Data envelopment analysis (DEA) is a non-parametric programming technique for evaluating efficiency of a set of homogenous decision making units (DMUs) with multiple inputs and multiple outputs. It has been proven to be an effective approach in identifying the best practice frontiers and ranking the DMUs. DEA has been extensively applied in performance evaluation and benchmarking of schools, hospitals, bank branches, production plants, and so on (Charnes, Cooper, Lewin, & Seiford, 1994). However, traditional DEA models, such as CCR model in Charnes, Cooper, and Rhodes (1978) can simply classify the DMUs into two groups, namely efficient DMUs and inefficient DMUs. Moreover, it is often possible in traditional DEA models that some inefficient DMUs are in fact better overall performers than some efficient ones. This is because of the unrestricted weight flexibility problem in DEA by being involved in an unreasonable self-rated scheme (Dyson and Thannassoulis, 1988 and Wong and Beasley, 1990). The DMU under evaluation heavily weighs few favorable measures and ignores other inputs and outputs in order to maximize its own DEA efficiency.
The cross efficiency method was developed as a DEA extension technique that could be utilized to identify efficient DMUs and to rank DMUs using cross efficiency scores that are linked to all DMUs (Sexton, Silkman, & Hogan, 1986). The main idea of the cross evaluation method is to use DEA in a peer evaluation instead of a self evaluation. There are at least three main advantages for cross-evaluation method. Firstly, it provides a unique ordering among the DMUs (Sexton et al., 1986). Secondly, it eliminates unrealistic weight schemes without requiring the elicitation of weight restrictions from application area experts (Anderson, Hollingsworth, & Inman, 2002). Finally, the cross evaluation method can effectively differentiate between good and poor performers (Boussofiane, Dyson, & Thanassoulis, 1991). Therefore the cross-evaluation method has been widely used for ranking performance of DMUs, for example, efficiency evaluations of nursing homes (Sexton et al., 1986), selection of a flexible manufacturing system (Shang & Sueyoshi, 1995), justification of advanced manufacturing technology (Talluri & Yoon, 2000), diagnosing best intelligent mailer (Kabassi, Virvou, & Despotis, 2003), and so on.
Although average cross efficiency has been widely used, there are still several disadvantages for utilizing the final average cross efficiency to evaluate and rank DMUs, like the losing association with the weights by averaging among the cross efficiencies (Despotis, 2002), which means that this method cannot clearly provide the weights to help decision makers improve their performance, especially, the average cross efficiency measure is not good enough since it is not a Pareto solution. Considering the shortcomings above, Wu, Liang, and Yang (2009) eliminate the average assumption for determining the ultimate cross efficiency scores, and DMUs are considered as the players in a cooperative game, in which the characteristic function values of coalitions are defined to compute the Shapley value of each DMU, and the common weights associate with the imputation of the Shapley values are used to determine the ultimate cross efficiency scores.
In the current paper, we will propose an approach based on information entropy theory instead of calculating the average cross efficiency scores. This approach has several advantages, for example, in this method, the most productive scale size (MPSS) units (Cooper, Seiford, & Tone, 2000) get the best rank and the interior points of the smallest production possibility sets (PPSs) which are inefficient in all models lie at the end of the ranking list (Soleimani & Zarepisheh, 2009). The rest of this paper is organized as follows: Section 2 introduces the cross efficiency evaluation method. The new method using Shannon entropy is proposed in Section 3. Section 4 gives an illustrative example, and conclusion and remarks are shown in Section 5.
