Production-Inventory games: a new class of totally balanced combinatorial optimization games

TL;DR

The paper addresses cooperative cost-sharing in dynamic production-inventory settings with discrete demand and backlogging by introducing production-inventory games (PI-games). It encodes each coalition's optimal cost as $c(S)=val(LPI(S))$ and leverages LP duality to prove that PI-games are totally balanced, with the Owen set collapsing to a singleton — the Owen point — which can be realized through a population-monotonic allocation scheme. The authors derive an explicit formula for the Owen point and show its relationship to classical allocation rules such as the Shapley value and nucleolus, while also providing a constructive PMAS to implement it. This yields stable, computation-friendly cost-sharing allocations for multi-agent production-inventory cooperation and clarifies structural properties of the core in this domain.

Abstract

In this paper we introduce a new class of cooperative games that arise from production-inventory problems. Several agents have to cover their demand over a finite time horizon and shortages are allowed. Each agent has its own unit production, inventory-holding and backlogging cost. Cooperation among agents is given by sharing production processes and warehouse facilities: agents in a coalition produce with \ the cheapest production cost and store with the cheapest inventory cost. We prove that the resulting cooperative game is totally balanced and the Owen set reduces to a singleton: the Owen point. Based on this type of allocation we find a population monotonic allocation scheme for this class of games. Finally, we point out the relationship of the Owen point with other well-known allocation rules such as the nucleolus and the Shapley value.

Theorems & Definitions (11)

• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Theorem 4.1
• Example 4.2
• Proposition 4.3
• Example 4.4
• Theorem 5.1
• Theorem 5.2
• Definition 5.3