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

Abstract

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.

Inventory Control with Multiple Setup Costs

“…These papers include analytical results that are more broadly applicable than our setting, but we restrict our attention to the case of identical fixed costs. Caliskan-Demirag et al (2010) use the notion of the (C, K)-convexity (Shaoxiang, 2004) to obtain a result on after-ordering inventory levels that is related to Theorem 7 of our paper. Their result can be thought of as an extension of Shaoxiang and Lambrecht (1996), where one can order at most one batch per period.…”

“…We note that (8) is similar to a capacitated inventory model with a fixed ordering cost studied in the literature, for example, Shaoxiang and Lambrecht (1996) and Shaoxiang (2004), but an important difference arises from the cost structure since (7) is not a typical "fixed ordering cost".…”

Optimal Inventory Control with Dual‐Sourcing, Heterogeneous Ordering Costs and Order Size Constraints

“…This is what makes the dual‐supplier problem with order size constraint different from the existing inventory problems, and is also the reason why the previous techniques based on K ‐convexity or a generalized notion of K ‐convexity of cost functions (e.g., Fox et al. , Gallego and Scheller‐Wolf , Shaoxiang ) cannot solve our problem.…”

“…Our basic model with order size constraints on only one supplier can be viewed as generalizing existing single supplier models with or without fixed costs and with or without capacity constraints; see, for example, Scarf (), Federgruen and Zipkin (, ), Shaoxiang and Lambrecht (), Gallego and Scheller‐Wolf (), and Shaoxiang (). Note that problems with both fixed costs and order size constraints, even with a single supplier, are notoriously difficult to analyze, as the optimal policy does not assume a simple form and the optimal cost function does not possess the K ‐convexity property observed in systems without order size constraints.…”

Optimal Policy for a Periodic-Review Inventory System Under a Supply Capacity Contract

“…) Our model is related to inventory systems with finite ordering capacity. Articles on this topic include Federgruen and Zipkin (1986), who study systems without a fixed cost, and Shaoxiang and Lambrecht (1996), Gallego and Scheller-Wolf (2000), and Shaoxiang (2004), who study the same system but with a fixed cost. The latter three works provide partial characterizations of the optimal policy, which specify the optimal order quantity except for one region.…”