On the spectrum of achievable targets in cross-efficiency evaluation and the associated secondary goal models

The cross-efficiency (CE) evaluation method was introduced to improve the discriminatory power of DEA and eliminate unrealistic DEA weighting schemes. One important issue in CE evaluation is the non-uniqueness of the CE scores. Several secondary goal models based on different targets for cross-efficiencies (CEs) of each DMU with respect to other DMUs were proposed to address this issue. However, the suggested targets, fixed value 1 and the CCR efficiency score, are not achievable for all CEs. Moreover, the proposed secondary goal models based on these targets are sensitive to outlier DMUs, and may generate unrealistic CE scores. In this manuscript, we prove that the spectrum of achievable targets of CEs can be obtained using the most resonated appreciative (MRA) model, proposed by Oral et al. (2015), and the least resonated appreciative (LRA) model that we introduce. To this end, we propose a general secondary goal model using multi-objective programming and show that the CEs generated using MRA (LRA) model for each DMU is greater (less) than the corresponding CEs obtained by any other model that can be derived from the proposed benevolent (aggressive) general model. Using this achievable spectrum, we then propose several benevolent, aggressive and neutral secondary goal, and a weighted average CE evaluation model. Using two real examples, we compare the results of the proposed CE methods with those obtained from several other CE methods. Our data analyses indicate that our proposed methods are less sensitive to outliers, less biased towards 1, has better discriminatory power and can identify pseudo-efficient DMUs.