Exploiting Packing Components in General-Purpose Integer Programming Solvers
- 825 Downloads
The problem of packing boxes into a large box is often only a part of a complex problem. For example in furniture supply chain applications, one needs to decide what trucks to use to transport furniture between production sites and distribution centres and stores, such that the furniture fits inside. Such problems are often formulated and sometimes solved using general-purpose integer programming solvers.
This chapter studies the problem of identifying a compact formulation of the multi-dimensional packing component in a general instance of integer linear programming, reformulating it using the discretisation of Allen–Burke–Mareček, and solving the extended reformulation. Results on instances of up to 10,000,000 boxes are reported.
KeywordsPacking Multi-dimensional packing Integer programming Structure exploitation
The views expressed in this chapter are personal views of the author and should not be construed as suggestions as to the product road map of IBM products. Some of the supporting code has been developed by Allen  for  and can be downloaded at http://discretisation.sf.net (September 30th, 2014). This material is loosely based upon an otherwise unpublished Chapter 8 of the dissertation of , but has been extended and improved greatly thanks to the comments of two anonymous referees, both in terms of the contents and the presentation. The author is most grateful for the referees’ thoughtful suggestions.
- 2.Allen, S.D.: Algorithms and data structures for three-dimensional packing. Ph.D. thesis, University of Nottingham (2012)Google Scholar
- 4.Amor, H.B., de Carvalho, J.V.: Cutting stock problems. In: Column Generation, pp. 131–161. Springer, New York (2005)Google Scholar
- 7.Bienstock, D., Zuckerberg, M.: Solving lp relaxations of large-scale precedence constrained problems. In: Eisenbrand, F., Shepherd, F. (eds.) Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, vol. 6080, pp. 1–14. Springer, Berlin/Heidelberg (2010)CrossRefGoogle Scholar
- 10.de Queiroz, T.A., Miyazawa, F.K., Wakabayashi, Y., Xavier, E.C.: Algorithms for 3d guillotine cutting problems: unbounded knapsack, cutting stock and strip packing. Comput. Oper. Res. 39(2), 200–212 (2012)Google Scholar
- 11.Fasano, G.: Cargo analytical integration in space engineering: a three-dimensional packing model. In: Ciriani, T.A., Gliozzi, S., Johnson, E.L., Tadei, R. (eds.) Operational Research in Industry, pp. 232–246. Purdue University Press, West Lafayette, IN (1999)Google Scholar
- 14.Fasano, G.: Erratum to: Chapter 3 model reformulations and tightening. In: Solving Non-standard Packing Problems by Global Optimization and Heuristics. Springer Briefs in Optimization, pp. E1–E2. Springer, New York (2014)Google Scholar
- 15.Fasano, G.: Model reformulations and tightening. In: Solving Non-standard Packing Problems by Global Optimization and Heuristics. Springer Briefs in Optimization, pp. 27–37. Springer, New York (2014)Google Scholar
- 16.Fasano, G.: Tetris-like items. In: Solving Non-standard Packing Problems by Global Optimization and Heuristics. Springer Briefs in Optimization, pp. 7–26. Springer, New York (2014)Google Scholar
- 23.Junqueira, L., Morabito, R., Yamashita, D.S., Yanasse, H.H.: Optimization models for the three-dimensional container loading problem with practical constraints. In: Fasano, G., Pintér, János D. (eds.) Modeling and Optimization in Space Engineering. Springer Optimization and Its Applications, vol.73, pp. 271–293. Springer, New York (2013)Google Scholar
- 26.Mareček, J.: Exploiting structure in integer programs. Ph.D. thesis, University of Nottingham (2012)Google Scholar
- 28.Moreno, E., Espinoza, D., Goycoolea, M.: Large-scale multi-period precedence constrained knapsack problem: a mining application. Electron. Notes Discret. Math. 36, 407–414 (2010) [ISCO 2010 - International Symposium on Combinatorial Optimization]Google Scholar
- 30.Pan, Y., Shi, L.: On the equivalence of the max-min transportation lower bound and the time-indexed lower bound for single-machine scheduling problems. Math. Program. 110(3, Ser. A), 543–559 (2007)Google Scholar
- 34.van den Akker, J.M.: LP-based solution methods for single-machine scheduling problems. Dissertation, Technische Universiteit Eindhoven, Eindhoven (1994)Google Scholar