IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v88y2018i3d10.1007_s00186-018-0642-4.html
   My bibliography  Save this article

Approximation algorithms for k-echelon extensions of the one warehouse multi-retailer problem

Author

Listed:
  • Gautier Stauffer

    (Kedge Business School)

Abstract

In this paper, we consider k-echelon extensions of the deterministic one warehouse multi-retailer problem. We give constant factor approximation algorithms for some of these extensions when k is fixed. We focus first on the case without backorders and we give a $$(2k-1)$$ ( 2 k - 1 ) -approximation algorithm under general assumptions on the evolution of the holding costs as products move toward the final customers. We then improve this result to a k-approximation when the holding costs are monotonically non-increasing or non-decreasing (which is a natural situation in practice). Finally we address problems with backorders: we give a 3-approximation for the one-warehouse multi-retailer problem with backlog and a k-approximation algorithm for the k-level Joint Replenishment Problem with backlog (a variant where inventory can only be kept at the final retailers). Ours results are the first constant approximation algorithms for those problems. In addition, we demonstrate the potential of our approach on a practical case. Our preliminary experiments show that the average optimality gap is around 15%.

Suggested Citation

  • Gautier Stauffer, 2018. "Approximation algorithms for k-echelon extensions of the one warehouse multi-retailer problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(3), pages 445-473, December.
  • Handle: RePEc:spr:mathme:v:88:y:2018:i:3:d:10.1007_s00186-018-0642-4
    DOI: 10.1007/s00186-018-0642-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00186-018-0642-4
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s00186-018-0642-4?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Gayon, J.-P. & Massonnet, G. & Rapine, C. & Stauffer, G., 2016. "Constant approximation algorithms for the one warehouse multiple retailers problem with backlog or lost-sales," European Journal of Operational Research, Elsevier, vol. 250(1), pages 155-163.
    2. Andrew J Mason, 2012. "OpenSolver - An Open Source Add-in to Solve Linear and Integer Progammes in Excel," Operations Research Proceedings, in: Diethard Klatte & Hans-Jakob Lüthi & Karl Schmedders (ed.), Operations Research Proceedings 2011, edition 127, pages 401-406, Springer.
    3. Awi Federgruen & Michal Tzur, 1999. "Time‐partitioning heuristics: Application to one warehouse, multiitem, multiretailer lot‐sizing problems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 46(5), pages 463-486, August.
    4. Leroy B. Schwarz, 1973. "A Simple Continuous Review Deterministic One-Warehouse N-Retailer Inventory Problem," Management Science, INFORMS, vol. 19(5), pages 555-566, January.
    5. Retsef Levi & Robin Roundy & David Shmoys & Maxim Sviridenko, 2008. "A Constant Approximation Algorithm for the One-Warehouse Multiretailer Problem," Management Science, INFORMS, vol. 54(4), pages 763-776, April.
    6. Khouja, Moutaz & Goyal, Suresh, 2008. "A review of the joint replenishment problem literature: 1989-2005," European Journal of Operational Research, Elsevier, vol. 186(1), pages 1-16, April.
    7. Gautier Stauffer, 2012. "Using the economical order quantity formula for inventory control in one‐warehouse multiretailer systems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 59(3‐4), pages 285-297, April.
    8. Zuo‐Jun Max Shen & Jia Shu & David Simchi‐Levi & Chung‐Piaw Teo & Jiawei Zhang, 2009. "Approximation algorithms for general one‐warehouse multi‐retailer systems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 56(7), pages 642-658, October.
    9. Arthur F. Veinott, 1969. "Minimum Concave-Cost Solution of Leontief Substitution Models of Multi-Facility Inventory Systems," Operations Research, INFORMS, vol. 17(2), pages 262-291, April.
    10. Robin Roundy, 1985. "98%-Effective Integer-Ratio Lot-Sizing for One-Warehouse Multi-Retailer Systems," Management Science, INFORMS, vol. 31(11), pages 1416-1430, November.
    11. Robin Roundy, 1986. "A 98%-Effective Lot-Sizing Rule for a Multi-Product, Multi-Stage Production / Inventory System," Mathematics of Operations Research, INFORMS, vol. 11(4), pages 699-727, November.
    12. Willard I. Zangwill, 1969. "A Backlogging Model and a Multi-Echelon Model of a Dynamic Economic Lot Size Production System--A Network Approach," Management Science, INFORMS, vol. 15(9), pages 506-527, May.
    13. Stephen F. Love, 1972. "A Facilities in Series Inventory Model with Nested Schedules," Management Science, INFORMS, vol. 18(5-Part-1), pages 327-338, January.
    14. Jean-Philippe Gayon & Guillaume Massonnet & Christophe Rapine & Gautier Stauffer, 2017. "Fast Approximation Algorithms for the One-Warehouse Multi-Retailer Problem Under General Cost Structures and Capacity Constraints," Mathematics of Operations Research, INFORMS, vol. 42(3), pages 854-875, August.
    15. Retsef Levi & Robin O. Roundy & David B. Shmoys, 2006. "Primal-Dual Algorithms for Deterministic Inventory Problems," Mathematics of Operations Research, INFORMS, vol. 31(2), pages 267-284, May.
    Full references (including those not matched with items on IDEAS)

    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. Jean-Philippe Gayon & Guillaume Massonnet & Christophe Rapine & Gautier Stauffer, 2017. "Fast Approximation Algorithms for the One-Warehouse Multi-Retailer Problem Under General Cost Structures and Capacity Constraints," Mathematics of Operations Research, INFORMS, vol. 42(3), pages 854-875, August.
    2. Li, Xiuhui & Wang, Qinan, 2007. "Coordination mechanisms of supply chain systems," European Journal of Operational Research, Elsevier, vol. 179(1), pages 1-16, May.
    3. Adeinat, Hamza & Pazhani, Subramanian & Mendoza, Abraham & Ventura, Jose A., 2022. "Coordination of pricing and inventory replenishment decisions in a supply chain with multiple geographically dispersed retailers," International Journal of Production Economics, Elsevier, vol. 248(C).
    4. Tamar Cohen-Hillel & Liron Yedidsion, 2018. "The Periodic Joint Replenishment Problem Is Strongly 𝒩𝒫-Hard," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1269-1289, November.
    5. Oğuz Solyalı & Haldun Süral, 2012. "The one-warehouse multi-retailer problem: reformulation, classification, and computational results," Annals of Operations Research, Springer, vol. 196(1), pages 517-541, July.
    6. Jesus Cunha & Rafael Melo, 2016. "On reformulations for the one-warehouse multi-retailer problem," Annals of Operations Research, Springer, vol. 238(1), pages 99-122, March.
    7. Danny Segev, 2014. "An Approximate Dynamic-Programming Approach to the Joint Replenishment Problem," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 432-444, May.
    8. Kimms, Alf & Drexl, Andreas, 1996. "Multi-level lot sizing: A literature survey," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 405, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    9. Xuefei Shi & Haiyan Wang, 2022. "Design of the cost allocation rule for joint replenishment to an overseas warehouse with a piecewise linear holding cost rate," Operational Research, Springer, vol. 22(5), pages 4905-4929, November.
    10. Awi Federgruen & Michal Tzur, 1999. "Time‐partitioning heuristics: Application to one warehouse, multiitem, multiretailer lot‐sizing problems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 46(5), pages 463-486, August.
    11. Brahimi, Nadjib & Absi, Nabil & Dauzère-Pérès, Stéphane & Nordli, Atle, 2017. "Single-item dynamic lot-sizing problems: An updated survey," European Journal of Operational Research, Elsevier, vol. 263(3), pages 838-863.
    12. Lehilton L. C. Pedrosa & Maxim Sviridenko, 2018. "Integrated Supply Chain Management via Randomized Rounding," INFORMS Journal on Computing, INFORMS, vol. 30(1), pages 124-136, February.
    13. Daniel Adelman & Diego Klabjan, 2005. "Duality and Existence of Optimal Policies in Generalized Joint Replenishment," Mathematics of Operations Research, INFORMS, vol. 30(1), pages 28-50, February.
    14. Jiawei Zhang, 2009. "Cost Allocation for Joint Replenishment Models," Operations Research, INFORMS, vol. 57(1), pages 146-156, February.
    15. Erenguc, S. Selcuk & Simpson, N. C. & Vakharia, Asoo J., 1999. "Integrated production/distribution planning in supply chains: An invited review," European Journal of Operational Research, Elsevier, vol. 115(2), pages 219-236, June.
    16. Wang, Qinan & Chay, Yiowmin & Wu, Zhang, 2011. "Streamlining inventory flows with time discounts to improve the profits of a decentralized supply chain," International Journal of Production Economics, Elsevier, vol. 132(2), pages 230-239, August.
    17. Jesus O. Cunha & Rafael A. Melo, 2016. "On reformulations for the one-warehouse multi-retailer problem," Annals of Operations Research, Springer, vol. 238(1), pages 99-122, March.
    18. Bouchery, Yann & Ghaffari, Asma & Jemai, Zied & Dallery, Yves, 2012. "Including sustainability criteria into inventory models," European Journal of Operational Research, Elsevier, vol. 222(2), pages 229-240.
    19. Chu, Chi-Leung & Leon, V. Jorge, 2009. "Scalable methodology for supply chain inventory coordination with private information," European Journal of Operational Research, Elsevier, vol. 195(1), pages 262-279, May.
    20. Wolsey, Laurence A., 1995. "Progress with single-item lot-sizing," European Journal of Operational Research, Elsevier, vol. 86(3), pages 395-401, November.

    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:spr:mathme:v:88:y:2018:i:3:d:10.1007_s00186-018-0642-4. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.