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A Network-Based Primal-Dual Heuristic for the Solution of Multicommodity Network Flow Problems

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  • Cynthia Barnhart

    (Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

  • Yosef Sheffi

    (Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

Abstract

In this paper, we present a primal-dual, heuristic solution approach for large-scale multicommodity network flow problems. The original problem is solved indirectly by repeatedly solving restated feasibility problems. Restrictions on problem size imposed by exact methods are overcome by solving the restated problems with a pure network-based heuristic procedure. To control the heuristic solution process, the network-based procedure is embedded within an iterative primal-dual framework. At each iteration, feasible dual solutions are generated, the dual objective function value strictly ascends, and primal solutions that are measurably closer to feasibility are determined. The algorithm terminates when the heuristic network-based procedure cannot determine an improved primal solution or when optimality is achieved. To demonstrate the effectiveness of the network-based solution strategy, a large-scale freight assignment problem encountered in the trucking industry is formulated as a multicommodity network flow problem. Two linear programming based, exact solution strategies (a primal-dual algorithm and a price-directive algorithm) are unable to achieve even an initial solution for this problem due to excessive memory requirements. The network-based heuristic, however, determines an optimal solution. We compare the performance of the new heuristic with that of the exact procedures using a set of smaller test problems. The effects of problem formulation and congestion are evaluated for each of the alternative solution strategies.

Suggested Citation

  • Cynthia Barnhart & Yosef Sheffi, 1993. "A Network-Based Primal-Dual Heuristic for the Solution of Multicommodity Network Flow Problems," Transportation Science, INFORMS, vol. 27(2), pages 102-117, May.
    • Handle: RePEc:inm:ortrsc:v:27:y:1993:i:2:p:102-117
    DOI: 10.1287/trsc.27.2.102
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    Cited by:

    1. Kraft, Edwin R. & Srikar, Bellur N. & Phillips, Robert L., 2000. "Revenue Management in Railroad Applications," Journal of the Transportation Research Forum, Transportation Research Forum, vol. 39(1).
    2. 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.
    3. Rina R. Schneur & James B. Orlin, 1998. "A Scaling Algorithm for Multicommodity Flow Problems," Operations Research, INFORMS, vol. 46(2), pages 231-246, April.
    4. Pengfei Zhang & Neng Fan, 2017. "Analysis of budget for interdiction on multicommodity network flows," Journal of Global Optimization, Springer, vol. 67(3), pages 495-525, March.
    5. Shaileshh Bojja Venkatakrishnan & Mohammad Alizadeh & Pramod Viswanath, 2018. "Costly circuits, submodular schedules and approximate Carathéodory Theorems," Queueing Systems: Theory and Applications, Springer, vol. 88(3), pages 311-347, April.
    6. Hongbin Liu & Guopeng Song & Tianyu Liu & Bo Guo, 2022. "Multitask Emergency Logistics Planning under Multimodal Transportation," Mathematics, MDPI, vol. 10(19), pages 1-25, October.
    7. 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.
    8. 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.
    9. Hall, Randolph W., 1999. "Stochastic freight flow patterns: implications for fleet optimization," Transportation Research Part A: Policy and Practice, Elsevier, vol. 33(6), pages 449-465, August.
    10. Dong Liang & Wilbert E. Wilhelm, 2013. "Dual‐ascent and primal heuristics for production‐assembly‐distribution system design," Naval Research Logistics (NRL), John Wiley & Sons, vol. 60(1), pages 1-18, February.
    11. 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.
    12. Raymond K. Cheung & B. Muralidharan, 2000. "Dynamic Routing for Priority Shipments in LTL Service Networks," Transportation Science, INFORMS, vol. 34(1), pages 86-98, February.
    13. Lin, Cheng-Chang, 2001. "The freight routing problem of time-definite freight delivery common carriers," Transportation Research Part B: Methodological, Elsevier, vol. 35(6), pages 525-547, July.
    14. K L Poh & K W Choo & C G Wong, 2005. "A heuristic approach to the multi-period multi-commodity transportation problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 56(6), pages 708-718, June.

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