The cross-efficiency evaluation model with judgmental information: Cooperative game and optimal coordination perspective

In the actual efficiency evaluation process, it frequently occurs that the quantitative input and output data of each decision-making unit is difficult to obtain due to the limitation in data measurement and the absence of prior information. Instead, the form of judgment information can be selected to express the input-output state of the evaluated decision-making units compared with the reference level. Traditional cross-efficiency evaluation methods do not involve this case and cannot be well used in this situation. Based on the above problems, a multiplicative cross-efficiency evaluation model with judgmental information is constructed, and the optimal adjustment model is provided by introducing deviation variables to deal with the inconsistent issue of data features. In addition, for the case of multiple optimal solutions, the maximizing and minimizing secondary objective models of the multiplicative cross-efficiency are established respectively to reasonably identify the corresponding optimal weights. Simultaneously, in order to effectively depict the interaction between decision making units, the characteristic functions of JM-cross efficiency cooperative game and the multiplicative Shapley value are established based on benevolent and aggressive cross-efficiency matrices. Moreover, in order to avoid the extreme ranking results that are oriented by high impact decision-making units, a series of optimization coordination models are developed, and an optimal efficiency ranking vector generation method with strong applicability and decision flexibility is proposed. The optimal efficiency ranking vector not only creates a precise match with the given Shapley index of the multiplicative cross-efficiency game, but also bounds between the minimum and the maximum average efficiency. Finally, the validity and rationality of the discussed models are illustrated by a concrete example.