On the hardness of palletizing bins using FIFO queues

Abstract

We study the combinatorial FIFO Stack-Up problem. In the delivery industry, bins have to be moved from conveyor belts onto pallets. Given is a list Q of k sequences of labeled bins and a positive integer p. The goal is to palletize the bins by iteratively removing the first bin of one of the k sequences and putting it onto a pallet located at one of p stack-up places. Each of these pallets has to contain bins of only one label, bins of different labels have to be placed on different pallets. After all bins of one label have been removed from the given sequences, the corresponding stack-up place becomes available for a pallet of bins of another label. The FIFO Stack-Up problem asks whether such a processing is possible. The FIFO Stack-Up problem is known to be NP-complete even if the sequences of Q contain together at most cQ=6 bins per pallet. This implies that the problem is also hard if all bins of every pallet are distributed to at most dQ=6 sequences. In this paper we strengthen the hardness to the cases cQ=5 and dQ=3. We also show the hardness for the practical case where the number of sequences is smaller than the number of pallets, and we point out differences to related problems like 3D Bin Packing and Train Marshalling.