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Source Sink Flows with Capacity Installation in Batches

Author

Listed:
  • Itzhak Gilboa

    (Kellogg [Northwestern] - Kellogg School of Management [Northwestern University, Evanston] - Northwestern University [Evanston])

  • Sunil Chopra

    (Kellogg [Northwestern] - Kellogg School of Management [Northwestern University, Evanston] - Northwestern University [Evanston])

  • S. Trilochan Sastri

Abstract

We consider the problem of sending flow from a source to a destination where there are flow costs on each arc and fixed costs toward the purchase of capacity. Capacity can be purchased in batches of C units on each arc. We show the problem to be NP-hard in general. If d is the quantity to be shipped from the source to the destination, we give an algorithm that solves the problem in time polynomial in the size of the graph but exponential in [d / C]. Thus, for bounded values of [d / C] the problem can be solved in polynomial time. This is useful since a simple heuristic gives a very good approximation of the optimal solution for large values of [d / C]. We also show a similar result to hold for the case when there are no flow costs but capacity can be purchased either in batches of 1 unit or C units. The results characterizing optimal solutions with a minimum number of free arcs are used to obtain extended formulations in each of the two cases. The LP-relaxations of the extended formulations are shown to be stronger than the natural formulations considered by earlier authors, even with a family of strong valid inequalities added.

Suggested Citation

  • Itzhak Gilboa & Sunil Chopra & S. Trilochan Sastri, 1998. "Source Sink Flows with Capacity Installation in Batches," Post-Print hal-00753123, HAL.
    • Handle: RePEc:hal:journl:hal-00753123
    DOI: 10.1016/S0166-218X(98)00024-9
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    Citations

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    Cited by:

    1. Babonneau, F. & Vial, J.-P., 2010. "A partitioning algorithm for the network loading problem," European Journal of Operational Research, Elsevier, vol. 204(1), pages 173-179, July.
    2. F. Sibel Salman & R. Ravi & John N. Hooker, 2008. "Solving the Capacitated Local Access Network Design Problem," INFORMS Journal on Computing, INFORMS, vol. 20(2), pages 243-254, May.
    3. Ada Suk‐fung Ng & Trilochan Sastry & Janny M.Y. Leung & X.Q. Cai, 2004. "On the uncapacitated K‐commodity network design problem with zero flow‐costs," Naval Research Logistics (NRL), John Wiley & Sons, vol. 51(8), pages 1149-1172, December.
    4. Andrew P. Armacost & Cynthia Barnhart & Keith A. Ware, 2002. "Composite Variable Formulations for Express Shipment Service Network Design," Transportation Science, INFORMS, vol. 36(1), pages 1-20, February.
    5. Liang Chen & Wei-Kun Chen & Mu-Ming Yang & Yu-Hong Dai, 2021. "An exact separation algorithm for unsplittable flow capacitated network design arc-set polyhedron," Journal of Global Optimization, Springer, vol. 81(3), pages 659-689, November.

    More about this item

    Keywords

    min cost flow; Integer programming;

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