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. Afterwards, we extend our analysis to games resulting from several approximations of demand including normal approximation. To establish these results, we also derive several interesting properties (e.g. subhomogeneity of degree one) of the performance measures (i.e., expected number of inventory and backorders, and average number of backorders). Finally, we derive an allocation scheme, which can be computed easily, for newsvendor games with normally distributed demand and service level constraint as no-stock out probability