Abstract
Bin packing with fragmentable items is a variant of the classic bin packing problem where items may be cut into smaller fragments. The objective is to minimize the number of item fragments, or equivalently, to minimize the number of cuts, for a given number of bins. Models based on packing fragmentable items are useful for representing finite shared resources. In this article, we present improvements to approximation and metaheuristic algorithms to obtain an optimality-preserving optimization algorithm with polynomial complexity, worst-case performance guarantees and parametrizable running time. We also present a new family of fast lower bounds and prove their worst-case performance ratios. We evaluate the performance and quality of the algorithm and the best lower bound through a series of computational experiments on representative problem instances. For the studied problem sets, one consisting of 180 problems with up to 20 items and another consisting of 450 problems with up to 1024 items, the lower bound performs no worse than 5 / 6. For the first problem set, the algorithm found an optimal solution in 92 % of all 1800 runs. For the second problem set, the algorithm found an optimal solution in 99 % of all 4500 runs. No run lasted longer than 220 ms.
Similar content being viewed by others
References
Byholm, B.: fragbinpacking-optimizer v1.3 (2017a). https://doi.org/10.5281/zenodo.1068975
Byholm, B.: fragbinpacking-problems v1.1 (2017b). https://doi.org/10.5281/zenodo.253942
Byholm, B.: fragbinpacking-results v1.3 (2017c). https://doi.org/10.5281/zenodo.1068972
Casazza, M., Ceselli, A.: Mathematical programming algorithms for bin packing problems with item fragmentation. Comput. Oper. Res. 46(C), 1–11 (2014). https://doi.org/10.1016/j.cor.2013.12.008
Casazza, M., Ceselli, A.: Exactly solving packing problems with fragmentation. Comput. Oper. Res. 75(C), 202–213 (2016). https://doi.org/10.1016/j.cor.2016.06.007
Falkenauer, E.: The grouping genetic algorithms—widening the scope of the GAs. Belg. J. Oper. Res. Stat. Comput. Sci. 33(1), 2 (1992)
Falkenauer, E.: A hybrid grouping genetic algorithm for bin packing. J. Heuristics 2(1), 5–30 (1996). https://doi.org/10.1007/BF00226291
Fekete, S.P., Schepers, J.: New classes of fast lower bounds for bin packing problems. Math. Program. 91(1), 11–31 (2001). https://doi.org/10.1007/s101070100243
Gajentaan, A., Overmars, M.H.: On a class of \({O}(n^2)\) problems in computational geometry. Comput. Geom. 45(4), 140–152 (2012). https://doi.org/10.1016/j.comgeo.2011.11.006
Garey, M.R., Johnson, D.S.: Computers and Intractability. W. H. Freeman & Co., New York (1990)
Jones, M.C.: Kumaraswamy’s distribution: a beta-type distribution with some tractability advantages. Stat. Methodol. 6(1), 70–81 (2009). https://doi.org/10.1016/j.stamet.2008.04.001
LeCun, B., Mautor, T., Quessette, F., Weisser, M.A.: Bin packing with fragmentable items: presentation and approximations. Theor. Comput. Sci. 602, 50–59 (2015). https://doi.org/10.1016/j.tcs.2015.08.005
Mandal, C.A., Chakrabarti, P.P., Ghose, S.: Complexity of fragmentable object bin packing and an application. Comput. Math. Appl. 35(11), 91–97 (1998). https://doi.org/10.1016/S0898-1221(98)00087-X
Quiroz Castellanos, M., Cruz Reyes, L., Torres Jiménez, J., Gómez Santillán, C., Fraire Huacuja, H.J., Alvim, A.C.: A grouping genetic algorithm with controlled gene transmission for the bin packing problem. Comput. Oper. Res. 55, 52–64 (2015). https://doi.org/10.1016/j.cor.2014.10.010
Acknowledgements
Benjamin Byholm received scholarships from the Nokia Foundation and the Finnish Foundation for Technology Promotion. This was part of the N4S project.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Byholm, B., Porres, I. Fast algorithms for fragmentable items bin packing. J Heuristics 24, 697–723 (2018). https://doi.org/10.1007/s10732-018-9375-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10732-018-9375-z