Advertisement

OR Spectrum

, Volume 39, Issue 2, pp 541–556 | Cite as

Stabilized column generation for the temporal knapsack problem using dual-optimal inequalities

  • Timo Gschwind
  • Stefan Irnich
Regular Article

Abstract

We present two new methods to stabilize column-generation algorithms for the temporal knapsack problem (TKP). Caprara et al. (INFORMS J Comp 25(3):560–571, 2013] were the first to suggest the use of branch-and-price algorithms for Dantzig–Wolfe reformulations of the TKP. Herein, the respective pricing problems are smaller-sized TKP that can be solved with a general-purpose MIP solver or by dynamic programming. Our stabilization methods are tailored to the TKP as they use (deep) dual-optimal inequalities, that is, inequalities known to be fulfilled by all (at least some) optimal dual solutions to the linear relaxation. Extensive computational tests reveal that both new stabilization techniques are helpful. Several previously unsolved instances are now solved to proven optimality.

Keywords

Column generation Dual inequalities Stabilization 

Notes

Acknowledgments

This research was funded by the Deutsche Forschungsgemeinschaft (DFG) under Grant No. IR 122/6-1.

Supplementary material

291_2016_463_MOESM1_ESM.pdf (254 kb)
Supplementary material 1 (pdf 253 KB)

References

  1. Bartlett M, Frisch A, Hamadi Y, Miguel I, Tarim S, Unsworth C (2005) The temporal knapsack problem and its solution. In: Barták R, Milano M (eds) Integration of AI and OR techniques in constraint programming for combinatorial optimization problems, Lecture notes in computer science, vol 3524, Springer, Berlin, pp 34–48, doi: 10.1007/11493853_5
  2. Ben Amor H, Desrosiers J, Valério de Carvalho JM (2006) Dual-optimal inequalities for stabilized column generation. Oper Res 54(3):454–463. doi: 10.1287/opre.1060.0278 CrossRefGoogle Scholar
  3. Caprara A, Malaguti E, Toth P (2011) A freight service design problem for a railway corridor. Transp Sci 45(2):147–162. doi: 10.1287/trsc.1100.0348 CrossRefGoogle Scholar
  4. Caprara A, Furini F, Malaguti E (2013) Uncommon Dantzig-Wolfe reformulation for the temporal knapsack problem. INFORMS J Comput 25(3):560–571CrossRefGoogle Scholar
  5. Caprara A, Furini F, Malaguti E, Traversi E (2016) Solving the temporal knapsack problem via recursive Dantzig–Wolfe reformulation. Inf Process Lett 116(5):379–386. doi: 10.1016/j.ipl.2016.01.008 CrossRefGoogle Scholar
  6. Desaulniers G, Desrosiers J, Solomon M (eds) (2005) Column generation. Springer, New YorkGoogle Scholar
  7. Desrosiers J, Gauthier JB, Lübbecke ME (2014) Row-reduced column generation for degenerate master problems. Eur J Oper Res 236(2):453–460. doi: 10.1016/j.ejor.2013.12.016 CrossRefGoogle Scholar
  8. Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91(2):201–213. doi: 10.1007/s101070100263 CrossRefGoogle Scholar
  9. du Merle O, Villeneuve D, Desrosiers J, Hansen P (1999) Stabilized column generation. Discrete Math 194:229–237CrossRefGoogle Scholar
  10. Gauthier JB, Desrosiers J, Lübbecke ME (2016) Tools for primal degenerate linear programs. EURO J Transport Logist 5(2):161–204. doi: 10.1007/s13676-015-0077-5
  11. Gilmore P, Gomory R (1961) A linear programming approach to the cutting-stock problem. Oper Res 9:849–859CrossRefGoogle Scholar
  12. Gschwind T, Irnich S (2016) Dual inequalities for stabilized column generation revisited. INFORMS J Comput 28(1):175–194CrossRefGoogle Scholar
  13. Hiriart-Urruty JB, Lemaréchal C (1993) Convex analysis and minimization algorithms, part 2: advanced theory and bundle methods, Grundlehren der mathematischen Wissenschaften, vol 306. Springer, BerlinGoogle Scholar
  14. Kellerer H, Pferschy U, Pisinger D (2004) Knapsack problems. Springer, BerlinCrossRefGoogle Scholar
  15. Lee C, Park S (2011) Chebyshev center based column generation. Discrete Appl Math 159(18):2251–2265. doi: 10.1016/j.dam.2011.08.009 CrossRefGoogle Scholar
  16. Lübbecke M, Desrosiers J (2005) Selected topics in column generation. Oper Res 53(6):1007–1023CrossRefGoogle Scholar
  17. Marsten R, Hogan W, Blankenship J (1975) The boxstep method for large-scale optimization. Oper Res 23:389–405CrossRefGoogle Scholar
  18. Poggi de Aragao M, Uchoa E (2003) Integer program reformulation for robust branch-and-cut-and-price algorithms. In: Proc. Conf. Math. Program in Rio: A Conference in Honour of Nelson Maculan, Rio de Janeiro, Brazil, pp 56–61Google Scholar
  19. Rousseau LM, Gendreau M, Feillet D (2007) Interior point stabilization for column generation. Oper Res Lett 35(5):660–668. doi: 10.1016/j.orl.2006.11.004 CrossRefGoogle Scholar
  20. Valério de Carvalho JM (2005) Using extra dual cuts to accelerate column generation. INFORMS J Comput 17(2):175–182CrossRefGoogle Scholar
  21. Vanderbeck F (2005) Implementing mixed integer column generation. In: Desaulniers G, Desrosiers J, Solomon M (eds) Column generation. Springer, New York, pp 331–358 (chap 12)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Chair of Logistics Management, Gutenberg School of Management and EconomicsJohannes Gutenberg University MainzMainzGermany

Personalised recommendations