Optimal inventory and timing decisions for emergency shipments

Emergency shipments provide a powerful mechanism to alleviate the risk of imminent stock-outs, and hence, can result in substantial benefits. Customer satisfaction and increased service level are immediate consequences of utilizing emergency shipments. In this article, we study a periodic review inventory problem of a retailer, which uses a base-stock policy to replenish its inventory from a supplier with infinite capacity. The retailer can also place emergency shipment orders in each period. The goal is to determine the optimal base-stock level, the optimal period length as well as the optimal timing and size of the emergency shipment orders such that the total expected operating cost composed of expected replenishment, emergency shipment, holding and backordering costs is minimized. Different from classic periodic review inventory models, we consider time-dependent holding and shortage costs. The timing decision is of particular interest since it provides valuable information to the supplier. Our analytical results reveal the complex behavior of the cost function. We prove the existence of the optimal solution and develop a heuristic algorithm to find a near-optimal solution. We compare our policy with simpler policies and with a policy that relies on the inventory level to decide whether to trigger emergency orders. In our analysis we study the case with a fixed period length as well. Our results indicate that the benefit of using the proposed policy depends on emergency shipment costs, backordering costs and demand variability. We observe that when the period length is a decision variable, in the case of small emergency shipment costs, the retailer needs to focus on the timing of emergency shipments, and when the period length is fixed and the demand variation is high, the size of the emergency shipments should be the focus. Our results indicate that using an inventory level-dependent emergency shipment policy is not beneficial especially when the period length is a decision variable and the emergency order lead time is long.