Polymatroid Optimization, Submodularity, and Joint Replenishment Games

Authors:
Simai He;Jiawei Zhang;Shuzhong Zhang
Affiliations:
Department of Management Sciences, City University of Hong Kong, Kowloon Tong, Hong Kong;Department of Information, Operations, and Management Sciences, Stern School of Business, New York University, New York, New York 10012;Industrial and Systems Engineering Program, University of Minnesota, Minneapolis, Minnesota 55455
Venue:
Operations Research
Year:
2012

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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].