The (r,Q) Inventory Model with Lost Sales : A backlog analogy

The continuous-review (r,Q) inventory model with lost sales and stochastic lead-time is considered. It is assumed that Qr, so that at most a single order may be outstanding at any time. This model is much treated in the literature, beginning with Hadley and Whitin (1963) and there are many heuristics, but no simple exact algorithm. We derive the model for both continuous and discrete (Poisson) demand and stochastic lead-time, and devise a simple iterative algorithm that is certain to locate the global minimum if the lead-time demand distribution function is log-concave. It iterates between a square-root formula for Q and a simple equation for r. The algorithm is applicable to discrete demand when amended by a final comparison between two adjacent policies. We finally show that the model is applicable without the constraint Qr, but then lead-time demand is a function of r and Q, the investigation of which is left to future research.