IDEAS home Printed from https://ideas.repec.org/a/inm/ormoor/v45y2020i2p547-575.html
   My bibliography  Save this article

Synthesis and Generalization of Structural Results in Inventory Management: A Generalized Convexity Property

Author

Listed:
  • Awi Federgruen

    (Graduate School of Business, Columbia University, New York, New York 10027;)

  • Zhe Liu

    (Graduate School of Business, Columbia University, New York, New York 10027;)

  • Lijian Lu

    (Shanghai SIYE Capital, Shanghai, China)

Abstract

We address a general periodic review inventory control model with the simultaneous presence of the following complications: (a) bilateral inventory adjustment options, via procurement orders and salvage sales or returns to the supplier; (b) fixed costs associated with procurement orders and downward inventory adjustments (via salvage sales or returns); and (c) capacity limits associated with upward or downward inventory adjustments. We characterize the optimal adjustment strategy, both for finite and infinite horizon periodic review models, by showing that in each period the inventory position line is to be partitioned into (maximally) five regions. Our results are obtained by identifying a novel generalized convexity property for the value functions, which we refer to as strong ( C 1 K 1 , C 2 K 2 )-convexity. To our knowledge, we recover most existing structural results for models with exogenous demands as special cases of a unified analysis.

Suggested Citation

  • 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.
    • Handle: RePEc:inm:ormoor:v:45:y:2020:i:2:p:547-575
    DOI: 10.1287/moor.2019.1001
    as

    Download full text from publisher

    File URL: https://doi.org/10.1287/moor.2019.1001
    Download Restriction: no
    File URL: https://libkey.io/10.1287/moor.2019.1001?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Ye Lu & Miao Song & Yi Yang, 2018. "Approximation Approaches for Inventory Systems with General Production/Ordering Cost Structures," Production and Operations Management, Production and Operations Management Society, vol. 27(3), pages 417-432, March.
    2. Ye Lu & Miao Song & Yi Yang, 2016. "Joint Inventory and Pricing Coordination with Incomplete Demand Information," Production and Operations Management, Production and Operations Management Society, vol. 25(4), pages 701-718, April.
    3. Qing Ye & Izak Duenyas, 2007. "Optimal Capacity Investment Decisions with Two-Sided Fixed-Capacity Adjustment Costs," Operations Research, INFORMS, vol. 55(2), pages 272-283, April.
    4. L. Joseph Thomas, 1974. "Technical Note—Price and Production Decisions with Random Demand," Operations Research, INFORMS, vol. 22(3), pages 513-518, June.
    5. John Semple, 2007. "Note: Generalized Notions of Concavity with an Application to Capacity Management," Operations Research, INFORMS, vol. 55(2), pages 284-291, April.
    6. Ye Lu & Miao Song, 2014. "Inventory Control with a Fixed Cost and a Piecewise Linear Convex Cost," Production and Operations Management, Production and Operations Management Society, vol. 23(11), pages 1966-1984, November.
    7. Eugene A. Feinberg & Mark E. Lewis, 2007. "Optimality Inequalities for Average Cost Markov Decision Processes and the Stochastic Cash Balance Problem," Mathematics of Operations Research, INFORMS, vol. 32(4), pages 769-783, November.
    8. Sundaram,Rangarajan K., 1996. "A First Course in Optimization Theory," Cambridge Books, Cambridge University Press, number 9780521497190, October.
    9. 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.
    10. Xin Chen & David Simchi-Levi, 2004. "Coordinating Inventory Control and Pricing Strategies with Random Demand and Fixed Ordering Cost: The Infinite Horizon Case," Mathematics of Operations Research, INFORMS, vol. 29(3), pages 698-723, August.
    11. Guillermo Gallego & Özalp Özer, 2001. "Integrating Replenishment Decisions with Advance Demand Information," Management Science, INFORMS, vol. 47(10), pages 1344-1360, October.
    12. 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.
    13. Xavier Vives, 2001. "Oligopoly Pricing: Old Ideas and New Tools," MIT Press Books, The MIT Press, edition 1, volume 1, number 026272040x, April.
    14. Awi Federgruen & Aliza Heching, 1999. "Combined Pricing and Inventory Control Under Uncertainty," Operations Research, INFORMS, vol. 47(3), pages 454-475, June.
    15. 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.
    16. Sundaram,Rangarajan K., 1996. "A First Course in Optimization Theory," Cambridge Books, Cambridge University Press, number 9780521497701, October.
    17. William D. Whisler, 1967. "A Stochastic Inventory Model for Rented Equipment," Management Science, INFORMS, vol. 13(9), pages 640-647, May.
    18. Edwin H. Neave, 1970. "The Stochastic Cash Balance Problem with Fixed Costs for Increases and Decreases," Management Science, INFORMS, vol. 16(7), pages 472-490, March.
    19. Chen Shaoxiang & M. Lambrecht, 1996. "X-Y Band and Modified ( s , S ) Policy," Operations Research, INFORMS, vol. 44(6), pages 1013-1019, December.
    20. Gallego, Guillermo & Scheller-Wolf, Alan, 2000. "Capacitated inventory problems with fixed order costs: Some optimal policy structure," European Journal of Operational Research, Elsevier, vol. 126(3), pages 603-613, November.
    21. George M. Constantinides & Scott F. Richard, 1978. "Existence of Optimal Simple Policies for Discounted-Cost Inventory and Cash Management in Continuous Time," Operations Research, INFORMS, vol. 26(4), pages 620-636, August.
    22. 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.
    23. 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.
    24. Xin Chen & David Simchi-Levi, 2004. "Coordinating Inventory Control and Pricing Strategies with Random Demand and Fixed Ordering Cost: The Finite Horizon Case," Operations Research, INFORMS, vol. 52(6), pages 887-896, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Yuan, Quan & Hua, Zhongsheng & Shen, Bin, 2021. "An automated system of emissions permit trading for transportation firms," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 152(C).
    2. Awi Federgruen & Zhe Liu & Lijian Lu, 2022. "Dual sourcing: Creating and utilizing flexible capacities with a second supply source," Production and Operations Management, Production and Operations Management Society, vol. 31(7), pages 2789-2805, July.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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. Sandun C. Perera & Suresh P. Sethi, 2023. "A survey of stochastic inventory models with fixed costs: Optimality of (s, S) and (s, S)‐type policies—Continuous‐time case," Production and Operations Management, Production and Operations Management Society, vol. 32(1), pages 154-169, January.
    3. 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.
    4. 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.
    5. Sandun C. Perera & Suresh P. Sethi, 2023. "A survey of stochastic inventory models with fixed costs: Optimality of (s, S) and (s, S)‐type policies—Discrete‐time case," Production and Operations Management, Production and Operations Management Society, vol. 32(1), pages 131-153, January.
    6. 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.
    7. Yonit Barron, 2022. "A probabilistic approach to the stochastic fluid cash management balance problem," Annals of Operations Research, Springer, vol. 312(2), pages 607-645, May.
    8. Jingchen Wu & Xiuli Chao, 2014. "Optimal Control of a Brownian Production/Inventory System with Average Cost Criterion," Mathematics of Operations Research, INFORMS, vol. 39(1), pages 163-189, February.
    9. Liu Shuren & Tang Pei, 2014. "The Stochastic Cash Balance Problem with Fixed Costs: The Risk-averse Case," Journal of Systems Science and Information, De Gruyter, vol. 2(6), pages 520-531, December.
    10. Lijian Lu & Xiaoming Yan, 2016. "Capacity investment decisions under risk aversion," Naval Research Logistics (NRL), John Wiley & Sons, vol. 63(3), pages 218-235, April.
    11. Wen Chen & Ying He, 2022. "Dynamic pricing and inventory control with delivery flexibility," Annals of Operations Research, Springer, vol. 317(2), pages 481-508, October.
    12. Chao, Xiuli & Chen, Hong & Zheng, Shaohui, 2008. "Joint replenishment and pricing decisions in inventory systems with stochastically dependent supply capacity," European Journal of Operational Research, Elsevier, vol. 191(1), pages 142-155, November.
    13. 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.
    14. 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.
    15. Awi Federgruen & Nan Yang, 2014. "Infinite Horizon Strategies for Replenishment Systems with a General Pool of Suppliers," Operations Research, INFORMS, vol. 62(1), pages 141-159, February.
    16. Eugene A. Feinberg & Mark E. Lewis, 2007. "Optimality Inequalities for Average Cost Markov Decision Processes and the Stochastic Cash Balance Problem," Mathematics of Operations Research, INFORMS, vol. 32(4), pages 769-783, November.
    17. 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.
    18. 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.
    19. Kaijie Zhu & Ulrich W. Thonemann, 2009. "Coordination of pricing and inventory control across products," Naval Research Logistics (NRL), John Wiley & Sons, vol. 56(2), pages 175-190, March.
    20. Gurkan, M. Edib & Tunc, Huseyin & Tarim, S. Armagan, 2022. "The joint stochastic lot sizing and pricing problem," Omega, Elsevier, vol. 108(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:ormoor:v:45:y:2020:i:2:p:547-575. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.