Asymptotic Optimality of Order-Up-To Replenishment Policies for Serial Inventory Systems with Lost Sales

We study the optimal policy for a serial inventory system under periodic review when excess demand at the retailer (i.e., the most downstream stage) is lost. We focus on “high service level environments” (i.e., systems where the cost of a lost sale is high compared to inventory holding costs). These environments are typical of products whose margins are high relative to their holding costs. When excess demand is backordered, the optimal policy is a base-stock policy with base-stock levels calculated using the algorithm of Clark and Scarf [1960]. In this paper, we first propose to use this algorithm in lost-sales inventory systems by setting the backorder cost parameter in the algorithm to be equal to the lost-sales cost parameter. We show that the resulting base-stock policy is asymptotically optimal as the lost-sales penalty cost parameter grows. We also show that this result is robust in the following sense: There is a large family of choices for the backorder cost parameter used in the algorithm such that the asymptotic optimality continues to hold. Next, we propose a specific choice for the backorder cost parameter based on a power approximation formula. While the theoretical results guarantee asymptotical optimality as the service level approaches one, our computational investigation of problem instances with moderately high values for the service level (75%-99%) shows that the cost of the best base-stock policy is, on an average, 1.6% higher than the cost of the optimal policy, and the cost of the base-stock policy based on the power approximation is only 0.2% higher than that of the best base-stock policy. Subject Classification: Inventory/Production: Lost Sales, Multi-echelon, Optimal Policy ∗Haskayne School of Business, University of Calgary, Calgary, Canada. E-mail: [email protected]. †Sauder School of Business, The University of British Columbia, Vancouver, BC Canada V6T 1Z4. Email: [email protected]. ‡School of Management, The University of Texas at Dallas, Richardson, TX 75080, USA. E-mail: [email protected].