Logical efficiency decomposition for general two-stage systems in view of cross efficiency

Abstract

Abstract Most of the previous cross efficiency evaluation methods consider the evaluated system as a “black box” system. This probably leads to misleading results. For example, the system is efficient but all of its divisions are inefficient. This paper firstly investigates the cross efficiency for a general two-stage system. In such system, some outputs of one stage are taken as inputs of the other stage, and some other outputs directly become the final products. Since the optimal weight vector for the self-evaluation of a decision making unit (DMU) may be non-unique, we apply the leader-follower method for the decomposition of the system s efficiency and then adopt the optimal weights to obtain the peer-evaluation. Then, multiplicative hesitant fuzzy elements (MHFEs) are applied to denote the ratio of all possible cross efficiencies of any two DMUs. Based on MHFEs, multiplicative hesitant fuzzy preference relations (MHFPRs) are used to express these relationships comprehensively. Furthermore, the optimization models for deriving acceptably consistent MHFPRs with the minimum total adjustment and the minimum number of adjusted elements are built when the consistency of MHFPRs is unacceptable. An algorithm for decomposing the overall efficiency according to the cross efficiency and acceptably consistent MHFPRs in the setting of the general two-stage system is provided. Finally, our method is applied to the analysis of nine top universities in China.