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Train scheduling for minimizing passenger waiting time with time-dependent demand and skip-stop patterns: Nonlinear integer programming models with linear constraints

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Cited by:

  1. Yuan, Jiawei & Gao, Yuan & Li, Shukai & Liu, Pei & Yang, Lixing, 2022. "Integrated optimization of train timetable, rolling stock assignment and short-turning strategy for a metro line," European Journal of Operational Research, Elsevier, vol. 301(3), pages 855-874.
  2. Pu, Song & Zhan, Shuguang, 2021. "Two-stage robust railway line-planning approach with passenger demand uncertainty," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 152(C).
  3. Pei, Mingyang & Lin, Peiqun & Du, Jun & Li, Xiaopeng & Chen, Zhiwei, 2021. "Vehicle dispatching in modular transit networks: A mixed-integer nonlinear programming model," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 147(C).
  4. Jianqiang Wang & Wenlong Zhao & Chenglin Liu & Zhipeng Huang, 2023. "A System Optimization Approach for Trains’ Operation Plan with a Time Flexible Pricing Strategy for High-Speed Rail Corridors," Sustainability, MDPI, vol. 15(12), pages 1-22, June.
  5. Han Zheng & Junhua Chen & Zhaocha Huang & Jianhao Zhu, 2022. "Joint Optimization of Multi-Cycle Timetable Considering Supply-to-Demand Relationship and Energy Consumption for Rail Express," Mathematics, MDPI, vol. 10(21), pages 1-29, November.
  6. He, Deqiang & Yang, Yanjie & Chen, Yanjun & Deng, Jianxin & Shan, Sheng & Liu, Jianren & Li, Xianwang, 2020. "An integrated optimization model of metro energy consumption based on regenerative energy and passenger transfer," Applied Energy, Elsevier, vol. 264(C).
  7. Huanyin Su & Shuting Peng & Shanglin Mo & Kaixin Wu, 2022. "Neural Network-Based Hybrid Forecasting Models for Time-Varying Passenger Flow of Intercity High-Speed Railways," Mathematics, MDPI, vol. 10(23), pages 1-21, December.
  8. Yin, Jiateng & Wang, Miao & D’Ariano, Andrea & Zhang, Jinlei & Yang, Lixing, 2023. "Synchronization of train timetables in an urban rail network: A bi-objective optimization approach," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 174(C).
  9. Zhao, Yaqiong & Li, Dewei & Yin, Yonghao & Zhao, Xiaoli, 2023. "Integrated optimization of demand-driven timetable, train formation plan and rolling stock circulation with variable running times and dwell times," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 171(C).
  10. Yuzhao Zhang & Jianqiang Wang & Wenjuan Cai, 2019. "Passengers’ Demand Characteristics Experimental Analysis of EMU Trains with Sleeping Cars in Northwest China," Sustainability, MDPI, vol. 11(19), pages 1-17, September.
  11. Limsawasd, Charinee & Athigakunagorn, Nathee & Khathawatcharakun, Phattadon & Boonmee, Atiwat, 2022. "Skip-Stop Strategy Patterns optimization to enhance mass transit operation under physical distancing policy due to COVID-19 pandemic outbreak," Transport Policy, Elsevier, vol. 126(C), pages 225-238.
  12. Pan, Hanchuan & Yang, Lixing & Liang, Zhe, 2023. "Demand-oriented integration optimization of train timetabling and rolling stock circulation planning with flexible train compositions: A column-generation-based approach," European Journal of Operational Research, Elsevier, vol. 305(1), pages 184-206.
  13. Yin, Jiateng & D’Ariano, Andrea & Wang, Yihui & Yang, Lixing & Tang, Tao, 2021. "Timetable coordination in a rail transit network with time-dependent passenger demand," European Journal of Operational Research, Elsevier, vol. 295(1), pages 183-202.
  14. Chen, Zhiwei & Li, Xiaopeng & Zhou, Xuesong, 2019. "Operational design for shuttle systems with modular vehicles under oversaturated traffic: Discrete modeling method," Transportation Research Part B: Methodological, Elsevier, vol. 122(C), pages 1-19.
  15. Yan, Fei & Goverde, Rob M.P., 2019. "Combined line planning and train timetabling for strongly heterogeneous railway lines with direct connections," Transportation Research Part B: Methodological, Elsevier, vol. 127(C), pages 20-46.
  16. Tian, Xiaopeng & Niu, Huimin, 2020. "Optimization of demand-oriented train timetables under overtaking operations: A surrogate-dual-variable column generation for eliminating indivisibility," Transportation Research Part B: Methodological, Elsevier, vol. 142(C), pages 143-173.
  17. Xin Zhang & Lei Nie & Xin Wu & Yu Ke, 2020. "How to Optimize Train Stops under Diverse Passenger Demand: a New Line Planning Method for Large-Scale High-Speed Rail Networks," Networks and Spatial Economics, Springer, vol. 20(4), pages 963-988, December.
  18. Chen, Zebin & Li, Shukai & D’Ariano, Andrea & Yang, Lixing, 2022. "Real-time optimization for train regulation and stop-skipping adjustment strategy of urban rail transit lines," Omega, Elsevier, vol. 110(C).
  19. Shuo Zhao & Xiwei Mi & Zhenyi Li, 2019. "A Stop-Probability Approach for O-D Service Frequency on High-Speed Railway Lines," Sustainability, MDPI, vol. 11(24), pages 1-21, December.
  20. Wang, Chao & Ma, Changxi & Xu, Xuecai(Daniel), 2020. "Multi-objective optimization of real-time customized bus routes based on two-stage method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).
  21. Wang, Entai & Yang, Lixing & Yin, Jiateng & Zhang, Jinlei & Gao, Ziyou, 2024. "Passenger-oriented rolling stock scheduling in the metro system with multiple depots: Network flow based approaches," Transportation Research Part B: Methodological, Elsevier, vol. 180(C).
  22. Polinder, G.-J. & Cacchiani, V. & Schmidt, M.E. & Huisman, D., 2020. "An iterative heuristic for passenger-centric train timetabling with integrated adaption times," ERIM Report Series Research in Management ERS-2020-006-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
  23. Ali Shahabi & Sadigh Raissi & Kaveh Khalili-Damghani & Meysam Rafei, 2021. "Designing a resilient skip-stop schedule in rapid rail transit using a simulation-based optimization methodology," Operational Research, Springer, vol. 21(3), pages 1691-1721, September.
  24. Luan, Xiaojie & Corman, Francesco, 2022. "Passenger-oriented traffic control for rail networks: An optimization model considering crowding effects on passenger choices and train operations," Transportation Research Part B: Methodological, Elsevier, vol. 158(C), pages 239-272.
  25. Niu, Huimin & Zhou, Xuesong & Tian, Xiaopeng, 2018. "Coordinating assignment and routing decisions in transit vehicle schedules: A variable-splitting Lagrangian decomposition approach for solution symmetry breaking," Transportation Research Part B: Methodological, Elsevier, vol. 107(C), pages 70-101.
  26. Shi, Jungang & Yang, Lixing & Yang, Jing & Gao, Ziyou, 2018. "Service-oriented train timetabling with collaborative passenger flow control on an oversaturated metro line: An integer linear optimization approach," Transportation Research Part B: Methodological, Elsevier, vol. 110(C), pages 26-59.
  27. Fan, Wenbo & Ran, Yu, 2021. "Planning skip-stop services with schedule coordination," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 145(C).
  28. Sadrani, Mohammad & Tirachini, Alejandro & Antoniou, Constantinos, 2022. "Vehicle dispatching plan for minimizing passenger waiting time in a corridor with buses of different sizes: Model formulation and solution approaches," European Journal of Operational Research, Elsevier, vol. 299(1), pages 263-282.
  29. Fei Dou & Huiru Zhang & Haodong Yin & Yun Wei & Yao Ning, 2022. "An Optimization Method of Urban Rail Train Operation Scheme Based on the Control of Load Factor," Sustainability, MDPI, vol. 14(21), pages 1-13, October.
  30. Yang, Xin & Chen, Anthony & Ning, Bin & Tang, Tao, 2016. "A stochastic model for the integrated optimization on metro timetable and speed profile with uncertain train mass," Transportation Research Part B: Methodological, Elsevier, vol. 91(C), pages 424-445.
  31. Kang, Liujiang & Li, Hao & Sun, Huijun & Wu, Jianjun & Cao, Zhiguang & Buhigiro, Nsabimana, 2021. "First train timetabling and bus service bridging in intermodal bus-and-train transit networks," Transportation Research Part B: Methodological, Elsevier, vol. 149(C), pages 443-462.
  32. Gkiotsalitis, K. & Alesiani, F., 2019. "Robust timetable optimization for bus lines subject to resource and regulatory constraints," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 128(C), pages 30-51.
  33. Huang, Yu & Zhou, Wenliang & Qin, Jin & Deng, Lianbo, 2023. "Optimization of energy-efficiency train schedule considering passenger demand and rolling stock circulation plan of subway line," Energy, Elsevier, vol. 275(C).
  34. Zhichao Cao & Zhenzhou Yuan & Silin Zhang, 2016. "Performance Analysis of Stop-Skipping Scheduling Plans in Rail Transit under Time-Dependent Demand," IJERPH, MDPI, vol. 13(7), pages 1-23, July.
  35. Li, Changle & Ma, Jiao & Luan, Tom H. & Zhou, Xun & Xiong, Lei, 2018. "An incentive-based optimizing strategy of service frequency for an urban rail transit system," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 118(C), pages 106-122.
  36. Liu, Jiangtao & Zhou, Xuesong, 2016. "Capacitated transit service network design with boundedly rational agents," Transportation Research Part B: Methodological, Elsevier, vol. 93(PA), pages 225-250.
  37. Chai, Simin & Yin, Jiateng & D’Ariano, Andrea & Liu, Ronghui & Yang, Lixing & Tang, Tao, 2024. "A branch-and-cut algorithm for scheduling train platoons in urban rail networks," Transportation Research Part B: Methodological, Elsevier, vol. 181(C).
  38. 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.
  39. 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.
  40. Xue, Hongjiao & Jia, Limin & Li, Jian & Guo, Jianyuan, 2022. "Jointly optimized demand-oriented train timetable and passenger flow control strategy for a congested subway line under a short-turning operation pattern," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 593(C).
  41. Kang, Liujiang & Meng, Qiang, 2017. "Two-phase decomposition method for the last train departure time choice in subway networks," Transportation Research Part B: Methodological, Elsevier, vol. 104(C), pages 568-582.
  42. Liu, Xiaoyu & Dabiri, Azita & Xun, Jing & De Schutter, Bart, 2023. "Bi-level model predictive control for metro networks: Integration of timetables, passenger flows, and train speed profiles," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 180(C).
  43. 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.
  44. Chen, Jingxu & Liu, Zhiyuan & Zhu, Senlai & Wang, Wei, 2015. "Design of limited-stop bus service with capacity constraint and stochastic travel time," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 83(C), pages 1-15.
  45. Huang, Yeran & Mannino, Carlo & Yang, Lixing & Tang, Tao, 2020. "Coupling time-indexed and big-M formulations for real-time train scheduling during metro service disruptions," Transportation Research Part B: Methodological, Elsevier, vol. 133(C), pages 38-61.
  46. Cacchiani, Valentina & Qi, Jianguo & Yang, Lixing, 2020. "Robust optimization models for integrated train stop planning and timetabling with passenger demand uncertainty," Transportation Research Part B: Methodological, Elsevier, vol. 136(C), pages 1-29.
  47. Wu, Yinghui & Yang, Hai & Zhao, Shuo & Shang, Pan, 2021. "Mitigating unfairness in urban rail transit operation: A mixed-integer linear programming approach," Transportation Research Part B: Methodological, Elsevier, vol. 149(C), pages 418-442.
  48. Shi, Jungang & Yang, Jing & Yang, Lixing & Tao, Lefeng & Qiang, Shengjie & Di, Zhen & Guo, Junhua, 2023. "Safety-oriented train timetabling and stop planning with time-varying and elastic demand on overcrowded commuter metro lines," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 175(C).
  49. Yang, Lin & Gao, Yuan & D’Ariano, Andrea & Xu, Suxiu, 2024. "Integrated optimization of train timetable and train unit circulation for a Y-type urban rail transit system with flexible train composition mode," Omega, Elsevier, vol. 122(C).
  50. Yang, Xin & Chen, Anthony & Ning, Bin & Tang, Tao, 2017. "Bi-objective programming approach for solving the metro timetable optimization problem with dwell time uncertainty," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 97(C), pages 22-37.
  51. Wenliang Zhou & Wenzhuang Fan & Xiaorong You & Lianbo Deng, 2019. "Demand-Oriented Train Timetabling Integrated with Passenger Train-Booking Decisions," Sustainability, MDPI, vol. 11(18), pages 1-34, September.
  52. Gao, Yuan & Kroon, Leo & Schmidt, Marie & Yang, Lixing, 2016. "Rescheduling a metro line in an over-crowded situation after disruptions," Transportation Research Part B: Methodological, Elsevier, vol. 93(PA), pages 425-449.
  53. Zhou, Wenliang & Tian, Junli & Xue, Lijuan & Jiang, Min & Deng, Lianbo & Qin, Jin, 2017. "Multi-periodic train timetabling using a period-type-based Lagrangian relaxation decomposition," Transportation Research Part B: Methodological, Elsevier, vol. 105(C), pages 144-173.
  54. Zhang, Yongxiang & Peng, Qiyuan & Lu, Gongyuan & Zhong, Qingwei & Yan, Xu & Zhou, Xuesong, 2022. "Integrated line planning and train timetabling through price-based cross-resolution feedback mechanism," Transportation Research Part B: Methodological, Elsevier, vol. 155(C), pages 240-277.
  55. Xu, Guangming & Zhong, Linhuan & Hu, Xinlei & Liu, Wei, 2022. "Optimal pricing and seat allocation schemes in passenger railway systems," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 157(C).
  56. Julia Lange & Frank Werner, 2018. "Approaches to modeling train scheduling problems as job-shop problems with blocking constraints," Journal of Scheduling, Springer, vol. 21(2), pages 191-207, April.
  57. Erfan Hassannayebi & Seyed Hessameddin Zegordi & Mohammad Reza Amin-Naseri & Masoud Yaghini, 2018. "Optimizing headways for urban rail transit services using adaptive particle swarm algorithms," Public Transport, Springer, vol. 10(1), pages 23-62, May.
  58. Chen, Yao & An, Kun, 2021. "Integrated optimization of bus bridging routes and timetables for rail disruptions," European Journal of Operational Research, Elsevier, vol. 295(2), pages 484-498.
  59. Tatsuki Yamauchi & Mizuyo Takamatsu & Shinji Imahori, 2023. "Optimizing train stopping patterns for congestion management," Public Transport, Springer, vol. 15(1), pages 1-29, March.
  60. Blanco, Víctor & Conde, Eduardo & Hinojosa, Yolanda & Puerto, Justo, 2020. "An optimization model for line planning and timetabling in automated urban metro subway networks. A case study," Omega, Elsevier, vol. 92(C).
  61. Kang, Liujiang & Zhu, Xiaoning & Sun, Huijun & Puchinger, Jakob & Ruthmair, Mario & Hu, Bin, 2016. "Modeling the first train timetabling problem with minimal missed trains and synchronization time differences in subway networks," Transportation Research Part B: Methodological, Elsevier, vol. 93(PA), pages 17-36.
  62. Chen, Zhiwei & Li, Xiaopeng & Zhou, Xuesong, 2020. "Operational design for shuttle systems with modular vehicles under oversaturated traffic: Continuous modeling method," Transportation Research Part B: Methodological, Elsevier, vol. 132(C), pages 76-100.
  63. Xu, Xiaoming & Li, Chung-Lun & Xu, Zhou, 2021. "Train timetabling with stop-skipping, passenger flow, and platform choice considerations," Transportation Research Part B: Methodological, Elsevier, vol. 150(C), pages 52-74.
  64. Zhang, Ping & Sun, Huijun & Qu, Yunchao & Yin, Haodong & Jin, Jian Gang & Wu, Jianjun, 2021. "Model and algorithm of coordinated flow controlling with station-based constraints in a metro system," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 148(C).
  65. Hu, Yuting & Li, Shukai & Dessouky, Maged M. & Yang, Lixing & Gao, Ziyou, 2022. "Computationally efficient train timetable generation of metro networks with uncertain transfer walking time to reduce passenger waiting time: A generalized Benders decomposition-based method," Transportation Research Part B: Methodological, Elsevier, vol. 163(C), pages 210-231.
  66. Xie, J. & Wong, S.C. & Zhan, S. & Lo, S.M. & Chen, Anthony, 2020. "Train schedule optimization based on schedule-based stochastic passenger assignment," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 136(C).
  67. Han, Zhenyu & Han, Baoming & Li, Dewei & Ning, Shangbin & Yang, Ruixia & Yin, Yonghao, 2021. "Train timetabling in rail transit network under uncertain and dynamic demand using Advanced and Adaptive NSGA-II," Transportation Research Part B: Methodological, Elsevier, vol. 154(C), pages 65-99.
  68. Zhang, Di & Gao, Yuan & Yang, Lixing & Cui, Lixin, 2024. "Timetable synchronization of the last several trains at night in an urban rail transit network," European Journal of Operational Research, Elsevier, vol. 313(2), pages 494-512.
  69. Xu, Guangming & Liu, Wei & Wu, Runfa & Yang, Hai, 2021. "A double time-scale passenger assignment model for high-speed railway networks with continuum capacity approximation," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 150(C).
  70. Pan Shang & Yu Yao & Liya Yang & Lingyun Meng & Pengli Mo, 2021. "Integrated Model for Timetabling and Circulation Planning on an Urban Rail Transit Line: a Coupled Network-Based Flow Formulation," Networks and Spatial Economics, Springer, vol. 21(2), pages 331-364, June.
  71. 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.
  72. Liu, Renming & Li, Shukai & Yang, Lixing, 2020. "Collaborative optimization for metro train scheduling and train connections combined with passenger flow control strategy," Omega, Elsevier, vol. 90(C).
  73. Tangjian Wei & Feng Shi & Guangming Xu, 2019. "Estimation of Time-Varying Passenger Demand for High Speed Rail System," Complexity, Hindawi, vol. 2019, pages 1-24, March.
  74. Xu, Guangming & Zhong, Linhuan & Liu, Wei & Guo, Jing, 2024. "A flexible train composition strategy with extra-long trains for high-speed railway corridors with time-varying demand," Transportation Research Part B: Methodological, Elsevier, vol. 179(C).
  75. Shuo Zhao & Jinfei Wu & Zhenyi Li & Ge Meng, 2022. "Train Operational Plan Optimization for Urban Rail Transit Lines Considering Circulation Balance," Sustainability, MDPI, vol. 14(9), pages 1-21, April.
  76. Xu, Guangming & Liu, Yihan & Gao, Yihan & Liu, Wei, 2023. "Integrated optimization of train stopping plan and seat allocation scheme for railway systems under equilibrium travel choice and elastic demand," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 177(C).
  77. Zhichao Cao & Yuqing Wang & Zihao Yang & Changjun Chen & Silin Zhang, 2023. "Timetable Rescheduling Using Skip-Stop Strategy for Sustainable Urban Rail Transit," Sustainability, MDPI, vol. 15(19), pages 1-29, October.
  78. Shang, Pan & Li, Ruimin & Guo, Jifu & Xian, Kai & Zhou, Xuesong, 2019. "Integrating Lagrangian and Eulerian observations for passenger flow state estimation in an urban rail transit network: A space-time-state hyper network-based assignment approach," Transportation Research Part B: Methodological, Elsevier, vol. 121(C), pages 135-167.
  79. Jiang, Feng & Cacchiani, Valentina & Toth, Paolo, 2017. "Train timetabling by skip-stop planning in highly congested lines," Transportation Research Part B: Methodological, Elsevier, vol. 104(C), pages 149-174.
  80. Zhang, Quan & Li, Xuan & Yan, Tao & Lu, Lili & Shi, Yang, 2022. "Last train timetabling optimization for minimizing passenger transfer failures in urban rail transit networks: A time period based approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 605(C).
  81. Jiateng Yin & Lixing Yang & Xuesong Zhou & Tao Tang & Ziyou Gao, 2019. "Balancing a one‐way corridor capacity and safety‐oriented reliability: A stochastic optimization approach for metro train timetabling," Naval Research Logistics (NRL), John Wiley & Sons, vol. 66(4), pages 297-320, June.
  82. 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.
  83. Liang, Jinpeng & Zang, Guangzhi & Liu, Haitao & Zheng, Jianfeng & Gao, Ziyou, 2023. "Reducing passenger waiting time in oversaturated metro lines with passenger flow control policy," Omega, Elsevier, vol. 117(C).
  84. Subrata Mitra & Balram Avittathur, 2018. "Application of linear programming in optimizing the procurement and movement of coal for an Indian coal-fired power-generating company," DECISION: Official Journal of the Indian Institute of Management Calcutta, Springer;Indian Institute of Management Calcutta, vol. 45(3), pages 207-224, September.
  85. Yiduo Huang & Zuojun Max Shen, 2021. "Optimizing timetable and network reopen plans for public transportation networks during a COVID19-like pandemic," Papers 2109.03940, arXiv.org.
  86. Chen, Junlan & Pu, Ziyuan & Guo, Xiucheng & Cao, Jieyu & Zhang, Fang, 2023. "Multiperiod metro timetable optimization based on the complex network and dynamic travel demand," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 611(C).
  87. Yang, Songpo & Liao, Feixiong & Wu, Jianjun & Timmermans, Harry J.P. & Sun, Huijun & Gao, Ziyou, 2020. "A bi-objective timetable optimization model incorporating energy allocation and passenger assignment in an energy-regenerative metro system," Transportation Research Part B: Methodological, Elsevier, vol. 133(C), pages 85-113.
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