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, E(s, 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 ≥ 0 given, E(s, Δ) turns out to be convex in s, so that the corresponding optimal reorder point, s₁(Δ), can be characterized easily. However, E(s₁(Δ), Δ) is not in general unimodal on Δ ≥ 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′(s₁(Δ), Δ) = 0, Δ ≥ 0, is both necessary and sufficient for a global minimum (E(s₁(Δ), Δ) is pseudoconvex on Δ ≥ 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.