A Dual of a Dynamic Inventory Control Model: The Deterministic and Stochastic Case

Abstract

We consider the single-commodity, N-stage, nonstationary production-inventory control model with deterministic or stochastic demands, and assume for simplicity that the production cost, the holding cost and the shortage cost are linear. It is well-known that the optimal control is determined by optimal inventory levels, which can be calculated by dynamic programming. In this paper we dualize the problem. In the deterministic case, the optimal dual variables are intimately related with the lengths of shortest paths in the underlying network. In the stochastic case they can be considered as a price system related with the condition that all productions must be nonanticipative. The dual problem too can be solved by dynamic programming. Actually, this method is equivalent to solving the original problem in differentiable form. Since all costs are linear we get explicit expressions for the dual solution in terms of probabilities of certain events.