Cross-Efficiency in Fuzzy Data Envelopment Analysis (FDEA): Some Proposals

Different techniques have been proposed in the literature to rank decision making units (DMUs) in the context of Fuzzy Data Envelopment Analysis. In our opinion, those that result from using a ranking method to order the fuzzy efficiencies obtained are susceptible to a serious criticism: they are not based on objective criteria. Cross-efficiency evaluation was introduced as an extension of DEA aimed at ranking the DMUs. This methodology has found a significant number of applications and has been extensively investigated. In this chapter, we discuss some difficulties that arise with the definition of fuzzy cross-efficiencies and we propose a fuzzy cross-efficiency evaluation based on the FDEA model by Guo and Tanaka. Such model relies on the dual multiplier formulation of the CCR model and the fuzzy efficiency of a given DMU is defined in a ratio form in terms of the input and output weights obtained. This allows us to define the cross-efficiencies in an analogous manner to that of the fuzzy efficiency. The resulting cross-efficiencies are consistent in the sense that the cross-efficiency of a given DMU, calculated with its own input and output weights, is equal to the relative efficiency of this unit. We illustrate our methodology with an example.