Cooperation in Spare Parts Systems with Expedited Deliveries

Abstract

We study cooperation between a group of companies, who need spare parts to maintain their expensive equipment, by pooling their spare parts 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 first show that under standard cost structure, the system with expedited deliveries is equivalent to the well studied (S-1,S) inventory system with backlogging. Afterwards, we derive several properties of the performance measures (i.e., expected number of inventory and back orders, and average number of back orders) that play an important role in subsequent study of the cores of the corresponding games. We study three cooperation scenarios: (i) cooperation by inventory pooling, (ii) cooperation under optimal inventory investment, and (iii) cooperation under service level constraint. For many of these games, we show that the cores are non-empty by identifying a PMAS. For the games with empty cores, we present examples. Afterwards, we extend our analysis to games with joint capacity investment and the games resulting from several approximations of the demand including normal approximation. Finally, we derive an easy to calculate allocation scheme for news-vendor games with normally distributed demand.