On the Optimality of (σ, S) Policies

Abstract

A standard periodic review, stochastic, dynamic multiproduct inventory model is considered in which the ordering cost consists of linear portions for each product and a setup cost. This setup cost is incurred if an order is placed for any number of products. There is no separate setup cost incurred for each product ordered. This basic problem has been studied by Johnson (Johnson, E. L. 1967. Optimality and computation of (σ, S) policies in the multi item-infinite horizon inventory problem. Management Sci. 13 475–491.) and Wheeler (Wheeler, A. 1968. Multiproduct inventory models with set-up. Technical report, Stanford University, Stanford.) among others. We extend their results by providing general conditions for the existence of an optimal policy and by further characterizing the optimal policy. In particular, we show that there exists an optimal (σ, S) policy. Such a policy does not order when the initial stock level is in σ and orders up to the vector level S otherwise (provided that such an order is feasible). Furthermore, the set σ is an upper layer (equivalently, an increasing set) with respect to a certain partial ordering. Such a policy reduces to an (s, S) policy in the single product case, in which case our conditions are very similar to those of Veinott (Veinott, A. F., Jr. 1966. On the optimality of (s, S) inventory policies. New conditions and a new proof. SIAM J. Appl. Math. 14 1067–1083.) and amount to a special case of the model of Schäl (Schäl, M. 1976. On the optimality of (s, S) policies in dynamic inventory models with finite horizon. SIAM J. Appl. Math. 30 518–537.). Our analysis is based in part on a generalization of quasi-convex functions (in the terminology of Porteus [Porteus, E. L. 1971. On the optimality of generalized (s, S) policies. Management Sci. 17 411–426.]). Finite and infinite horizon results are given.