IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v60y2012i1p92-105.html
   My bibliography  Save this article

Lower Bounds and Heuristics for Supply Chain Stock Allocation

Author

Listed:
  • Johan Marklund

    (Department of Industrial Management and Logistics, Lund University, 22100 Lund, Sweden)

  • Kaj Rosling

    (School of Technology, Linnæus University, SE-35195 Växjö, Sweden)

Abstract

Assume that m periods with stochastic demand remain until the next replenishment arrives at a central warehouse. How should the available inventory be allocated among N retailers? This paper presents a new policy and a new lower bound for the expected cost of this problem. The lower bound becomes tight as N (rightarrow) (infinity) . The infinite horizon problem then decomposes into N independent m -period problems with optimal retailer ship-up-to levels that decrease over the m periods, and the warehouse is optimally replenished by an order-up-to level that renders zero (local) warehouse safety stock at the end of each replenishment cycle. Based on the lower bound solution, we suggest a heuristic for finite N . In a numerical study it outperforms the heuristic by Jackson [Jackson, P. L. 1988. Stock allocation in a two-echelon distribution system or what to do until your ship comes in. Management Sci. 34 (7) 880--895], and the new lower bound improves on Clark and Scarf's [Clark, A. J., H. Scarf. 1960. Optimal policies for a multi-echelon inventory problem. Management Sci. 6 (4) 475--490] bound when N is not too small. Moreover, the warehouse zero-safety-stock heuristic is comparable to Clark and Scarf's warehouse policy for lead times that are not too long. The suggested approach is quite general and may be applied to other logistical problems. In the present application it retains some of the risk-pooling benefits of holding central warehouse stock.

Suggested Citation

  • Johan Marklund & Kaj Rosling, 2012. "Lower Bounds and Heuristics for Supply Chain Stock Allocation," Operations Research, INFORMS, vol. 60(1), pages 92-105, February.
    • Handle: RePEc:inm:oropre:v:60:y:2012:i:1:p:92-105
    DOI: 10.1287/opre.1110.1009
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.1110.1009
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.1110.1009?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. Alan S. Manne, 1958. "Programming of Economic Lot Sizes," Management Science, INFORMS, vol. 4(2), pages 115-135, January.
    2. Andrew J. Clark & Herbert Scarf, 2004. "Optimal Policies for a Multi-Echelon Inventory Problem," Management Science, INFORMS, vol. 50(12_supple), pages 1782-1790, December.
    3. Özalp Özer, 2003. "Replenishment Strategies for Distribution Systems Under Advance Demand Information," Management Science, INFORMS, vol. 49(3), pages 255-272, March.
    4. Gérard P. Cachon & Marshall Fisher, 2000. "Supply Chain Inventory Management and the Value of Shared Information," Management Science, INFORMS, vol. 46(8), pages 1032-1048, August.
    5. Peter L. Jackson & John A. Muckstadt, 1989. "Risk pooling in a two‐period, two‐echelon inventory stocking and allocation problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 36(1), pages 1-26, February.
    6. Diks, E. B. & de Kok, A. G., 1999. "Computational results for the control of a divergent N-echelon inventory system," International Journal of Production Economics, Elsevier, vol. 59(1-3), pages 327-336, March.
    7. van der Heijden, M. C. & Diks, E. B. & de Kok, A. G., 1997. "Stock allocation in general multi-echelon distribution systems with (R, S) order-up-to-policies," International Journal of Production Economics, Elsevier, vol. 49(2), pages 157-174, April.
    8. Verrijdt, J. H. C. M. & de Kok, A. G., 1996. "Distribution planning for a divergent depotless two-echelon network under service constraints," European Journal of Operational Research, Elsevier, vol. 89(2), pages 341-354, March.
    9. Ashok Kumar & Leroy B. Schwarz & James E. Ward, 1995. "Risk-Pooling Along a Fixed Delivery Route Using a Dynamic Inventory-Allocation Policy," Management Science, INFORMS, vol. 41(2), pages 344-362, February.
    10. Kevin H. Shang & Sean X. Zhou, 2010. "Optimal and Heuristic Echelon ( r, nQ, T ) Policies in Serial Inventory Systems with Fixed Costs," Operations Research, INFORMS, vol. 58(2), pages 414-427, April.
    11. Awi Federgruen & Paul Zipkin, 1984. "Approximations of Dynamic, Multilocation Production and Inventory Problems," Management Science, INFORMS, vol. 30(1), pages 69-84, January.
    12. Guillermo Gallego & Paul Zipkin, 1999. "Stock Positioning and Performance Estimation in Serial Production-Transportation Systems," Manufacturing & Service Operations Management, INFORMS, vol. 1(1), pages 77-88.
    13. Sven Axsäter & Johan Marklund, 2008. "Optimal Position-Based Warehouse Ordering in Divergent Two-Echelon Inventory Systems," Operations Research, INFORMS, vol. 56(4), pages 976-991, August.
    14. Yossi Aviv & Awi Federgruen, 2001. "Capacitated Multi-Item Inventory Systems with Random and Seasonally Fluctuating Demands: Implications for Postponement Strategies," Management Science, INFORMS, vol. 47(4), pages 512-531, April.
    15. Xiuli Chao & Sean X. Zhou, 2009. "Optimal Policy for a Multiechelon Inventory System with Batch Ordering and Fixed Replenishment Intervals," Operations Research, INFORMS, vol. 57(2), pages 377-390, April.
    16. Bollapragada, rinivas & Akella, Ram & Srinivasan, Ramesh, 1998. "Centralized ordering and allocation policies in a two-echelon system with non-identical warehouses," European Journal of Operational Research, Elsevier, vol. 106(1), pages 74-81, April.
    17. Awi Federgruen & Paul Zipkin, 1984. "Computational Issues in an Infinite-Horizon, Multiechelon Inventory Model," Operations Research, INFORMS, vol. 32(4), pages 818-836, August.
    18. Edward J. McGavin & James E. Ward & Leroy B. Schwarz, 1997. "Balancing Retailer Inventories," Operations Research, INFORMS, vol. 45(6), pages 820-830, December.
    19. van der Heijden, Matthieu C., 1999. "Multi-echelon inventory control in divergent systems with shipping frequencies," European Journal of Operational Research, Elsevier, vol. 116(2), pages 331-351, July.
    20. van Houtum, G. J. & Inderfurth, K. & Zijm, W. H. M., 1996. "Materials coordination in stochastic multi-echelon systems," European Journal of Operational Research, Elsevier, vol. 95(1), pages 1-23, November.
    21. H. Jönsson & E. A. Silver, 1987. "Analysis of a Two-Echelon Inventory Control System with Complete Redistribution," Management Science, INFORMS, vol. 33(2), pages 215-227, February.
    22. Gérard P. Cachon, 2001. "Exact Evaluation of Batch-Ordering Inventory Policies in Two-Echelon Supply Chains with Periodic Review," Operations Research, INFORMS, vol. 49(1), pages 79-98, February.
    23. Paul Zipkin, 1984. "On the Imbalance of Inventories in Multi-Echelon Systems," Mathematics of Operations Research, INFORMS, vol. 9(3), pages 402-423, August.
    24. James A. Rappold & John A. Muckstadt, 2000. "A computationally efficient approach for determining inventory levels in a capacitated multiechelon production‐distribution system," Naval Research Logistics (NRL), John Wiley & Sons, vol. 47(5), pages 377-398, August.
    25. Sven Axsäter & Johan Marklund & Edward A. Silver, 2002. "Heuristic Methods for Centralized Control of One-Warehouse, N-Retailer Inventory Systems," Manufacturing & Service Operations Management, INFORMS, vol. 4(1), pages 75-97, October.
    26. Edward J. McGavin & Leroy B. Schwarz & James E. Ward, 1993. "Two-Interval Inventory-Allocation Policies in a One-Warehouse N-Identical-Retailer Distribution System," Management Science, INFORMS, vol. 39(9), pages 1092-1107, September.
    27. Peter L. Jackson, 1988. "Stock Allocation in a Two-Echelon Distribution System Or "What to Do Until Your Ship Comes In"," Management Science, INFORMS, vol. 34(7), pages 880-895, July.
    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. de Kok, Ton & Grob, Christopher & Laumanns, Marco & Minner, Stefan & Rambau, Jörg & Schade, Konrad, 2018. "A typology and literature review on stochastic multi-echelon inventory models," European Journal of Operational Research, Elsevier, vol. 269(3), pages 955-983.
    2. Pinto, Roberto, 2016. "Stock rationing under a profit satisficing objective," Omega, Elsevier, vol. 65(C), pages 55-68.
    3. Santiago R. Balseiro & David B. Brown & Chen Chen, 2021. "Dynamic Pricing of Relocating Resources in Large Networks," Management Science, INFORMS, vol. 67(7), pages 4075-4094, July.
    4. Huaxiao Shen & Tian Tian & Han Zhu, 2019. "A Two-Echelon Inventory System with a Minimum Order Quantity Requirement," Sustainability, MDPI, vol. 11(18), pages 1-22, September.
    5. Felix Papier, 2016. "Supply Allocation Under Sequential Advance Demand Information," Operations Research, INFORMS, vol. 64(2), pages 341-361, April.
    6. Edirisinghe, Chanaka & Atkins, Derek, 2017. "Lower bounding inventory allocations for risk pooling in two-echelon supply chains," International Journal of Production Economics, Elsevier, vol. 187(C), pages 159-167.
    7. Darmawan, Agus & Wong, Hartanto & Thorstenson, Anders, 2021. "Supply chain network design with coordinated inventory control," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 145(C).
    8. Olof Stenius & Ayşe Gönül Karaarslan & Johan Marklund & A. G. de Kok, 2016. "Exact Analysis of Divergent Inventory Systems with Time-Based Shipment Consolidation and Compound Poisson Demand," Operations Research, INFORMS, vol. 64(4), pages 906-921, August.
    9. Topan, E. & van der Heijden, M.C., 2020. "Operational level planning of a multi-item two-echelon spare parts inventory system with reactive and proactive interventions," European Journal of Operational Research, Elsevier, vol. 284(1), pages 164-175.
    10. Awi Federgruen & C. Daniel Guetta & Garud Iyengar, 2018. "Two‐echelon distribution systems with random demands and storage constraints," Naval Research Logistics (NRL), John Wiley & Sons, vol. 65(8), pages 594-618, December.
    11. Mila Nambiar & David Simchi‐Levi & He Wang, 2021. "Dynamic Inventory Allocation with Demand Learning for Seasonal Goods," Production and Operations Management, Production and Operations Management Society, vol. 30(3), pages 750-765, March.
    12. Berling, Peter & Marklund, Johan, 2013. "A model for heuristic coordination of real life distribution inventory systems with lumpy demand," European Journal of Operational Research, Elsevier, vol. 230(3), pages 515-526.
    13. Stenius, Olof & Marklund, Johan & Axsäter, Sven, 2018. "Sustainable multi-echelon inventory control with shipment consolidation and volume dependent freight costs," European Journal of Operational Research, Elsevier, vol. 267(3), pages 904-916.

    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. de Kok, Ton & Grob, Christopher & Laumanns, Marco & Minner, Stefan & Rambau, Jörg & Schade, Konrad, 2018. "A typology and literature review on stochastic multi-echelon inventory models," European Journal of Operational Research, Elsevier, vol. 269(3), pages 955-983.
    2. Huaxiao Shen & Tian Tian & Han Zhu, 2019. "A Two-Echelon Inventory System with a Minimum Order Quantity Requirement," Sustainability, MDPI, vol. 11(18), pages 1-22, September.
    3. Howard, Christian & Marklund, Johan, 2011. "Evaluation of stock allocation policies in a divergent inventory system with shipment consolidation," European Journal of Operational Research, Elsevier, vol. 211(2), pages 298-309, June.
    4. Olof Stenius & Ayşe Gönül Karaarslan & Johan Marklund & A. G. de Kok, 2016. "Exact Analysis of Divergent Inventory Systems with Time-Based Shipment Consolidation and Compound Poisson Demand," Operations Research, INFORMS, vol. 64(4), pages 906-921, August.
    5. Sven Axsäter & Johan Marklund & Edward A. Silver, 2002. "Heuristic Methods for Centralized Control of One-Warehouse, N-Retailer Inventory Systems," Manufacturing & Service Operations Management, INFORMS, vol. 4(1), pages 75-97, October.
    6. Lagodimos, A.G. & Koukoumialos, S., 2008. "Service performance of two-echelon supply chains under linear rationing," International Journal of Production Economics, Elsevier, vol. 112(2), pages 869-884, April.
    7. Agrawal, Vipul & Chao, Xiuli & Seshadri, Sridhar, 2004. "Dynamic balancing of inventory in supply chains," European Journal of Operational Research, Elsevier, vol. 159(2), pages 296-317, December.
    8. Geert-Jan van Houtum & Alan Scheller-Wolf & Jinxin Yi, 2007. "Optimal Control of Serial Inventory Systems with Fixed Replenishment Intervals," Operations Research, INFORMS, vol. 55(4), pages 674-687, August.
    9. Ankur Goel & Genaro J. Gutierrez, 2011. "Multiechelon Procurement and Distribution Policies for Traded Commodities," Management Science, INFORMS, vol. 57(12), pages 2228-2244, December.
    10. van der Heijden, Matthieu C., 1999. "Multi-echelon inventory control in divergent systems with shipping frequencies," European Journal of Operational Research, Elsevier, vol. 116(2), pages 331-351, July.
    11. Peter L. Jackson & John A. Muckstadt & Yuexing Li, 2019. "Multiperiod Stock Allocation via Robust Optimization," Management Science, INFORMS, vol. 65(2), pages 794-818, February.
    12. Johan Marklund, 2011. "Inventory control in divergent supply chains with time‐based dispatching and shipment consolidation," Naval Research Logistics (NRL), John Wiley & Sons, vol. 58(1), pages 59-71, February.
    13. Yossi Aviv & Awi Federgruen, 2001. "Design for Postponement: A Comprehensive Characterization of Its Benefits Under Unknown Demand Distributions," Operations Research, INFORMS, vol. 49(4), pages 578-598, August.
    14. De‐bi Cao & Edward A. Silver, 2005. "A dynamic allocation heuristic for centralized safety stock," Naval Research Logistics (NRL), John Wiley & Sons, vol. 52(6), pages 513-526, September.
    15. Berling, Peter & Marklund, Johan, 2013. "A model for heuristic coordination of real life distribution inventory systems with lumpy demand," European Journal of Operational Research, Elsevier, vol. 230(3), pages 515-526.
    16. Qinan Wang & Sven Axsäter, 2013. "Fixed‐interval joint‐replenishment policies for distribution systems with multiple retailers and stochastic demand," Naval Research Logistics (NRL), John Wiley & Sons, vol. 60(8), pages 637-651, December.
    17. Diks, E. B. & de Kok, A. G. & Lagodimos, A. G., 1996. "Multi-echelon systems: A service measure perspective," European Journal of Operational Research, Elsevier, vol. 95(2), pages 241-263, December.
    18. Wang, Qinan & Wan, Guangyu, 2023. "Fixed-interval order-up-to policies and myopic optimal warehouse stock allocation for one-warehouse multiple-retailer systems," European Journal of Operational Research, Elsevier, vol. 309(3), pages 1112-1124.
    19. Stenius, Olof & Marklund, Johan & Axsäter, Sven, 2018. "Sustainable multi-echelon inventory control with shipment consolidation and volume dependent freight costs," European Journal of Operational Research, Elsevier, vol. 267(3), pages 904-916.
    20. Protopappa-Sieke, Margarita & Sieke, Marcel A. & Thonemann, Ulrich W., 2016. "Optimal two-period inventory allocation under multiple service level contracts," European Journal of Operational Research, Elsevier, vol. 252(1), pages 145-155.

    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:oropre:v:60:y:2012:i:1:p:92-105. 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.