2 research outputs found
Integer Programming Models and Parameterized Algorithms for Controlling Palletizers
We study the combinatorial FIFO Stack-Up problem, where 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. The FIFO Stack-Up problem asks whether
there is some processing of the sequences of bins such that at most p stack-up
places are used. In this paper we strengthen the hardness of the FIFO Stack-Up
by considering practical cases and the distribution of the pallets onto the
sequences. We introduce a digraph model for this problem, the so called
decision graph, which allows us to give a breadth first search solution.
Further we apply methods to solve hard problems to the FIFO Stack-Up problem.
In order to evaluate our algorithms, we introduce a method to generate random,
but realistic instances for the FIFO Stack-Up problem. Our experimental study
of running times shows that the breadth first search solution on the decision
graph combined with a cutting technique can be used to solve practical
instances on several thousands of bins of the FIFO Stack-Up problem. Further we
analyze two integer programming approaches implemented in CPLEX and GLPK. As
expected CPLEX can solve the instances much faster than GLPK and our pallet
solution approach is much better than the bin solution approach.Comment: 27 pages, 7 figures. arXiv admin note: text overlap with
arXiv:1307.191
Complexity of the FIFO Stack-Up Problem
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, ..., 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.Comment: 18 pages, 7 figure