Optimal Joint Replenishment and Transshipment Policies in a Multi-Period Inventory System with Lost Sales

Mismatch between supply and demand when the uncertainty of the demand is high and the supply lead time is relatively long, such as seasonal good markets, can result in high overstocking and understocking costs. In this paper we propose transshipment as a powerful mechanism to mitigate the mismatch between the supply and demand. We consider a finite horizon multi-period inventory system where in each period two retailers have the option to replenish their inventory from a supplier (if there is any supply) or via transshipment from the other retailer. Each retailer observes nonnegative stochastic demand with general distribution in each period and incurs overstocking/understocking costs as well as costs for replenishment and transshipment that may be time dependent. We study a stochastic control problem where the objective is to determine the optimal joint replenishment and transshipment policies so as to minimize the total expected cost over the season. We characterize the structure of the optimal policy and show that unlike the known order-up-to level inventory policy, the optimal ordering policy in each period is determined based on two switching curves. Similarly, the optimal transshipment policy is also identified by two switching curves. These four curves together partition the optimal joint ordering and transshipment polices to eight regions where in each region the optimal policy is an order-up-to-curve policy. We demonstrate that the structure of the optimal policy holds for any known sequence and combination of ordering and transshipment over time.