No abstract

No abstract

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.

We derive time-dependent and stationary distributions of inventory position and on-hand inventory under the (s, S) policy for a continuous review inventory system with general inter-arrival and demand distributions and a constant lead time. Some results are also obtained for the characterization of the optimal policies.

We consider a product that is sold under one of the two common warranty policies. Under the “pro rata” warranty policy, a failed item is replaced by a new one or is repaired at a cost prorated to the age of the failed item. Under the “free replacement” warranty policy, replacements or repairs during the warranty period are provided by the supplier free of charge to the buyer. Assuming that successive failure times form a renewal process, we derive moments of the total replacement cost for both policies during the product life cycle (0, t]. We also provide an extension to time-varying failure time distributions in the case of the pro rata warranty policy.