Journal of Global Optimization

, Volume 74, Issue 3, pp 467–476 | Cite as

Large proper gaps in bin packing and dual bin packing problems

  • Vadim M. Kartak
  • Artem V. RipattiEmail author


We consider the one-dimensional skiving stock problem, also known as the dual bin packing problem, with the aim of maximizing the best known dual and proper dual gaps. We apply the methods of computational search of large gaps initially developed for the one-dimensional cutting stock problem, which is related to the bin packing problem. The best known dual gap is raised from 1.0476 to 1.1795. The proper dual gap is improved to 1.1319. We also apply a number of new heuristics developed for the skiving stock problem back to the cutting stock problem, raising the largest known proper gap from 1.0625 to 1.1.


Skiving stock problem Cutting stock problem Bin packing problem Dual bin packing problem Integer round-up property Equivalence of classes Proper gap 



The authors would like to thank Dresden Technical University for providing computing power and Damir Akhmetzyanov, who helped to improve English in this paper.


  1. 1.
    Alvim, A.C.F., Ribeiro, C.C., Glover, F., Aloise, D.J.: A hybrid improvement heuristic for the one-dimensional bin packing problem. J. Heuristics 10(2), 205–229 (2004)CrossRefGoogle Scholar
  2. 2.
    Assmann, S.F., Johnson, D.S., Kleitman, D.J., Leung, J.Y.T.: On a dual version of the one-dimensional bin packing problem. J. Algorithms 5(4), 502–525 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Baum, S., Trotter Jr., L.: Integer rounding for polymatroid and branching optimization problems. SIAM J. Algebr. Discret. Methods 2(4), 416–425 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bruno, J., Downey, P.: Probabilistic bounds for dual bin-packing. Acta Inform. 22(3), 333–345 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Caprara, A., Dell’Amico, M., Díaz, J.C.D., Iori, M., Rizzi, R.: Friendly bin packing instances without integer round-up property. Math. Program. 150(1), 5–17 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Csirik, J., Frenk, J.B.G., Galambos, G., Kan, A.H.G.R.: Probabilistic analysis of algorithms for dual bin packing problems. J. Algorithms 12(2), 189–203 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Gilmore, P., Gomory, R.: A linear programming approach to the cutting-stock problem. Oper. Res. 9(6), 849–859 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Kartak, V.M., Ripatti, A.V., Scheithauer, G., Kurz, S.: Minimal proper non-IRUP instances of the one-dimensional cutting stock problem. Discret. Appl. Math. 187, 120–129 (2015). (Complete) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Labbé, M., Laporte, G., Martello, S.: An exact algorithm for the dual bin packing problem. Oper. Res. Lett. 17(1), 9–18 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Marcotte, O.: The cutting stock problem and integer rounding. Math. Program. 33(1), 82–92 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Marcotte, O.: An instance of the cutting stock problem for which the rounding property does not hold. Oper. Res. Lett. 4(5), 239–243 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Martinovic, J., Scheithauer, G.: Integer rounding and modified integer rounding for the skiving stock problem. Discret. Optim. 21, 118–130 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Martinovic, J., Scheithauer, G.: The proper relaxation and the proper gap of the skiving stock problem. Math. Meth. Oper. Res. 84(3), 527–548 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Nitsche, C., Scheithauer, G., Terno, J.: Tighter relaxations for the cutting stock problem. Eur. J. Oper. Res. 112(3), 654–663 (1999)zbMATHCrossRefGoogle Scholar
  15. 15.
    Peeters, M.: One dimensional cutting and packing: new problems and algorithms. Ph.D. thesis, Katholieke Universiteit Leuven (2002)Google Scholar
  16. 16.
    Peeters, M., Degraeve, Z.: Branch-and-price algorithms for the dual bin packing and maximum cardinality bin packing problem. Eur. J. Oper. Res. 170(2), 416–439 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Rietz, J., Dempe, S.: Large gaps in one-dimensional cutting stock problems. Discret. Appl. Math. 156(10), 1929–1935 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Scheithauer, G., Terno, J.: About the gap between the optimal values of the integer and continuous relaxation one-dimensional cutting stock problem. In: Operations Research Proceedings 1991, pp. 439–444. Springer (1992)Google Scholar
  19. 19.
    Scheithauer, G., Terno, J.: The modified integer round-up property of the one-dimensional cutting stock problem. Eur. J. Oper. Res. 84(3), 562–571 (1995)zbMATHCrossRefGoogle Scholar
  20. 20.
    Scheithauer, G., Terno, J.: Theoretical investigations on the modified integer round-up property for the one-dimensional cutting stock problem. Oper. Res. Lett. 20(2), 93–100 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Stanley, R.P.: Weyl groups, the hard Lefschetz theorem, and the Sperner property. SIAM J. Matrix Anal. Appl. 1, 168–184 (1980)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Sullivan, B.: On a conjecture of Andrica and Tomescu. J. Integer Seq. 16, 3 (2013). (Article 13.3.1) MathSciNetzbMATHGoogle Scholar
  23. 23.
    Vijayakumar, B., Parikh, P.J., Scott, R., Barnes, A., Gallimore, J.: A dual bin-packing approach to scheduling surgical cases at a publicly-funded hospital. Eur. J. Oper. Res. 224(3), 583–591 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Zak, E.J.: The skiving stock problem as a counterpart of the cutting stock problem. Int. Trans. Oper. Res. 10(6), 637–650 (2003)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Bashkir State Pedagogical University named after M. AkmullahUfaRussia
  2. 2.Ufa State Aviation Technical UniversityUfaRussia

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