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A comparative study of the arcflow model and the one-cut model for one-dimensional cutting stock problems

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  • Martinovic, J.
  • Scheithauer, G.
  • Valério de Carvalho, J.M.

Abstract

We consider the one-dimensional cutting stock problem which consists in determining the minimum number of given large stock rolls that has to be cut to satisfy the demands of certain smaller item lengths. Besides the standard pattern-based approach of Gilmore and Gomory, containing an exponential number of variables, several pseudo-polynomial formulations were proposed in the last decades. Much research has dealt with arcflow models, their relationship to the standard model, and possible reduction methods, whereas the one-cut approach has not attracted that much scientific interest yet.

Suggested Citation

  • Martinovic, J. & Scheithauer, G. & Valério de Carvalho, J.M., 2018. "A comparative study of the arcflow model and the one-cut model for one-dimensional cutting stock problems," European Journal of Operational Research, Elsevier, vol. 266(2), pages 458-471.
    • Handle: RePEc:eee:ejores:v:266:y:2018:i:2:p:458-471
    DOI: 10.1016/j.ejor.2017.10.008
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    References listed on IDEAS

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    1. Silva, Elsa & Alvelos, Filipe & Valério de Carvalho, J.M., 2010. "An integer programming model for two- and three-stage two-dimensional cutting stock problems," European Journal of Operational Research, Elsevier, vol. 205(3), pages 699-708, September.
    2. Nicos Christofides & Charles Whitlock, 1977. "An Algorithm for Two-Dimensional Cutting Problems," Operations Research, INFORMS, vol. 25(1), pages 30-44, 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. 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.
    5. Martinovic, J. & Scheithauer, G., 2016. "Integer linear programming models for the skiving stock problem," European Journal of Operational Research, Elsevier, vol. 251(2), pages 356-368.
    6. 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.
    7. Kallrath, Julia & Rebennack, Steffen & Kallrath, Josef & Kusche, Rüdiger, 2014. "Solving real-world cutting stock-problems in the paper industry: Mathematical approaches, experience and challenges," European Journal of Operational Research, Elsevier, vol. 238(1), pages 374-389.
    8. 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.
    9. 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.
    10. 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.
    11. John Martinovic & Guntram Scheithauer, 2016. "The proper relaxation and the proper gap of the skiving stock problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(3), pages 527-548, December.
    12. 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.
    13. Stadtler, Hartmut, 1990. "A one-dimensional cutting stock problem in the aluminium industry and its solution," European Journal of Operational Research, Elsevier, vol. 44(2), pages 209-223, January.
    14. Harald Dyckhoff, 1981. "A New Linear Programming Approach to the Cutting Stock Problem," Operations Research, INFORMS, vol. 29(6), pages 1092-1104, December.
    15. Alves, Claudio & Valerio de Carvalho, J.M., 2007. "Accelerating column generation for variable sized bin-packing problems," European Journal of Operational Research, Elsevier, vol. 183(3), pages 1333-1352, December.
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    Cited by:

    1. John Martinovic & Markus Hähnel & Guntram Scheithauer & Waltenegus Dargie & Andreas Fischer, 2019. "Cutting stock problems with nondeterministic item lengths: a new approach to server consolidation," 4OR, Springer, vol. 17(2), pages 173-200, June.
    2. Wang, Danni & Xiao, Fan & Zhou, Lei & Liang, Zhe, 2020. "Two-dimensional skiving and cutting stock problem with setup cost based on column-and-row generation," European Journal of Operational Research, Elsevier, vol. 286(2), pages 547-563.
    3. John Martinovic, 2022. "A note on the integrality gap of cutting and skiving stock instances," 4OR, Springer, vol. 20(1), pages 85-104, March.
    4. de Lima, Vinícius L. & Alves, Cláudio & Clautiaux, François & Iori, Manuel & Valério de Carvalho, José M., 2022. "Arc flow formulations based on dynamic programming: Theoretical foundations and applications," European Journal of Operational Research, Elsevier, vol. 296(1), pages 3-21.
    5. Dell’Amico, Mauro & Delorme, Maxence & Iori, Manuel & Martello, Silvano, 2019. "Mathematical models and decomposition methods for the multiple knapsack problem," European Journal of Operational Research, Elsevier, vol. 274(3), pages 886-899.
    6. Maxence Delorme & Manuel Iori, 2020. "Enhanced Pseudo-polynomial Formulations for Bin Packing and Cutting Stock Problems," INFORMS Journal on Computing, INFORMS, vol. 32(1), pages 101-119, January.
    7. John Martinovic & Markus Hähnel & Guntram Scheithauer & Waltenegus Dargie, 2022. "An introduction to stochastic bin packing-based server consolidation with conflicts," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(2), pages 296-331, July.
    8. John Martinovic & Guntram Scheithauer, 2018. "Combinatorial investigations on the maximum gap for skiving stock instances of the divisible case," Annals of Operations Research, Springer, vol. 271(2), pages 811-829, December.

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