Polymatroid Optimization, Submodularity, and Joint Replenishment Games

• 成果类型: Article
• 署名作者: He, Simai; Zhang, Jiawei; Zhang, Shuzhong
• 署名单位: City University of Hong Kong; New York University; University of Minnesota System; University of Minnesota Twin Cities
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.1110.1000
• 发表日期: 2012
• 页码: 128-137
• 关键词: resource-allocation; cost allocation; heuristics

摘要

In this paper we consider the problem of maximizing a separable concave function over a polymatroid. More specifically, we study the submodularity of its optimal objective value in the parameters of the objective function. This question is interesting in its own right and is encountered in many applications. But our research has been motivated mainly by a cooperative game associated with the well-known joint replenishment model. By applying our general results on polymatroid optimization, we prove that this cooperative game is submodular (i.e., its characteristic cost function is submodular) if the joint setup cost is a normalized and nondecreasing submodular function. Furthermore, the same result holds true for a more general one-warehouse multiple retailer game, which affirmatively answers an open question posed by Anily and Haviv [Anily, S., M. Haviv. 2007. The cost allocation problem for the first order interaction joint replenishment model. Oper Res. 55(2) 292-302].