Cooperation in Spare Parts Systems with Penalty Cost Per Unit Backlogged

We study cooperation between a group of companies, who keep spare parts to maintain their expensive equipments, by pooling their inventory. To establish a stable cooperation, the cost of running the pooled inventory should be allocated between the participating members in a fair fashion. To address this cost allocation problem, we borrow concepts from cooperative game theory and examine the core of the resulting cooperative games. We especially focus on systems with high penalty cost per unit backlogged, which is modelled as a well known (S-1,S) inventory system with backlogging. We study two basic cooperation scenarios; (i) cooperation under optimal inventory investment and (ii) cooperation by inventory pooling, and two extensions of scenario (i) (i.e., cooperation with joint capacity investment and cooperation under service level constraints). For many of the corresponding games, we show that the cores are non-empty by identifying a core allocation, which is easy to compute and, for some, a population monotonic allocation (PMAS). For the games with empty cores, we present examples explaining why the core of these games can be empty. To establish these results, we also derive several interesting properties (e.g. subhomogeneity of degree one) of the main performance measure (i.e., average number of backorders)