Fuzzy measures for fuzzy cross efficiency in data envelopment analysis

The cross-efficiency evaluation measures the efficiencies of decision-making units (DMUs) through both self- and peer-evaluation methods. Since the cross-efficiency is effective in discriminating among DMUs, this evaluation technique has been widely used in many applications. In the real world, there are cases in which the observations are difficult to measure precisely. The existing approaches of fuzzy cross-efficiency evaluation employ the secondary goal approach to determine the weights for measuring fuzzy cross-efficiencies. However, the different approaches for determining the weights may produce different fuzzy cross-efficiencies. In this paper, we propose a novel method that considers all possible weights of all the DMUs simultaneously to calculate the fuzzy cross-efficiency directly, and the choice of weights is not required. Since the α-level-based approach is one of the most popular approaches for developing fuzzy data envelopment analysis models, this approach is employed to formulate the proposed fuzzy cross-efficiency evaluation. A pair of linear programs is developed to calculate the fuzzy cross-efficiency. At a specific α-level, solving the pair of linear programs generates the lower bound and upper bound of the fuzzy efficiency score. The illustrated examples show that the fuzzy cross-efficiency evaluation method proposed in this paper has discriminative power in ranking the DMUs when the data are fuzzy numbers.