Unitary Owen points in cooperative lot-sizing models with backlogging

TL;DR

This paper advances cooperative lot-sizing with backlogging by introducing unitary Owen points, a dual-based pricing mechanism that extends the Owen point to SI-games with heterogeneous costs. It proves that SI-games are totally balanced via Pochet–Wolsey duality, and provides necessary and sufficient conditions (involving weights $eta_t^S$) under which unitary Owen points lie in the core, supported by extensive simulations. A fundamental link to production-inventory (PI) games is established through surplus games, showing that SI-core allocations correspond to Owen points of PI-situations derived from surplus-core allocations, with nucleolus correspondence. The approach yields a practical framework for fair cost allocations and offers a new avenue to leverage well-understood PI-game results to analyze SI-games.

Abstract

Cooperative lot-sizing models with backlogging and heterogeneous costs are studied in Guardiola et al. (2020). In this model several firms participate in a consortium aiming at satisfying their demand over the planing horizon with minimal operation cost. Each firm uses the best ordering channel and holding technology provided by the participants in the consortium. The authors show that there are always fair allocations of the overall operation cost among the firms so that no group of agents profit from leaving the consortium. This paper revisits those cooperative lot-sizing models and presents a new family of cost allocations, the unitary Owen points. This family is an extension of the Owen set which enjoys very good properties in production-inventory proble, introduced by Guardiola et al. (2008). Necessary and sufficient conditions are provided for the unitary Owen points to be fair allocations. In addition, we provide empirical evidence, throughout simulation, showing that the above condition is fulfilled in most cases. Additionally, a relationship between lot-sizing games and a certain family of production-inventory games, through Owen's points of the latter, is described. This interesting relationship enables to easily construct a variety of fair allocations for cooperative lot-sizing models.

Theorems & Definitions (15)

• Theorem 2.1
• Definition 3.1
• Definition 3.2
• Example 3.3
• Proposition 3.4
• Theorem 3.5
• Example 3.6
• Example 3.7
• Definition 4.1
• Definition 4.2