A DEA ranking method based on cross-efficiency intervals and signal-to-noise ratio

Cross-efficiency evaluation is a useful approach to ranking decision making units (DMUs) in data envelopment analysis (DEA). The possible existence of multiple optimal weights for the DEA may reduce the usefulness of the cross-efficiency evaluation since the ranking is according to the choice of weights that different DMUs make. Most of existing approaches for cross-efficiency evaluation employ the average cross-efficiency to further discriminate among the DEA efficient units or focus on how to determine input and output weights uniquely, but lay little emphasis on the consideration of the ranges and variances of cross-efficiencies as alternative ranking factors. In this paper we consider cross-efficiency intervals and their variances for ranking DMUs. The aggressive and benevolent formulations are taken into account at the same time. Consequently, a number of cross-efficiency intervals is obtained for each DMU. The signal-to-noise (SN) ratio, originally designed for optimizing the robustness of a process, is constructed as a numerical index for ranking DMUs. A nonlinear fractional program with bound constraints is formulated to find the optimal value of the SN ratio. By model reduction and variable substitution, this nonlinear fractional program is transformed into a quadratic one for deriving the global optimum solution. With the derived SN ratios, we are able to fully rank all DMUs accordingly. Two examples are given to illustrate the effectiveness of the methodology proposed in this paper.