Abstract
Scheduling over scenarios is one of the latest approaches in modelling scheduling problems including uncertainty. However, to the best of our knowledge, scenarios have never been applied to the bin packing problem, so here we introduce the bin packing problem with scenarios. In this model, we have a list of items with sizes between 0 and 1, and each item is assigned to one or more scenarios. In reality, the items of only one scenario will occur, but this chosen scenario is unknown at the time of packing, so the algorithms have to examine all scenarios. This means that the items have to be packed into bins such that for any scenario, the total size of the items in this scenario is at most 1 in each bin. The objective of the standard bin packing problem is to minimize the number of bins. Here, we introduce some extensions of the objective function to the scenario based model, and we present our competitive analysis of some online bin packing algorithms adapted to scenarios.
Similar content being viewed by others
References
Balogh J, Békési J, Galambos G (2012) New lower bounds for certain classes of bin packing algorithms. Theor Comput Sci 440:1–13
Ben-Tal A, Nemirovski A (2000) Robust solutions of linear programming problems contaminated with uncertain data. Math Prog 88(3):411–424
Bertsimas D (1988) Probabilistic combinatorial optimization problems. Ph.D. thesis, Massachusetts Institute of Technology
Coffman EG Jr, Csirik J, Galambos G, Martello S, Vigo D (2013) Bin packing approximation algorithms: survey and classification. In: Pardalos PM, Du DZ, Graham RL (eds) Handbook of combinatorial optimization. Springer, New York, pp 455–531
Dósa G, Sgall J (2013) First fit bin packing: a tight analysis. In: LIPIcs-Leibniz international proceedings in informatics, vol. 20. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik
Feuerstein E, Marchetti-Spaccamela A, Schalekamp F, Sitters R, van der Ster S, Stougie L, van Zuylen A (2017) Minimizing worst-case and average-case makespan over scenarios. J Sched 20:545. https://doi.org/10.1007/s10951-016-0484-y
Johnson DS (1974) Fast algorithms for bin packing. J Comput Syst Sci 8(3):272–314
Johnson DS, Demers A, Ullman JD, Garey MR, Graham RL (1974) Worst-case performance bounds for simple one-dimensional packing algorithms. SIAM J Comput 3(4):299–325
Kall P, Wallace SW, Kall P (1994) Stochastic programming. Springer, New York
Kasperski A, Zieliński P (2011) Bottleneck combinatorial optimization problems with fuzzy scenarios. In: Nonlinear mathematics for uncertainty and its applications, Springer, pp 197–204
Kasperski A, Zieliński P (2016) Single machine scheduling problems with uncertain parameters and the OWA criterion. J Schedul 19(2):177–190
Kasperski A, Kurpisz A, Zieliński P (2012) Parallel machine scheduling under uncertainty. In: International conference on information processing and management of uncertainty in knowledge-based systems, Springer, pp 74–83
Kasperski A, Kurpisz A, Zieliński P (2013) Approximating the min–max (regret) selecting items problem. Inf Process Lett 113(1):23–29
Kataoka S (1963) A stochastic programming model. Econometrica 31(1/2):181–196
Lee CC, Lee DT (1985) A simple on-line bin-packing algorithm. J ACM (JACM) 32(3):562–572
Prékopa A (2013) Stochastic programming, vol 324. Springer, New York
Rischke R (2014) Two-stage robust combinatorial optimization with priced scenarios. In: Operations research proceedings 2013, Springer, pp 377–382
Soyster AL (1973) Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper Res 21(5):1154–1157
Acknowledgements
We were proud to have the opportunity to work with Csanád Imreh, who originally came up with the idea of using scenarios in the bin packing problem, and started investigating this model. After his sudden unexpected death, we carried on researching this problem. It was a pleasure to have known such a devoted researcher. We are also grateful to János Csirik, who helped us and provided useful remarks and suggestions. János Balogh was supported by the European Union, co-financed by the European Social Fund (EFOP-3.6.3-VEKOP-16-2017-00002).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bódis, A., Balogh, J. Bin packing problem with scenarios. Cent Eur J Oper Res 27, 377–395 (2019). https://doi.org/10.1007/s10100-018-0574-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10100-018-0574-3