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

Luis Guardiola, Ana Meca, Justo Puerto

TL;DR

The paper studies cooperative cost-sharing in finite-horizon lot-sizing with heterogeneous costs by formulating setup-inventory ($SI$) games where coalitions share ordering, holding, and backlogging technologies. It proves that $SI$-games are balanced (in fact totally balanced), ensuring a nonempty core and stable coalitions, and introduces Extended Owen points as a tractable parametric family of core allocations. A key contribution is the analysis of consolidated periods, showing that in consolidated $SI$-games the extended Owen point lies in the core and can be realized via a population monotonic allocation scheme. The results provide explicit, scalable core allocations for heterogeneous-cost settings and illuminate how consolidation affects stability, with implications for fair cost-sharing in supply chains and coordination of inventory decisions.

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.

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

TL;DR

The paper studies cooperative cost-sharing in finite-horizon lot-sizing with heterogeneous costs by formulating setup-inventory () games where coalitions share ordering, holding, and backlogging technologies. It proves that -games are balanced (in fact totally balanced), ensuring a nonempty core and stable coalitions, and introduces Extended Owen points as a tractable parametric family of core allocations. A key contribution is the analysis of consolidated periods, showing that in consolidated -games the extended Owen point lies in the core and can be realized via a population monotonic allocation scheme. The results provide explicit, scalable core allocations for heterogeneous-cost settings and illuminate how consolidation affects stability, with implications for fair cost-sharing in supply chains and coordination of inventory decisions.

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.

Paper Structure

This paper contains 7 sections, 9 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Model
  3. Cooperation in lot-sizing with heterogeneous costs
  4. Extended Owen points
  5. Consolidated situations and stability
  6. Concluding Remarks
  7. Acknowledgments

Key Result

Lemma 3.1

Let $\mathcal{B}$ be a balanced collection of coalitions with balancing coefficients $\{\lambda _{S}\}_{S\in \mathcal{B}}$. Assume that for each coalition $S\in \mathcal{B}$, $\pi ^{S}$ is the optimal order plan for the problem $(T,d^{S},k^{N},h^{N},b^{N},p^{N})$. Let $r$ be the smallest positive in

Theorems & Definitions (14)

  • Lemma 3.1
  • Example 3.2
  • Theorem 3.3
  • Proposition 4.1
  • Example 4.2
  • Corollary 4.3
  • Example 4.4
  • Corollary 4.5
  • Definition 5.1
  • Lemma 5.2
  • ...and 4 more