Common revenue allocation in DMUs with two stages based on DEA cross-efficiency and cooperative game

In this paper, we examine two-stage production organizations as decision-making units (DMUs) that can collaborate to form alliances. We present a novel approach to transform a grand coalition of n DMUs with a two-stage structure into 2n single-stage sub-DMUs by extending the vectors of the initial input, intermediate product, and final output, thus creating a 2n*2n DEA cross-efficiency (CREE) matrix. By combining cooperative game theory with CREE and utilizing three cooperative game solution concepts, namely, the nucleolus, the least core and the Shapley value, a characteristic function is developed to account for two types of allocation, i.e., direct allocation and secondary allocation. Moreover, the super-additivity and the core non-emptiness properties are explored. It is found that the sum of the revenue allocated to all DMUs will remain constant at each stage regardless of the allocation manner and the cooperative solution concept selected. To illustrate the efficiency and practicality of the proposed approach, both a numerical example and an empirical application are provided.