Cooperation in lot-sizing problems with heterogeneous costs: the effect of consolidated periods

TL;DR

This paper studies cooperative lot-sizing among firms with heterogeneous costs, proposing fair and stable cost allocations, and analyzing how consolidated periods influence the stability of these allocations.

Contribution

It introduces a fair cost allocation method for cooperative lot-sizing with heterogeneous costs and examines the impact of consolidated periods on stability.

Findings

Existence of fair, stable cost allocations among firms.

A parametric family of cost allocations with stability conditions.

Consolidated periods affect the stability of cost-sharing arrangements.

Abstract

We consider a cooperative game defined by an economic lot-sizing problem with heterogeneous costs over a finite time horizon, in which each firm faces demand for a single product in each period and coalitions can pool orders. The model of cooperation works as follows: ordering channels and holding and backlogging technologies are shared among the members of the coalitions. This implies that each firm uses the best ordering channel and holding technology provided by the participants in the consortium. That is, they purchase, hold inventory, pay backlogged demand and make orders at the minimum cost of the coalition members. Thus, firms aim at satisfying their demand over the planing horizon with minimal operation cost. Our contribution is to show that there exist fair allocations of the overall operation cost among the firms so that no group of agents profit from leaving the consortium. Then we propose a parametric family of cost allocations and provide sufficient conditions for this to be a stable family against coalitional defections of firms. Finally, we focus on those periods of the time horizon that are consolidated and we analyze their effect on the stability of cost allocations.