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

TL;DR

This paper introduces a novel method combining DEA cross-efficiency and cooperative game theory to allocate revenue among two-stage DMUs, ensuring consistent total revenue distribution across different cooperative solutions.

Contribution

It develops a new approach transforming two-stage DMUs into single-stage subunits and integrates cooperative game concepts for revenue allocation.

Findings

Total revenue remains constant regardless of allocation method.

The approach ensures fair and stable revenue distribution.

Numerical and empirical examples validate the method's effectiveness.

Abstract

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.