ارزیابی بهره وری متقابل براساس واحدهای تصمیم گیری ایده آل و آنتی ایده آل

Cross-efficiency evaluation is an effective approach to ranking decision making units (DMUs) that utilize multiple inputs to produce multiple outputs. Its models can usually be developed in a way that is either aggressive or benevolent to other DMUs, depending upon the decision maker (DM)’s subjective preference to the two extreme cases. This paper proposes several new data envelopment analysis (DEA) models for cross-efficiency evaluation by introducing a virtual ideal DMU (IDMU) and a virtual anti-ideal DMU (ADMU). The new DEA models determine input and output weights from the point of view of distance from IDMU or ADMU without the need to be aggressive or benevolent to any DMUs. As a result, the cross-efficiencies measured by these new DEA models are neutral and more logical. Numerical examples are provided to illustrate the potential applications of these new DEA models and their effectiveness in ranking DMUs.

Data envelopment analysis (DEA) is a methodology for assessing the performances of a group of decision making units (DMUs) that utilize multiple inputs to produce multiple outputs. It measures the performances of the DMUs by maximizing the efficiency of every DMU, respectively, subject to the constraints that none of the efficiencies of the DMUs can be bigger than one. The efficiencies measured in this way are referred to as optimistic efficiencies or the best relative efficiencies. The way to measure the optimistic efficiencies of the DMUs is referred to as self-evaluation. If a DMU is self-evaluated to have an efficiency score of one, then it is said to be DEA efficient; otherwise, the DMU is said to be non-DEA efficient. DEA efficient units are usually thought to perform better than those non-DEA efficient units. The self-evaluation assigns the most favorable set of input and output weights to each DMU to maximize its optimistic efficiency, often leading to more than one DMU being evaluated as DEA efficient and unable to be discriminated. To improve the discrimination power of DEA, peer-evaluation was suggested as a supplement to the self-evaluation. The peer-evaluation assesses the performances of the DMUs not only in terms of their optimistic efficiencies, but also their cross-efficiencies computed using the weights determined by other peer DMUs. It is thus also called cross-efficiency evaluation. The cross-efficiency evaluation was first proposed by Sexton, Silkman, and Hogan (1986) and was later investigated by Doyle and Green, 1994 and Doyle and Green, 1995. It determines a unique set of input and output weights for each DMU and then calculates its efficiencies using all the sets of weights. Accordingly, each DMU will have multiple yet different efficiency scores, whose average reflects the overall performance of the DMU. Based on average cross-efficiencies, DMUs can be compared and ranked. Due to its strong discrimination power, the cross-efficiency evaluation has found extensive applications in the DEA literature. For example, Shang and Sueyoshi (1995) used the cross-efficiency evaluation for the selection of the most efficient flexible manufacturing systems (FMS). Green, Doyle, and Cook (1996) employed the cross-efficiency evaluation for preference voting and project ranking. Baker and Talluri (1997) applied the cross-efficiency evaluation for industrial robot selection. Sun (2002) utilized the cross-efficiency evaluation for differentiation between good and bad computer numerical control (CNC) machines. Ertay and Ruan (2005) took advantage of the cross-efficiency evaluation to determine the best labor assignment in cellular manufacturing system (CMS). Lu and Lo, 2007a and Lu and Lo, 2007b examined the economic–environmental performances of 31 regions in China by taking account of various environmental factors and then integrated the cross-efficiency evaluation with cluster analysis to construct a benchmark-learning roadmap for those inefficient regions to improve their efficiencies progressively. Wu et al., 2008 and Wu et al., 2009a made use of the cross-efficiency evaluation in conjunction with cluster analysis for Olympic ranking and benchmarking. Theoretical research on the cross-efficiency evaluation has also been investigated. For instance, Anderson, Hollingsworth, and Inman (2002) found the fixed weighting nature of the cross-efficiency evaluation where there is only one input and demonstrated with a numerical example how this unseen fixed set of weights might still be unrealistic. Sun and Lu (2005) presented a cross-efficiency profiling (CEP) model which evaluates each input with respect to the outputs that consume the input, separately. Bao, Chen, and Chang (2008) provided an alternative interpretation to the cross-efficiency evaluation from the point of view of slack analysis in DEA. Wu (2009) presented a revised benevolent cross-efficiency model and used the cross-efficiencies to construct a fuzzy preference relation for DEA ranking. Liang, Wu, Cook, and Zhu (2008a) proposed three alternative secondary goals for the cross-efficiency evaluation. Liang, Wu, Cook, and Zhu (2008b) also generalized the concept of cross-efficiency to game cross-efficiency and presented a convergent iterative algorithm to derive Nash equilibrium point. Wu, Liang, and Chen (2009) suggested a modified DEA game cross-efficiency model by appending an extra constraint to avoid producing negative cross-efficiencies under variable returns to scale (VRS) and applied it for Olympic rankings. Wu, Liang, and Yang (2009b) calculated ultimate efficiency scores by weighting cross-efficiency values rather than simply averaging them, where the weights for ultimate efficiency scores were determined by using the Shapley value in cooperative game. Wu, Liang, Yang, and Yan (2009) developed a bargaining game model for the cross-efficiency evaluation. In their bargaining game model, each DMU was seen as an independent player and the bargaining solution between the optimistic efficiency and the cross-efficiency were obtained by using the classical Nash bargaining game model. Wu, Liang, Zha, and Yang (2009) developed a mixed integer programming model for the cross-efficiency evaluation to find the best ranking order for each DMU. Recently, Wang and Chin (2010) proposed a neutral DEA model for the cross-efficiency evaluation and extended it to cross-weight evaluation. The neutral DEA model does not require the DM to make a difficult choice between aggressive and benevolent formulations. It determines the input and output weights only from the perspective of the DMU that is under evaluation, without being aggressive or benevolent to the other DMUs. The cross-efficiencies computed in this way turn out to be neutral and more logical. In this paper, we provide four more neutral DEA models for cross-efficiency evaluation from the perspective of multiple criteria decision analysis (MADA), which are built using ideal DMU and anti-ideal DMU. These new DEA models provide more insights into the cross-efficiency evaluation and enhance the theory and methodology of DEA. The remainder of the paper is organized as follows. Section 2 gives a brief introduction to the cross-efficiency evaluation and its main formulations. The new DEA models for cross-efficiency evaluation are developed in Section 3. Numerical examples are examined in Section 4. The paper concludes in Section 5.

Cross-efficiency evaluation is an important yet practical method for comparing and ranking DMUs. In this paper, we have proposed four new DEA models for cross-efficiency evaluation from the perspective of multiple criteria decision analysis by introducing two virtual DMUs, which are ideal DMU and anti-ideal DMU. These new DEA models determine the input and output weights by minimizing the distance of a DMU from IDMU, or maximizing the distance between the DMU and ADMU, or both of them simultaneously, without the use of the input and output information of the other DMUs as a secondary goal. As a result, the cross-efficiencies determined by these new DEA models are neutral and more logical than those by the aggressive or benevolent formulations. Numerical examples have been tested to show the potential applications of the new DEA models and their effectiveness in discriminating among DMUs. It is hoped that these new DEA models based on IDMU and ADMU can provide more insights into the cross-efficiency evaluation and enhance the theory and methodology of DEA.