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Exact solution approaches for bilevel assignment problems

Author

Listed:
  • Behdad Beheshti

    (University of Pittsburgh)

  • Oleg A. Prokopyev

    (University of Pittsburgh)

  • Eduardo L. Pasiliao

    (AFRL Munitions Directorate)

Abstract

We consider the bilevel assignment problem in which each decision maker (i.e., the leader and the follower) has its own objective function and controls a distinct set of edges in a given bipartite graph. The leader acts first by choosing some of its edges. Subsequently, the follower completes the assignment process. The edges selected by the leader and the follower are required to constitute a perfect matching. In this paper we propose an exact solution approach, which is based on a branch-and-bound framework and exploits structural properties of the assignment problem. Extensive computational experiments with linear sum and linear bottleneck objective functions are conducted to demonstrate the performance of the developed methods. While the considered problem is known to be NP-hard in general, we also describe some restricted cases that can be solved in polynomial time.

Suggested Citation

  • Behdad Beheshti & Oleg A. Prokopyev & Eduardo L. Pasiliao, 2016. "Exact solution approaches for bilevel assignment problems," Computational Optimization and Applications, Springer, vol. 64(1), pages 215-242, May.
    • Handle: RePEc:spr:coopap:v:64:y:2016:i:1:d:10.1007_s10589-015-9799-4
    DOI: 10.1007/s10589-015-9799-4
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    References listed on IDEAS

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    1. Renata M. Aiex & Mauricio G. C. Resende & Panos M. Pardalos & Gerardo Toraldo, 2005. "GRASP with Path Relinking for Three-Index Assignment," INFORMS Journal on Computing, INFORMS, vol. 17(2), pages 224-247, May.
    2. M. L. Balinski & R. E. Gomory, 1964. "A Primal Method for the Assignment and Transportation Problems," Management Science, INFORMS, vol. 10(3), pages 578-593, April.
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    4. Benoît Colson & Patrice Marcotte & Gilles Savard, 2007. "An overview of bilevel optimization," Annals of Operations Research, Springer, vol. 153(1), pages 235-256, September.
    5. Liu, Yi-Hsin & Spencer, Thomas H., 1995. "Solving a bilevel linear program when the inner decision maker controls few variables," European Journal of Operational Research, Elsevier, vol. 81(3), pages 644-651, March.
    6. Lieshout, P.M.D. & Volgenant, A., 2007. "A branch-and-bound algorithm for the singly constrained assignment problem," European Journal of Operational Research, Elsevier, vol. 176(1), pages 151-164, January.
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    Cited by:

    1. Gabriel Lopez Zenarosa & Oleg A. Prokopyev & Eduardo L. Pasiliao, 2021. "On exact solution approaches for bilevel quadratic 0–1 knapsack problem," Annals of Operations Research, Springer, vol. 298(1), pages 555-572, March.
    2. Jorge A. Sefair & J. Cole Smith, 2017. "Exact algorithms and bounds for the dynamic assignment interdiction problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 64(5), pages 373-387, August.
    3. M. Hosein Zare & Juan S. Borrero & Bo Zeng & Oleg A. Prokopyev, 2019. "A note on linearized reformulations for a class of bilevel linear integer problems," Annals of Operations Research, Springer, vol. 272(1), pages 99-117, January.

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