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A Primal Partitioning Solution for the Arc-Chain Formulation of a Multicommodity Network Flow Problem

Author

Listed:
  • Judith M. Farvolden

    (University of Toronto, Toronto, Ontario, Canada)

  • Warren B. Powell

    (Princeton University, Princeton, New Jersey)

  • Irvin J. Lustig

    (Princeton University, Princeton, New Jersey)

Abstract

We present a new solution approach for the multicommodity network flow problem (MCNF) based upon both primal partitioning and decomposition techniques, which simplifies the computations required by the simplex method. The partitioning is performed on an arc-chain incidence matrix of the MCNF, similar within a change of variables to the constraint matrix of the master problem generated in a Dantzig-Wolfe decomposition, to isolate a very sparse, near-triangular working basis of greatly reduced dimension. The majority of the simplex operations performed on the partitioned basis are simply additive and network operations specialized for the nine possible pivot types identified. The columns of the arc-chain incidence matrix are generated by a dual network simplex method for updating shortest paths when link costs change.

Suggested Citation

  • Judith M. Farvolden & Warren B. Powell & Irvin J. Lustig, 1993. "A Primal Partitioning Solution for the Arc-Chain Formulation of a Multicommodity Network Flow Problem," Operations Research, INFORMS, vol. 41(4), pages 669-693, August.
    • Handle: RePEc:inm:oropre:v:41:y:1993:i:4:p:669-693
    DOI: 10.1287/opre.41.4.669
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    Citations

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    Cited by:

    1. F. Babonneau & O. du Merle & J.-P. Vial, 2006. "Solving Large-Scale Linear Multicommodity Flow Problems with an Active Set Strategy and Proximal-ACCPM," Operations Research, INFORMS, vol. 54(1), pages 184-197, February.
    2. P. Chardaire & G. P. McKeown & S. A. Verity-Harrison & S. B. Richardson, 2005. "Solving a Time-Space Network Formulation for the Convoy Movement Problem," Operations Research, INFORMS, vol. 53(2), pages 219-230, April.
    3. P. Chardaire & A. Lisser, 2002. "Simplex and Interior Point Specialized Algorithms for Solving Nonoriented Multicommodity Flow Problems," Operations Research, INFORMS, vol. 50(2), pages 260-276, April.
    4. Zhang, M. & Pel, A.J., 2016. "Synchromodal hinterland freight transport: Model study for the port of Rotterdam," Journal of Transport Geography, Elsevier, vol. 52(C), pages 1-10.
    5. Rina R. Schneur & James B. Orlin, 1998. "A Scaling Algorithm for Multicommodity Flow Problems," Operations Research, INFORMS, vol. 46(2), pages 231-246, April.
    6. Garg, Manish & Smith, J. Cole, 2008. "Models and algorithms for the design of survivable multicommodity flow networks with general failure scenarios," Omega, Elsevier, vol. 36(6), pages 1057-1071, December.
    7. Kraft, Edwin R., 2002. "Scheduling railway freight delivery appointments using a bid price approach," Transportation Research Part A: Policy and Practice, Elsevier, vol. 36(2), pages 145-165, February.
    8. Antonio Frangioni & Giorgio Gallo, 1999. "A Bundle Type Dual-Ascent Approach to Linear Multicommodity Min-Cost Flow Problems," INFORMS Journal on Computing, INFORMS, vol. 11(4), pages 370-393, November.
    9. Xin Wang & Teodor Gabriel Crainic & Stein W. Wallace, 2019. "Stochastic Network Design for Planning Scheduled Transportation Services: The Value of Deterministic Solutions," INFORMS Journal on Computing, INFORMS, vol. 31(1), pages 153-170, February.
    10. Rémy Dupas & Eiichi Taniguchi & Jean-Christophe Deschamps & Ali G. Qureshi, 2020. "A Multi-commodity Network Flow Model for Sustainable Performance Evaluation in City Logistics: Application to the Distribution of Multi-tenant Buildings in Tokyo," Sustainability, MDPI, vol. 12(6), pages 1-18, March.
    11. Kaj Holmberg & Di Yuan, 2003. "A Multicommodity Network-Flow Problem with Side Constraints on Paths Solved by Column Generation," INFORMS Journal on Computing, INFORMS, vol. 15(1), pages 42-57, February.
    12. A. Ouorou & P. Mahey & J.-Ph. Vial, 2000. "A Survey of Algorithms for Convex Multicommodity Flow Problems," Management Science, INFORMS, vol. 46(1), pages 126-147, January.
    13. Richard D. McBride, 1998. "Advances in Solving the Multicommodity-Flow Problem," Interfaces, INFORMS, vol. 28(2), pages 32-41, April.

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