Abstract

Abstract We consider the optimal control of a production-inventory system with a single product where items are produced one unit at a time. Upon arrival, customer orders can be fulfilled from existing inventory, if there is any, backordered, or rejected. Although customers are willing to wait if their orders are backlogged, they are not infinitely patient. In particular, customers cancel their orders if their waiting time in backlog exceeds a certain patience time. This patience time is random and varies from one customer to another. At each decision epoch, we must determine whether or not to produce an item and, should an order arise and there is no inventory on-hand, whether to reject the order or backorder it. Rejecting an order incurs a rejection cost. If an order is backordered but the order ends up being cancelled by the customer, we incur a cancellation cost. We can mitigate both costs by holding inventory but we incur an inventory holding cost. We formulate the problem as a Markov decision process. We show that the optimal policy can be described by two thresholds: a production base-stock level that determines when production takes place and an admission threshold that determines when orders should be accepted. We also characterize analytically the sensitivity of these thresholds to operating parameters, including the demand and production rates and the various cost parameters. Using the structure of the optimal policy, we formulate the dynamics of the corresponding production-inventory system as a Markov chain, which allows us to compute efficiently the performance of the system for any choice of base-stock level and admission threshold. In a numerical study, we compare the performance of the optimal policy against several other policies and show that those that ignore customer impatience can perform poorly.