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The constrained shortest path problem with stochastic correlated link travel times

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  • Wang, Li
  • Yang, Lixing
  • Gao, Ziyou

Abstract

This paper investigates the constrained shortest path problem in a transportation network where the link travel times are assumed to be random variables defined on the basis of joint probability mass functions. A 0–1 integer programming model is formulated to find the least expected travel time paths. Besides the flow balance and side constraints, the unique link selection constraint is particularly introduced to guarantee that only an optimal path can be finally generated. Then, a Lagrangian relaxation solution approach is provided to relax the hard constraints and decompose the relaxed model into two parts. An algorithmic framework, integrating the sub-gradient algorithm, label-correcting algorithm and K-shortest path algorithm, is designed to minimize the gap between the upper and lower bounds to find near-optimal solutions. An extension that considers the joint probability mass functions of link travel times varying with time is discussed. We decompose the time-dependent problem into three subproblems and solve it by the modified algorithmic framework. Computational tests are conducted on different scales of transportation networks. Experimental results in a large-scale network indicate that the proposed algorithm can quickly find the high-quality solutions with small relative gaps, demonstrating the effectiveness of the proposed approaches.

Suggested Citation

  • Wang, Li & Yang, Lixing & Gao, Ziyou, 2016. "The constrained shortest path problem with stochastic correlated link travel times," European Journal of Operational Research, Elsevier, vol. 255(1), pages 43-57.
    • Handle: RePEc:eee:ejores:v:255:y:2016:i:1:p:43-57
    DOI: 10.1016/j.ejor.2016.05.040
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    1. Marshall L. Fisher, 2004. "The Lagrangian Relaxation Method for Solving Integer Programming Problems," Management Science, INFORMS, vol. 50(12_supple), pages 1861-1871, December.
    2. Di Puglia Pugliese, Luigi & Guerriero, Francesca, 2013. "Shortest path problem with forbidden paths: The elementary version," European Journal of Operational Research, Elsevier, vol. 227(2), pages 254-267.
    3. Huang, He & Gao, Song, 2012. "Optimal paths in dynamic networks with dependent random link travel times," Transportation Research Part B: Methodological, Elsevier, vol. 46(5), pages 579-598.
    4. Nielsen, Lars Relund & Andersen, Kim Allan & Pretolani, Daniele, 2014. "Ranking paths in stochastic time-dependent networks," European Journal of Operational Research, Elsevier, vol. 236(3), pages 903-914.
    5. Raj A. Sivakumar & Rajan Batta, 1994. "The Variance-Constrained Shortest Path Problem," Transportation Science, INFORMS, vol. 28(4), pages 309-316, November.
    6. Ishwar Murthy & Sumit Sarkar, 1998. "Stochastic Shortest Path Problems with Piecewise-Linear Concave Utility Functions," Management Science, INFORMS, vol. 44(11-Part-2), pages 125-136, November.
    7. Rivera, Juan Carlos & Murat Afsar, H. & Prins, Christian, 2016. "Mathematical formulations and exact algorithm for the multitrip cumulative capacitated single-vehicle routing problem," European Journal of Operational Research, Elsevier, vol. 249(1), pages 93-104.
    8. Yang, Lixing & Zhang, Yan & Li, Shukai & Gao, Yuan, 2016. "A two-stage stochastic optimization model for the transfer activity choice in metro networks," Transportation Research Part B: Methodological, Elsevier, vol. 83(C), pages 271-297.
    9. Suvrajeet Sen & Rekha Pillai & Shirish Joshi & Ajay K. Rathi, 2001. "A Mean-Variance Model for Route Guidance in Advanced Traveler Information Systems," Transportation Science, INFORMS, vol. 35(1), pages 37-49, February.
    10. Nachtigall, K., 1995. "Time depending shortest-path problems with applications to railway networks," European Journal of Operational Research, Elsevier, vol. 83(1), pages 154-166, May.
    11. Gao, Song & Chabini, Ismail, 2006. "Optimal routing policy problems in stochastic time-dependent networks," Transportation Research Part B: Methodological, Elsevier, vol. 40(2), pages 93-122, February.
    12. Avella, Pasquale & Boccia, Maurizio & Sforza, Antonio, 2002. "A penalty function heuristic for the resource constrained shortest path problem," European Journal of Operational Research, Elsevier, vol. 142(2), pages 221-230, October.
    13. Yang, Lixing & Zhou, Xuesong, 2014. "Constraint reformulation and a Lagrangian relaxation-based solution algorithm for a least expected time path problem," Transportation Research Part B: Methodological, Elsevier, vol. 59(C), pages 22-44.
    14. H. Frank, 1969. "Shortest Paths in Probabilistic Graphs," Operations Research, INFORMS, vol. 17(4), pages 583-599, August.
    15. Michel Gamache & François Soumis & Gérald Marquis & Jacques Desrosiers, 1999. "A Column Generation Approach for Large-Scale Aircrew Rostering Problems," Operations Research, INFORMS, vol. 47(2), pages 247-263, April.
    16. Santos, Luis & Coutinho-Rodrigues, João & Current, John R., 2007. "An improved solution algorithm for the constrained shortest path problem," Transportation Research Part B: Methodological, Elsevier, vol. 41(7), pages 756-771, August.
    17. Taş, D. & Gendreau, M. & Dellaert, N. & van Woensel, T. & de Kok, A.G., 2014. "Vehicle routing with soft time windows and stochastic travel times: A column generation and branch-and-price solution approach," European Journal of Operational Research, Elsevier, vol. 236(3), pages 789-799.
    18. Bode, Claudia & Irnich, Stefan, 2014. "The shortest-path problem with resource constraints with (k,2)-loop elimination and its application to the capacitated arc-routing problem," European Journal of Operational Research, Elsevier, vol. 238(2), pages 415-426.
    19. Amir Eiger & Pitu B. Mirchandani & Hossein Soroush, 1985. "Path Preferences and Optimal Paths in Probabilistic Networks," Transportation Science, INFORMS, vol. 19(1), pages 75-84, February.
    20. Lavoie, Sylvie & Minoux, Michel & Odier, Edouard, 1988. "A new approach for crew pairing problems by column generation with an application to air transportation," European Journal of Operational Research, Elsevier, vol. 35(1), pages 45-58, April.
    21. Marshall L. Fisher, 2004. "Comments on ÜThe Lagrangian Relaxation Method for Solving Integer Programming ProblemsÝ," Management Science, INFORMS, vol. 50(12_supple), pages 1872-1874, December.
    22. Tong, Lu & Zhou, Xuesong & Miller, Harvey J., 2015. "Transportation network design for maximizing space–time accessibility," Transportation Research Part B: Methodological, Elsevier, vol. 81(P2), pages 555-576.
    23. Zhu, Xiaoyan & Wilhelm, Wilbert E., 2007. "Three-stage approaches for optimizing some variations of the resource constrained shortest-path sub-problem in a column generation context," European Journal of Operational Research, Elsevier, vol. 183(2), pages 564-577, December.
    24. Yang, Lixing & Zhou, Xuesong & Gao, Ziyou, 2014. "Credibility-based rescheduling model in a double-track railway network: a fuzzy reliable optimization approach," Omega, Elsevier, vol. 48(C), pages 75-93.
    25. George B. Dantzig, 1960. "On the Shortest Route Through a Network," Management Science, INFORMS, vol. 6(2), pages 187-190, January.
    26. Nie, Yu (Marco) & Wu, Xing, 2009. "Shortest path problem considering on-time arrival probability," Transportation Research Part B: Methodological, Elsevier, vol. 43(6), pages 597-613, July.
    27. Martins, Ernesto Queiros Vieira, 1984. "On a multicriteria shortest path problem," European Journal of Operational Research, Elsevier, vol. 16(2), pages 236-245, May.
    28. Yang, Lixing & Li, Keping & Gao, Ziyou & Li, Xiang, 2012. "Optimizing trains movement on a railway network," Omega, Elsevier, vol. 40(5), pages 619-633.
    29. Elimam, A. A. & Kohler, David, 1997. "Two engineering applications of a constrained shortest-path model," European Journal of Operational Research, Elsevier, vol. 103(3), pages 426-438, December.
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