Optimal (r,Q) inventory policies with Poisson demands and lost sales: Discounted and undiscounted cases

We consider a continuous review (r, Q) inventory system with Poisson demands and at most one order outstanding. The replenishment lead time is either constant or exponentially distributed. Demands not covered immediately from inventory are lost. Costs include a linear order cost with a fixed cost per order, and a fixed cost per unit lost sale. As regards inventory holding costs, the cost of capital often constitutes a major part. This paper focuses on these interest-related holding costs. In the undiscounted case, holding costs are linear and inventory performance is measured by the long-run average total cost incurred per unit time. In the discounted case, the performance measure is the expected present value of the ordering and lost sales costs. The cost associated with capital tied up in inventory is accounted for by an appropriate discount rate. We formulate an exact model and design a policy-iteration algorithm for the discounted case. Results on the form of an optimal replenishment policy are derived and the model is compared to a previously derived model for the undiscounted case. Numerical experiments are used to evaluate the difference between the optimal solutions with and without discounting. The effect of a stochastic lead time on this difference is also considered by comparing solutions with constant and exponential lead times. In general, the differences seem to be fairly small but exceptional cases exist when the service level is low.