A Dynamic Inventory Model with Stochastic Lead Times

A study is made of a dynamic inventory model with stochastic lead times. A probability model is developed for the arrival of outstanding orders in which it is assumed that orders do not cross in time and that the arrival probabilities are independent of the number and size of outstanding orders. With these assumptions, it is shown that the sequential multidimensional minimization problem normally associated with the random lead time model can be reduced to a sequence of one-dimensional minimizations. The minimizations are a function of a variable representing the sum of stock on hand plus all outstanding orders. Optimal ordering policies are characterized under the assumptions of convex expected holding and shortage costs, a linear ordering cost and a fixed setup cost (greater than or equal to zero) paid when the order is placed. These policies are shown to be quite similar to those obtained with deterministic lead times but some differences in the behavior of the single-period critical numbers (when the setup cost is zero) are noted.