Ranking DMUs by Comparing DEA Cross-Efficiency Intervals Using Entropy Measures

Abstract

Cross-efficiency evaluation, an extension of data envelopment analysis (DEA), can eliminate unrealistic weighing schemes and provide a ranking for decision making units (DMUs). In the literature, the determination of input and output weights uniquely receives more attentions. However, the problem of choosing the aggressive (minimal) or benevolent (maximal) formulation for decision-making might still remain. In this paper, we develop a procedure to perform cross-efficiency evaluation without the need to make any specific choice of DEA weights. The proposed procedure takes into account the aggressive and benevolent formulations at the same time, and the choice of DEA weights can then be avoided. Consequently, a number of cross-efficiency intervals is obtained for each DMU. The entropy, which is based on information theory, is an effective tool to measure the uncertainty. We then utilize the entropy to construct a numerical index for DMUs with cross-efficiency intervals. A mathematical program is proposed to find the optimal entropy values of DMUs for comparison. With the derived entropy value, we can rank DMUs accordingly. Two examples are illustrated to show the effectiveness of the idea proposed in this paper.

An Entropy-Based Cross-Efficiency under Variable Returns to Scale

“…Wang et al [24] explored the DEA entropy model to construct the cross-efficiency intervals for ranking DMUs. Lu and Liu [25] considered the benevolent and aggressive and formulations simultaneously for obtaining a number of cross-efficiency intervals, and the entropy was used to build a numerical index for the DMUs to be comparable. Rotela Junior et al [26] adopted a DEA model in portfolio optimization, and Shannon entropy was included to ensure an efficient asset diversification while return and portfolio risk were maximized and minimized, respectively.…”

“…Since we take into account the aggressive and benevolent cross-efficiency at the same time, a family of cross-efficiency intervals is derived for each DMU. The idea of Lu and Liu [25], which calculated the entropy value of the cross-efficiency interval, is adopted to construct a numerical index for DMUs with cross efficiency intervals.…”

“…With the smallest entropy (uncertainty) of each DMU, we can discriminate DMUs more easily. Based on Lu and Liu [25], Equation (13) can be transformed intô…”

Hybrid cluster analyzing and data envelopment analysis with interval data

“…Data envelop analysis (DEA) is a linear programming model and non-parametric approach that evaluates relative technical efficiencies of decision making units (DMUs) on the basis of multiple inputs and outputs by computing the ratio of weighted sum of their outputs to their inputs [1]. This technique has been used in many fields successfully with crisp values, however in real application there are inaccurate data similar to probabilistic, interval, ordinal, qualitative, or fuzzy.…”

“…This technique has been used in many fields successfully with crisp values, however in real application there are inaccurate data similar to probabilistic, interval, ordinal, qualitative, or fuzzy. Hence, some researchers conducted several theoretical development of DEA model with data such as interval [1]. Nevertheless, there are many models and techniques to solve this problem, but there is a new problem for ranking the efficient DMUs with interval data, so that in some researches DMUs were ranked by these ideal points [2,3].…”