Cooperation in Spare Parts Systems with Penalty Cost per Unit Backlogged

Inventory pooling is well known to reduce inventory costs and improve customer service. Although potential benefits are well understood, one important prerequisite to form a stable coalition between independent companies, who like to cooperate to benefit from inventory pooling, is agreeing on a cost allocation such that no group of companies would better off leaving the coalition. Existence of such stable allocations would depend on the scenarios that these companies are cooperating in. In this paper, we study cooperation between a group of companies, who keep spare parts to maintain their expensive equipment, by pooling their inventory in several practically motivated scenarios. To address the cost allocation problem, we borrow concepts from cooperative game theory and examine the core of the resulting cooperative games in each scenario. We especially focus on systems with high penalty cost per unit backlogged, which is modelled as a well known inventory system with backlogging. We study cooperation under four scenarios, namely cooperation under penalty cost, cooperation with joint capacity investment, cooperation under service level constraints and cooperation by inventory pooling. The first three scenarios consider cooperation by investing into a central pool of inventory optimally, whereas in the last scenario, companies cooperate by pooling their available inventory only. In most of the scenarios, we show the existence of stable cost allocations by identifying a core allocation, which is easy to compute and, in some, a population monotonic allocation (PMAS). In other scenarios, we present examples explaining why finding a stable allocation might not be possible by showing the core of the resulting games can be empty. To establish these results, we also derive several interesting properties ( subhomogeneity of degree one) of the main performance measure ( , average number of backorders).