Duality Approaches to Economic Lot-Sizing Games

Sharing common production, resources and services to reduce cost are important for not for profit operations due to limited and mission-oriented budget and effective cost allocation mechanisms are essential for encouraging effective collaborations. In this paper, we illustrate how rigorous methodologies can be developed to derive effective cost allocations to facilitate sustainable collaborations in not for profit operations by modeling the cost allocation problem arising from an economic lot-sizing (ELS) setting as a cooperative game. Specifically, we consider the economic lot-sizing (ELS) game with general concave ordering cost. In this cooperative game, multiple retailers form a coalition by placing joint orders to a single supplier in order to reduce ordering cost. When both the inventory holding cost and backlogging cost are linear functions, it can be shown that the core of this game is non-empty. The main contribution of this paper is to show that a core allocation can be computed in polynomial time under the assumption that all retailers have the same cost parameters. Our approach is based on linear programming (LP) duality. More specifically, we study an integer programming formulation for the ELS problem and show that its LP relaxation admits zero integrality gap, which makes it possible to analyze the ELS game by using LP duality. We show that there exists an optimal dual solution that defines an allocation in the core.