In data envelopment analysis, the existing methods for measuring the relative efficiencies of decision making units (DMUs) are to compare DMUs relative to the best or the worst of all DMUs. In this paper, we consider both the best DMU and the worst DMU as the reference DMUs and propose the normalized efficiency. Further, from the optimistic and pessimistic viewpoints, we construct two DEA models to obtain the upper and lower bounds of the normalized efficiency and then achieve an interval efficiency evaluation to rank all DMUs completely. Finally, two examples are presented to illustrate the performance of the interval efficiency evaluation.

Introduction

Data envelopment analysis (DEA) has been proved to be an effective approach for measuring the performance of a group of decision making units (DMUs) with multiple inputs and multiple outputs. For a DMU, the CCR efficiency, developed by Charnels, Cooper and Rhodes (1978), is achieved by maximizing the ratio of the weighted sum of its outputs to that of its inputs under the constraint that the ratio should not exceed one for every DMU. Accordingly, the CCR efficiency is regarded as the best relative efficiency. The CCR efficiency evaluation can classify all DMUs into two groups, namely CCR efficient units and CCR inefficient units, but cannot rank all the DMUs completely. In recent years, a variety of DEA methods have been proposed to rank the performance of all DMUs. Adler, Friedman, and Sinuany-Stern (2002) divided these ranking methods into six areas. Cross-efficiency evaluation and super efficiency evaluation are the first and the second areas. Cross-efficiency evaluation, first proposed by Sexton, Silkman, and Hogan (1986), requests each DMU not only to be self-evaluated but also to be peer-evaluated. Specifically, based on the CCR model, a DMU determines a set of weights to evaluate the other DMUs. Yet, due to the non-uniqueness of the CCR optimal weights, the secondary goals have to be proposed to deal with the non-uniqueness issue. Doyle and Green (1994, 1995a) constructed several aggressive or benevolent cross-efficiency models. For more contributions to the cross-efficiency evaluation, readers are referred to the literature (Chen 2002; Contreras 2012; Liang, Wu, Cook, & Zhu, 2008a, 2008b; Oral, Amin, & Oukil, 2015; Wang & Chin 2010; Wu, Sun, & Liang, 2012; Yang, Ang, Xia, & Yang, 2012; Jeong and OK 2013; Hong and Jeong 2017). Andersen and Petersen (1993) considered a reference technology spanned by all the other DMUs except the evaluated DMU and then achieved the super efficiency evaluation. Indeed, when DMUs are evaluated, the reference technology is crucial. Doyle and Green (1995b) pointed out three reference points which occur naturally in everyday comparison, namely comparison relative to the best, to the average or to the worst of the rest. For each of these reference points of comparison, they presented two DEA models to obtain the best performance and the worst performance of each DMU, respectively, and then constructed the upper and lower bound evaluation. The traditional DEA models are usually built to achieve the best performance of DMUs from the optimistic viewpoint. Accordingly, the maximum ratio of the weighted sum of outputs to the weighted sum of inputs under some constraints is called the best relative efficiency or the optimistic efficiency. In fact, the worst relative efficiency or the pessimistic efficiency of a DMU, namely the minimum ratio of the weighted sum of outputs to that of inputs under some constraints, should still be paid enough attention to. The best relative efficiency and the worst relative efficiency measure the two kinds of extreme performances of a DMU. It is easily biased if considering only the best relative efficiency or the worst relative efficiency while neglecting the other one. Recently, many pessimistic DEA models have been presented to obtain the pessimistic efficiency of a DMU.