Strategies for the Probabilistic Lot-Sizing Problem with Service-Level Constraints

We formulate a stochastic-demand version of the single-stage lot-sizing problem with time-varying demand, incorporating a service-level constraint on the probability of a stockout. Three strategies are studied. The “static uncertainty” strategy, in which lot-sizing decisions for every period must be made at the beginning of period 1, is shown to yield an equivalent deterministic problem with time-varying demands for which optimal or good heuristic solutions exist. The procedure by which this equivalent problem is obtained is computationally simple. The “dynamic uncertainty” strategy allows subsequent lot sizes to be chosen on the basis of demands that have become known at a later point in time. The “static-dynamic” uncertainty approach combines features of the above two strategies and yields an equivalent linear program for any given order schedule. Relationships are suggested between these strategies and various aspects of rolling horizon production planning. Arguments are given that in such an environment, the static uncertainty strategy is the most straightforward to modify and “roll along” as new demands become known. Good results are found when this procedure is applied to some 300-period stochastic-demand problems using rolling horizons of between 2 and 12 periods in length.