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On the facet defining inequalities of the mixed-integer bilinear covering set

Author

Listed:
  • Hamidur Rahman

    (Indian Institute of Technology Bombay)

  • Ashutosh Mahajan

    (Indian Institute of Technology Bombay)

Abstract

We study the facet defining inequalities of the convex hull of a mixed-integer bilinear covering arising in trim-loss (or cutting stock) problem under the framework of disjunctive cuts. We show that all of them can be derived using a disjunctive procedure. Some of these are split cuts of rank one for a convex mixed-integer relaxation of the covering set, while others have rank at least two. For certain linear objective functions, the rank-one split cuts are shown to be sufficient for finding the optimal value over the convex hull of the covering set. A relaxation of the trim-loss problem has this property, and our computational results show that these rank-one inequalities find the lower bound quickly.

Suggested Citation

  • Hamidur Rahman & Ashutosh Mahajan, 2020. "On the facet defining inequalities of the mixed-integer bilinear covering set," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(3), pages 545-575, December.
    • Handle: RePEc:spr:mathme:v:92:y:2020:i:3:d:10.1007_s00186-020-00723-9
    DOI: 10.1007/s00186-020-00723-9
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    References listed on IDEAS

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    1. Hamidur Rahman & Ashutosh Mahajan, 2019. "Facets of a mixed-integer bilinear covering set with bounds on variables," Journal of Global Optimization, Springer, vol. 74(3), pages 417-442, July.
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    4. Harjunkoski, Iiro & Westerlund, Tapio & Porn, Ray & Skrifvars, Hans, 1998. "Different transformations for solving non-convex trim-loss problems by MINLP," European Journal of Operational Research, Elsevier, vol. 105(3), pages 594-603, March.
    5. Umetani, Shunji & Yagiura, Mutsunori & Ibaraki, Toshihide, 2003. "One-dimensional cutting stock problem to minimize the number of different patterns," European Journal of Operational Research, Elsevier, vol. 146(2), pages 388-402, April.
    6. NEMHAUSER, George L. & WOLSEY, Laurence A., 1990. "A recursive procedure to generate all cuts for 0-1 mixed integer programs," LIDAM Reprints CORE 894, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. Santanu S. Dey & Quentin Louveaux, 2011. "Split Rank of Triangle and Quadrilateral Inequalities," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 432-461, August.
    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.
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