Almost 40 years ago, H. Scarf established the optimal (s,S) policy structure for the periodic review inventory problem with fixed ordering costs and no capacity constraint. Since then, the capacitated problem has resisted characterization. In the present paper we partially bridge this gap; using a generalization of Scarf's K-convexity we show that the optimal capacitated policy has an (s,S)-like structure. To do so we divide the parameter space into four regions: In two of these regions the optimal policy is completely specified, while in the other two, it is partially specified. We complement these findings with a computational study. This study suggests that a still simpler optimal policy structure exists.
