Evaluation of Decision Making Units in the Presence. of Fuzzy and Non-discretionary

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1 Applied Mathematical Sciences, Vol. 7, 2013, no. 28, HIKARI Ltd, Evaluation of Decision Making Units in the Presence of Fuzzy and Non-discretionary Neda Fathi and Mohammad Izadikhah Department of Mathematics, Arak Branch, Islamic Azad University, Arak, Iran Copyright 2013 Neda Fathi and Mohammad Izadikhah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In original data envelopment analysis (DEA) models, input and output are measured by exact values on a ratio scale. However, this assumption may not be always valid. For example some data may be only known as in forms of fuzzy and non discretionary data. In this paper we try to suggest a generalized model when some inputs and outputs are fuzzy and non-discretionary data. Keywords: Data envelopment analysis, fuzzy data, non-discretionary data. 1. Introduction Data envelopment analysis (DEA),is currently a popular technique for analyzing technical efficiency. DEA originally proposed by Charnes et al.[1]is a non-parametric technique for measuring and evaluating the relative efficiencies of a set of entities, called decision making units (DMUs),with the common inputs and outputs. Often the assumption of homogeneous environments is violated and factors that describe the differences in the environments need to be including in the analysis. These factors, and other factors outside the control of the DMUs, are frequently called nondiscretionary factors instances from the DEA literature include snowfall or weather in evaluating the efficiency of maintenance units. Therefore, some papers were presented on the theorical development of this technique with non-discretionary data. Many researchers addressed the problem with non-discretionary data (Banker and

2 1388 N. Fathi and M. Izadikhah Morey[2], Ruggiero[3],[7], Golany and Rool[4], Ray[5], Fried et al.[6],[8], Muñiz[9], Yang and Paradi [10]). In the recent years, in different applications of DEA, inputs and outputs have been observed whose values are indefinite. Such data are called inaccurate. Inaccurate data can be probabilistic, interval, ordinal, qualitative, or fuzzy. Therefore, some papers were presented on the theorical development of this technique with fuzzy data. Recently, many researchers addressed the problem with fuzzy data (Cooper et al.[11][12][13]; Kao and Liu[16]; Entani et al.[14]; Guo and Tanaka[15]; Lertworasirikul et al.[17]). In this paper, we introduce DEA model in the presence of non-discretionary and fuzzy data and therefore a method for evaluation DMUs with non-discretionary and fuzzy data is proposed. The rest of paper is as follows: In section 2, first we review the standard DEA model then and consider CCR model for dealing with non-discretionary data. In other part, we review CCR model for dealing with fuzzy data. In section 3, we will discuss the proposed model with incorporating both non-discretionary and fuzzy data into CCR model. In section 4, we obtain the efficiency score of DMUs through Numerical Example with nondiscretionary and fuzzy data. Conclusion are given is section Preliminaries In this section we look briefly concepts used in this paper Standard DEA model Assume that there are n DMUs, where each DMU (j=1,,n),uses m different inputs, (i=1,,m), to produce s different outputs, (r=1,,s).we assume that the data set are positive. We denote by (r=1,,s) the level of the r-th output from unit j(j=1,,n)and by (i=1,,n)the level of the i-th input to the j-th unit. Let jo be the evaluated unit. To obtain efficiency of DMU we use the CCR model, which is as follows. Where and are the weights associated, respectively with input i and output r..: 1 0,1,, u,v DEA and non-discretionary data Now we assume that for each DMU from 1 to n, some of outputs and input are non-discretionary. So the CCR model for evaluating DMUs with non-discretionary data is as follows: (2.1)

3 Evaluation of decision making units 1389 s.t: 0,j1, n 1,i DI, 0,i NI,r Do, 0,r No That we define DI and NI refer to discretionary and non-discretionary inputs and DO and NO refer to discretionary and non-discretionary outputs DEA and fuzzy data Now we assume that for each DMU from 1 to n, some of outputs and input are fuzzy. So the CCR model for evaluating DMUs with fuzzy data is as follows: max ~ s.t: ~ ~ 0,1,, ~ 1 ~, (2.2) (2.3) Where: ~,, ), ~,,,1 ~ 1,1,1,and {EI},{FI} refer to exact and fuzzy inputs indices.{eo},{fo} refer to exact and fuzzy outputs indices. Now, we calculate -cut of each fuzzy number and rearrange this model in the fallowing form: max s.t: , 0 1,, But L can obtain value greater than 1 and it is possible that obtained efficiency become gather than 1. So we write the last constraint in the fallowing form: (2.4) αl1 So, let L=1 and we don t have (*) unequal. The above model is non-linear. Substituting for and for and operationting two last constraints at and, we have final model. For solving first we give to values between [0,1] then solve this model. s.t: 0 1,, 1 ;r FO ;i FI, (2.5)

4 1390 N. Fathi and M. Izadikhah 3. DEA model with both non-discretionary and ordinal data In this section, we propose a CCR model in the presence of non-discretionary and fuzzy data as follows: max ~ ~ ~ s.t: ~ ~ ~ ~ 0,1,, ~ 1 ~,i DEI,, 0,i NEI,NFI,r DEoDFO, 0,r NEo,NFO Where: ~,,, ~,, (3.1),1 ~ 1,1,1 and {DEI},{DFI} refer to discretionary exact and fuzzy inputs indexes.{nei},{nfi} refer to nondiscretionary exact and fuzzy inputs indices. {DEO},{DFO} refer to discretionary exact and fuzzy outputs indexes. {NEO},{NFO} refer to non-discretionary exact and fuzzy outputs indices. Now, we calculate -cut of each fuzzy number and rearrange this model in the fallowing form: max s.t: 0 1,, ,i DEI,, 0,i NEI,NFI,r DEoDFO, 0,r NEo,NFO But L can obtain value greater than 1 and it is possible that obtained efficiency become gather than 1.so we write the last constraint in the fallowing form: (3.2) So, let L=1 and we don t have (*) unequal. The above model is non-linear. Substituting for and for and operationting two last constraints at and, we have the final model. For solving, first we give to values between[0, 1] then solve this model. max s.t: 0 1,, 1 ; FO (3.3)

5 Evaluation of decision making units 1391 ;i FI,i DEI,, 0,i NEI,NFI,r DEoDFO, 0,r NEo,NFO 4. Numerical example To illustrate the above, consider the fuzzy and non-discretionary data setting of Table 1. Table 1 reports the data. and as two inputs and, and as the three outputs where is fuzzy input and is non-discretionary output. Table 2 shows the efficiency when they evaluated by model (3.3). Table 1: Related attributes for 5 DMUs input 1 input 2 output 1 output 2 output (3,4,5) (9.4,9.8,10.2) (2.1,2.2,2.3) (3.7,4.2,4.7) (4.9,5.6,6.3) Conclusion In some situations, some inputs and outputs of DMUs are non-discretionary and fuzzy data. In this paper, we introduced these data and proposed a method by CCR model, for efficiency evaluation of DMUs with non-discretionary and fuzzy data. The proposed model uses the linear programming problem for efficiency evaluation of DMUs. This statement is seen in the example. Table 2: efficiency evaluated for the 5DMUs References [1] A.Charnes,W.W.Cooper,E.Rhodes,Measuring the efficiency of decision making units, Euro.J.Operat. Res.2(1978)

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