The stochastic dynamic production/inventory lot-sizing problem with service-level constraints

This paper addresses the multi-period single-item inventory lot-sizing problem with stochastic demands under the “static–dynamic uncertainty” strategy of Bookbinder and Tan. In the static–dynamic uncertainty strategy, the replenishment periods are fixed at the beginning of the planning horizon, but the actual orders are determined only at those replenishment periods and will depend upon the demand that is realised. Their solution heuristic was a two-stage process of firstly fixing the replenishment periods and then secondly determining what adjustments should be made to the planned orders as demand was realised. We present a mixed integer programming formulation that determines both in a single step giving the optimal solution for the “static–dynamic uncertainty” strategy. The total expected inventory holding, ordering and direct item costs during the planning horizon are minimised under the constraint that the probability that the closing inventory in each time period will not be negative is set to at least a certain value. This formulation includes the effect of a unit variable purchase/production cost, which was excluded by the two-stage Bookbinder–Tan heuristic. An evaluation of the accuracy of the heuristic against the optimal solution for the case of a zero unit purchase/production cost is made for a wide variety of demand patterns, coefficients of demand variability and relative holding cost to ordering cost ratios. The practical constraint of non-negative orders and the existence of the unit variable cost mean that the replenishment cycles cannot be treated independently and so the problem cannot be solved as a stochastic form of the Wagner–Whitin problem, applying the shortest route algorithm.