تحلیل پوششی داده های فازی: یک رویکرد ارزش مورد انتظار فازی

Performance assessment often has to be conducted under uncertainty. This paper proposes a “fuzzy expected value approach” for data envelopment analysis (DEA) in which fuzzy inputs and fuzzy outputs are first weighted, respectively, and their expected values then used to measure the optimistic and pessimistic efficiencies of decision making units (DMUs) in fuzzy environments. The two efficiencies are finally geometrically averaged for the purposes of ranking and identifying the best performing DMU. The proposed fuzzy expected value approach and its resultant models are illustrated with three numerical examples, including the selection of a flexible manufacturing system (FMS).

Traditional data envelopment analysis (DEA) (Charnes, Cooper, & Rhodes, 1978) requires crisp input and output data, which may not always be available in real word applications. Significant efforts have been made to handle fuzzy input and fuzzy output data in DEA. For example, Sengupta (1992) incorporated fuzziness into DEA by defining tolerance levels for both the objective function and violations of constraints and proposed a fuzzy mathematical programming approach. Triantis and Girod (1998) transformed fuzzy input and fuzzy output data into crisp data using membership function values and suggested a mathematical programming approach in which efficiency scores were computed for different values of membership functions and then averaged. Guo and Tanaka (2001) converted fuzzy constraints such as fuzzy equalities and fuzzy inequalities into crisp constraints by predefining a possibility level and using the comparison rule for fuzzy numbers and presented a fuzzy CCR model. León, Liern, Ruiz, and Sirvent (2003) suggested a fuzzy BCC model based on the same idea.

In view of the fact that precise input and output data may not always be available in real world performance assessments due to the existence of uncertainty, we have proposed in this paper a fuzzy expected value approach for fuzzy DEA to conduct performance assessments in fuzzy environments from different perspectives. The fuzzy expected value approach transforms fuzzy input and fuzzy output data into two total expected values for inputs and outputs, respectively, based on which two pairs of fuzzy expected value models have been constructed to measure the optimistic and the pessimistic efficiencies of DMUs by using fuzzy or crisp weights. The two extreme efficiencies have then been integrated through their geometric average to measure the overall performances of the DMUs and identify the best performing DMU. The fuzzy expected value approach and the resultant models have finally been tested with three numerical examples including the selection of a FMS.