An approximation algorithm for the stack-up problem

Rethmann, Jochen; Wanke, Egon

doi:10.1007/s001860050085

Cited by 6 publications

(6 citation statements)

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“…Many facts are known about single-line stack-up systems [26,27,28]. In [26] it is shown that the single-line stack-up decision problem is NP-complete, but can be solved efficiently if the storage capacity s or the number of available stack-up places p is a fixed constant.…”

Section: Stack−up Places Storage Area Main Conveyor Beltmentioning

confidence: 99%

“…In [26] it is shown that the single-line stack-up decision problem is NP-complete, but can be solved efficiently if the storage capacity s or the number of available stack-up places p is a fixed constant. The problem remains NP-complete as shown in [27], even if the sequence contains at most 9 bins per pallet. In [27], a polynomial-time off-line approximation algorithm for minimizing the storage capacity s is introduced.…”

Section: Stack−up Places Storage Area Main Conveyor Beltmentioning

confidence: 99%

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Integer Programming Models and Parameterized Algorithms for Controlling Palletizers

Gurski¹,

Rethmann²,

Wanke³

2015

Preprint

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We study the combinatorial FIFO Stack-Up problem, where bins have to be stacked-up from conveyor belts onto pallets. This is done by palletizers or palletizing robots. 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 stack-up place becomes available for a pallet of bins of another label. All bins have the same size. 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 shown in [20] 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 of running time O(n 2 •(m+2) k ), where m represents the number of pallets and n denotes the total number of bins in all sequences. Further we apply methods to solve hard problems to the FIFO Stack-Up problem. Therefor we consider restricted versions of the problem, two integer programming models, exponential time algorithms, parameterized algorithms, and approximation algorithms. 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 our 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.

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Section: Stack−up Places Storage Area Main Conveyor Beltmentioning

confidence: 99%

Section: Stack−up Places Storage Area Main Conveyor Beltmentioning

confidence: 99%

Integer Programming Models and Parameterized Algorithms for Controlling Palletizers

Gurski¹,

Rethmann²,

Wanke³

2015

Preprint

Self Cite

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“…Many facts are known about single-line stack-up systems [12,13,14]. In [12] it is shown that the single-line stack-up decision problem is NP-complete, but can be solved efficiently if the storage capacity s or the number of available stack-up places p is fixed.…”

Section: Introductionmentioning

confidence: 99%

“…In [12] it is shown that the single-line stack-up decision problem is NP-complete, but can be solved efficiently if the storage capacity s or the number of available stack-up places p is fixed. The problem remains NP-complete as shown in [13], even if the sequence contains at most 9 bins per pallet. In [13], a polynomial-time off-line approximation algorithm for minimizing the storage capacity s is introduced.…”

Section: Introductionmentioning

confidence: 99%

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On the complexity of the FIFO stack-up problem

2015

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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.

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