Cross-efficiency evaluation capable of dealing with negative data: A directional distance function based approach

Abstract

Cross-efficiency evaluation is a commonly used technique for ranking decision making units (DMUs) in data envelopment analysis while it is just applicable to the non-negative data set. Based on the range directional measure (RDM), we explore the duality relation regarding the maximum slacks formulation of RDM and define cross-efficiencies on the basis of the equivalences with some fractional programming problems. The newly defined cross-efficiency evaluation is under the variable returns to scale condition. It can handle negative values in inputs and outputs well and the resulting cross-efficiencies as well as cross-efficiency scores are between 0 and 1 for all DMUs. It provides self- and peer-evaluations of DMUs by measuring the inefficiency in both inputs and outputs simultaneously. In addition, we propose a corresponding model that implements the benevolent criterion to select weights. A numerical example demonstrates the validity and effectiveness of the proposed cross-efficiency evaluation in dealing with negative data.