Directed pathwidth and palletizers

Abstract

In delivery industry, bins have to be stacked-up from conveyor belts onto pallets. Given k sequences of labeled bins and a positive integer p. The goal is to stack-up the bins by iteratively removing the first bin of one of the k sequences and put 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 place becomes available for a pallet of bins of another label. In this paper we introduce a graph model for this problem, the so called sequence graph, which allows us to 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.