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 \(q_1, \ldots , q_k\) 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 \(p-1\). 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.
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Acknowledgments
We thank Prof. Dr. Christian Ewering who shaped our interest into controlling stack-up systems. Furthermore, we are very grateful to Bertelsmann Distribution GmbH in Gütersloh, Germany, for providing the possibility to get an insight into the real problematic nature and for providing real data instances.We would also like to thank the referees for their valuable comments and suggestions, which improved the presentation of this paper.
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A short version of this paper appeared in the Proceedings of the International Conference on Operations Research (OR 2013) (Gurski et al. 2014).
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Gurski, F., Rethmann, J. & Wanke, E. On the complexity of the FIFO stack-up problem. Math Meth Oper Res 83, 33–52 (2016). https://doi.org/10.1007/s00186-015-0518-9
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DOI: https://doi.org/10.1007/s00186-015-0518-9