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On the complexity of the FIFO stack-up problem

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  • Frank Gurski
  • Jochen Rethmann
  • Egon Wanke

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. Copyright Springer-Verlag Berlin Heidelberg 2016

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:p:33-52
    DOI: 10.1007/s00186-015-0518-9
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    References listed on IDEAS

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    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. 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.
    3. 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.
    4. 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.
    5. 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.
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    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.

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