The Optimality of (s, S) Policies for a Stochastic Inventory Problem with Proportional and Lump-Sum Penalty Cost

In this paper we consider a single product multi-period inventory problem for which the penalty cost consists of two parts, a lump-sum portion which is independent of the size of the shortage and a portion which is linear in the size of the shortage. We show that for all nonincreasing demand density functions, the expected total cost function is K-convex and hence, there is an optimal policy for the n-period problem that is (s, S).