Data Envelopment Analysis with Fuzzy Parameters: An Interactive Approach

Data envelopment analysis (DEA) is a methodology for measuring the relative efficiencies of a set of decision making units (DMUs) that use multiple inputs to produce multiple outputs. In the conventional DEA, all the data assume the form of specific numerical values. However, the observed values of the input and output data in real-life problems are sometimes imprecise or vague. Previous methods have not considered the preferences of the decision makers (DMs) in the evaluation process. This paper proposes an interactive evaluation process for measuring the relative efficiencies of a set of DMUs in fuzzy DEA with consideration of the DMs’ preferences. The authors construct a linear programming (LP) model with fuzzy parameters and calculate the fuzzy efficiency of the DMUs for different α levels. Then, the DM identifies his or her most preferred fuzzy goal for each DMU under consideration. A modified Yager index is used to develop a ranking order of the DMUs. This study allows the DMs to use their preferences or value judgments when evaluating the performance of the DMUs. efficiency. A DMU is considered efficient when no other DMU can produce more outputs using an equal or lesser amount of inputs. The DEA generalizes the usual efficiency measurement from a single-input single-output ratio to a multiple-input multiple-output ratio by using a ratio of the weighted sum of outputs to the weighted sum of inputs. The traditional DEA methods such as CCR (Charnes et al., 1978) and BCC (Banker et al., 1984) require accurate measurement of both the inputs and outputs. However, the real evaluation of the DMUs often DOI: 10.4018/joris.2011070103 40 International Journal of Operations Research and Information Systems, 2(3), 39-53, July-September 2011 Copyright © 2011, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. exhibit imprecision and great uncertainty. In general, as the system’s complexity increases, exact evaluation of data becomes extremely difficult. In addition, the traditional DEA models generally do not consider the decision maker’s (DM’s) preferences or value judgments. Although a few researchers have considered the DM’s preferences, their model requires precise and exact measurement of both the input and output data (Joro et al., 2003; Wong et al., 2009; Yang et al., 2009). In this study, we propose an interactive evaluation process for measuring the relative efficiencies of a set of DMUs in fuzzy DEA with consideration of the DMs’ preferences. We construct a linear programming (LP) model with fuzzy parameters and calculate the fuzzy efficiency of the DMUs for different α levels. Then, the DM identifies his/her most preferred fuzzy goal for each DMU under consideration. A modified Yager index is introduced and used to develop a ranking order of the DMUs. The main thrust of this study is to allow the DMs to use their preferences or value judgments when evaluating the performance of the DMUs. This paper is organized into seven sections. The next section presents a brief review of the existing literature on fuzzy DEA followed by a discussion of fuzzy number rankings. Then, we present an overview of fuzzy DEA. Following this overview, we illustrate the details of the proposed framework followed by a numerical example in order to demonstrate the applicability of the proposed framework and also to exhibit the efficacy of the procedures and algorithms. Finally, we conclude with our conclusions and future research directions. FUZZY DEA LITERATURE REVIEW In the conventional DEA, all the data assume the form of specific numerical values. However, the observed values of the input and output data in real-life problems are sometimes imprecise or vague. Imprecise evaluations may be the result of unquantifiable, incomplete and non obtainable information. The “Stochastic approach” and the “fuzzy approach” are two existence approaches for modeling uncertainty in the DEA literature. The stochastic approach involves specifying a probability distribution function (e.g., normal) for the error process (Sengupta, 1992). However, as pointed out by Sengupta (1992), the stochastic approach has two drawbacks associated with modeling the uncertainty in DEA problems: a. Small sample sizes in DEA make it difficult to use stochastic models, and b. In stochastic approaches, the DM is required to assume a specific error distribution (e.g., normal or exponential) to derive specific results. However, this assumption may not be realistic because on an a priori basis there is very little empirical evidence to choose one type of distribution over another. Some researchers have proposed various fuzzy methods for dealing with the impreciseness and ambiguity in DEA. Fuzzy set algebra developed by Zadeh (1965) is the formal body of theory that allows the treatment of imprecise estimates in uncertain environments. Sengupta (1992) proposed a fuzzy mathematical programming approach by incorporating fuzzy input and output data into a DEA model and defining tolerance levels for the objective function and constraint violations. Triantis and Girod (1998) proposed a mathematical programming approach by transforming fuzziness into a DEA model using membership function values. Guo and Tanaka (2001), León et al. (2003) and Lertworasirikul et al. (2003a) proposed three similar fuzzy DEA models by considering the uncertainties in fuzzy objectives and fuzzy constraints using the possibility approach. Lertworasirikul et al. (2003b) proposed a fuzzy DEA model using the credibility approach where fuzzy variables were replaced by expected credits according to the credibility measures. Lertworasirikul et al. (2003c) further extended the fuzzy DEA through the possibility and credibility approaches. International Journal of Operations Research and Information Systems, 2(3), 39-53, July-September 2011 41 Copyright © 2011, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. Kao and Liu (2000b) transformed fuzzy input and output data into intervals by using α-level sets. The α-level set approach was extended by Saati et al. (2002), who defined the fuzzy DEA model as a possibilistic-programming problem and transformed it into an interval programming. Hatami-Marbini et al. (2009a) proposed a four-phase fuzzy DEA framework based on the theory of displaced ideal. Entani et al. (2002) extended the α-level set research by changing fuzzy input and output data into intervals. Dia (2004) proposed a fuzzy DEA model where a fuzzy aspiration level and a safety α-level were used to transform the fuzzy DEA model into a crisp DEA. Wang et al. (2005) also used the α-level set approach to change fuzzy data into intervals. Saati and Memariani (2005) further extended the α-level set approach so that all DMUs could be evaluated by using a common set of weights under a given α-level set. Soleimani-damaneh et al. (2006) addressed some of the limitations of the fuzzy DEA models proposed by Kao and Liu (2000a), León et al. (2003) and Lertworasirikul et al. (2003a) and suggested a fuzzy DEA model to produce crisp efficiencies. Liu (2008) and Liu and Chuang (2009) also extended the α-level set approach by proposing the assurance region approach in the fuzzy DEA model. Hatami-Marbini and Saati (2009) improved a fuzzy BCC (Banker et al., 1984) model which considered fuzziness in the input and output data as well as the decision variable. The thrust of this study is to allow the DM to use his or her preferences or value judgments when evaluating the performance of the DMUs. In the DEA framework proposed in this study, the DM defines uncertain data by means of language statements and determines a preferred fuzzy goal based on the obtained fuzzy efficiencies of the DMUs for different levels. In other words, the DM can interactively impact the ranking of the DMUs with his/her most preferred goal. Although all the efficiencies of the DMUs obtained from the fuzzy DEA model are mathematically acceptable, a DM can utilize his/her preferences in relation to the conflicting objectives and select his/her most preferred DMU. Finally, the fuzzy efficiency value and the fuzzy goal stated by the DM for each DMU are aggregated and used to rank the DMUs on the basis of their efficiencies. The interactive fuzzy DEA approach proposed in this study is based on the recently developed fuzzy mathematical programming approach by Jiménez et al. (2007). Jiménez et al.’s method breakdowns in some cases because of the Yager index (Yager, 1979) used in their method. We propose a modification to the Jiménez et al.’s (2007) method to overcome this problem. THE FUZZY NUMBER RANKINGS A fuzzy number  A = (a, b, c, d), is called a generalized trapezoidal fuzzy number with membership function m  A which has the following properties (Dubois & Prade, 1978):