A Model for Ranking Decision Making Units in Data Envelopment Analysis

A Model for Ranking Decision Making Units in Data Envelopment Analysis S. Saati M. M. Zarafat Angiz L. A. Memariani G. R. Jahanshahloo Riassunto: DEA (Data Evaluation Analysis) valuta le prestazioni di Unitci decisional2 (Decision Making Units, DMUs). La riformulazione dei programmi lineari per il ranking delle DMU in DEA omettendo le corrispondenti colonne nella matrice tecnologica porta a dificolta teoriche e pratiche. Uno di questi problemi si manifesta quando il problema corrispondente diviene infeasible. Questo articolo suggerisce una semplice ma importante modijica per il modello MAJ e dimostra che la versione modijicata 6 sempre feasible, e il suo ranking sta in {0,1]. A diflerenza dei precedenti modelli, il modello 6 simultaneamente orientato all'input e alljoutput. Per dimostrare il concetto sono forniti alcuni esempi numerici. Department of Mathematics, Science & Research Branch, Islamic Azad University, Tehran, Iran Department of Mathematics, Firooz-Kooh Branch, Islamic Azad University, Firooz-Kooh, Iran Department of Industrial Engineering, Tarbiat Modarres University, Tehran, Iran. E-mail: memarqmodares. ac. ir Department of Mathematics, Teacher Training University, Tehran, Iran Ricerca Operativa vo1.31 n.97

48 S. Saati el al. - Ranking decision making units in data envelopment analfisis Abstract: Data Envelopment Analysis (DEA) evaluates the performance of Decision Making Units (DMUs). Reformulating the linear programes for ranking DMUs in DEA by omitting the corresponding column in technological matrix makes some theorical and applied dificulties. One of the these problems arrise when the corresponding problem becomes infeasible. This paper suggestes a simple but very important modification for MAJ model and proves that the modified version is always feasible and the ranking lies in {0,1]. Unlike the previous models, this model is both input and output oriented, simultaneously. To demonstrate the concept, some numerical ezamples are solved. Keywords: Data Envelopment Analysis, Eficiency, Ranking 1. Introduction Data Envelopment Analysis (DEA) proposed by Charnes et al. [4] (CCR model) and developed by Banker et al. [2] (BCC model) is an approach for evaluating the efficiencies of Decision Making Units (DMUs). DEA does not require any a priori weight for inputs and outputs. Outcome of DEA models is an efficiency score equal to one to efficient DMUs and less than one to inefficienct DMUs. So, for inefficient DMUs a ranking is given but efficient DMUs cannot be ranked. Some methods for ranking efficient DMUs are proposed, but these methods break down in some cases. Cross-efficiencies in DEA is one method that could be utilized to identify good overall performers and effectively rank DMUs [ll]. Crossefficiency methods evaluate the performance of a DMU with respect to the optimal input and output weights of other DMUs. A limitation in using this method is that the factor weights obtained from CCR model may not be unique. The existance of alternative optimal solution in efficiency evaluation of DMUs, coases some difficulties. Some techniques have been proposed for obtaining robust factor weights for use in the contruction of cross-efficiencies method. Doyle and green [6] have developed a set of formulations for this purpose. Sarkis and Talluri [lo] extended the Cross-efficiency method to include both,cardinal and ordinal input and output factors, which is based on the work by Cook et al. [5]. They proposed a combination of models that allowed for effective ranking of

S. Saati et al. - Ranking decision making units in data envelopment analysis 49 DMUs in the presence of both quantitative as well as qualitative factors. These models are also based on cross-evaluations in DEA. Other ranking method that do not specifically include cross-efficiencies was proposed by Rousseau and Semple [9]. Rousseau and Semple approached the same problem as a two-person ratio efficiency game. Their formulation provides a unique set of weights in a single phase as opposed to the two-phase approaches presented above. The revision of DEA models by omitting the corresponding column of DMU under consideration in technological matrix, has been proposed by Andersen and Petersen [I] (AP model). Banker et al, [3] showed that for some data, AP model is infeasible. In Dula et al. [7] some theorems are proved to show the cases in which the DEA revised models by constant, variable, increasing and decreasing Returns to Scale (RTS) by omitting a 6olumn of technological matrix are feasible or infeasible. In Mehrabian et al. [8] (MAJ model) a model is suggested to deal with zero data. However, MAJ model is break down in some cases which is shown by a theorem in [8]. But this method also breakes down in some cases. All these methods evaluate the efficiency in either input or output oriented fashion. In this paper, an alternative definition of efficiency and some models with constant, variable, increasing and decreasing returns to scale for evaluating and ranking DMUs are proposed. One of the most important characteristic of these models is their ability in evaluating a DMU in input and output orientation, simultaneously. The paper is organized as follows: Section 2 provides a short background about DEA standard, MAJ and revised models. The suggested models are presented in section 3 and the feasibility of them are discussed. In section 4, abilities of proposed models are shown by solving some numerical examples. Section 6 closes with conclusion. 2. Background Suppose that there are n DMUs for efficiency evaluation, and each DMU consumes m inputs to produce s outputs. Particular, DMUp

50 S. Saati et al. - Ranking decision making units in data envelopment analysis consumes xip (i = 1,..., m), the amount of input i, to produce yrp (r = 1,..., m), the amount of output r. In the model formulation, Xj and Yj (j = 1,..., n) denote the nonnegative vectors of input and output values for DMUj, respectively. Associated to each DEA models, there are a Production Possibility Set (PPS). Definition: The production possibility set is the set, which defined as: T = {(X, Y)J The output vector Y > 0 can be produced from input vector X 2 0). The PPS associated to constant returns to scale models in DEA is expressed as follows: By adding the following constraints to Tc, the PPS associated to variable, increasing and decreasing returns to scale models, respectively, in DEA are obtained: To measure the efficiency of DMU,, i.e. DMU under consideration, the following models associated to T, are considered: min 9 max 4

S. Saati et al. - Ranking decision making units in data envelopment analysis 51 n By adding the constraints of returns to scale C hi < 1 to (2), j= 1 (J the models with variable, increasing and decreasing returns to scale are obtained. Omitting the corresponding column to DMUp in the technological matrix of (2), results the following revised models (AP model): (Input Orientation) (Output Orientation), min 8 max 4 n n By adding the following revised constraints of returns to scale: to (3), the revised models with variable, increasing and decreasing returns to scale are obtained. AP model, in some cases, breaks down with zero data and may be unstable because of extreme sensitivity to small variations in the data when some DMUs have relatively small values for some of its inputs. These cases are discussed in detail in [7], [a] and [12]. Another model for ranking DMUs is proposed in [a] (MAJ model).

52 S. Saati et al. - Ranking decision making units in data envelopment analysis x Figure I - The projection schema in proposed model This model is as follows: min cp = wp + 1 n where w, is a free variable and 1 is vector of ones. As noted in [a], MAJ model is feasible for evaluation of DMU, with vector Y, 2 0 if and only if for each r (r = 1,..., s), either y,, = 0 or there exists a DMUj, j # p, such that y,j # 0. This assumption is not sufficient for feasibility of the AP model. In [7], a list of cases that leads AP model to the infeasible problems are presented. The following example shows that in some cases, MAJ model can be infeasible. Example 1 Consider 3 DMUs with 2 inputs and 2 outputs as table 1. The MAJ and AP models can not evaluate D2, since its corresponding problems become infeasible.

S. Saati et al. - Ranking decision making units in data envelopment analysis 53 DMU 1 I1 I2 01 02 1 AP MAJ Dl 1 4 2 0 1 1 1.30 1.13 Table 1 - Data and results for ezarnple I 3. Proposed models This section presents some models which can evaluate the efficiency of'dmus in input and output orientation, simultaneously. These models by decreasing inputs and increasing the outputs of the DMU under consideration by equal sizes, project it on the frontier. The simultaneously changes in input and output are equal in size because otherwise due to giving different preferences to them; the problem becomes a multi objective programming one and hence yielding a complex situation. The proposed model by production possibility set T, for evaluating DMU, is as follows: min cp = wp + 1 where wp is a free variable and 1 is vector of ones. Figure 1 shows the projection by (5). By adding the constraints of returns to scale in (5), the models by variable, increasing and decreasing returns to scale are obtained. Since the inputs and outputs are not homogeneous and scale of objective functions in proposed models is depended on the units of measurement of input and output data, unit dependence is obtained by normalization

54 S. Saati et al. - Ranking decision making units in data envelopment analysis e.g. dividing each input and output to the largest of them as one of the techniqus for normalization. Theorem The proposed models are always feasible and their optimal values are in (0,1]. Proof: A, = 0 (j = 1,..., n, j # p), A, = 1 and w, = 0 is a feasible point for proposed models. Therefore, the proposed models are always feasible. This feasible point implies that p* 5 1 where, p* is the optimal value of (5). Suppose that wp 5-1. Then, x,, + w, 5 0, since the inputs are normalized. This is a contradiction, since it was shown that the proposed models are feasible. Therefore, UI, > -1 and p* E (0,ll.O Omitting the column corresponding to DMU,, the DMU under consideration in the technological matrix of (5), the ranking model is obtained as follows: min p = wp + 1 Adding the revised constraints of returns to scale, modified for (6) variable, increasing and decreasing returns to scale models. The resulted models are always feasible, since

S. Saati et al. - Ranking decision making units in data envelopment analysis 55 and is a feasible point for these models. 4. Numerical example Some numerical examples are presented in [7], which DEA revised models are unable to evaluate them. These examples alongwith an example which MAJ model can not solve it, are presented in the following, and is shown that the suggested model is able to evaluate them. Example 2 (The case of constant RTS) In table 2, 8 DMUs with 4 inputs and 2 outputs are considered. The DEA revised constant returns to scale model (3) can not evaluate D6, D7 and D8 but, new revised constant returns to scale model presents an efficiency scale equal to 1.30, 1.26 and 1.11 for them, respectively. - DMU - Dl D2 D3 D4 D5 D6 D7 D8 - I1 I2 I3 I4 01 02 7 1 5 6 1 1 3 0 4 1 1 1 4 5 3 0 1 1 2 9 7 3 1 1 5 0 6 0 0 1 1 0 0 5 1 1 0 0 2 0 0 1 6 0 3 0 1 1 Proposed MAJ AP model 0.89 0.89 0.40 1.09 1.12 1.50 1.07 1.80 1.50 1.00 1.00 1.00 1.00 1.00 0.33 1.30 1.44 infea. 1.26 1.35 inre.. 1.11 1.13 mfea. Table I. Data and results for example I Example 3 (The case of variable and increasing RTS) Table 3 presents the data for DEA analysis using the example illustrates without zeros in the data. Using the input oriented variable

56 S. S ad et al. - Ranking decision making units in data envelopment analysis returns to scale model for D4, we find that the efficiency score is one. However, the MAJ and AP linear programming problems with D4 are infeasible. With the suggested revised model its efficiency value is 1.18. The same thing accurs with the increasing returns to scale model. F D4 model 1.17 infea. imfea. 1 3 1.18 infea. inlea. Table 3. Data and results for example 3 Example 4 (The case of output orientation) 4 DMUs with 2 inputs and one output are considered in table 4. The revised model with decreasing returns to scale in output orientation is infeasible for D2, D3 and D4 but the proposed model evaluates them as Table 4. model infea. infea. 1.20 infea. infes. D4 2 1.00 idea. idea Table 4. Data and results for example 4 Example 5 (The case of MAJ infeasibility) Consider the data of example 1. As noted, MAJ model could not evaluate D2, since its problem is infeasible, but proposed model take an efficiency value equal to 2.00 for this unit.

S. Saati et al. - Ranking decision making units in data envelopment analysis 57 model 2.00 inres. inres. 1.14 2.00 1.20 Table 5. Data and results for example 5 5. Conclusion Ranking of DMUs in DEA is an important phase for efficiency evaluation of DMUs. DEA techniques generally do not rank the 'efficient DMUs. Andersen and Petersen [I] presented an extension of the basic DEA methodology which has the desirable feauture of ranking not only the inefficient DMUs, but also the efficient ones as well. However, their basic approach and its variants have several conceptual problems. Another ranking method is proposed by Mehrabian et al. [8] (MAJ model). This model is also have some difficulties. This paper suggests some models which remove the difficulties about infeasibility of AP and MAJ models. As proved, these models are always feasible. Another advantage of our model is its projection method on efficiency frontier. AP and MAJ models project the DMUs in input or output orientation but our method do it in both orientation. This is useful for decision makers which can vary simultaneously their inputs and outputs. Acknowledgment Useful comments from anonymous referee are gratefully acknowledged.

58 S. Saati et al. - Ranking decision making units in data envelopment analysis References [l] ANDERN, P. and N. C. PETERSEN, (1993), "A Procedure for Ranking Efficient Units in Data Envelopment Analysis", Management Science, Vol. 39, No. 10, pp. 1261-1264. [2] BANKER, R. D., A. CHARNES and W. W. COOPER, (1984), "Some Methods for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis", Management Science, Vol. 30, No. 9, pp. 1078-1092. [3] BANKER, R. D., J. L. GIFFORT, "Relative Efficiency Analysis", Unpublished manuscript (A 1987 version appeared as a Working Paper, Schools of Urban and Public Affairs, Carnegie-Millon University). [4] CHARNES, A., W. W. COOPER and E. RHODES, (1978), "Measuring the Efficiency of Decision Making Units", EJOR, Vol. 2, No. 6, pp. 429-444. [5] COOK, W. D., M. KRESS and L. M. SEIFORD, (1996), "Data Envelopment Analysis in the presence of both quantitative and qualitative factors", Journal of the Operational Research Society, Vol. 74, No. 7, pp. 945-953. [6] DOYLE, J. and R. GREEN, (1994), "Efficiency and cross-efficiency in DEA: Derivations, meanings and uses", Journal of the Operational Research Society, Vol. 45, No. 5, pp. 567-578. [7] DULA, J.H. and B. L. HICKMAN, (1997), "Effects of Excluding the Column Being Scored from the DEA Envelopment LP Technology Matrix", Oper. Res. Society, Vol. 48, pp. 1001-1012. [8] MEHRABIAN, S., M. ALIREZAEE and G. JAHANSHAHLOO, (1999), "A Complete Efficiency Ranking of Decision Making Units in Data Envelopment Analysis", Computational Optimization and Applications, Vol. 14, pp. 261-266. [9] ROUSSEAU, J. J. and J. H. SEMPLE, (1995), "Two-Person Ratio Efficiency Games", Management Science, Vol. 41, No. 3, pp. 435-441. [lo] SARKIS, J. and S. TALLURI, (1999), "A Decision Model for Evaluation of Flexible Manufacturing Systems in the Presence of Both Cardinal and Ordinal Factors", International Journal of Production Research, Vol. 37, No. 13, pp. 2927-2938. [ll] SEXTON, T. R., R. H. SILKMAN and A. HOGAN, (1986), "Data Envelopment Analysis: Critique and Extensions", In R. H. Silkman (Ed.), Measuring Eficiency: An Assessment of Data Envelopment Analysis, Publication no. 32 in the series New Direction of Program Evaluation, Jossey Bass, San Francisco.

S. Saati et al. - Ranking decision making units in data envelopment analysis 59 [12] ZARAFAT A. L., M., G. JAHANSHAHLOO and S. SAATI M., (1999), "A Note on Andersen-Petersen Model for Ranking Decision Making Units in DEA", J. of Sci., Islamic Aaad Uni., (in persian), Vol. 9, No. 31-32, pp. 2371-2381.