The Infinite Horizon Periodic Review Problem with Setup Costs and Capacity Constraints: A Partial Characterization of the Optimal Policy

The one-item, periodic review production and inventory system has been extensively studied in literature. Theories have been established for various basic constructs of the system of either finite or infinite horizon, except for the case where production capacity is finite and production cost contains a fixed (as well as a variable) component. It was conjectured in earlier research papers that the modified (s, S) policy would be optimal to the finite-capacity, fixed-cost model in infinite horizon. This paper studies the long-run limiting behavior of such systems. It proves that the limiting cost function exists, and there exist stationary policies that are optimal in the long run. The optimal policy, however, is not of the modified (s, S) type in general, but continues to exhibit the X-Y band structure: Whenever the inventory level drops below X, order up to capacity; when the inventory level is above Y, do nothing. When the inventory level is between X and Y, however, the ordering pattern seems to be changing from problem to problem. Nevertheless, based on a concept called (C, K)-convexity, introduced in this paper, the X-Y band is shown to be no more than one capacity of width. One calculation for the bounds on such X and Y boundaries that are tight in some cases is also provided. By exploring the X-Y band structure, a linear program model is proposed to find the optimal policy completely. Finally, an attempt is made to compare “the best modified (s, S) policy” with the optimal one, and a numerical example indicates that the deviation may be more than 11% in cost performance.