Planning Horizons for a Stochastic Lead-Time Inventory Model

The planning-horizon literature has thus far focused on the analysis of deterministic lead-time inventory systems. We show that under certain conditions several planning-horizon theorems hold in the stochastic lead-time case as well. Specifically, demands are assumed non-interchangeable and deterministic; thus production runs are assigned to sets of specific demands and can only be used in the satisfaction of those demands. When holding, backlogging, setup (ordering), and variable production costs are stationary over time, and when the lead-time distribution is invariant over time, it is always optimal to produce for sets of consecutive integral demands. Production dates for specific shipments follow from convexity properties. Building upon these results, we prove several planning-horizon theorems (of the Wagner-Whitin and Zabel varieties). A forward dynamic programming recursion is given. These results are shown to generalize those of the basic dynamic lot-size model. We present a numerical example that illustrates the sensitivity of the optimal policy to changes in lead-time variance.