Polymatroid optimization, submodularity, and joint replenishment games

Simai He, Jiawei Zhang, Shuzhong Zhang

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

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].

Original languageEnglish (US)
Pages (from-to)128-137
Number of pages10
JournalOperations research
Volume60
Issue number1
DOIs
StatePublished - Jan 1 2012

Keywords

  • Cooperative games
  • Joint replenishment problem
  • Polymatroid optimization
  • Separable concave function

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