptions is violated, the efficiency estimates of these DMUs will be biased. 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. An approach based on system cluster analysis is proposed to discriminate these non-homogeneous DMUs. The DMUs utilizing similar technologies are aggregated in a cluster. A simulation analysis is employed to illustrate and verify our approach. Simulated results indicate that the efficiency estimates evaluated after clustering analysis reflect true efficiency better. Key words: Data Envelopment Analysis; Technology Difference; Cluster Analysis

3 . 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., 978). During the recent years, the theories and applications of DEA have been extensively developed (Banker et al., 984; Anderson et al., 00; Cook and Green, 005). A general precondition of DEA is that all DMUs to be evaluated are homogeneous (Charnes et al., 978), 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., 00; Haas and Murphy, 003). Some authors have done some research on non-homogeneity in DEA. Anderson et al. (00) tested the null hypothesis that the distributions of efficiency scores for equity and hybrid Real Estate Investment Trusts (REITs) are the same. Their results reect the null hypothesis, implying that the two types of REITs may operate under different technologies and differing environments. As a result, it is not appropriate to pool the data of the two groups when measuring efficiency. In a study of 44 bank branches, Soteriou and Zenios (999) 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. (998) 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 3

4 higher level. These researches separate non-homogeneous DMUs into different groups and evaluate them in different models. Similar method can also be find in Athanassopoulos and Thanassoulis(995), Soteriou and Zenios (999); Cook et al.(998); Cook and Green (005). 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, 986; Ruggiero, 996; Hua et al. 005). 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., 00), we suggest that, two DMUs are homogeneous in technology should satisfy the following three conditions: () undertaking similar activities; () 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. This study firstly analyzes the impact of internal technology difference of DMUs on the evaluated efficiencies, and illustrates theoretically that it is unreasonable to pool DMUs with obvious technology difference when measuring efficiencies. In order to solve this problem, this paper proposes an approach based on system cluster analysis to discriminate non-homogeneous DMUs before utilizing DEA techniques. 4

5 The simulation results show that our method can effectively discriminate DMUs utilizing different technologies. The remainder of the paper is organized as follows: Section 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 approach to discriminate DMUs utilizing different technology difference; Section 5 describes the simulation data and result for illustrating and verifying our approach; Section 6 summarizes the result and concludes the paper.. The Data Envelopment Analysis 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. Suppose we have a set of n peer DMUs, which produce multiple outputs y (,,..., n) =, by utilizing multiple inputs x (,,..., n) =. 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, =,,..., } C n = =, () n n n T = {( x, y) η x x, λ y y, λ =, λ 0, =,,..., n} V. () = = = 5

6 TC implies that the activities of all DMUs are under constant returns to scale. T V implies that the activities of all DMUs are under variable returns to scale. The CCR model (Charnes et al, 978) is a typical CRS DEA model, and the BCC model (Banker et al, 984) is a typical VRS DEA model. When a DMU 0 is under evaluation by the CCR model, we have: Min θ subect to n = n = λ 0, =,,, 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: Min φ subect to n = n = n = λ = λ 0, =,,, n λ x λ y φx y 0 0 (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. 6

7 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 =, ) as the maximal output of DMU which is possible for each of different production functions, and similarly index the α α α input vectors x ( x, x ) = in order to evaluate the technical efficiency under each of these two different sets of production possibilities. Hence we have y f( x, x ) = (5) y f( x, x ) = (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, 7

8 etc, designated as asterisks, are producing with technology (5). DMU E, F, G, H etc, designated as dots, are producing with technology (6). Figure 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 piece-wise line E F G H., and the DMUs in the second category are enveloped by the dotted x A E F B Q C G D H O x Legend: = DMU under the first technology = DMU under the second technology FIGURE. DMUs Using Different Technologies As showed by figure, 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 8

9 technologies, it is technical inefficient. In fact, keeping the level of output remain the same, DMU F can possibly decrease its inputs x and x 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. An approach to discriminate non-homogeneous DMUs From above analysis, we can see that the true efficiencies of DMUs cannot be reflected properly if the reference set includes DMUs using different technologies. Separating DMUs utilizing different technologies before evaluation may be a good method to evaluate the group of DMUs. However, it is often unknown that whether the DMUs under evaluation are homogeneous or not. Since basically none of two DMUs use exactly the same technology (technologies can be described as production functions), we will udge DMUs which produce with the same production function as homogeneous in technology, and DMUs which produce with different production function as non-homogeneous in technology. In this paper, we propose an approach to 9

10 identify whether the DMUs involved are homogeneous in technology, and if not, classify DMUs using different technologies. The approach is based on System Cluster Analysis method. After clustering, DMUs in the same cluster are homogenous in technology, which means that they produce with similar technology; DMUs in different clusters are non-homogenous in technology, which means that they produce with much different technologies. System Cluster Analysis (SCA) is one of most used cluster analysis methods. It aggregates individuals into a subset step by step, until all individuals are aggregated into one set. It is applicable for whatever variables or samples with data characteristics. Frequently used SCA methods include nearest neighbor, furthest neighbor, within-group linkage, centroid clustering, median clustering and Ward s method. It is more important to ascertain reasonable clustering variables than to choose a clustering method. The clustering variables should contain some characteristics of the samples to be clustered; herein the characteristics are the technology information of DMUs. In the following text, we analytically choose the suitable clustering variables for technologies depicted by Cobb-Douglas production functions. c c Consider a type of C-D functions with two inputs and one output, like y = Ax x. x, xand y represent two inputs and one output respectively. The parameter A is a constant coefficient to each production function. The parameters and c are input-output elasticities of c x and x respectively. While c + c <, the production function corresponds to decreasing returns-to-scale; while c + c =, the production 0

11 function corresponds to constant returns-to-scale; while c + c >, the production function corresponds to increasing returns-to-scale. According to Ruggiero (996), the above production function presents the relationship between inputs and output under efficient production process. It gives the maximal output obtainable from these input values. However, varieties of factors like inputs waste, management inefficiency, et al. make the production process not completely efficient. The more real production function can be written as c c ( r) Ax y = x, (7) where r is the inefficiency of each DMU during its production process, hence ( r ) is the efficiency. Inputs and output in this function are observations. Different technologies correspond to different A, and c. In fact, the production function of each DMU is unknown or at least unobtainable, so the clustering variables can only be formed through these observations and should reflect the difference of and c in each production function. c c Take natural logarithm to each side of formula (7) and we get equation (8). ( ) ln y = ln A+ ln r + cln x + c ln (8) x Dividing each ln y by ln x and ln x respectively, we get the following two variables: ( r) ln x ln y ln A ln v = = + + c+ c ln x ln x ln x ln x (9) v ( r) ln x ln y ln A ln = = + + c + c (0) ln x ln x ln x ln x

12 Different production functions have different values of A, or c, which make the two variables much different, even when given the same inputs values and same efficiencies. The difference of inputs value could also influence the values of these two variables; however, the influence can be less than that from difference of production functions. If two production functions differ much, the difference will probably be reflected on variables and v. v c In the economic view, the value of and v of each DMU could be interpreted as v the contribution of input x and x on the output y, respectively. The larger the value of or v is, the more advanced technology the DMU use on the v corresponding input. Now we can classify DMUs with technologies differ much into different groups and aggregate DMUs with similar technologies into the same group in the way of cluster analysis, based on the difference of variables and v of each DMU. v In the case of DMUs with multi-input and multi-output, the approach is similar as long as each output can be expressed by a C-D type production function. The only difference lies in that the technology difference is reflected on more than two variables, i.e. the natural logarithms of each output to each input together reflect the technology difference of DMUs. One point should be kept in mind is that the more variables used in cluster analysis, the more time would be spent. However, in case of too many variables, principal component analysis could be utilized to reduce the number of clustering indexes. After the suitable clustering variables have been set, the system cluster analysis

13 could do well on clustering DMUs under different technologies. In this paper, we adopt nearest neighbor cluster analysis. One of the principles that must be followed in clustering analysis is that any cluster should be distinct from the clusters around it, and the number of elements in each cluster should not be too large. We have the following criterion of udging homogeneity in technology of DMUs. Criterion. For DMUs utilizing C-D production functions, if they can be divided into two or more groups after cluster analysis using the clustering variables v and v, none of which contains more than 80% of all the DMUs, the whole DMU set are non-homogeneous in technology. Generally speaking, the number of DMUs in reference set when evaluating efficiencies should be more than three times the sum of the number of inputs and outputs (Banker et al., 984). If not, the DMUs in this set could be regarded as singular points, and be aggregated into the nearest group to be evaluated. If the DMU set can be obviously aggregated into two or more clusters based on the proposed method, and none of which contains more than 80% of all the DMUs, then we can sufficiently say that these DMUs are non-homogeneous in technologies and should be divided into homogeneous groups, after which DEA models can be used in each group. The approach to discriminate such potentially non-homogeneous DMUs can be 3

14 sum up as following steps: Step : Compute the clustering variables and v as equations (8)(9)(0) based on v observations of inputs and outputs. Step : Cluster the DMUs using ordinary SCA method. Identify whether the DMUs are non-homogeneous or not in technologies according to the Criterion. If they are non-homogeneous, turn to Step 3; otherwise evaluate them together. Step 3: Evaluate the efficiencies of DMUs in each group separately using DEA methods. 5. Simulation analysis To illustrate the impact of internal technology difference of DMUs on the evaluated efficiencies, and show the validity of our method, a simulation study is performed assuming two inputs, x and x, are used to produce one output y. Suppose there are 300 DMUs, produce following three C-D production functions as in equations ()()(3), 00 DMUs for each. ( r) x /3 y = x, () ( ) y = r x x, () ( ) / / y = r x x. (3) /4 /3 All inputs are randomly generated from a random uniform distribution for 300 observations according to the following intervals: x : (0,30) x : (40,00) 4

15 The true level of technical inefficiencies r is generated from the following distribution: r N(0,0.036) The value of r is further restricted by setting r equal to zero if the value of r generated is greater than To facilitate the validity test of our method, these 300 DMUs are evaluated in three ways: () non-clustering : pool all the 300 DMUs in the reference set when evaluating the efficiency of each DMU. () Ideal clustering : cluster the 300 DMUs into three groups based on three production functions and evaluate efficiency in the group separately. (3) SCA clustering : cluster the 300 DMUs using the index proposed and nearest neighbor SCA method and evaluate efficiency of each DMU only comparing to the DMUs in the same group. Compare the difference between efficiency estimate of each DMU and its true efficiency in the three different ways. According to Ruggiero (996), two measures are utilized to compare the difference of evaluated efficiencies. One is the correlation coefficient; the larger this value is the better the evaluation is. The other is the difference of means between the evaluated efficiencies and the true efficiencies of all DMUs; the smaller this value is the better the evaluation is. According to the generated Dendrogram and the scatter plot, it is appropriate to aggregate 300 DMUs into three clusters. The scatter plot of the 300 DMUs is showed in Figure. 5

16 Legend: = DMU v ln y ln x = v FIGURE. Scatter Plot of the 300 DMUs = ln ln y x BCC model is utilized for all the evaluation. The comparison of the difference between efficiency estimate of each DMU and its true efficiency is showed in the following table. Different ways TABLE Correlation coefficients Pearson Spearman Mean efficiencies Difference depart from mean true efficiencies* Non-clustering Ideal clustering 0.857(**) 0.84(**) SCA clustering 0.85(**) 0.803(**) * The mean value of true efficiencies is ** Correlation is significant at the 0.0 level (-tailed). Comparing the results between non-clustering and ideal clustering, we find out the Pearson coefficient rise up by 998% when the evaluations are done in each ideal group, the counterpart of Spearman coefficient also rise up by 67%, and the 6

17 difference of means between the evaluated efficiencies and the true efficiencies in the ideal clustering way is only 7.8% of the non-clustering way. The mean efficiencies of DMUs evaluated by non-clustering method are significantly lower than the mean true efficiencies of these DMUs. This result is in accord with the expectation we did in section that pooling non-homogeneous DMUs together when evaluating efficiencies will underestimate the true efficiencies of these DMUs. This illustration strongly shows that evaluating the efficiencies of DMUs with different technologies separately do a much better ob than evaluating them together. The improvement of efficiency evaluation brought by the proposed SCA approach is significant as well. The Pearson coefficient rise up by 944% when the evaluations are done in groups formed by SCA clustering, compared to the evaluation in non-clustering way. The Spearman coefficient also rises up by 637%, and the difference of means between the evaluated efficiencies and the true efficiencies in the SCA clustering way is only 6.9% of the non-clustering way. The evaluation result of proposed approach is much close to that of ideal clustering way. The efficiencies evaluated after SCA clustering are much close to the true technical efficiencies of each DMU, so they can better reflecting the true efficiencies of these non-homogeneous DMUs. The simulation shows that, pooling DMUs with different technologies together when evaluating efficiencies will deteriorate the estimated efficiencies. The proposed approach based on system cluster analysis can effectively discriminate non-homogeneous DMUs utilizing different technologies implied by different C-D 7

18 production functions. The efficiency evaluation done after clustering analysis could properly reflect the true efficiencies of DMUs and provide decision makers with useful information. More simulations show that, the larger the difference among technologies, the larger the improvement we can get from the proposed approach. Similar simulations have been done to test the impact of specific SCA methods, e.g., centroid clustering, median clustering and Ward s method, on the estimated efficiencies. The simulation results showed that as long as the index proposed in this paper, the impact of different SCA methods is not significant. 6. Conclusion When DEA method is used to evaluate the efficiencies of DMUs, the technology difference of DMUs may have an important impact on the evaluation. If the DMUs under evaluation are utilizing much different technologies, the evaluation should be done in each group separately. This paper analyzes the impact of technology difference of DMUs on the efficiency evaluation and proposes an approach to discriminate these non-homogeneous DMUs. This approach is based on system cluster analysis, utilizing clustering variables formed by the observations of inputs and outputs. The simulation analysis verified the reasonability, validity and feasibility of the proposed approach. The clustering index proposed in this paper is applicable to C-D production functions. It is an interesting problem to udge homogeneity when input/output data of DMUs are generated from other way than Cobb-Douglas production functions. 8

19 Another research area is how we can compensate the non-homogeneity caused by technology difference in evaluating non-homogeneous DMUs, In other words, whether we can adust the scores of DMUs or input/output data to account for the impact of technology difference so that we can evaluate all these DMUs together? These will be important research subects in the future. References Anderson, R. I., R. Fok, T. Springer, J. Webb. 00. Technical efficiency and economies of scale: A non-parametric analysis of REIT operating efficiency. European Journal of Operational Research, 39, Athanassopoulos, A.D., E. Thanassoulis Separating market efficiency from profitability and its implications for planning. Journal of Operational Research Society, 46, 0-34 Banker, R.D., A. Charnes, W.W. Cooper Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science, 30(9), Banker, R.D., R. Morey Efficiency analysis for exogenously fixed inputs and outputs. Operations Research, 34,

20 Charnes, A., W.W. Cooper, E. Rhodes Measuring the efficiency of decision making units. European Journal of Operational Research,, Cook, W.D., D. Chai, J. Doyle, R. Green Hierarchies and Groups in DEA. Journal of Productivity Analysis, 0, Cook, W. D., R. Green Evaluating power plant efficiency: a hierarchical model. Computers & Operations Research, 3, Dyson, R.G., R. Allen, A.S. Camanho, V.V. Podinovski, C.S. Sarrico, E.A. Shale. 00. Pitfalls and protocols in DEA. European Journal of Operational Research, 3, Haas, D.A., F.H. Murphy Compensating for non-homogeneity in decision-making units in data envelopment analysis. European Journal of Operational Research, 44, 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. 0

21 Oum, T.H., C. Yu Economic efficiency of railways and implications for public policy. Journal of Transport Economics and Policy, 8, 38. Ruggiero J On the measurement of technical efficiency in the public sector. European Journal of Operational Research, 90, Soteriou, A., S.A. Zenios Operations, quality, and profitability in the provision of banking services. Management Science, 45, 38.