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Microscopic optimization model and algorithm for integrating train timetabling and track maintenance task scheduling

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  • Zhang, Yongxiang
  • D'Ariano, Andrea
  • He, Bisheng
  • Peng, Qiyuan

Abstract

This paper addresses the problem of improving the integration between passenger timetabling and track maintenance scheduling. We propose a microscopic optimization model and an iterative algorithm for solving this problem efficiently. Block sections are considered as the basic microscopic elements for train movements in a railway network. A mixed-integer linear programming formulation is proposed for the integrated optimization problem in which train timing, sequencing and routing are the timetabling variables, while timing and sequencing of maintenance tasks are the track maintenance variables. The objective function is to minimize the total train travel time and the maintenance tardiness cost. The constraints proposed in this work address the practical specifications of the INFORMS RAS 2016 Problem Solving Competition (2016 PSC). In this context, the main decision variables are the entrance and exit times of the trains on each block section plus the start and end times of each maintenance task. Since the integrated optimization problem is strongly NP-hard, an iterative algorithm is proposed to compute near-optimal solutions in a short computation time. The algorithm is based on a decomposition of the overall problem into sub-problems related to train scheduling and/or routing with or without track maintenance task scheduling. The connecting information between the two sub-problems concerns the selected train routes plus the start and end times of the maintenance tasks. Computational experiments are performed on a set of realistic railway instances, which were introduced during the 2016 PSC. The iterative algorithm outperforms a standard MILP solver and the first-place team of this competition in terms of both solution quality and time to deliver the new best-known solutions. The scalability of the iterative algorithm is investigated when increasing the number of trains and track maintenance tasks.

Suggested Citation

  • Zhang, Yongxiang & D'Ariano, Andrea & He, Bisheng & Peng, Qiyuan, 2019. "Microscopic optimization model and algorithm for integrating train timetabling and track maintenance task scheduling," Transportation Research Part B: Methodological, Elsevier, vol. 127(C), pages 237-278.
    • Handle: RePEc:eee:transb:v:127:y:2019:i:c:p:237-278
    DOI: 10.1016/j.trb.2019.07.010
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    1. Wang, Pengling & Goverde, Rob M.P., 2019. "Multi-train trajectory optimization for energy-efficient timetabling," European Journal of Operational Research, Elsevier, vol. 272(2), pages 621-635.
    2. Wang, Pengling & Goverde, Rob M.P., 2017. "Multi-train trajectory optimization for energy efficiency and delay recovery on single-track railway lines," Transportation Research Part B: Methodological, Elsevier, vol. 105(C), pages 340-361.
    3. Luan, Xiaojie & Wang, Yihui & De Schutter, Bart & Meng, Lingyun & Lodewijks, Gabriel & Corman, Francesco, 2018. "Integration of real-time traffic management and train control for rail networks - Part 2: Extensions towards energy-efficient train operations," Transportation Research Part B: Methodological, Elsevier, vol. 115(C), pages 72-94.
    4. Li, Shukai & Dessouky, Maged M. & Yang, Lixing & Gao, Ziyou, 2017. "Joint optimal train regulation and passenger flow control strategy for high-frequency metro lines," Transportation Research Part B: Methodological, Elsevier, vol. 99(C), pages 113-137.
    5. Corman, Francesco & D'Ariano, Andrea & Pacciarelli, Dario & Pranzo, Marco, 2010. "A tabu search algorithm for rerouting trains during rail operations," Transportation Research Part B: Methodological, Elsevier, vol. 44(1), pages 175-192, January.
    6. Corman, Francesco & D’Ariano, Andrea & Marra, Alessio D. & Pacciarelli, Dario & Samà, Marcella, 2017. "Integrating train scheduling and delay management in real-time railway traffic control," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 105(C), pages 213-239.
    7. G Budai & D Huisman & R Dekker, 2006. "Scheduling preventive railway maintenance activities," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 57(9), pages 1035-1044, September.
    8. Wang, Yihui & D’Ariano, Andrea & Yin, Jiateng & Meng, Lingyun & Tang, Tao & Ning, Bin, 2018. "Passenger demand oriented train scheduling and rolling stock circulation planning for an urban rail transit line," Transportation Research Part B: Methodological, Elsevier, vol. 118(C), pages 193-227.
    9. Richard Lusby & Jesper Larsen & David Ryan & Matthias Ehrgott, 2011. "Routing Trains Through Railway Junctions: A New Set-Packing Approach," Transportation Science, INFORMS, vol. 45(2), pages 228-245, May.
    10. Pellegrini, Paola & Marlière, Grégory & Rodriguez, Joaquin, 2014. "Optimal train routing and scheduling for managing traffic perturbations in complex junctions," Transportation Research Part B: Methodological, Elsevier, vol. 59(C), pages 58-80.
    11. Scheepmaker, Gerben M. & Goverde, Rob M.P. & Kroon, Leo G., 2017. "Review of energy-efficient train control and timetabling," European Journal of Operational Research, Elsevier, vol. 257(2), pages 355-376.
    12. Andrea D’Ariano & Marco Pranzo, 2009. "An Advanced Real-Time Train Dispatching System for Minimizing the Propagation of Delays in a Dispatching Area Under Severe Disturbances," Networks and Spatial Economics, Springer, vol. 9(1), pages 63-84, March.
    13. Corman, F. & D’Ariano, A. & Pacciarelli, D. & Pranzo, M., 2012. "Optimal inter-area coordination of train rescheduling decisions," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 48(1), pages 71-88.
    14. Vansteenwegen, Pieter & Dewilde, Thijs & Burggraeve, Sofie & Cattrysse, Dirk, 2016. "An iterative approach for reducing the impact of infrastructure maintenance on the performance of railway systems," European Journal of Operational Research, Elsevier, vol. 252(1), pages 39-53.
    15. Huang, Yeran & Yang, Lixing & Tang, Tao & Gao, Ziyou & Cao, Fang, 2017. "Joint train scheduling optimization with service quality and energy efficiency in urban rail transit networks," Energy, Elsevier, vol. 138(C), pages 1124-1147.
    16. Yin, Jiateng & Tang, Tao & Yang, Lixing & Gao, Ziyou & Ran, Bin, 2016. "Energy-efficient metro train rescheduling with uncertain time-variant passenger demands: An approximate dynamic programming approach," Transportation Research Part B: Methodological, Elsevier, vol. 91(C), pages 178-210.
    17. Meng, Lingyun & Zhou, Xuesong, 2019. "An integrated train service plan optimization model with variable demand: A team-based scheduling approach with dual cost information in a layered network," Transportation Research Part B: Methodological, Elsevier, vol. 125(C), pages 1-28.
    18. Jean-François Cordeau & Paolo Toth & Daniele Vigo, 1998. "A Survey of Optimization Models for Train Routing and Scheduling," Transportation Science, INFORMS, vol. 32(4), pages 380-404, November.
    19. G. Caimi & F. Chudak & M. Fuchsberger & M. Laumanns & R. Zenklusen, 2011. "A New Resource-Constrained Multicommodity Flow Model for Conflict-Free Train Routing and Scheduling," Transportation Science, INFORMS, vol. 45(2), pages 212-227, May.
    20. Alberto Caprara & Matteo Fischetti & Paolo Toth, 2002. "Modeling and Solving the Train Timetabling Problem," Operations Research, INFORMS, vol. 50(5), pages 851-861, October.
    21. D'Ariano, Andrea & Pacciarelli, Dario & Pranzo, Marco, 2007. "A branch and bound algorithm for scheduling trains in a railway network," European Journal of Operational Research, Elsevier, vol. 183(2), pages 643-657, December.
    22. Yang, Lixing & Li, Keping & Gao, Ziyou & Li, Xiang, 2012. "Optimizing trains movement on a railway network," Omega, Elsevier, vol. 40(5), pages 619-633.
    23. Leonardo Lamorgese & Carlo Mannino, 2015. "An Exact Decomposition Approach for the Real-Time Train Dispatching Problem," Operations Research, INFORMS, vol. 63(1), pages 48-64, February.
    24. Luan, Xiaojie & Wang, Yihui & De Schutter, Bart & Meng, Lingyun & Lodewijks, Gabriel & Corman, Francesco, 2018. "Integration of real-time traffic management and train control for rail networks - Part 1: Optimization problems and solution approaches," Transportation Research Part B: Methodological, Elsevier, vol. 115(C), pages 41-71.
    25. Zhou, Xuesong & Zhong, Ming, 2007. "Single-track train timetabling with guaranteed optimality: Branch-and-bound algorithms with enhanced lower bounds," Transportation Research Part B: Methodological, Elsevier, vol. 41(3), pages 320-341, March.
    26. Burdett, R.L. & Kozan, E., 2010. "A disjunctive graph model and framework for constructing new train schedules," European Journal of Operational Research, Elsevier, vol. 200(1), pages 85-98, January.
    27. Törnquist, Johanna & Persson, Jan A., 2007. "N-tracked railway traffic re-scheduling during disturbances," Transportation Research Part B: Methodological, Elsevier, vol. 41(3), pages 342-362, March.
    28. Bešinović, Nikola & Goverde, Rob M.P. & Quaglietta, Egidio & Roberti, Roberto, 2016. "An integrated micro–macro approach to robust railway timetabling," Transportation Research Part B: Methodological, Elsevier, vol. 87(C), pages 14-32.
    29. Andrea D'Ariano & Francesco Corman & Dario Pacciarelli & Marco Pranzo, 2008. "Reordering and Local Rerouting Strategies to Manage Train Traffic in Real Time," Transportation Science, INFORMS, vol. 42(4), pages 405-419, November.
    30. Samà, Marcella & Pellegrini, Paola & D’Ariano, Andrea & Rodriguez, Joaquin & Pacciarelli, Dario, 2016. "Ant colony optimization for the real-time train routing selection problem," Transportation Research Part B: Methodological, Elsevier, vol. 85(C), pages 89-108.
    31. Mu, Shi & Dessouky, Maged, 2011. "Scheduling freight trains traveling on complex networks," Transportation Research Part B: Methodological, Elsevier, vol. 45(7), pages 1103-1123, August.
    32. Zhou, Leishan & Tong, Lu (Carol) & Chen, Junhua & Tang, Jinjin & Zhou, Xuesong, 2017. "Joint optimization of high-speed train timetables and speed profiles: A unified modeling approach using space-time-speed grid networks," Transportation Research Part B: Methodological, Elsevier, vol. 97(C), pages 157-181.
    33. Samà, Marcella & D’Ariano, Andrea & D’Ariano, Paolo & Pacciarelli, Dario, 2017. "Scheduling models for optimal aircraft traffic control at busy airports: Tardiness, priorities, equity and violations considerations," Omega, Elsevier, vol. 67(C), pages 81-98.
    34. Meng, Lingyun & Zhou, Xuesong, 2014. "Simultaneous train rerouting and rescheduling on an N-track network: A model reformulation with network-based cumulative flow variables," Transportation Research Part B: Methodological, Elsevier, vol. 67(C), pages 208-234.
    35. Yang, Lixing & Qi, Jianguo & Li, Shukai & Gao, Yuan, 2016. "Collaborative optimization for train scheduling and train stop planning on high-speed railways," Omega, Elsevier, vol. 64(C), pages 57-76.
    36. Rodriguez, Joaquín, 2007. "A constraint programming model for real-time train scheduling at junctions," Transportation Research Part B: Methodological, Elsevier, vol. 41(2), pages 231-245, February.
    37. Yin, Jiateng & Yang, Lixing & Tang, Tao & Gao, Ziyou & Ran, Bin, 2017. "Dynamic passenger demand oriented metro train scheduling with energy-efficiency and waiting time minimization: Mixed-integer linear programming approaches," Transportation Research Part B: Methodological, Elsevier, vol. 97(C), pages 182-213.
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    20. Zhengwen Liao, 2023. "Rescheduling Out-of-Gauge Trains with Speed Restrictions and Temporal Blockades on the Opposite-Direction Track," Mathematics, MDPI, vol. 11(12), pages 1-26, June.

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