Projected Inventory-Level Policies for Lost Sales Inventory Systems: Asymptotic Optimality in Two Regimes

We consider the canonical periodic review lost sales inventory system with positive lead times and stochastic i.i.d. demand under the average cost criterion. We introduce a new policy that places orders such that the expected inventory level at the time of arrival of an order is at a fixed level and call it the projected inventory-level policy. We prove that this policy has a cost rate superior to the equivalent system where excess demand is backordered instead of lost and therefore, is asymptotically optimal as the cost of losing a sale approaches infinity under mild distributional assumptions. We further show that this policy dominates the constant-order policy for any finite lead time and therefore, is asymptotically optimal as the lead time approaches infinity for the case of exponentially distributed demand per period. Numerical results show that this policy also performs superior relative to other policies.