Lost-Sales Problems with Stochastic Lead Times: Convexity Results for Base-Stock Policies

We consider a single-location inventory system with periodic review and stochastic demand. It places replenishment orders to raise the inventory position—that is, inventory on hand plus inventory in transit—to exactly S at the beginning of every period. The lead time associated with each of these orders is random. However, the lead-time process is such that these orders do not cross. Demand that cannot be met with inventory available on hand is lost permanently. We state and prove some sample-path properties of lost sales, inventory on hand at the end of a period, and inventory position at the end of a period as functions of S. The main result is the convexity of the expected discounted sum of holding and lost-sales costs as a function of S. This result justifies the use of common search procedures or linear programming methods to determine optimal base-stock levels for inventory systems with lost sales and stochastic lead times. It should be noted that the class of base-stock policies is suboptimal for such systems, and we are primarily interested in them because of their widespread use.