An optimistic-pessimistic game cross-efficiency method based on a Gibbs entropy model for ranking decision making units

The game cross-efficiency method, a commonly utilized approach for ranking decision-making units in tie-breaking scenarios, is based on secondary goals. However, in certain data envelopment analysis ranking problems, the classical game cross-efficiency method may fail to differentiate all decision-making units effectively. To address this limitation, it is prudent to explore the development of a new method that can enhance the ranking performance of the classical game cross-efficiency approach. In this study, we propose a novel Gibbs entropy linear programming model that integrates both optimistic and pessimistic perspectives of the classical game cross-efficiency method for data envelopment analysis ranking problems. To validate the reliability and utility of our proposed method, we present three examples: the six nursing homes problem, numerical example 2, and an application involving twenty Thai provinces with cash crop data. The reliability of the proposed method is assessed using Spearman’s correlation coefficient (rs) on the numerical examples. The results demonstrate that the rs values for both the proposed method and the classical game crossefficiency method, specifically for the six nursing homes problem, numerical example 2, and the application involving twenty Thai provinces, are determined to be rs=0.998, 0.998, and 0.986 respectively.