Optimal Inventories for an (S − 1, S) System with No Backorders

This paper considers the problem of finding optimal stock levels for an inventory system with Poisson demands, arbitrary replacement time distribution, and emergency handling of shortages at a premium cost. The ordering policy is assumed to be of the (S − 1, S) type, i.e., a replacement item is ordered as soon as a unit of stock is used. A criterion for selecting the optimal stock level is formulated by minimizing the expected cost per unit time in steady state. Based on the fact that the probability of shortage for this model is given by the Erlang loss formula, it is possible to derive an approximate stocking level formula, which appears to be accurate over a wide range of parameter values. Comparisons with other related models and determination of achieved service levels are also discussed.