IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v83y2016i1d10.1007_s00186-015-0518-9.html
   My bibliography  Save this article

On the complexity of the FIFO stack-up problem

Author

Listed:
  • Frank Gurski

    (University of Düsseldorf, Institute of Computer Science)

  • Jochen Rethmann

    (Niederrhein University of Applied Sciences)

  • Egon Wanke

    (University of Düsseldorf, Institute of Computer Science)

Abstract

We study the combinatorial FIFO stack-up problem. In delivery industry, bins have to be stacked-up from conveyor belts onto pallets with respect to customer orders. Given k sequences $$q_1, \ldots , q_k$$ q 1 , … , q k of labeled bins and a positive integer p, the aim is to stack-up the bins by iteratively removing the first bin of one of the k sequences and put it onto an initially empty pallet of unbounded capacity located at one of p stack-up places. Bins with different pallet labels have to be placed on different pallets, bins with the same pallet label have to be placed on the same pallet. After all bins for a pallet have been removed from the given sequences, the corresponding stack-up place will be cleared and becomes available for a further pallet. The FIFO stack-up problem is to find a stack-up sequence such that all pallets can be build-up with the available p stack-up places. In this paper, we introduce two digraph models for the FIFO stack-up problem, namely the processing graph and the sequence graph. We show that there is a processing of some list of sequences with at most p stack-up places if and only if the sequence graph of this list has directed pathwidth at most $$p-1$$ p - 1 . This connection implies that the FIFO stack-up problem is NP-complete in general, even if there are at most 6 bins for every pallet and that the problem can be solved in polynomial time, if the number p of stack-up places is assumed to be fixed. Further the processing graph allows us to show that the problem can be solved in polynomial time, if the number k of sequences is assumed to be fixed.

Suggested Citation

  • Frank Gurski & Jochen Rethmann & Egon Wanke, 2016. "On the complexity of the FIFO stack-up problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(1), pages 33-52, February.
  • Handle: RePEc:spr:mathme:v:83:y:2016:i:1:d:10.1007_s00186-015-0518-9
    DOI: 10.1007/s00186-015-0518-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00186-015-0518-9
    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/s00186-015-0518-9?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. Rethmann, J. & Wanke, E., 2001. "Stack-up algorithms for palletizing at delivery industry," European Journal of Operational Research, Elsevier, vol. 128(1), pages 74-97, January.
    2. Jochen Rethmann & Egon Wanke, 2000. "An approximation algorithm for the stack-up problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 51(2), pages 203-233, April.
    3. Frank Gurski & Jochen Rethmann & Egon Wanke, 2014. "Moving Bins from Conveyor Belts onto Pallets Using FIFO Queues," Operations Research Proceedings, in: Dennis Huisman & Ilse Louwerse & Albert P.M. Wagelmans (ed.), Operations Research Proceedings 2013, edition 127, pages 185-191, Springer.
    4. de Koster, Rene, 1994. "Performance approximation of pick-to-belt orderpicking systems," European Journal of Operational Research, Elsevier, vol. 72(3), pages 558-573, February.
    5. Rethmann, J. & Wanke, E., 1997. "Storage controlled pile-up systems, theoretical foundations," European Journal of Operational Research, Elsevier, vol. 103(3), pages 515-530, 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. Frank Gurski & Carolin Rehs & Jochen Rethmann & Egon Wanke, 2019. "Controlling distribution conveyors and multiline palletizers: theoretical foundations and online algorithms," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 41(2), pages 581-611, June.

    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. Frank Gurski & Jochen Rethmann & Egon Wanke, 2016. "On the complexity of the FIFO stack-up problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(1), pages 33-52, February.
    2. Frank Gurski & Carolin Rehs & Jochen Rethmann & Egon Wanke, 2019. "Controlling distribution conveyors and multiline palletizers: theoretical foundations and online algorithms," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 41(2), pages 581-611, June.
    3. Rethmann, J. & Wanke, E., 2001. "Stack-up algorithms for palletizing at delivery industry," European Journal of Operational Research, Elsevier, vol. 128(1), pages 74-97, January.
    4. Espelage, W. & Wanke, E., 2005. "A linear time approximation algorithm for movement minimization in conveyor flow shop processing," European Journal of Operational Research, Elsevier, vol. 160(2), pages 485-500, January.
    5. Rethmann, J. & Wanke, E., 1997. "Storage controlled pile-up systems, theoretical foundations," European Journal of Operational Research, Elsevier, vol. 103(3), pages 515-530, December.
    6. Rafael Diaz, 2016. "Using dynamic demand information and zoning for the storage of non-uniform density stock keeping units," International Journal of Production Research, Taylor & Francis Journals, vol. 54(8), pages 2487-2498, April.
    7. Grzegorz Tarczyński, 2023. "Linear programming models for optimal workload and batching in pick-and-pass warehousing systems," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 33(3), pages 141-158.
    8. Pan, Jason Chao-Hsien & Shih, Po-Hsun & Wu, Ming-Hung, 2015. "Order batching in a pick-and-pass warehousing system with group genetic algorithm," Omega, Elsevier, vol. 57(PB), pages 238-248.
    9. Gu, Jinxiang & Goetschalckx, Marc & McGinnis, Leon F., 2010. "Research on warehouse design and performance evaluation: A comprehensive review," European Journal of Operational Research, Elsevier, vol. 203(3), pages 539-549, June.
    10. Yu, Mengfei & de Koster, René B.M., 2009. "The impact of order batching and picking area zoning on order picking system performance," European Journal of Operational Research, Elsevier, vol. 198(2), pages 480-490, October.
    11. Yu, M. & de Koster, M.B.M., 2007. "Performance Approximation and Design of Pick-and-Pass Order Picking Systems," ERIM Report Series Research in Management ERS-2007-082-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    12. Stein, William E. & Seale, Darryl A. & Rapoport, Amnon, 2003. "Analysis of heuristic solutions to the best choice problem," European Journal of Operational Research, Elsevier, vol. 151(1), pages 140-152, November.
    13. de Koster, M.B.M. & Le-Duc, T. & Roodbergen, K.J., 2006. "Design and Control of Warehouse Order Picking: a literature review," ERIM Report Series Research in Management ERS-2006-005-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    14. de Koster, Rene & Le-Duc, Tho & Roodbergen, Kees Jan, 2007. "Design and control of warehouse order picking: A literature review," European Journal of Operational Research, Elsevier, vol. 182(2), pages 481-501, October.
    15. J. P. van der Gaast & M. B. M. de Koster & I. J. B. F. Adan, 2018. "Conveyor Merges in Zone Picking Systems: A Tractable and Accurate Approximate Model," Service Science, INFORMS, vol. 52(6), pages 1428-1443, December.
    16. van der Gaast, Jelmer Pier & Weidinger, Felix, 2022. "A deep learning approach for the selection of an order picking system," European Journal of Operational Research, Elsevier, vol. 302(2), pages 530-543.
    17. N Anken & J-P Gagliardi & J Renaud & A Ruiz, 2011. "Space allocation and aisle positioning for an industrial pick-to-belt system," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 62(1), pages 38-49, January.
    18. Jelmer P. van der Gaast & René B. M. de Koster & Ivo J. B. F. Adan & Jacques A. C. Resing, 2020. "Capacity Analysis of Sequential Zone Picking Systems," Operations Research, INFORMS, vol. 68(1), pages 161-179, January.
    19. Raji Alahmad & Kazuo Ishii, 2021. "A Puzzle-Based Sequencing System for Logistics Items," Logistics, MDPI, vol. 5(4), pages 1-18, October.

    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:83:y:2016:i:1:d:10.1007_s00186-015-0518-9. 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.