Inventory Centralization in a Newsvendor Setting When Shortage Costs Differ: Priorities and Costs Allocation

Abstract

Risk pooling is a practical managerial tool which can reduce the consequences of the uncertainty involved in a system. In operations management, it is often achieved by consolidating a product with a random demands into one location, which is known to be beneficial. The basic assumption that underlies most previous research is that the cost parameters (overage and underage cost per unit) of all populations are identical, and therefore are equal to those of the centralized system. But in many contexts, underage cost per unit is not independent of the type of customer. This work generalizes the centralized inventory model so that one group of retailers differs from another in the underage cost per unit. In such a system, the proper allocation of the centralized inventory among the groups is a challenge. When the inventory is not allocated optimally, the expected cost of the centralized system may exceed that of the decentralized one. We define a priority rule for allocating the pooled inventory and prove that giving absolute priority to the population whose underage cost is higher (“preferred population”) is optimal. Under this policy, we model the pooled inventory system with priorities and prove its advantage over the un-pooled system. We then prove the advantage of the pooled inventory system with absolute priority from each retailer’s point of view, meaning that the core of the cooperative inventory game is not empty. Thus, with appropriate cost allocation, it is better to join the pool even if you were to become a low-priority customer. Finally, we introduce a pooled inventory model where the inventory is allocated according to each retailer contribution to the system, which is defined as the number of units it produces and deposits in the central warehouse. We use game theory concepts to model this system where each player’s strategy is the number of units it contributes to the system. We prove the existence and uniqueness of the Nash equilibrium and characterize each player’s strategy according to it.