Exploiting Packing Components in General-Purpose Integer Programming Solvers

  • Jakub MarečekEmail author
Part of the Springer Optimization and Its Applications book series (SOIA, volume 105)


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.


Packing 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 [2] for [3] and can be downloaded at (September 30th, 2014). This material is loosely based upon an otherwise unpublished Chapter  8 of the dissertation of [26], but has been extended and improved greatly thanks to the comments of two anonymous referees, both in terms of the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.


  1. 1.
    Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1(1), 1–41 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Allen, S.D.: Algorithms and data structures for three-dimensional packing. Ph.D. thesis, University of Nottingham (2012)Google Scholar
  3. 3.
    Allen, S.D., Burke, E.K., Mareček, J.: A space-indexed formulation of packing boxes into a larger box. Oper. Res. Lett. 40, 20–24 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Amor, H.B., de Carvalho, J.V.: Cutting stock problems. In: Column Generation, pp. 131–161. Springer, New York (2005)Google Scholar
  5. 5.
    Baldi, M.M., Perboli, G., Tadei, R.: The three-dimensional knapsack problem with balancing constraints. Appl. Math. Comput. 218(19), 9802–9818 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Beasley, J.E.: An exact two-dimensional non-guillotine cutting tree search procedure. Oper. Res. 33(1), 49–64 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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
  8. 8.
    Chen, C.S., Lee, S.M., Shen, Q.S.: An analytical model for the container loading problem. Eur. J. Oper. Res. 80(1), 68–76 (1995)CrossRefzbMATHGoogle Scholar
  9. 9.
    Chlebík, M., Chlebíková, J.: Hardness of approximation for orthogonal rectangle packing and covering problems. J. Discrete Algorithms 7(3), 291–305 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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. 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
  12. 12.
    Fasano, G.: A mip approach for some practical packing problems: balancing constraints and tetris-like items. Q. J. Belg. Fr. Ital. Oper. Res. Soc. 2(2), 161–174 (2004)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Fasano, G.: Mip-based heuristic for non-standard 3d-packing problems. Q. J. Belg. Fr. Ital. Oper. Res. Soc. 6(3), 291–310 (2008)MathSciNetzbMATHGoogle Scholar
  14. 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. 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. 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
  17. 17.
    Fasano, G., Pintér, J.: Modeling and Optimization in Space Engineering. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  18. 18.
    Gendreau, M., Iori, M., Laporte, G., Martello, S.: A tabu search algorithm for a routing and container loading problem. Transp. Sci. 40(3), 342–350 (2006)CrossRefGoogle Scholar
  19. 19.
    Gilmore, P.C., Gomory, R.E.: Multistage cutting stock problems of two and more dimensions. Oper. Res. 13(1), 94–120 (1965)CrossRefzbMATHGoogle Scholar
  20. 20.
    Herz, J.C.: Recursive computational procedure for two-dimensional stock cutting. IBM J. Res. Dev. 16(5), 462–469 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Iori, M., Salazar-González, J.J., Vigo, D.: An exact approach for the vehicle routing problem with two-dimensional loading constraints. Transp. Sci. 41(2), 253–264 (2007)CrossRefGoogle Scholar
  22. 22.
    Junqueira, L., Morabito, R., Yamashita, D.S.: Three-dimensional container loading models with cargo stability and load bearing constraints. Comput. Oper. Res. 39(1), 74–85 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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
  24. 24.
    Koch, T., Achterberg, T., Andersen, E., Bastert, O., Berthold, T., Bixby, R.E., Danna, E., Gamrath, G., Gleixner, A.M., Heinz, S., Lodi, A., Mittelmann, H., Ralphs, T., Salvagnin, D., Steffy, D.E., Wolter, K.: MIPLIB 2010. Math. Program. Comput. 3(2), 103–163 (2011)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Madsen, O.B.G.: Glass cutting in a small firm. Math. Program. 17(1), 85–90 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mareček, J.: Exploiting structure in integer programs. Ph.D. thesis, University of Nottingham (2012)Google Scholar
  27. 27.
    Martello, S., Pisinger, D., Vigo, D.: The three-dimensional bin packing problem. Oper. Res. 48(2), 256–267 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 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
  29. 29.
    Padberg, M.: Packing small boxes into a big box. Math. Meth. Oper. Res. 52(1), 1–21 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 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
  31. 31.
    Papadimitriou, C.H.: Worst-case and probabilistic analysis of a geometric location problem. SIAM J. Comput. 10(3), 542–557 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Papadimitriou, C.H.: On the complexity of unique solutions. J. Assoc. Comput. Mach. 31(2), 392–400 (1984)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Sousa, J.P., Wolsey, L.A.: A time indexed formulation of non-preemptive single machine scheduling problems. Math. Program. 54, 353–367 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    van den Akker, J.M.: LP-based solution methods for single-machine scheduling problems. Dissertation, Technische Universiteit Eindhoven, Eindhoven (1994)Google Scholar
  35. 35.
    Zemel, E.: Probabilistic analysis of geometric location problems. SIAM J. Algebraic Discret. Meth. 6(2), 189–200 (1985)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.IBM Research – IrelandDublin 15The Republic of Ireland

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