Measuring Efficiency and Ranking Fully Fuzzy DEA

In this current literature, there are several models of Fully Fuzzy Data Envelopment Analysis (FFDEA) where inputsoutputs data and weights were fuzzy numbers. The main purpose of this study is to evaluate the performance of a set of Decision Making Units (DMUs) and fully ranking DMUs in fully fuzzified environment. In this paper, an FFDEA problem is discussed, and a method for solving this FFDEA problem is also proposed. This method based on the multi-objective linear programming and the simplex method is proposed for computing an optimal fuzzy solution to a FFDEA problem in which fuzzy ranking functions are not used. We acquire fuzzy efficiency scores with solving FFDEA. Afterwards, we have used a ranking function to rank these fuzzy scores. A numerical example is used to demonstrate and compare the results with those obtained using alternative approaches. Measuring Efficiency and Ranking Fully Fuzzy DEA Sayedali Khaleghi, Abbasali Noura* and Farhad Hosseinzadeh Lotfi Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran; alimath1390@gmail.com


Introduction
Data Envelopment Analysis (DEA) is systematic approach for the analyzing the performance of organizations and operational processes. It can evaluate the relative efficiencies of homogeneous Decision Making Units (DMUs) without knowing production functions, only by using input and output data. The first model od DEA introduced by charnes et al 4 . To see the other classic models of DEA, the readers can see 3,5,7,15 . The basic DEA results group the DMUs into two sets: one set is efficient DMUs and the other is inefficient DMUs. In many cases, it is necessary to give a full ranking of the DMUs. For this purpose, different methods with different properties to achieve full ranking of DMUs have been proposed. Sexton et al. proposed the ranking of DMUs based on a cross efficiency 17 . The benchmarking, initially developed by Torgersen was employed to rank all the efficient DMUs 18 . Andersen and petersen first developed the most popular ranking method called superefficiency 19 . Cooper and Tone ranked the DMUs according to scalar measuring of inefficiency in DEA, based on the slack variables 6 . Liu and Peng proposed Common Weights Analysis (CWA) to determine a set of indices for common weights to rank efficient DMUs of DEA 9 . In many situations, such as manufacturing system, a production process or service system, inputs and outputs are volatile and complex so that it is difficult to measure them in an accurate way. Instead the data can be given as a fuzzy variable. Many fuzzy approaches have been introduced in the DEA literature. Guo 26 . This paper will extend the CCR model to a fuzzy DEA model based on cedibility measure and then give a fuzzy ranking method all the DMUs with fuzzy inputs and outputs. The fuzzy ranking method was developed by Guo and Tanaka 23 . This approach provides fuzzy efficiency for an evaluated DMU at a specified α-level. Meilin Wen et al. proposed a new fuzzy DEA model based on credibility measure as well as a ranking method provided, they designed a hybrid algorithm combined with fuzzy simulation and genetic algorithm to compute the fuzzy model DEA 12 31,32 . To see the other FDEA models, the readers can see 8,10,11,16,19 . This paper is organized as follows: In section 2 we present the basic definitions of fuzzy arithmetic. In the next section, is defined ranking function. In section 4 the FFDEA is introduced and solve using the above mentioned multiplication and the properties of the presented linear ranking function with related theorems, are in introduced. Section 5, is devoted to a numerical example.

Ranking Function
A simple method for ordering fuzzy numbers consists in the definition of a ranking function F, mapping each fuzzy number to the real number R, where a natural order exists. Suppose S = {Ã 1 , Ã 2 , Ã 3 , ..., Ã n } is a set of n fuzzy numbers,and the ranking function F is a mapping from S to the real numbers R, i.e. F:S → R then for any distinct pair of fuzzy numbers Ã i , Ã j є S, the ranking function can be defined as, This implies for example, that if F(Ã i ) > F(Ã k ), the fuzzy number Ã i is numerically greater than fuzzy number Ã k the higher Ã i is, the larger F(Ã i ) is.
Here we introduce a linear ranking function that is similar to the ranking function 27 . For any arbitrary fuzzy number Ã = (A(r), Ā(r)), we use ranking function as follows: For triangular fuzzy number this reduces to: (1) ).

Numerical Example
In this section we have a numerical example from Guo and Tanaka 23 study that its data are on Table 1. By putting the data of Table 1 on the proposed model, we will have model (5). This model is written to calculate the first decision making units efficiency and 4 other models have to be written to calculate the efficiency of other units. Now, the MOLP problem to the given fully fuzzy DEA problem is given below: 4 v 1 (2) + 2.1 v 2 (2) ≤ 1; We consider the following LP problem (6)  4 v 1 (2) + 2.1 v 2 (2) ≤ 1; We applied fuzzy ranking function (1) to the results of Hatami-Marbini approach 33 , Kazemi and et al. approach 31 and our approach, the results are presented in Table 3.
If we consider the decimal numbers in seventh column of third Table, we will notice that DMU 2 , DMU 5 will be classified in first rank and DMU 4 in second rank. Also the DMU 3 , DMU 1 are located in third, fourth ranks, respectively. As regards all presented methods for evaluating of efficiency DMUs differ together, thus it's naturalized that the results of methods are nuanced together. But generally, as for Table 2 and Table 3 all methods are assessed DMU 2 , DMU 4 , DMU 5 best decision making units and after the units three and one take the score efficiency orders . Table  2 shown to conform both the results of this study and the results of other methods.

Conclusion
In this paper, an FFDEA problem has been presented. Also, an approach has been given to solve it. We are using the proposed method of Pandian 35 to find an optimal fuzzy solution to a FFDEA problem. The main advantage of the proposed method is that the FFDEA problems can be solved by any LP solver using the level-sum method since it's based on only simplex method. With using of this method, we obtain fuzzy efficiency scores and DMUs are ranked.