On the complexity of the FIFO stack-up problem

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 q1, . . . , 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.

Integer Programming Models and Parameterized Algorithms for Controlling Palletizers

“…In [20] we have shown the following correlation between the used number of stack-up places for a processing of an instance Q and the directed pathwidth of the sequence graph G Q .…”

“…That is, the number of stack-up places can be larger than or less than the number of buffer conveyors. We could show in [20] that the FIFO Stack-Up problem is NP-complete, but can be solved in polynomial time if the number k of sequences or the number p of stack-up places is fixed.…”

“…In Section 2, we give preliminaries and the problem statement. In Section 3, we recall the definition of a sequence graph from [20]. This digraph has a vertex for every pallet and an arc from pallet a to pallet b, if and only if in any processing pallet b can only be closed if pallet a has already been opened.…”

“…It has a vertex for every possible configuration of the system and an arc from configuration A to configuration B if configuration B can be obtained from configuration A by a single processing step. The algorithmic use of the processing graph was already mentioned in [20] and will be explained more in detail here in order to ease the understanding on the second digraph model, which is called the decision graph. It has a vertex for every decision configuration of the system, i.e.…”

“…Next we recall the connection between the used number of stack-up places for a processing of an instance Q and the directed pathwidth of the sequence graph G Q from [20], which is useful for our hardness results, our integer programming model for computing a pallet solution, and for several parameterized algorithms.…”