بهره وری فوق العاده بر پایه تابع فاصله جهت دار اصلاح شده

The problem of infeasibility arises in conventional radial super-efficiency data envelopment analysis (DEA) models under variable returns to scale (VRS). To tackle this issue, a Nerlove–Luenberger (N–L) measure of super-efficiency is developed based on a directional distance function. Although this N–L super-efficiency model does not suffer infeasibility problem as in the conventional radial super-efficiency DEA models, it can produce an infeasible solution in two special situations. The current paper proposes to modify the directional distance function by selecting proper feasible reference bundles so that the resulting N–L measure of super-efficiency is always feasible. As a result, our modified VRS super-efficiency model successfully addresses the infeasibility issues occurring either in conventional VRS models or the N–L super-efficiency model. Numerical examples are used to demonstrate our approach and compare results obtained from various super-efficiency measures.

Data envelopment analysis (DEA) is a method for measuring relative efficiency of peer decision making units (DMUs). In recent years, DEA has been applied to various settings, such as performance evaluations in Olympic Games [1], estimating the importance of objectives in agricultural economics [2], regional R & D investment evaluations in China [3], and bankruptcy assessment for corporations [4]. In an effort to differentiate the performance of efficient DMUs, Andersen and Petersen [5] develop a super-efficiency model based upon the constant returns to scale (CRS) model [6]. However, when the concept of super-efficiency is applied to the variable returns to scale (VRS) model [7], the resulting model must be infeasible for certain DMUs [8]. Infeasibility restricts a wider use of super-efficiency DEA. Recent years have seen several studies addressing the infeasibility issue and the development of new super-efficiency models. For example, Lovell and Rouse [9] modify the conventional radial super-efficiency model by scaling up the concerning input vector (in an input-oriented case), or by scaling down the concerning output vector (in an output-oriented case). Also under the VRS assumption, Chen [10] and [11] replaces inefficient observations by their respective efficient projections, and performs super-efficiency analysis with this revised data set. Cook et al. [12] show that for infeasibility cases, one needs to adjust both the input and output levels to move an efficient DMU under evaluation onto the frontier formed by the remaining DMUs. They develop a two-stage process to address the infeasibility issue. Lee et al. [13] develop an alternative two-stage process to addressing infeasibility issue in the conventional VRS super-efficiency models. On the other hand, based on the directional distance function [14], Ray [15] develops a procedure to obtain Nerlove–Luenberger (N–L) measure of super-efficiency in a single model to adjust both input and output levels. As a result, this N–L super-efficiency model does not pose a similar infeasibility problem in the conventional VRS super-efficiency models. However, Ray [15] points out that the N–L super-efficiency model fails in two special situations. First, no feasible solution exists if the zero input value is present in a DMU under evaluation and all other DMUs in the reference set are positive-valued in that input. Second, when an N–L super-efficiency score is greater than 2, the model will yield an efficient projection involving negative output quantities. In fact, zero data are problematic in any super-efficiency models. For example, Lee and Zhu [16] show that either the conventional VRS super-efficiency or the two-stage super-efficiency procedure in [12] will become infeasible when zero data are present. Therefore, it is necessary to address the two issues presented in [15]. The current paper shows that we can choose a proper reference input–output bundle in the directional distance function (DDF) [14], and modify Ray's DDF-based VRS super-efficiency model [15]. The new super-efficiency model successfully addresses the infeasibility issues occurring either in conventional VRS models or the N–L super-efficiency model. The remainder of this paper is organized as follows. Section 2 introduces the DDF and its previous applications in N–L efficiency and super-efficiency assessments. Section 3 proposes a modified DDF, based on which a new VRS DEA model is developed for super-efficiency measurement. This new super-efficiency model addresses the infeasibility issues occurring either in conventional VRS models or the N–L super-efficiency model. Section 4 illustrates the new approach by using the data set from [8]. Section 5 concludes with a summary of our contributions.

Conventional radial VRS super-efficiency models become infeasible in certain situations. Although the Nerlove–Luenberger (N–L) measure of super-efficiency [15] avoids such infeasibility issues under VRS condition, it would fail in two special cases. This paper modifies the DDF-based super-efficiency in [15] so that the newly-proposed approach is always feasible. We point out that the values of parameters a and b depend on the specific data set in order to make all DMUs feasible in our DDF-based model. This does not indicate that inefficient DMUs will affect the efficiency scores. Note that the parameter ranges are calculated based on extreme values of related input/output measures. Therefore, the values influencing the choice regions of parameters are the maximum or minimum input/output values from the whole data set, and are not relevant to the specific group of inefficient DMUs.