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Bin packing and cutting stock problems: Mathematical models and exact algorithms

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  • Delorme, Maxence
  • Iori, Manuel
  • Martello, Silvano

Abstract

We review the most important mathematical models and algorithms developed for the exact solution of the one-dimensional bin packing and cutting stock problems, and experimentally evaluate, on state-of-the art computers, the performance of the main available software tools.

Suggested Citation

  • Delorme, Maxence & Iori, Manuel & Martello, Silvano, 2016. "Bin packing and cutting stock problems: Mathematical models and exact algorithms," European Journal of Operational Research, Elsevier, vol. 255(1), pages 1-20.
    • Handle: RePEc:eee:ejores:v:255:y:2016:i:1:p:1-20
    DOI: 10.1016/j.ejor.2016.04.030
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    as
    1. Valerio de Carvalho, J. M., 2002. "LP models for bin packing and cutting stock problems," European Journal of Operational Research, Elsevier, vol. 141(2), pages 253-273, September.
    2. Michael Trick, 2003. "A Dynamic Programming Approach for Consistency and Propagation for Knapsack Constraints," Annals of Operations Research, Springer, vol. 118(1), pages 73-84, February.
    3. WOLSEY, Laurence A., 1977. "Valid inequalities, covering problems and discrete dynamic programs," LIDAM Reprints CORE 302, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Scholl, Armin & Klein, Robert & Jürgens, Christian, 1996. "BISON : a fast hybrid procedure for exactly solving the one-dimensional bin packing problem," Publications of Darmstadt Technical University, Institute for Business Studies (BWL) 49135, Darmstadt Technical University, Department of Business Administration, Economics and Law, Institute for Business Studies (BWL).
    5. Crainic, Teodor Gabriel & Perboli, Guido & Pezzuto, Miriam & Tadei, Roberto, 2007. "Computing the asymptotic worst-case of bin packing lower bounds," European Journal of Operational Research, Elsevier, vol. 183(3), pages 1295-1303, December.
    6. François Vanderbeck, 2000. "On Dantzig-Wolfe Decomposition in Integer Programming and ways to Perform Branching in a Branch-and-Price Algorithm," Operations Research, INFORMS, vol. 48(1), pages 111-128, February.
    7. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(4), pages 691-705, August.
    8. Wascher, Gerhard & Hau[ss]ner, Heike & Schumann, Holger, 2007. "An improved typology of cutting and packing problems," European Journal of Operational Research, Elsevier, vol. 183(3), pages 1109-1130, December.
    9. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(1), pages 225-228, February.
    10. Hatem Ben Amor & José Valério de Carvalho, 2005. "Cutting Stock Problems," Springer Books, in: Guy Desaulniers & Jacques Desrosiers & Marius M. Solomon (ed.), Column Generation, chapter 0, pages 131-161, Springer.
    11. Krzysztof C. Kiwiel, 2010. "An Inexact Bundle Approach to Cutting-Stock Problems," INFORMS Journal on Computing, INFORMS, vol. 22(1), pages 131-143, February.
    12. L. R. Ford, Jr. & D. R. Fulkerson, 1958. "A Suggested Computation for Maximal Multi-Commodity Network Flows," Management Science, INFORMS, vol. 5(1), pages 97-101, October.
    13. Robert W. Haessler, 1975. "Controlling Cutting Pattern Changes in One-Dimensional Trim Problems," Operations Research, INFORMS, vol. 23(3), pages 483-493, June.
    14. Fleszar, Krzysztof & Charalambous, Christoforos, 2011. "Average-weight-controlled bin-oriented heuristics for the one-dimensional bin-packing problem," European Journal of Operational Research, Elsevier, vol. 210(2), pages 176-184, April.
    15. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(5), pages 777-788, October.
    16. P. C. Gilmore & R. E. Gomory, 1961. "A Linear Programming Approach to the Cutting-Stock Problem," Operations Research, INFORMS, vol. 9(6), pages 849-859, December.
    17. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(1), pages 151-160, February.
    18. Belov, G. & Scheithauer, G., 2002. "A cutting plane algorithm for the one-dimensional cutting stock problem with multiple stock lengths," European Journal of Operational Research, Elsevier, vol. 141(2), pages 274-294, September.
    19. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(5), pages 879-883, October.
    20. L. V. Kantorovich, 1960. "Mathematical Methods of Organizing and Planning Production," Management Science, INFORMS, vol. 6(4), pages 366-422, July.
    21. Samuel Eilon & Nicos Christofides, 1971. "The Loading Problem," Management Science, INFORMS, vol. 17(5), pages 259-268, January.
    22. Harald Dyckhoff, 1981. "A New Linear Programming Approach to the Cutting Stock Problem," Operations Research, INFORMS, vol. 29(6), pages 1092-1104, December.
    23. Gleb Belov & Guntram Scheithauer & Cláudio Alves & José M. Valério de Carvalho, 2008. "Gomory Cuts from a Position-Indexed Formulation of 1D Stock Cutting," Springer Books, in: Andreas Bortfeldt & Jörg Homberger & Herbert Kopfer & Giselher Pankratz & Reinhard Strangmeier (ed.), Intelligent Decision Support, pages 3-14, Springer.
    24. G Scheithauer & J Terno & A Müller & G Belov, 2001. "Solving one-dimensional cutting stock problems exactly with a cutting plane algorithm," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 52(12), pages 1390-1401, December.
    25. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(2), pages 411-413, April.
    26. Dyckhoff, Harald, 1990. "A typology of cutting and packing problems," European Journal of Operational Research, Elsevier, vol. 44(2), pages 145-159, January.
    27. Unknown, 1986. "Letters," Choices: The Magazine of Food, Farm, and Resource Issues, Agricultural and Applied Economics Association, vol. 1(4), pages 1-9.
    28. Elhedhli, Samir, 2005. "Ranking lower bounds for the bin-packing problem," European Journal of Operational Research, Elsevier, vol. 160(1), pages 34-46, January.
    29. Haessler, Robert W. & Sweeney, Paul E., 1991. "Cutting stock problems and solution procedures," European Journal of Operational Research, Elsevier, vol. 54(2), pages 141-150, September.
    30. Martine Labbé & Gilbert Laporte & Hélène Mercure, 1991. "Capacitated Vehicle Routing on Trees," Operations Research, INFORMS, vol. 39(4), pages 616-622, August.
    31. Lodi, Andrea & Martello, Silvano & Monaci, Michele, 2002. "Two-dimensional packing problems: A survey," European Journal of Operational Research, Elsevier, vol. 141(2), pages 241-252, September.
    32. Philipp Rohlfshagen & John Bullinaria, 2010. "Nature inspired genetic algorithms for hard packing problems," Annals of Operations Research, Springer, vol. 179(1), pages 393-419, September.
    33. Hatem Ben Amor & Jacques Desrosiers & José Manuel Valério de Carvalho, 2006. "Dual-Optimal Inequalities for Stabilized Column Generation," Operations Research, INFORMS, vol. 54(3), pages 454-463, June.
    34. François Clautiaux & Cláudio Alves & José Valério de Carvalho, 2010. "A survey of dual-feasible and superadditive functions," Annals of Operations Research, Springer, vol. 179(1), pages 317-342, September.
    35. George B. Dantzig & Philip Wolfe, 1960. "Decomposition Principle for Linear Programs," Operations Research, INFORMS, vol. 8(1), pages 101-111, February.
    36. A. A. Farley, 1990. "A Note on Bounding a Class of Linear Programming Problems, Including Cutting Stock Problems," Operations Research, INFORMS, vol. 38(5), pages 922-923, October.
    37. Goulimis, Constantine, 1990. "Optimal solutions for the cutting stock problem," European Journal of Operational Research, Elsevier, vol. 44(2), pages 197-208, January.
    38. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(3), pages 427-432, June.
    39. François Clautiaux & Cláudio Alves & José Valério de Carvalho & Jürgen Rietz, 2011. "New Stabilization Procedures for the Cutting Stock Problem," INFORMS Journal on Computing, INFORMS, vol. 23(4), pages 530-545, November.
    40. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(4), pages 629-637, August.
    41. Belov, G. & Scheithauer, G., 2006. "A branch-and-cut-and-price algorithm for one-dimensional stock cutting and two-dimensional two-stage cutting," European Journal of Operational Research, Elsevier, vol. 171(1), pages 85-106, May.
    42. Mauro Dell’Amico & Silvano Martello, 1995. "Optimal Scheduling of Tasks on Identical Parallel Processors," INFORMS Journal on Computing, INFORMS, vol. 7(2), pages 191-200, May.
    43. Gau, T. & Wascher, G., 1995. "CUTGEN1: A problem generator for the standard one-dimensional cutting stock problem," European Journal of Operational Research, Elsevier, vol. 84(3), pages 572-579, August.
    44. Scheithauer, Guntram & Terno, Johannes, 1995. "The modified integer round-up property of the one-dimensional cutting stock problem," European Journal of Operational Research, Elsevier, vol. 84(3), pages 562-571, August.
    45. Holthaus, Oliver, 2002. "Decomposition approaches for solving the integer one-dimensional cutting stock problem with different types of standard lengths," European Journal of Operational Research, Elsevier, vol. 141(2), pages 295-312, September.
    46. Marco E. Lübbecke & Jacques Desrosiers, 2005. "Selected Topics in Column Generation," Operations Research, INFORMS, vol. 53(6), pages 1007-1023, December.
    47. Nitsche, Christoph & Scheithauer, Guntram & Terno, Johannes, 1999. "Tighter relaxations for the cutting stock problem," European Journal of Operational Research, Elsevier, vol. 112(3), pages 654-663, February.
    48. Kurt Eisemann, 1957. "The Trim Problem," Management Science, INFORMS, vol. 3(3), pages 279-284, April.
    49. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(6), pages 1195-1198, December.
    50. J Levine & F Ducatelle, 2004. "Ant colony optimization and local search for bin packing and cutting stock problems," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 55(7), pages 705-716, July.
    51. Bassem Jarboui & Saber Ibrahim & Abdelwaheb Rebai, 2010. "A new destructive bounding scheme for the bin packing problem," Annals of Operations Research, Springer, vol. 179(1), pages 187-202, September.
    52. Egon Balas, 1965. "An Additive Algorithm for Solving Linear Programs with Zero-One Variables," Operations Research, INFORMS, vol. 13(4), pages 517-546, August.
    53. Zeger Degraeve & Marc Peeters, 2003. "Optimal Integer Solutions to Industrial Cutting-Stock Problems: Part 2, Benchmark Results," INFORMS Journal on Computing, INFORMS, vol. 15(1), pages 58-81, February.
    54. Zeger Degraeve & Linus Schrage, 1999. "Optimal Integer Solutions to Industrial Cutting Stock Problems," INFORMS Journal on Computing, INFORMS, vol. 11(4), pages 406-419, November.
    55. P. C. Gilmore & R. E. Gomory, 1965. "Multistage Cutting Stock Problems of Two and More Dimensions," Operations Research, INFORMS, vol. 13(1), pages 94-120, February.
    56. Vahrenkamp, Richard, 1996. "Random search in the one-dimensional cutting stock problem," European Journal of Operational Research, Elsevier, vol. 95(1), pages 191-200, November.
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