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An EDA for the 2D knapsack problem with guillotine constraint

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Abstract

In this paper we present an evolutionary heuristic for the 2D knapsack problem with guillotine constraint. In this problem we have a set of rectangles and there is a profit for each rectangle. The goal is to cut a subset of rectangles without overlap from a rectangular strip of width W and height H, so that the total profit of the rectangles from the subset is maximal. The sides of the rectangles are parallel to the strip sides and every cutting is restricted by orthogonal guillotine-cuts. A guillotine-cut is parallel to the horizontal or vertical side of the strip and cuts the strip into two separated rectangular strips. Our algorithm is an estimation of distribution algorithm (EDA), where recombination and mutation evolutionary operators are replaced by probability estimation and sampling techniques. Our EDA works with two probability models. It improves the quality of the solutions with local search procedures. The algorithm was tested on well-known benchmark instances from the literature.

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Authors and Affiliations

  1. Faculty of Business Economics, University of Pecs, Pecs, Hungary

    István Borgulya

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  1. István Borgulya
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Correspondence to István Borgulya.

Appendix

Appendix

In the appendix the tables give the results of the test sets. In the tables we see the names of the instances or instance groups, the number of types of rectangles (m) the number of rectangles (n), the optimum or upper bound (opt/upper), the number of instances where the optimal solutions were found in an instance group (#opt) or the number of optimal solutions found at an instance in 10 run (Hits), the average best result in gap % of an instance group or the best profit found at an instance.

See Tables 3, 4, 5, 6, 7, 8, 9 and 10.

Table 3 The results of 2DKEDA on set1
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Table 4 The results of 2DKEDA on set2
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Table 5 The results of 2DKEDA on set3
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Table 6 The results of 2DKEDA on set5
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Table 7 The results of 2DKEDA on set6
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Table 8 The average best results (gap %) of 2DKEDA on set7 (ngcutfs1)
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Table 9 The average best results (gap %) of 2DKEDA on set7 (ngcutfs2)
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Table 10 The average best results (gap %) of 2DKEDA on set7 (ngcutfs3)
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Borgulya, I. An EDA for the 2D knapsack problem with guillotine constraint. Cent Eur J Oper Res 27, 329–356 (2019). https://doi.org/10.1007/s10100-018-0551-x

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