Optimal Capacity in a Coordinated Supply Chain Xiuli Chao, 1 Sridhar Seshadri, 2 Michael Pinedo 2 1 Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, Michigan 48109–2117 2 Stern School of Business, New York University, New York, New York 10012 Received 12 September 2004; revised 18 August 2007; accepted 2 November 2007 DOI 10.1002/nav.20271 Published online 9 January 2008 in Wiley InterScience (www.interscience.wiley.com). Abstract: We consider a supply chain in which a retailer faces a stochastic demand, incurs backorder and inventory holding costs and uses a periodic review system to place orders from a manufacturer. The manufacturer must fill the entire order. The manufacturer incurs costs of overtime and undertime if the order deviates from the planned production capacity. We determine the optimal capacity for the manufacturer in case there is no coordination with the retailer as well as in case there is full coordination with the retailer. When there is no coordination the optimal capacity for the manufacturer is found by solving a newsvendor problem. When there is coordination, we present a dynamic programming formulation and establish that the optimal ordering policy for the retailer is characterized by two parameters. The optimal coordinated capacity for the manufacturer can then be obtained by solving a nonlinear programming problem. We present an efficient exact algorithm and a heuristic algorithm for computing the manufacturer’s capacity. We discuss the impact of coordination on the supply chain cost as well as on the manufacturer’s capacity. We also identify the situations in which coordination is most beneficial. © 2008 Wiley Periodicals, Inc. Naval Research Logistics 55: 130–141, 2008 Keywords: optimal capacity; supply chain; coordination; dynamic programming; inventory systems 1. INTRODUCTION In both the periodic and the continuous review stochastic inventory models that appear in the literature, it is standard to assume that the manufacturer either has an extremely large capacity or that they can increase or decrease their capac- ity at will at a negligible cost. In fact, this assumption is so strongly ingrained in the analysis that even if the manufac- turer experiences difficulties in production (for example, due to the shortage of materials or due to a breakdown), the man- ufacturer is expected upon resumption of production not only to make up for the backlog immediately but also to restore the inventory to the prescribed levels at the very next shipment. However, we know from the aggregate production plan- ning literature that when overtime as well as undertime are allowed, there is an optimal rate at which production should be organized. In addition, there is anecdotal evidence that well-managed firms increase and decrease their capacity to match demand. Such fine tuning is achieved by resorting to overtime, short term subcontracting, capacity sharing with other manufacturers, etc. Presumably in these firms the opti- mal capacity is first chosen and adjustments through overtime and undertime are made from period to period in response to Correspondence to: Xiuli Chao (xchao@umich.edu) changes in the mix and the volume of demand. With increased visibility provided by collaborative planning technologies, as adopted by Cisco, Dell, and other companies, we argue that the manufacturer can plan and execute such short-term changes in capacity nowadays more easily than was possible in the past. However, as a consequence, suppliers are expected to fill orders completely even though their capacity is finite. This is in contrast to traditional production–inventory mod- els in which the manufacturer is assumed to have the option to backlog an order, for example, see Holt et al. [12]. There- fore, in this article we focus on the case where the production facility has no option but to fill the entire order. We seek the optimal rate of production with the assumption that this rate can be deviated from, but increasing or decreasing this rate requires an effort as well as more expensive resources. Several researchers have noted that joint capacity and inventory planning can reduce costs in a supply chain. The queueing theoretic work in this regard is summarized suc- cinctly in Buzacott and Shanthikumar [4]. In contrast to our work, capacities in queueing network models for manufac- turing systems are considered fixed at least for the short term. There are many articles that deal with capacity plan- ning problems that consider stochastic demand and allow overtime; for example, see the survey in [24]. These studies do not infer or use the structure of the optimal coordinated © 2008 Wiley Periodicals, Inc.