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Dynamic Shortest Paths in Acyclic Networks with Markovian Arc Costs

Author

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  • Harilaos N. Psaraftis

    (National Technical University of Athens, Athens, Greece)

  • John N. Tsitsiklis

    (Massachusetts Institute of Technology, Cambridge, Massachusetts)

Abstract

We examine shortest path problems in acyclic networks in which arc costs are known functions of certain environment variables at network nodes. Each of these variables evolves according to an independent Markov process. The vehicle can wait at a node (at a cost) in anticipation of more favorable arc costs. We first develop two recursive procedures for the individual arc case, one based on successive approximations, and the other on policy iteration. We also solve the same problem via parametric linear programming. We show that the optimal policy essentially classifies the state of the environment variable at a node into two categories: green states for which the optimal action is to immediately traverse the arc, and red states for which the optimal action is to wait. We then extend these concepts for the entire network by developing a dynamic programming procedure that solves the corresponding problem. The complexity of this method is shown to be O ( n 2 K + nK 3 ), where n is the number of network nodes and K is the number of Markov states at each node. We present examples and discuss possible research extensions.

Suggested Citation

  • Harilaos N. Psaraftis & John N. Tsitsiklis, 1993. "Dynamic Shortest Paths in Acyclic Networks with Markovian Arc Costs," Operations Research, INFORMS, vol. 41(1), pages 91-101, February.
    • Handle: RePEc:inm:oropre:v:41:y:1993:i:1:p:91-101
    DOI: 10.1287/opre.41.1.91
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    Cited by:

    1. N Shi & R K Cheung & H Xu & K K Lai, 2011. "An adaptive routing strategy for freight transportation networks," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 62(4), pages 799-805, April.
    2. Güner, Ali R. & Murat, Alper & Chinnam, Ratna Babu, 2017. "Dynamic routing for milk-run tours with time windows in stochastic time-dependent networks," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 97(C), pages 251-267.
    3. 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.
    4. Pramesh Kumar & Alireza Khani, 2021. "Adaptive Park-and-ride Choice on Time-dependent Stochastic Multimodal Transportation Network," Networks and Spatial Economics, Springer, vol. 21(4), pages 771-800, December.
    5. Roberto Tadei & Guido Perboli & Francesca Perfetti, 2017. "The multi-path Traveling Salesman Problem with stochastic travel costs," EURO Journal on Transportation and Logistics, Springer;EURO - The Association of European Operational Research Societies, vol. 6(1), pages 3-23, March.
    6. He Huang & Song Gao, 2018. "Trajectory-Adaptive Routing in Dynamic Networks with Dependent Random Link Travel Times," Transportation Science, INFORMS, vol. 52(1), pages 102-117, January.
    7. Barrett W. Thomas, 2007. "Waiting Strategies for Anticipating Service Requests from Known Customer Locations," Transportation Science, INFORMS, vol. 41(3), pages 319-331, August.
    8. Raymond K. Cheung & B. Muralidharan, 2000. "Dynamic Routing for Priority Shipments in LTL Service Networks," Transportation Science, INFORMS, vol. 34(1), pages 86-98, February.
    9. Ichoua, Soumia & Gendreau, Michel & Potvin, Jean-Yves, 2003. "Vehicle dispatching with time-dependent travel times," European Journal of Operational Research, Elsevier, vol. 144(2), pages 379-396, January.
    10. Elise D. Miller-Hooks & Hani S. Mahmassani, 2000. "Least Expected Time Paths in Stochastic, Time-Varying Transportation Networks," Transportation Science, INFORMS, vol. 34(2), pages 198-215, May.
    11. Jeffrey P. Kharoufeh & Natarajan Gautam, 2004. "A fluid queueing model for link travel time moments," Naval Research Logistics (NRL), John Wiley & Sons, vol. 51(2), pages 242-257, March.
    12. Matsubayashi, Nobuo & Nishino, Hisakazu, 1999. "An application of Lemke's method to a class of Markov decision problems," European Journal of Operational Research, Elsevier, vol. 116(3), pages 584-590, August.
    13. S. Waller & David Fajardo & Melissa Duell & Vinayak Dixit, 2013. "Linear Programming Formulation for Strategic Dynamic Traffic Assignment," Networks and Spatial Economics, Springer, vol. 13(4), pages 427-443, December.
    14. 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.
    15. Lei Gao & Dong Han, 2020. "Extreme Value Distributions for Two Kinds of Path Sums of Markov Chain," Methodology and Computing in Applied Probability, Springer, vol. 22(1), pages 279-294, March.
    16. Thomas, Barrett W. & White III, Chelsea C., 2007. "The dynamic shortest path problem with anticipation," European Journal of Operational Research, Elsevier, vol. 176(2), pages 836-854, January.
    17. Häme, Lauri & Hakula, Harri, 2013. "Dynamic journeying under uncertainty," European Journal of Operational Research, Elsevier, vol. 225(3), pages 455-471.
    18. Azadian, Farshid & Murat, Alper E. & Chinnam, Ratna Babu, 2012. "Dynamic routing of time-sensitive air cargo using real-time information," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 48(1), pages 355-372.
    19. Azaron, Amir & Kianfar, Farhad, 2003. "Dynamic shortest path in stochastic dynamic networks: Ship routing problem," European Journal of Operational Research, Elsevier, vol. 144(1), pages 138-156, January.
    20. Miller-Hooks, Elise & Mahmassani, Hani, 2003. "Path comparisons for a priori and time-adaptive decisions in stochastic, time-varying networks," European Journal of Operational Research, Elsevier, vol. 146(1), pages 67-82, April.
    21. Liu, Zeyu & Li, Xueping & Khojandi, Anahita, 2022. "The flying sidekick traveling salesman problem with stochastic travel time: A reinforcement learning approach," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 164(C).
    22. Levering, Nikki & Boon, Marko & Mandjes, Michel & Núñez-Queija, Rudesindo, 2022. "A framework for efficient dynamic routing under stochastically varying conditions," Transportation Research Part B: Methodological, Elsevier, vol. 160(C), pages 97-124.
    23. Barrett W. Thomas & Chelsea C. White, 2004. "Anticipatory Route Selection," Transportation Science, INFORMS, vol. 38(4), pages 473-487, November.
    24. Murthy, Ishwar & Sarkar, Sumit, 1997. "Exact algorithms for the stochastic shortest path problem with a decreasing deadline utility function," European Journal of Operational Research, Elsevier, vol. 103(1), pages 209-229, November.
    25. Stephen Boyles & S. Waller, 2011. "Optimal Information Location for Adaptive Routing," Networks and Spatial Economics, Springer, vol. 11(2), pages 233-254, June.

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