The ( r, Q) control of a periodic-review inventory system with continuous demand and lost sales

In this paper we consider a periodic review inventory model with lost sales during a stockout and with the constraint that at most one replenishment order may be outstanding at any time. Demands in successive review periods are independent, identically distributed variables from a continuous distribution. The fixed lead time is an integral number of review periods. We explore control policies of the ( r, Q) type – that is a replenishment order of size Q is placed when the inventory position (stock in hand plus stock on order) falls to or below the re-order level r. We use asymptotic renewal theory results to estimate the `undershoot’ of the re-order level r and also to estimate the cycle stockholding cost (which turns out to take a relatively simple form). Based on these approximations we set out a policy improvement solution methodology and illustrate this with some numerical examples for which demand is normally distributed. These numerical examples suggest that a relatively simple approach, based on the economic order quantity, can provide results which are very close to optimal.