Fuzzy Data Envelopment Analysis with SBM using α-level Fuzzy Approach

In this paper, a generalized ratio-type estimator based on ranked set sampling (RSS) is proposed for The applications of fuzzy analysis in data-oriented techniques are the challenging aspect in the field of applied operational research. The use of fuzzy set theoretic measure is explored here in the context of data envelopment analysis (DEA) where we are utilizing the fuzzy α-level approach in the three types of efficiency models. Namely, BCC models, SBM model and supper efficiency model in DEA. It was observed from the result that the fuzzy SBM model has good discrimination power over fuzzy BCC. On the other side, both the models fuzzy BCC and fuzzy SBM are not able to make the genuine ranking which is acceptable for all. So this weakness is overcome with the help of fuzzy super SBM model and all three models are applied to illustrate the types of decisions and solutions that are achievable when the data are vague and prior information is in imprecise. In this paper, we are considering that our inputs and outputs are not known with absolute precision in DEA and here, we using Fuzzy-DEA models based on an α-level fuzzy approach to assessing fuzzy data.


Introduction
Since DEA was proposed in 1978 and after that it has been got comprehensive attention in both theories as well as in application. DEA becomes an important analysis tool and research way in management science, operational research, system engineering, decision analysis, and economics. Performance analysis has become a vital part of the management practices in the banking industry. There are numerous applications using DEA models to estimate efficiency in banking, and most of them assume that inputs and outputs are known with absolute precision. The different modification was in the mathematical approaches of DEA, namely mixed orientation approach of DEA was given Qaiser et al. (2016). Two-stage production processes with double frontier were given by Arif et al. (2017). DEA as decision support system was given by Qaiser et al. (2017). But it is not always possible that our input and output data are known with absolute precision. DEA is based on the production process and the data of production processes cannot be precisely measured always since the uncertain theory has played an important role in the inputs and outputs. The fuzzy analysis is helpful for handling the different type of data, namely uncertainty data, interval data; identify the missing variable and high-frequency data. A possible path to handle input/output uncertainty in DEA relies on the use of probability distributions to model their inherent randomness. These distributions are subsequently employed in stochastic DEA models. However, these probability distributions require being somewhat estimable a priori or a posteriori, limiting the use of stochastic DEA models in cases where the event is unique or deterministic. Alternatively, however, uncertainty in input/output may be related to imprecision or vagueness, rather than to randomness. This being the case imprecision or vagueness in input/output values can be expressed by membership functions within the ambit of fuzzy logic.
The α-level approach is possibly the most popular, given the numerous papers produced using its variations, despite the fact that their models are not computationally efficient. This is so because α-level models demand more linear programs to be solved for each α value (Soleimani-damaneh, Abbasbandy et al., and Jahanshahlooet al., 2006) within the α-level approach, the FDEA model is first converted into a pair of parametric programs so that the lower and upper bounds of the efficiency scores can be computed next for a given value of α in Emrouznejad and Tavana (2014). The rationale behind the selection of then α-level approach in this study is related to a number of aspects. First, when using this approach, fuzzy inputs and outputs may be expressed as crisp numbers representing the limiting bounds of the intervals for different α-levels in Chen et al. (2013), thus allowing the uncertainty of the data collected from Mozambican banks to be easily modelled as triangular fuzzy numbers. Second, in the situation of various α levels for the inputs and the outputs, Fuzzy DES may be translated into traditional DEA (crisp) models in light of the extension principle, thus making solving their respective linear programs simpler (Yager, 1981;Zadeh, 1965a;Zimmerman, 1976). Third, owing to the input and output data being fuzzy numbers, the efficiency scores are also fuzzy numbers in Puri and Yadav (2013). Moreover, as long as the efficiency values considered here are the upper and lower "crisp" bounds computed for various α levels, the membership functions for the true fuzzy efficiency cannot be reconstructed, which has a number of implications on how fuzzy efficiencies should be ranked in Chen et al.( 2013); Puri and Yadav (2013); Hsiao et al. (2011). These bounds, however, can be treated as crisp values and incorporated into statistical modelling as efficiency scores subjected to certain fixed

DEA with Fuzzy Sets
The two fundamental DEA models, CCR (Charnes Cooper Rhodes DEA-mode) and BCC (Banker Charnes Cooper DEA-model) are based on the assumption of know, numerical, and fixed value of inputs and outputs. But this type of input-output data are not always possible, sometimes we are observing that our observed input-output data are imprecise or vague. In such situations, the traditional DEA models fail to get any information from such type of data. So, in order to overcome from such type of difficulties, we are making the use of fuzzy theory. Sengupta (1992) was first introducing a fuzzy mathematical programming approach into which fuzziness was incorporated into DEA methodology.
Let  Where " " denotes the fuzziness of input and outputs. The model (2.1) is an input-oriented fuzzy CCR model in multiplier form. Envelopment form of the above model is more feasible to solve, which can convert by making the use of duality theory. Thus envelopment form of model (2.1) is as:  is said to be inefficient. We can solve the above model into two phases, in phase-I, we are estimating the feasible optimal solution of fuzzy BCC-model by using fuzzy linear programming approaches after that we use the optimal value * k  in order to calculate the values of the slack. Phase II of fuzzy BCC-model is given below:

Fuzzy Approaches in DEA
The application of fuzzy set theory in DEA categorized into four approaches available in the literature which are discussed under as:

Tolerance approach
The Tolerance approach was one of the first fuzzy approaches in DEA and that was introduced by Sengupta (1992) later the same approach was improved by Kahraman and Tolga (1998). The main idea in this approach to incorporate with uncertainty input and output data in DEA models by defining the tolerance levels on set if constraints violations. This approach does not treat fuzzy coefficients directly but it is fuzzified the inequality or equality signs. Although in most production processes fuzziness is present both in terms of not meeting specific objectives and in terms of the imprecision of the data, the tolerance approach provides flexibility by relaxing the DEA relationships while the input and output coefficients are treated as crisp.

Fuzzy ranking approach
The fuzzy ranking approach is another and popular technique in fuzzy DEA was initially developed by Guo and Tanaka (2001). In this approach, the main idea is to estimate the fuzzy efficiency scores of the DMUs by using fuzzy linear programming which requires ranking the fuzzy set. Besides this, the approach is based on the fuzzy DEA models in which fuzzy constraints involves fuzzy equalities and fuzzy inequalities and converted into crisp constraints by predefining and possibility level by comparison rule of fuzzy numbers.

Possibility approach
The possibility approach is based on the fundamental principle of the possibility theory was imitated by Zadah (1977) and it was related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable. More specifically, if F is a fuzzy subset of a universe of discourse , where X is variable taking value in U , induces a possibility distribution X  which equates the possibility of X taking the value ) (u to u F  the compatibility of U with F. in this X becomes a fuzzy variable which is associated with a probability distribution see (Zadah 1977). The proposed possibility CCR model was developed by Lertworasirikul et al. (2003) where they applied the concept of chance-constrained programming (CCP) and the possibility of fuzzy events are represented by the following: are predetermined admissible levels of possibility. The envelopment from of the model (3.1) after using the CC-transformation and then use of duality in the above model which required a form of fuzzy CRR is as: The extension of the model (3.2) was in Lertworasirikul et al. (2003) in case of VRS case for developing fuzzy BCC which as: are predetermined admissible levels of possibility.

α-Level Approach
The level −  approach is the most used method in fuzzy DEA. This is evident by the large number published reported work of FDEA based on level −  approach. The mathematical idea in this approach is to convert the FDEA model into a pair of parametric programming in order to find the lower and upper bounds of the level −  membership functions of the efficiency score. Girod (1992) used the approach proposed by Carlsson and Korhonen (1986) to formulate the fuzzy BCC and free disposal hull (FDH) models which were radial measures of efficiency. In this model, the inputs could fluctuate between risk-free (upper) and impossible (lower) bounds and the outputs could fluctuate between risk-free (lower) and impossible (upper) bounds. The method was approximately the membership function of the fuzzy efficiency measures by applying the level −  approach. Then the extension of the same approach was done by Zadeh (1976) and Zimmermenn (1996) and they transformed the fuzzy DEA model to a pair of parametric mathematical programs and used the ranking fuzzy numbers methods proposed by Chen and Klein (1997) for estimating the efficiency measures of DMUs. The optimal solution of the fuzzy BCC model (7.5) can be estimated by using a twolevel mathematical model. The two levels are using to calculate the lower and upper bounds of the fuzzy DEA model and efficiency scores for a specific level −  as follow: Similarly, upper bounds of fuzzy DEA model and efficiency scores for a specific level −  as follow: the fuzzy inputs and fuzzy outputs respectively. The two-level mathematical model can be simplified to the conventional one-level model as follows: Similarly, upper bounds of fuzzy DEA model are given as: The above membership function is built by solving the lower and upper bounds DEA-model is derived for a particular case, where the inputs and outputs are triangular fuzzy numbers is as given below: Then Liu (2008) proposed two parametric mathematical programs are as given below: . , , 0 , Ahn Y. Hyo, Gulbadian F. Dar,Shariq A. Bhat,Arif M. Tali, Yasir H  respectively. It is assumed that the data set is known and strictly positive. In Tone (2001) proposed a non-oriented and non-radial DEA technique called slack based (SBM) measure through which we can Minimizing the inefficiency rate directly by make the user input and output slacks. The mathematical formulation of SBM is given below: The above model is based on the assumption that input and output data are fixed, known and strictly positive. This is not possible always in the real world. Sometimes input and output data are known, positive but fuzzy type. In such situations, the model ( . In order to solve the fuzzy input and fuzzy outputs, we are defining the corresponding membership functions are level −  the form of the fuzzy inputs and fuzzy outputs respectively. . The super-efficiency can be calculated by using the mathematical model as given below, under the assumption that k DMU should be efficient.

Fuzzy SBM model for supper efficiency in DEA
We can use of level −  the fuzzy approach for estimating lower and upper bounds of networking fuzzy SBM model. In all the above-mentioned approaches of handling the fuzzy data in DEA, the role of the fuzzy arithmetic is very much important in DEA terminology. The fuzziness in DEA crates complexity in order to use fuzzy equalities and fuzzy inequalities i.e. (  = , , ).

Numerical illustration
In this section, we present a numerical example based on 24 banks as DMUs using Assets, Expense, and Deposit as inputs while as fees, amount of loans, and amount of investments are used as output. In order to illustrate the use of the methodology proposed here like fuzzy BCC, fuzzy SBM and fuzzy Super SMB. We treat the two outputs amount of loans and investments are as fuzzy items and analyze the efficiency of the banking sector with triangles fuzzy functions from the fuzzy theory. The data are taken from (Li, 2003), and recorded in Table 1.       Efficient but performing weakly. In other words, we can say that there is a chance to improve efficiency by reducing the input and output slacks of DMU 6 and 12 respectively. Thus it was clear that fuzzy SBM has good discrimination power over fuzzy BCC. The results of Table 3 are the lower bounds of efficiency scores in fuzzy SBM model at different levels of . Table 4 shows the upper bounds of fuzzy SBM model at different levels of . In addition, we also observed that DMU2 is full efficient at  =1, but when  =0 it showing inefficiency. Thus in conventional DEA DMU2 is fully efficient, however, Bank 2 is not affected by its overdue loans ratio, and as such, its efficiency score is an overestimation.   123 1.122 1.121 1.121 1.120 1.119 1.118 1.118 1.117 1.116 1.115 The results of fuzzy BCC model and fuzzy SMB model are having the same shortcoming and we cannot rank the DMUs, because in both the models the efficiency is denoted as 1. Therefore, I order to overcome from this weakness, we are using fuzzy supper efficiency model based on the slacks. Fuzzy supper SBM excludes all inefficient DMUs and rank all those efficient DMUs, whose efficiency score is equal to unity in the fuzzy SBM model. Such that all those DMUs will be rank easily. In Table 6, we are showing the supper efficiency lower bounds of efficient DMUs at different levels of  and supper efficiency upper bounds at same levels  are present in the Table8.