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New models for the robust shortest path problem: complexity, resolution and generalization

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  • Virginie Gabrel
  • Cécile Murat
  • Lei Wu

Abstract

In optimization, it is common to deal with uncertain and inaccurate factors which make it difficult to assign a single value to each parameter in the model. It may be more suitable to assign a set of values to each uncertain parameter. A scenario is defined as a realization of the uncertain parameters. In this context, a robust solution has to be as good as possible on a majority of scenarios and never be too bad. Such characterization admits numerous possible interpretations and therefore gives rise to various approaches of robustness. These approaches differ from each other depending on models used to represent uncertain factors, on methodology used to measure robustness, and finally on analysis and design of solution methods. In this paper, we focus on the application of a recent criterion for the shortest path problem with uncertain arc lengths. We first present two usual uncertainty models: the interval model and the discrete scenario set model. For each model, we then apply a criterion, called bw-robustness (originally proposed by B. Roy) which defines a new measure of robustness. According to each uncertainty model, we propose a formulation in terms of large scale integer linear program. Furthermore, we analyze the theoretical complexity of the resulting problems. Our computational experiments perform on a set of large scale graphs. By observing the results, we can conclude that the approved solvers, e.g. Cplex, are able to solve the mathematical models proposed which are promising for robustness analysis. In the end, we show that our formulations can be applied to the general linear program in which the objective function includes uncertain coefficients. Copyright Springer Science+Business Media, LLC 2013

Suggested Citation

  • Virginie Gabrel & Cécile Murat & Lei Wu, 2013. "New models for the robust shortest path problem: complexity, resolution and generalization," Annals of Operations Research, Springer, vol. 207(1), pages 97-120, August.
  • Handle: RePEc:spr:annopr:v:207:y:2013:i:1:p:97-120:10.1007/s10479-011-1004-2
    DOI: 10.1007/s10479-011-1004-2
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    References listed on IDEAS

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    1. V Gabrel & C Murat, 2010. "Robustness and duality in linear programming," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 61(8), pages 1288-1296, August.
    2. Roy, Bernard, 2010. "Robustness in operational research and decision aiding: A multi-faceted issue," European Journal of Operational Research, Elsevier, vol. 200(3), pages 629-638, February.
    3. Montemanni, Roberto, 2006. "A Benders decomposition approach for the robust spanning tree problem with interval data," European Journal of Operational Research, Elsevier, vol. 174(3), pages 1479-1490, November.
    4. Dimitris Bertsimas & Melvyn Sim, 2004. "The Price of Robustness," Operations Research, INFORMS, vol. 52(1), pages 35-53, February.
    5. Alper Atamtürk & Muhong Zhang, 2007. "Two-Stage Robust Network Flow and Design Under Demand Uncertainty," Operations Research, INFORMS, vol. 55(4), pages 662-673, August.
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    Cited by:

    1. Juan Carlos Espinoza Garcia & Laurent Alfandari, 2018. "Robust location of new housing developments using a choice model," Annals of Operations Research, Springer, vol. 271(2), pages 527-550, December.
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    3. Sharf, Miel & Romm, Iliya & Palman, Michael & Zelazo, Daniel & Cukurel, Beni, 2022. "Economic dispatch of a single micro gas turbine under CHP operation with uncertain demands," Applied Energy, Elsevier, vol. 309(C).
    4. Gabrel, Virginie & Murat, Cécile & Thiele, Aurélie, 2014. "Recent advances in robust optimization: An overview," European Journal of Operational Research, Elsevier, vol. 235(3), pages 471-483.
    5. Espinoza Garcia, Juan Carlos & Alfandari, Laurent, 2015. "Robust location of new housing developments using a choice model," ESSEC Working Papers WP1521, ESSEC Research Center, ESSEC Business School.
    6. Jinzuo Guo & Tianyu Liu & Guopeng Song & Bo Guo, 2024. "Solving the Robust Shortest Path Problem with Multimodal Transportation," Mathematics, MDPI, vol. 12(19), pages 1-14, September.
    7. Tereza Sedlářová Nehézová & Michal Škoda & Robert Hlavatý & Helena Brožová, 2022. "Fuzzy and robust approach for decision-making in disaster situations," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 30(2), pages 617-645, June.
    8. Juan Carlos Espinoza Garcia & Laurent Alfandari, 2015. "Robust location of new housing developments using a choice model," Working Papers hal-01230621, HAL.
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    10. Amadeu Coco & João Júnior & Thiago Noronha & Andréa Santos, 2014. "An integer linear programming formulation and heuristics for the minmax relative regret robust shortest path problem," Journal of Global Optimization, Springer, vol. 60(2), pages 265-287, October.
    11. Dan A. Iancu & Nikolaos Trichakis, 2014. "Pareto Efficiency in Robust Optimization," Management Science, INFORMS, vol. 60(1), pages 130-147, January.

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