An inventory model with Poisson demands and emergency orders

Abstract

In this paper we explicitly consider the opportunity to use an emergency supply mode to hedge against demand uncertainty when replenishing a single-item inventory. Normal orders with a relatively long and constant lead time are controlled by a standard ( R, Q) policy. These orders can only be issued when no other orders are outstanding. When a normal order is outstanding, emergency orders are controlled by a reorder point r(j) and an order-up-to level u(j), where j is a measure of the time remaining until the normal order is delivered. The emergency orders have a short lead time and they may also have different ordering costs compared to normal orders. Demands not satisfied immediately from inventory are backordered. We formulate a long-run average cost model that includes ordering costs for the two types of orders, backordering costs, and holding costs. Backordering costs are considered both as a unit-based cost and as a cost rate. A tailor-made policy-iteration algorithm is designed and utilized to minimize the inventory cost rate with state-dependent emergency orders. For comparisons, an algorithm for finding a best simple emergency-order policy is also implemented. Simulation is used to check the validity of our model. Numerical results are presented for a set of parameter variations and compared to results without emergency orders as well as to results from an earlier model in the literature. Our results show that substantial cost savings might be obtained by using emergency orders, especially when backordering costs are high. However, the marginal gains are fairly small if state-dependent emergency orders are used instead of a simple emergency-order policy.