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Modeling and Heuristic Worst-Case Performance Analysis of the Two-Level Network Design Problem

Author

Listed:
  • Anantaram Balakrishnan

    (Sloan School of Management, M. I. T., Cambridge, Massachusetts 02139)

  • Thomas L. Magnanti

    (Sloan School of Management, M. I. T., Cambridge, Massachusetts 02139)

  • Prakash Mirchandani

    (Katz Graduate School of Business, University of Pittsburgh, Pittsburgh, Pennsylvania 15260)

Abstract

This paper studies a multi-facility network synthesis problem, called the Two-level Network Design (TLND) problem, that arises in the topological design of hierarchical communication, transportation, and electric power distribution networks. We are given an undirected network containing two types of nodes---primary and secondary---and fixed costs for installing either a primary or a secondary facility on each edge. Primary nodes require higher grade interconnections than secondary nodes, using the more expensive primary facilities. The TLND problem seeks a minimum cost connected design that spans all the nodes, and connects primary nodes via edges containing primary facilities; the design can use either primary or secondary edges to connect the secondary nodes. The TLND problem generalizes the well-known Steiner network problem and the hierarchical network design problem. In this paper, we study the relationship between alternative model formulations for this problem (e.g., directed and undirected models), and analyze the worst-case performance for a composite TLND heuristic based upon solving embedded subproblems (e.g., minimum spanning tree and either Steiner tree or shortest path subproblems). When the ratio of primary to secondary costs is the same for all edges and when we solve the embedded subproblems optimally, the worst-case performance ratio of the composite TLND heuristic is 4/3. This result applies to the hierarchical network design problem with constant primary-to-secondary cost ratio since its subproblems are shortest path and minimum spanning tree problems. For more general situations, we express the TLND heuristic worst-case ratio in terms of the performance of any heuristic used to solve the embedded Steiner tree subproblem. A companion paper develops and tests a dual ascent procedure that generates tight upper and lower bounds on the optimal value of a multi-level extension of this problem.

Suggested Citation

  • Anantaram Balakrishnan & Thomas L. Magnanti & Prakash Mirchandani, 1994. "Modeling and Heuristic Worst-Case Performance Analysis of the Two-Level Network Design Problem," Management Science, INFORMS, vol. 40(7), pages 846-867, July.
  • Handle: RePEc:inm:ormnsc:v:40:y:1994:i:7:p:846-867
    DOI: 10.1287/mnsc.40.7.846
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    Cited by:

    1. Chardy, M. & Costa, M.-C. & Faye, A. & Trampont, M., 2012. "Optimizing splitter and fiber location in a multilevel optical FTTH network," European Journal of Operational Research, Elsevier, vol. 222(3), pages 430-440.
    2. Amiri, Ali, 2001. "The multi-hour service system design problem," European Journal of Operational Research, Elsevier, vol. 128(3), pages 625-638, February.
    3. Gouveia, Luis, 1996. "Multicommodity flow models for spanning trees with hop constraints," European Journal of Operational Research, Elsevier, vol. 95(1), pages 178-190, November.
    4. Amiri, Ali, 1998. "The design of service systems with queueing time cost, workload capacities and backup service," European Journal of Operational Research, Elsevier, vol. 104(1), pages 201-217, January.
    5. van de Leensel, R.L.J.M. & Flippo, O.E. & Koster, Arie M.C.A. & Kolen, A.W.J., 1996. "A dynamic programming algorithm for the local access network expansion problem," Research Memorandum 027, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    6. Obreque, Carlos & Donoso, Macarena & Gutiérrez, Gabriel & Marianov, Vladimir, 2010. "A branch and cut algorithm for the hierarchical network design problem," European Journal of Operational Research, Elsevier, vol. 200(1), pages 28-35, January.
    7. Costa, Alysson M. & França, Paulo M. & Lyra Filho, Christiano, 2011. "Two-level network design with intermediate facilities: An application to electrical distribution systems," Omega, Elsevier, vol. 39(1), pages 3-13, January.
    8. Eduardo Álvarez-Miranda & Ivana Ljubić & S. Raghavan & Paolo Toth, 2015. "The Recoverable Robust Two-Level Network Design Problem," INFORMS Journal on Computing, INFORMS, vol. 27(1), pages 1-19, February.
    9. Flippo, Olaf E. & Kolen, Antoon W. J. & Koster, Arie M. C. A. & van de Leensel, Robert L. M. J., 2000. "A dynamic programming algorithm for the local access telecommunication network expansion problem," European Journal of Operational Research, Elsevier, vol. 127(1), pages 189-202, November.
    10. Gouveia, Luis & Janssen, Eric, 1998. "Designing reliable tree networks with two cable technologies," European Journal of Operational Research, Elsevier, vol. 105(3), pages 552-568, March.
    11. Cruz, F. R. B. & Smith, J. MacGregor & Mateus, G. R., 1999. "Algorithms for a multi-level network optimization problem," European Journal of Operational Research, Elsevier, vol. 118(1), pages 164-180, October.
    12. Gouveia, Luis & Leitner, Markus, 2017. "Design of survivable networks with vulnerability constraints," European Journal of Operational Research, Elsevier, vol. 258(1), pages 89-103.
    13. Gollowitzer, Stefan & Gouveia, Luis & Ljubić, Ivana, 2013. "Enhanced formulations and branch-and-cut for the two level network design problem with transition facilities," European Journal of Operational Research, Elsevier, vol. 225(2), pages 211-222.
    14. Souza, Fernanda S.H. & Gendreau, Michel & Mateus, Geraldo R., 2014. "Branch-and-price algorithm for the Resilient Multi-level Hop-constrained Network Design," European Journal of Operational Research, Elsevier, vol. 233(1), pages 84-93.
    15. Masashi Miyagawa, 2009. "Optimal hierarchical system of a grid road network," Annals of Operations Research, Springer, vol. 172(1), pages 349-361, November.

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