Deriving near-optimality conditions using approximation procedures for an inventory control system with lost sales

Inventory theory often assumes that demands during out-of-stock periods are backordered, although lost demands are arguably more realistic, especially in retail settings. This is, to a large extent, due to the fact that lost-sales systems are harder to analyse, especially if multiple orders can be outstanding. This paper considers the classic single-item, single-location, continuous review (R, Q) inventory model with lost sales under a cost minimization objective, where reorder level R is optimized given a fixed order quantity Q. Using approximation procedures, we derive a near-optimality condition that can be used to determine the reorder level. This can be seen as a modification and generalisation of the near-optimality condition presented by Hadley and Whitin (1963, Eq. (4-22)). A numerical analysis for a considerable set of instances shows that the new near-optimality condition outperforms that of Hadley and Whitin (1963), reducing the cost gap vs. the optimal solution from 0.12% to 0.01% under the assumptions of smooth demand (i.e. continuous or unit-sized demand) and at most a single order outstanding, and from 0.32% to 0.08% over all considered instances, including ones with multiple orders outstanding and lumpy demand. We also compare to settings where the cost expression uses the loss functions by Rosling (2002). This is optimal under the mentioned assumptions and has a cost gap of 0.24% over all considered instances. Since the conditions and the complex cost expression by Rosling (2002) are not easy to implement in real life, we also test simpler approximations based on Hadley and Whitin's (1963) and the New near-optimality condition that take the quantile of a Normal distribution. These have cost gaps to the optimal solution of 0.71% and 0.32%, respectively.