Unitary Owen Points in Cooperative Lot-Sizing Models with Backlogging

Abstract

This paper analyzes cost sharing in uncapacitated lot-sizing models with backlogging and heterogeneous costs. It is assumed that several firms participate in a consortium aiming at satisfying their demand over the planning horizon with minimal operating cost. Each individual firm has its own ordering channel and holding technology, but cooperation with other firms consists in sharing that information. Therefore, the firms that cooperate can use the best ordering channels and holding technology among members of the consortium. This mode of cooperation is stable. in that allocations of the overall operating cost exist, so that no group of agents benefit from leaving the consortium. Our contribution in the current paper is to present a new family of cost sharing allocations with good properties for enforcing cooperation: the unitary Owen points. Necessary and sufficient conditions are provided for the unitary Owen points to belong to the core of the cooperative game. In addition, we provide empirical evidence, through simulation, showing that, in randomly-generated situations, the above condition is fulfilled in 99% of the 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 easily constructing a variety of coalitionally stable allocations for cooperative lot-sizing models.

The effect of consolidated periods in heterogeneous lot-sizing games

“…The main result in that paper is that the cooperative games induced by ELS-situations enjoy a nonempty core. Subsequently, (Guardiola et al 2008(Guardiola et al , 2009(Guardiola et al , 2021b present a new class of totally balanced combinatorial optimization games: production-inventory games (henceforth, PI-games). PI-games bring the essentials of inventory and combinatorial optimization games.…”

“…Notice that if all setup costs are zero, the problem we are dealing with is a PIproblem (see Guardiola et al 2009Guardiola et al , 2021b. Moreover, although the model assumes that companies produce their demand, we can interchangeably consider the case where demand is satisfied either by producing or purchasing.…”