Stability in supply chain networks: An approach by discrete convex analysis

Ostrovsky generalized the stable marriage model of Gale and Shapley to a model on an acyclic directed graph, and showed the existence of a chain stable allocation under the conditions called same- side substitutability and cross-side complementarity. In this paper, we extend Ostrovsky's model and the concepts of same-side substitutability and cross-side complementarity by using value functions which are defined on integral vectors and allow indifference. We give a characterization of chain stability under the extended versions of same-side substitutability and cross-side complementarity, and develop an algorithm which always finds a chain stable allocation. We also verify that twisted M ♮ -concave functions, which are variants of M ♮ -concave functions central to discrete convex analysis, satisfy these extended conditions. For twisted M ♮ -concave value functions of the agents, we analyze the time-complexity of our algorithm.