Optimality of (s,S)(s,S) policies for a stochastic inventory model with proportional and lump-sum shortage costs

This paper is concerned with the optimality of an (s,S)(s,S) policy for a single-item infinite-horizon inventory model when the penalty cost is made up of two parts: a lump-sum cost independent of the amount of the shortage and a variable cost proportional to the amount of the shortage. Using a Quasi-Variational Inequality (QVI) approach, an (s,S)(s,S) policy is shown to be optimal under some mild technical conditions.