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

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  • Chen Shaoxiang

    (Nanyang Business School, Nanyang Technological University, Singapore 639798)

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.

Suggested Citation

  • Chen Shaoxiang, 2004. "The Infinite Horizon Periodic Review Problem with Setup Costs and Capacity Constraints: A Partial Characterization of the Optimal Policy," Operations Research, INFORMS, vol. 52(3), pages 409-421, June.
    • Handle: RePEc:inm:oropre:v:52:y:2004:i:3:p:409-421
    DOI: 10.1287/opre.1030.0104
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    References listed on IDEAS

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    1. Arthur F. Veinott, Jr. & Harvey M. Wagner, 1965. "Computing Optimal (s, S) Inventory Policies," Management Science, INFORMS, vol. 11(5), pages 525-552, March.
    2. Guy T. de Ghellinck & Gary D. Eppen, 1967. "Linear Programming Solutions for Separable Markovian Decision Problems," Management Science, INFORMS, vol. 13(5), pages 371-394, January.
    3. A. Federgruen & P. Zipkin, 1986. "An Inventory Model with Limited Production Capacity and Uncertain Demands II. The Discounted-Cost Criterion," Mathematics of Operations Research, INFORMS, vol. 11(2), pages 208-215, May.
    4. Arthur F. Veinott, Jr., 1966. "The Status of Mathematical Inventory Theory," Management Science, INFORMS, vol. 12(11), pages 745-777, July.
    5. DE GHELLINCK, Guy T. & EPPEN, Gary D., 1967. "Linear programming solutions for separable Markovian decision problems," LIDAM Reprints CORE 6, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Donald L. Iglehart, 1963. "Optimality of (s, S) Policies in the Infinite Horizon Dynamic Inventory Problem," Management Science, INFORMS, vol. 9(2), pages 259-267, January.
    7. Chen Shaoxiang & M. Lambrecht, 1996. "X-Y Band and Modified ( s , S ) Policy," Operations Research, INFORMS, vol. 44(6), pages 1013-1019, December.
    8. A. Federgruen & P. Zipkin, 1986. "An Inventory Model with Limited Production Capacity and Uncertain Demands I. The Average-Cost Criterion," Mathematics of Operations Research, INFORMS, vol. 11(2), pages 193-207, May.
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    Citations

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    Cited by:

    1. Peng Hu & Ye Lu & Miao Song, 2019. "Joint Pricing and Inventory Control with Fixed and Convex/Concave Variable Production Costs," Production and Operations Management, Production and Operations Management Society, vol. 28(4), pages 847-877, April.
    2. Ozgun Caliskan-Demirag & Youhua Chen & Yi Yang, 2013. "Production-inventory control policy under warm/cold state-dependent fixed costs and stochastic demand: partial characterization and heuristics," Annals of Operations Research, Springer, vol. 208(1), pages 531-556, September.
    3. Liqing Zhang & Sıla Çetinkaya, 2017. "Stochastic Dynamic Inventory Problem Under Explicit Inbound Transportation Cost and Capacity," Operations Research, INFORMS, vol. 65(5), pages 1267-1274, October.
    4. Li Chen & Hau L. Lee, 2012. "Bullwhip Effect Measurement and Its Implications," Operations Research, INFORMS, vol. 60(4), pages 771-784, August.
    5. Osman Alp & Woonghee Tim Huh & Tarkan Tan, 2014. "Inventory Control with Multiple Setup Costs," Manufacturing & Service Operations Management, INFORMS, vol. 16(1), pages 89-103, February.
    6. Rossi, Roberto & Chen, Zhen & Tarim, S. Armagan, 2024. "On the stochastic inventory problem under order capacity constraints," European Journal of Operational Research, Elsevier, vol. 312(2), pages 541-555.
    7. Awi Federgruen & Zhe Liu & Lijian Lu, 2020. "Synthesis and Generalization of Structural Results in Inventory Management: A Generalized Convexity Property," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 547-575, May.
    8. repec:dgr:rugsom:10005 is not listed on IDEAS
    9. Ozgun Caliskan-Demirag & Youhua (Frank) Chen & Yi Yang, 2012. "Ordering Policies for Periodic-Review Inventory Systems with Quantity-Dependent Fixed Costs," Operations Research, INFORMS, vol. 60(4), pages 785-796, August.
    10. Nicky D. Van Foreest & Jacob Wijngaard, 2014. "On Optimal Policies for Production-Inventory Systems with Compound Poisson Demand and Setup Costs," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 517-532, May.
    11. Foreest, N. D. van & Wijngaard, J., 2010. "On the Optimal Policy for the Single-product Inventory Problem with Set-up Cost and a Restricted Production Capacity," Research Report 10005, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    12. Chen, Junlin & Zhao, Xiaobo & Zhou, Yun, 2012. "A periodic-review inventory system with a capacitated backup supplier for mitigating supply disruptions," European Journal of Operational Research, Elsevier, vol. 219(2), pages 312-323.
    13. Xiuli Chao & Paul H. Zipkin, 2008. "Optimal Policy for a Periodic-Review Inventory System Under a Supply Capacity Contract," Operations Research, INFORMS, vol. 56(1), pages 59-68, February.
    14. Woonghee Tim Huh & Ganesh Janakiraman & Mahesh Nagarajan, 2011. "Average Cost Single-Stage Inventory Models: An Analysis Using a Vanishing Discount Approach," Operations Research, INFORMS, vol. 59(1), pages 143-155, February.
    15. Yang, Yi & Yuan, Quan & Xue, Weili & Zhou, Yun, 2014. "Analysis of batch ordering inventory models with setup cost and capacity constraint," International Journal of Production Economics, Elsevier, vol. 155(C), pages 340-350.
    16. Chen, Zhen & Rossi, Roberto, 2021. "A dynamic ordering policy for a stochastic inventory problem with cash constraints," Omega, Elsevier, vol. 102(C).

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