Dynamic lot sizing with random demand and non-stationary costs

We describe a formulation of the dynamic lot sizing problem when demand is random and the costs are non-stationary. Assuming that the distribution of the cumulative demand is known for each period and that all unsatisfied demand is backordered, the problem can be modeled as a mixed integer nonlinear program. An optimal solution algorithm is developed that resembles the Wagner-Whitin algorithm for the deterministic problem but with some additional feasibility constraints. We derive two important properties of the optimal solution. The first increases the computational efficiency of the solution algorithm. The second property demonstrates that the lot sizes used in the rolling-horizon implementation of this algorithm are bounded below by the optimal lot sizes for a stochastic dynamic programming formulation.