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Variable Disaggregation in Network Flow Problems with Piecewise Linear Costs

Author

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  • Keely L. Croxton

    (Fisher College of Business, The Ohio State University, 518 Fisher Hall, 2100 Neil Avenue, Columbus, Ohio 43210-1144)

  • Bernard Gendron

    (Département d’informatique et de recherche opérationnelle and Centre de recherche sur les transports, Université de Montréal, C.P. 6128, succ. Centre-ville, Montreal, Quebec, Canada H3C 3J7)

  • Thomas L. Magnanti

    (School of Engineering and Sloan School of Management, Massachusetts Institute of Technology, Room 1-206, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307)

Abstract

We study mixed-integer programming formulations, based upon variable disaggregation, for generic multicommodity network flow problems with nonconvex piecewise linear costs, a problem class that arises frequently in many application domains in telecommunications, transportation, and logistics. We present several structural results for these formulations, and we analyze the results of extensive experiments on a large set of instances with various characteristics. In particular, we show that the linear programming relaxation of an extended disaggregated model approximates the objective function by its lower convex envelope in the space of commodity flows. Together, the theoretical and computational results allow us to suggest which formulation might be the most appropriate, depending on the characteristics of the problem instances.

Suggested Citation

  • Keely L. Croxton & Bernard Gendron & Thomas L. Magnanti, 2007. "Variable Disaggregation in Network Flow Problems with Piecewise Linear Costs," Operations Research, INFORMS, vol. 55(1), pages 146-157, February.
    • Handle: RePEc:inm:oropre:v:55:y:2007:i:1:p:146-157
    DOI: 10.1287/opre.1060.0314
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    References listed on IDEAS

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    Cited by:

    1. Paraskevopoulos, Dimitris C. & Gürel, Sinan & Bektaş, Tolga, 2016. "The congested multicommodity network design problem," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 85(C), pages 166-187.
    2. Hu, Qian & Lim, Andrew & Zhu, Wenbin, 2015. "The two-dimensional vector packing problem with piecewise linear cost function," Omega, Elsevier, vol. 50(C), pages 43-53.
    3. Hanbazazah, Abdulkader S. & Abril, Luis & Erkoc, Murat & Shaikh, Nazrul, 2019. "Freight consolidation with divisible shipments, delivery time windows, and piecewise transportation costs," European Journal of Operational Research, Elsevier, vol. 276(1), pages 187-201.
    4. Bernard Gendron & Luis Gouveia, 2017. "Reformulations by Discretization for Piecewise Linear Integer Multicommodity Network Flow Problems," Transportation Science, INFORMS, vol. 51(2), pages 629-649, May.
    5. Juan Pablo Vielma & Shabbir Ahmed & George Nemhauser, 2010. "Mixed-Integer Models for Nonseparable Piecewise-Linear Optimization: Unifying Framework and Extensions," Operations Research, INFORMS, vol. 58(2), pages 303-315, April.
    6. Ali, Agha Iqbal & O'Connor, Debra J., 2010. "The impact of distribution system characteristics on computational tractability," European Journal of Operational Research, Elsevier, vol. 200(2), pages 323-333, January.
    7. Fortz, Bernard & Gouveia, Luís & Joyce-Moniz, Martim, 2017. "Models for the piecewise linear unsplittable multicommodity flow problems," European Journal of Operational Research, Elsevier, vol. 261(1), pages 30-42.
    8. Naoto Katayama, 2017. "A Combined Matheuristic for the Piecewise Linear Multicommodity Network Flow Problem," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 34(06), pages 1-24, December.
    9. Arslan, Okan & Archetti, Claudia & Jabali, Ola & Laporte, Gilbert & Grazia Speranza, Maria, 2020. "Minimum cost network design in strategic alliances," Omega, Elsevier, vol. 96(C).
    10. Ya Ping Fang & Kaiwen Meng & Xiao Qi Yang, 2012. "Piecewise Linear Multicriteria Programs: The Continuous Case and Its Discontinuous Generalization," Operations Research, INFORMS, vol. 60(2), pages 398-409, April.
    11. Samir Elhedhli & Zichao Li & James H. Bookbinder, 2017. "Airfreight forwarding under system-wide and double discounts," EURO Journal on Transportation and Logistics, Springer;EURO - The Association of European Operational Research Societies, vol. 6(2), pages 165-183, June.
    12. Jose L. Andrade-Pineda & David Canca & Pedro L. Gonzalez-R, 2017. "On modelling non-linear quantity discounts in a supplier selection problem by mixed linear integer optimization," Annals of Operations Research, Springer, vol. 258(2), pages 301-346, November.
    13. Nowak, Maciek & Hewitt, Mike & Bachour, Hussam, 2019. "Mileage bands in freight transportation," European Journal of Operational Research, Elsevier, vol. 272(2), pages 549-564.
    14. F. J. Hwang & Yao-Huei Huang, 2021. "An effective logarithmic formulation for piecewise linearization requiring no inequality constraint," Computational Optimization and Applications, Springer, vol. 79(3), pages 601-631, July.
    15. Pan, Shenle & Ballot, Eric & Fontane, Frédéric, 2013. "The reduction of greenhouse gas emissions from freight transport by pooling supply chains," International Journal of Production Economics, Elsevier, vol. 143(1), pages 86-94.
    16. Tue R. L. Christensen & Kim Allan Andersen & Andreas Klose, 2013. "Solving the Single-Sink, Fixed-Charge, Multiple-Choice Transportation Problem by Dynamic Programming," Transportation Science, INFORMS, vol. 47(3), pages 428-438, August.
    17. SteadieSeifi, M. & Dellaert, N.P. & Nuijten, W. & Van Woensel, T. & Raoufi, R., 2014. "Multimodal freight transportation planning: A literature review," European Journal of Operational Research, Elsevier, vol. 233(1), pages 1-15.

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