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Cross-efficiency evaluation is an effective way of ranking decision making units (DMUs) in data envelopment analysis (DEA). Existing approaches for cross-efficiency evaluation are mainly focused on the calculation of cross-efficiency matrix, but pay little attention to the aggregation of the efficiencies in the cross-efficiency matrix. The most widely used approach is to aggregate the efficiencies in each row or column in the cross-efficiency matrix with equal weights into an average cross-efficiency score for each DMU and view it as the overall performance measurement of the DMU. This paper focuses on the aggregation process of the efficiencies in the cross-efficiency matrix and proposes the use of ordered weighted averaging (OWA) operator weights for cross-efficiency aggregation. The use of OWA operator weights for cross-efficiency aggregation allows the decision maker (DM)’s optimism level towards the best relative efficiencies, characterized by an orness degree, to be taken into consideration in the final overall efficiency assessment and particularly in the selection of the best DMU.

Cross-efficiency evaluation, proposed by Sexton at al. [1], is an effective way of ranking decision making units (DMUs). It allows the overall efficiencies of the DMUs to be evaluated through both self- and peer-evaluations. The self-evaluation allows the efficiencies of the DMUs to be evaluated with the most favorable weights so that each of them can achieve its best possible relative efficiency, whereas the peer-evaluation requires the efficiency of each DMU to be evaluated with the weights determined by the other DMUs. The self-evaluated efficiency and peer-evaluated efficiencies of each DMU are then averaged as the overall efficiency of the DMU. Since its remarkable discrimination power, the cross-efficiency evaluation has found significant number of applications in a wide variety of areas such as the selections of R&D projects [2] and [3], flexible manufacturing systems (FMSs) [4], industrial robots [5] and computer numerical control (CNC) machines [6], preference voting and project ranking [7] and [8], cellular layouts evaluation [9], overall efficient electricity distribution sectors identification [10], the best labor assignment determination in cellular manufacturing system (CMS) [11], economic-environmental performance assessment [12] and [13], Olympic ranking and benchmarking [14], [15] and [16], the efficiency evaluation of information sharing in supply chains [17], etc. Besides a large number of applications, theoretical research has also been extensively conducted on the cross-efficiency evaluation. For example, Doyle and Green [18] and [19] examined in detail the idea of cross-efficiency, both mathematically and intuitively, and presented mathematical formulations of intuitive meanings for possible implementations of aggressive and benevolent cross-efficiencies. Liang et al. [20] proposed the concept of game cross-efficiency and developed a game cross-efficiency model which treats each DMU as a player that seeks to maximize its own efficiency under the condition that the cross-efficiency of each of the other DMUs does not deteriorate. The game cross-efficiency model was later extended to variable returns to scale (VRS) by Wu et al. [16]. Interested readers may refer to Anderson et al. [21] for the fixed weighting nature of the cross-efficiency evaluation in the case of single input and multiple outputs, Sun and Lu [22] for cross-efficiency profiling (CEP) analysis, Bao et al. [23] for an alternative interpretation to the cross-efficiency evaluation from the viewpoint of slack analysis in DEA, Liang et al. [24] for alternative secondary goals for the cross-efficiency evaluation, Wang and Chin [25] and [26] for more alternative DEA models for cross-efficiency and cross-weight evaluations, Wu [27] for the construction of fuzzy preference relation using cross-efficiencies, Wu et al. [28] for considering the performance rankings of the DMUs as a secondary goal, and Ramón et al. [29] for the choice of weights profiles in cross-efficiency evaluations. Existing researches on the cross-efficiency evaluation are mainly focused on either its applications or the calculation of cross-efficiency matrix. Little attention has been paid to the aggregation process of cross-efficiencies. The most extensively used approach is to aggregate cross-efficiencies with equal weights. Our literature review reveals that only Wu et al. [30] and [31] determined ultimate cross-efficiency by weighting n cross-efficiency scores rather than simply averaging them. The weights they utilized for aggregation were determined in terms of the nucleolus solution and the Shapley value in cooperative game, respectively. In our view, the use of equal weights for cross-efficiency aggregation has a significant drawback. That is self-evaluated efficiencies are much less weighted than peer-evaluated efficiencies. This is because each DMU has only one self-evaluated efficiency value, but multiple peer-evaluated efficiency values. When they are simply averaged together, the weight assigned to the self-evaluated efficiency is only 1/n if there are n DMUs to be evaluated, whereas the remaining weights (n−1)/n are all given to those peer-evaluated efficiencies. Quiet obviously, self-evaluated efficiencies fail to play a sufficient role in the final overall assessment and ranking. More importantly, the use of equal weights for aggregation has no way to take into consideration the decision maker (DM)’s subjective preferences on the best relative efficiencies in the final overall assessment and ranking. Besides, the assignment of a fixed but different weight to each DMU as done in Wu et al. [30] and [31] suffers from another significant drawback. That is the self-evaluated efficiency of each DMU plays a distinct role in the final overall assessment and ranking due to the fact that the self-evaluated efficiencies lie on the leading diagonal of the cross-efficiency matrix and the weights assigned to them differ from one DMU to another. To overcome these drawbacks, we propose the use of ordered weighted averaging (OWA) operator weights for aggregating cross-efficiencies. The use of OWA operator weights for the cross-efficiency aggregation allows the weights to be reasonably allocated between self- and peer-evaluated efficiencies in terms of the DM’s optimism level, characterized by an orness degree. By adjusting the DM’s optimism level, self-evaluated efficiencies can play a desirable role in the final overall efficiency assessment or ranking of the DMUs. The remainder of the paper is organized as follows: Section 2 briefly reviews OWA operators and their weight determination methods. Section 3 introduces the cross-efficiency evaluation in DEA. Section 4 proposes the use of OWA operator weights for cross-efficiency aggregation and discusses various possible preferences of the DM. Section 5 provides an illustrative example to demonstrate the potential applications of OWA operator weights in cross-efficiency aggregation. Section 6 concludes the paper.

As a pragmatic and advanced management decision tool, DEA has been widely used for performance evaluation [38] and [39], productivity analysis [40], [41] and [42], resource allocation [43], production planning [44], and so on. The cross-efficiency evaluation is an important method for ranking DMUs in DEA. Existing approaches for DEA cross-efficiency evaluation focus mainly on the calculation of cross-efficiency matrix while pay little attention to the aggregation of cross-efficiencies. Contrary to these approaches, we have focused our attention in this paper on the aggregation process of cross-efficiencies and proposed the use of OWA operator weights for cross-efficiency aggregation, which allows the DM’s optimism level towards the best relative efficiencies or subjective preferences on different efficiencies to be taken into consideration in the final overall efficiency assessment. The traditional equal aggregation in the cross-efficiency evaluation turns out to be a special case of the use of OWA operator weights for cross-efficiency aggregation. We have discussed various subjective preferences of the DM and derived corresponding weight formulations. We have also provided an illustrative example about the selection of industrial robots to demonstrate the potential applications of OWA operator weights in cross-efficiency aggregation. Our demonstration reveals that the best selection of robots (DMUs) is not fixed. The selection varies with the DM’s optimism level. Therefore, it is very essential that the DM’s subjective preference or optimism level should be taken into consideration in the final overall efficiency assessment and particularly in the final selection of the best DMU. Finally, we point out that the use of benevolent or aggressive cross-efficiencies for aggregation may result in different conclusions since they are usually not the same. To avoid this problem from happening, we suggest the use of neutral cross-efficiencies suggested by Wang and Chin [26] for aggregation. This can avoid making a difficult choice between aggressive and benevolent cross-efficiencies. Another possible way of handling this problem is the combinational use of benevolent and aggressive cross-efficiencies and aggregating them together after re-ordering them in descending order. More approaches for dealing with this non-uniqueness problem can be further investigated in the future.