Two-stage network data envelopment analysis production games

DEA (Data Envelopment Analysis) production games combine DEA theory with cooperation games and assess the benefits to production organizations with single-stage structure. However, in practical production problems, the production organizations are always with network structures. The structure of the production organization not only affects its own benefits, but also relates to the cooperation among organizations. Therefore, it is necessary to study DEA production games with network structures. In this paper, we consider the production organizations with two-stage processes, wherein the organizations are assumed to possess available resources and own technologies. The technology level of each organization is reflected by the observed units based on the network DEA (NDEA) production possibility set. Suppose that the organizations can cooperate through the ways of pooling the initial resources and (or) sharing the technology in each production process. According to the different cooperation styles of each stage in the alliance, seven types of cooperation among organizations are considered. The models of maximizing the revenues of coalitions, namely the NDEA production games, are established corresponding to the seven types, by which the maximal revenue for each coalition can be calculated. We prove that two-stage DEA production games have the super-additive property, and can be expressed as linear programming games. Hence, they are equivalent to the linear production games, and they are totally balanced. Therefore, the proposed cooperative games have a non-empty core, and hence have nucleolus, and the Owen set belongs to the core. In addition, based on the basic conceptions of the nucleolus and the Owen set, the revenue can be allocated among organizations in the alliance. Finally, a numerical example and an empirical application to 17 bank branches of the China Construction Bank in the Anhui Province are presented to illustrate the applicability of the proposed approach, and the relationship between the cooperative manners and the revenue allocation is reflected in analytical results.