On the Objective Function Behavior in (s, S) Inventory Models

The most common measure of effectiveness used in determining the optimal s, S inventory policies is the total cost function per unit time, Es, Δ, Δ = S-s. In stationary analysis, this function is constructed through the limiting distribution of on-hand inventory, and it involves some renewal-theoretic elements. For Δ â‰¥ 0 given, Es, Δ turns out to be convex in s, so that the corresponding optimal reorder point, s1Δ, can be characterized easily. However, Es1Δ, Δ is not in general unimodal on Δ â‰¥ 0. This requires the use of complicated search routines in computations, as there is no guarantee that a local minimum is global. Both for periodic and continuous review systems with constant lead times, full backlogging and linear holding and shortage costs, we prove in this paper that E's1Δ, Δ = 0, Δ â‰¥ 0, is both necessary and sufficient for a global minimum Es1Δ, Δ is pseudoconvex on Δ â‰¥ 0 if the underlying renewal function is concave. The optimal stationary policy can then be computed efficiently by a one-dimensional search routine. The renewal function in question is that of the renewal process of periodic demands in the periodic review model and of demand.sizes in the continuous review model.