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Single-commodity robust network design problem: Complexity, instances and heuristic solutions

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  • Álvarez-Miranda, Eduardo
  • Cacchiani, Valentina
  • Lodi, Andrea
  • Parriani, Tiziano
  • Schmidt, Daniel R.

Abstract

We study a single-commodity Robust Network Design problem (RND) in which an undirected graph with edge costs is given together with a discrete set of balance matrices, representing different supply/demand scenarios. In each scenario, a subset of the nodes is exchanging flow. The goal is to determine the minimum cost installation of capacities on the edges such that the flow exchange is feasible for every scenario. Previously conducted computational investigations on the problem motivated the study of the complexity of some special cases and we present complexity results on them, including hypercubes. In turn, these results lead to the definition of new instances (random graphs with {−1,0,1} balances) that are computationally hard for the natural flow formulation. These instances are then solved by means of a new heuristic algorithm for RND, which consists of three phases. In the first phase the graph representing the network is reduced by heuristically deleting a subset of the arcs, and a feasible solution is built. The second phase consists of a neighborhood search on the reduced graph based on a Mixed-Integer (Linear) Programming (MIP) flow model. Finally, the third phase applies a proximity search approach to further improve the solution, taking into account the original graph. The heuristic is tested on the new instances, and the comparison with the solutions obtained by Cplex on a natural flow formulation shows the effectiveness of the proposed method.

Suggested Citation

  • Álvarez-Miranda, Eduardo & Cacchiani, Valentina & Lodi, Andrea & Parriani, Tiziano & Schmidt, Daniel R., 2014. "Single-commodity robust network design problem: Complexity, instances and heuristic solutions," European Journal of Operational Research, Elsevier, vol. 238(3), pages 711-723.
    • Handle: RePEc:eee:ejores:v:238:y:2014:i:3:p:711-723
    DOI: 10.1016/j.ejor.2014.04.023
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    References listed on IDEAS

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    1. Dimitris Bertsimas & Melvyn Sim, 2004. "The Price of Robustness," Operations Research, INFORMS, vol. 52(1), pages 35-53, February.
    2. Sara Mattia, 2013. "The robust network loading problem with dynamic routing," Computational Optimization and Applications, Springer, vol. 54(3), pages 619-643, April.
    3. Edward Rothberg, 2007. "An Evolutionary Algorithm for Polishing Mixed Integer Programming Solutions," INFORMS Journal on Computing, INFORMS, vol. 19(4), pages 534-541, November.
    4. ORTEGA , Francisco & WOLSEY, Laurence A., 2003. "A branch-and-cut algorithm for the single-commodity, uncapacitated, fixed-charge network flow problem," LIDAM Reprints CORE 1611, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    2. Ashwin Arulselvan & Mohsen Rezapour, 2017. "Exact Approaches for Designing Multifacility Buy-at-Bulk Networks," INFORMS Journal on Computing, INFORMS, vol. 29(4), pages 597-611, November.
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    4. Matthews, Logan R. & Gounaris, Chrysanthos E. & Kevrekidis, Ioannis G., 2019. "Designing networks with resiliency to edge failures using two-stage robust optimization," European Journal of Operational Research, Elsevier, vol. 279(3), pages 704-720.

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