# An Integrated Model for Demand Forecasting and Train Stop Planning for High-Speed Rail

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Literature

#### 2.1. Train Stop Planning Under Given Demand

#### 2.2. Modal Choice

#### 2.3. Integrated Model of Modal Choice and Transportation Services

## 3. An Integrated Model for Modal Choice and Train Stop Planning

#### 3.1. Train Stop Planning Model

**Assumption**

**1.**

**Assumption**

**2.**

#### 3.2. Modal Choice Model

#### 3.3. An Integrated Model

## 4. Solution Procedure

## 5. Case Study

#### 5.1. Data Collection

#### 5.2. Beijing–Shanghai HSR

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Station | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.05 | 0.22 | 0.18 | 0.26 | 0.97 | 0.97 | 0.97 | 0.95 | 0.94 | 0.97 | 0.99 | 0.98 | 0.96 | 0.94 | 0.97 | 0.96 | 0.98 | 0.97 | 0.94 | 0.96 | 0.94 | 0.99 |

2 | 0.03 | 0.07 | 0.16 | 0.14 | 0.50 | 0.97 | 0.95 | 0.98 | 0.95 | 0.96 | 0.99 | 0.95 | 1.00 | 0.99 | 0.97 | 0.99 | 1.00 | 0.98 | 0.99 | 0.98 | 0.98 | |

3 | 0.09 | 0.16 | 0.14 | 0.94 | 0.99 | 0.96 | 0.99 | 0.99 | 0.66 | 0.94 | 0.97 | 0.96 | 0.99 | 0.95 | 0.98 | 0.97 | 0.96 | 0.97 | 1.00 | 0.94 | ||

4 | 0.16 | 0.44 | 0.32 | 1.00 | 0.95 | 0.94 | 0.98 | 0.57 | 0.98 | 0.79 | 0.97 | 0.96 | 0.96 | 0.95 | 0.95 | 0.99 | 0.95 | 0.95 | 0.95 | |||

5 | 0.13 | 0.96 | 0.90 | 0.97 | 0.74 | 0.95 | 0.99 | 0.94 | 0.98 | 0.99 | 0.96 | 0.98 | 0.94 | 0.99 | 0.97 | 0.95 | 0.97 | 1.00 | ||||

6 | 0.04 | 0.29 | 0.25 | 0.46 | 1.00 | 0.57 | 0.95 | 0.98 | 0.94 | 0.99 | 0.98 | 0.94 | 0.99 | 0.96 | 0.95 | 0.96 | 0.99 | |||||

7 | 0.08 | 0.16 | 0.16 | 0.94 | 0.33 | 0.96 | 0.33 | 0.31 | 1.00 | 0.96 | 1.00 | 0.96 | 0.97 | 0.99 | 1.00 | 0.97 | ||||||

8 | 0.16 | 0.19 | 0.21 | 0.48 | 0.97 | 0.96 | 0.99 | 0.95 | 0.98 | 0.94 | 1.00 | 0.97 | 0.97 | 0.94 | 0.94 | |||||||

9 | 0.10 | 0.25 | 0.86 | 0.49 | 0.98 | 0.33 | 0.99 | 0.96 | 0.96 | 0.50 | 0.98 | 1.00 | 0.98 | 0.95 | ||||||||

10 | 0.27 | 0.20 | 0.33 | 0.20 | 0.35 | 0.99 | 0.95 | 0.99 | 0.94 | 0.96 | 0.96 | 0.98 | 0.97 | |||||||||

11 | 0.36 | 0.89 | 0.24 | 0.48 | 0.50 | 0.99 | 0.95 | 0.97 | 0.98 | 0.96 | 1.00 | 0.95 | ||||||||||

12 | 0.19 | 0.09 | 0.33 | 1.00 | 0.31 | 0.49 | 0.27 | 0.47 | 1.00 | 0.96 | 0.97 | |||||||||||

13 | 0.17 | 0.24 | 0.73 | 0.96 | 0.50 | 0.42 | 0.26 | 0.99 | 0.97 | 0.94 | ||||||||||||

14 | 0.10 | 0.57 | 0.80 | 0.91 | 0.53 | 0.95 | 0.94 | 0.53 | 0.98 | |||||||||||||

15 | 0.35 | 0.32 | 0.59 | 0.96 | 0.98 | 0.98 | 0.37 | 0.99 | ||||||||||||||

16 | 0.18 | 0.32 | 0.99 | 0.98 | 0.98 | 0.94 | 0.95 | |||||||||||||||

17 | 0.10 | 0.09 | 0.34 | 0.49 | 0.96 | 1.00 | ||||||||||||||||

18 | 0.07 | 0.14 | 0.52 | 0.24 | 0.98 | |||||||||||||||||

19 | 0.09 | 0.15 | 0.18 | 0.94 | ||||||||||||||||||

20 | 0.18 | 0.11 | 0.97 | |||||||||||||||||||

21 | 0.03 | 0.09 | ||||||||||||||||||||

22 | 0.03 |

**Table A2.**Growth rate of the maximum passenger flow compared with the current passenger flow in one direction (%).

Station | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.02 | −0.27 | −1.73 | 2.51 | 5.18 | 7.99 | 16.80 | 10.99 | 13.03 | 9.60 | 21.99 | 23.21 | 0.00 | 24.98 | 12.62 | 0.00 | 0.00 | 15.17 | 15.11 | 19.74 | 9.60 | 20.20 |

2 | 0.15 | 0.00 | −2.11 | 1.68 | 1.08 | 0.00 | 12.64 | 4.78 | 5.60 | 1.65 | 3.28 | 0.00 | 0.60 | 20.08 | 0.00 | 0.00 | 13.64 | 0.70 | 1.71 | 11.76 | 11.75 | |

3 | 0.20 | 0.00 | 11.20 | 5.86 | 3.00 | 3.47 | 22.62 | 16.18 | 3.52 | 1.30 | 0.00 | 0.61 | 16.11 | 0.85 | 83.33 | 1.18 | 1.61 | 1.44 | 2.51 | 15.21 | ||

4 | 6.19 | 0.05 | 8.16 | 8.43 | 1.84 | 2.82 | 16.63 | 9.22 | 16.83 | 0.00 | 3.68 | 2.85 | 0.37 | 63.07 | 3.28 | 0.73 | 1.78 | 0.57 | 2.80 | |||

5 | 0.02 | 1.43 | 0.00 | 2.47 | 1.86 | 2.93 | 0.07 | 4.89 | 0.00 | 3.19 | 5.34 | 2.05 | 0.99 | 26.32 | 4.03 | 2.24 | 0.68 | 1.04 | ||||

6 | 0.07 | −0.27 | 9.35 | 2.90 | 13.59 | 6.04 | 7.61 | 0.26 | 22.52 | 15.28 | 6.74 | 17.60 | 1.04 | 61.65 | 3.82 | 4.74 | 11.44 | |||||

7 | 0.15 | −0.22 | 4.39 | 17.14 | 10.29 | 4.39 | 0.00 | 14.46 | 14.76 | 2.33 | 3.48 | 0.29 | 1.93 | 51.53 | 2.47 | 2.30 | ||||||

8 | 0.05 | 0.07 | 0.93 | 0.07 | 1.63 | 0.00 | 1.90 | 3.83 | 0.00 | 1.89 | 0.00 | 0.00 | 0.11 | 6.25 | 1.20 | |||||||

9 | 0.16 | −1.48 | 5.36 | 1.59 | 0.00 | 7.04 | 6.04 | 11.00 | 1.91 | 1.06 | 76.42 | 2.53 | 53.30 | 7.49 | ||||||||

10 | 0.01 | 8.84 | 1.73 | 0.00 | 22.13 | 3.51 | 18.08 | 1.60 | 8.12 | 17.05 | 4.81 | 51.43 | 5.37 | |||||||||

11 | 0.01 | 6.47 | 0.00 | 5.87 | 13.29 | 2.94 | 9.65 | 8.01 | 26.57 | 12.16 | 42.89 | 24.80 | ||||||||||

12 | 0.03 | 0.00 | 3.48 | 5.66 | 2.95 | 0.00 | 15.96 | 3.30 | 7.19 | 2.22 | 6.79 | |||||||||||

13 | 0.87 | −0.62 | 5.11 | 3.34 | 2.21 | 12.42 | 1.22 | 48.11 | 12.50 | 11.31 | ||||||||||||

14 | 0.67 | 0.56 | 0.00 | 2.65 | 1.00 | 2.05 | 3.53 | 3.43 | 3.88 | |||||||||||||

15 | 0.98 | 7.28 | 2.76 | 2.57 | 4.02 | 6.17 | 4.18 | 11.25 | ||||||||||||||

16 | 1.07 | 0.10 | 3.94 | 5.46 | 18.84 | 10.87 | 18.16 | |||||||||||||||

17 | 0.49 | −0.19 | −0.67 | 0.53 | −0.51 | 11.26 | ||||||||||||||||

18 | 0.36 | −1.11 | 1.60 | 4.55 | 1.93 | |||||||||||||||||

19 | 0.21 | 11.40 | 5.70 | 6.41 | ||||||||||||||||||

20 | 0.85 | 6.31 | 7.89 | |||||||||||||||||||

21 | 0.50 | −0.26 | ||||||||||||||||||||

22 | 0.97 |

## Appendix B

## References

- Caseetta, E.; Coppola, P. High Speed Rail (HSR) Induced Demand Models. Available online: https://www.sciencedirect.com/science/article/pii/S1877042814000482 (accessed on 3 March 2019).
- Borjesson, M. Forecasting demand for high speed rail. Transp. Res. Pt. A-Policy Pract.
**2014**, 70, 81–92. [Google Scholar] [CrossRef] [Green Version] - Cordone, R.; Redaelli, F. Optimizing the demand captured by a railway system with a regular timetable. Transp. Res. Pt. B-Methodol.
**2011**, 45, 430–446. [Google Scholar] [CrossRef] - Chang, Y.H.; Yeh, C.H.; Shen, C.C. A multiobjective model for passenger train services planning: application to Taiwan’s high-speed rail line. Transp. Res. Pt. B-Methodol.
**2000**, 34, 91–106. [Google Scholar] [CrossRef] - van Hoesel, C.; Goossens, J.; Kroon, L. Optimising Halting Station of Passenger Railway Lines. Available online: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.528.4097&rep=rep1&type=pdf (accessed on 3 March 2019).
- Goossens, J.W.; van Hoesel, S.; Kroon, L. On solving multi-type railway line planning problems. Eur. J. Oper. Res.
**2006**, 168, 403–424. [Google Scholar] [CrossRef] - Shi, F.; Deng, L.; Huo, L. Bi-level programming model and algorithm of passenger train operation plan. China Railw. Sci.
**2007**, 28, 110–116. [Google Scholar] - Deng, L.; Shi, F.; Zhou, W. Stop schedule plan optimization for passenger train. China Railw. Sci.
**2009**, 30, 102–107. [Google Scholar] - Ulusoy, Y.Y.; Chien, S.I.J.; Wei, C.H. Optimal All-Stop, Short-turn, and express transit services under heterogeneous demand. Transp. Res. Rec.
**2010**, 8–18. [Google Scholar] [CrossRef] - Wang, L.; Jia, L.M.; Qin, Y.; Xu, J.; Mo, W.T. A two-layer optimization model for high-speed railway line planning. J. Zhejiang Univ.-Sci. A
**2011**, 12, 902–912. [Google Scholar] [CrossRef] - Fu, H.L.; Nie, L.; Sperry, B.R.; He, Z.H. Train stop scheduling in a high-speed rail network by utilizing a two-stage approach. Math. Probl. Eng.
**2012**, 2012. [Google Scholar] [CrossRef] - Huang, J.; Peng, Q. Two-stage optimization algorithm for stop schedule plan of high-speed train. J. Southwest Jiaotong Univ.
**2012**, 47, 484–489. [Google Scholar] - Jong, J.C.; Suen, C.S.; Chang, S.K. Decision support system to optimize railway stopping patterns application to taiwan high-speed rail. Transp. Res. Rec.
**2012**, 2289, 24–33. [Google Scholar] [CrossRef] - Fu, H.L.; Sperry, B.R.; Nie, L. Operational impacts of using restricted passenger flow assignment in high-speed train stop scheduling problem. Math. Probl. Eng.
**2013**, 2013. [Google Scholar] [CrossRef] - Li, D.-W.; Han, B.-M.; Li, X.-J.; Zhang, H.-J. High-speed railway stopping schedule optimization model based on node service. J. China Railw. Soc.
**2013**, 35, 1–5. [Google Scholar] [PubMed] - Park, B.H.; Seo, Y.I.; Hong, S.P.; Rho, H.L. Column generation approach to line planning with various halting patterns—application to the Korean high-speed railway. Asia Pac. J. Oper. Res.
**2013**, 30. [Google Scholar] [CrossRef] - Zhang, X.; Fu, H.; Tong, L. Optimizing the high speed train stop schedule using flexible stopping patterns combination. In Proceedings of the 2014 17th IEEE International Conference on Intelligent Transportation Systems (ITSC 2014), Qingdao, China, 8–11 October 2014; pp. 2398–2403. [Google Scholar]
- Wang, Z.-P.; Luo, X. Stopping schedule optimization of express/local trains in urban rail transit. J. South China Univ. Technol.
**2015**, 43, 91–98. [Google Scholar] - Lai, Y.C.; Shih, M.C.; Chen, G.H. Development of efficient stop planning optimization process for high-speed rail systems. J. Adv. Transp.
**2016**, 50, 1802–1819. [Google Scholar] [CrossRef] - Yang, L.X.; Qi, J.G.; Li, S.K.; Gao, Y. Collaborative optimization for train scheduling and train stop planning on high-speed railways. Omega-Int. J. Manage. Sci.
**2016**, 64, 57–76. [Google Scholar] [CrossRef] - Yue, Y.X.; Wang, S.F.; Zhou, L.S.; Tong, L.; Saat, M.R. Optimizing train stopping patterns and schedules for high-speed passenger rail corridors. Transp. Res. Pt. C-Emerg. Technol.
**2016**, 63, 126–146. [Google Scholar] [CrossRef] - Qi, J.G.; Li, S.K.; Gao, Y.; Yang, K.; Liu, P. Joint optimization model for train scheduling and train stop planning with passengers distribution on railway corridors. J. Oper. Res. Soc.
**2018**, 69, 556–570. [Google Scholar] [CrossRef] - Ben-Akiva, M.; Lerman, S.R. Discrete Choice Analysis: Theory And Application To Travel Demand; MIT Press: Cambridge, MA, USA, 1985. [Google Scholar]
- Cascetta, E.; Nuzzolo, A.; Russo, F.; Vitetta, A. A modified logit route choice model overcoming path overlapping problems. Specification and some calibration results for interurban networks. In Proceedings of the 13th International Symposium on Transportation and Traffic Theory, Lyon, France, 24–26 July 1996. [Google Scholar]
- Vovsha, P. Application of cross-nested logit model to mode choice in Tel Aviv, Israel, metropolitan area. Trans. Res. Rec.
**1997**, 1607, 6–15. [Google Scholar] [CrossRef] - Vovsha, P.; Bekhor, S. Link-nested logit model of route choice—Overcoming route overlapping problem. In Forecasting, Travel Behavior, And Network Modeling; National Academy of Sciences: Washington, DC USA, 1998; pp. 133–142. [Google Scholar]
- Ben-Akiva, M.; Bierlaire, M. Discrete choice methods and their applications to short term travel decisions. In Handbook of Transportation Science; Springer: Boston, MA, USA, 1999; pp. 5–33. [Google Scholar]
- Ramming, M.S. Network knowledge and route choice. Unpublished. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, 2001. [Google Scholar]
- Prashker, J.N.; Bekhor, S. Route choice models used in the stochastic user equilibrium problem: A review. Transp. Rev.
**2004**, 24, 437–463. [Google Scholar] [CrossRef] - Arentze, T.A.; Molin, E.J.E. Travelers’ preferences in multimodal networks: Design and results of a comprehensive series of choice experiments. Transp. Res. Pt. A-Policy Pract.
**2013**, 58, 15–28. [Google Scholar] [CrossRef] - Cascetta, E.; Carteni, A. The hedonic value of railways terminals. A quantitative analysis of the impact of stations quality on travellers behaviour. Transp. Res. Pt. A-Policy Pract.
**2014**, 61, 41–52. [Google Scholar] [CrossRef] - Baek, J.; Sohn, K. An investigation into passenger preference for express trains during peak hours. Transportation
**2016**, 43, 623–641. [Google Scholar] [CrossRef] - Li, Z.C.; Sheng, D. Forecasting passenger travel demand for air and high-speed rail integration service: A case study of Beijing-Guangzhou corridor, China. Transp. Res. Pt. A-Policy Pract.
**2016**, 94, 397–410. [Google Scholar] [CrossRef] - Sohoni, A.V.; Thomas, M.; Rao, K.V.K. Mode shift behavior of commuters due to the introduction of new rail transit mode. In Transportation Research Procedia; Elsevier: Amsterdam, The Netherlands, 2017; pp. 2607–2622. [Google Scholar]
- Carteni, A.; Pariota, L.; Henke, I. Hedonic value of high-speed rail services: Quantitative analysis of the students’ domestic tourist attractiveness of the main Italian cities. Transp. Res. Pt. A-Policy Pract.
**2017**, 100, 348–365. [Google Scholar] [CrossRef] - Mattson, J.; Hough, J.; Varma, A. Estimating demand for rural intercity bus services. Res. Transp. Econ.
**2018**, 71, 68–75. [Google Scholar] [CrossRef] - Borndorfer, R.; Karbstein, M.; Pfetsch, M.E. Models for fare planning in public transport. Discret Appl. Math.
**2012**, 160, 2591–2605. [Google Scholar] [CrossRef] [Green Version] - Espinosa-Aranda, J.L.; Garcia-Rodenas, R.; Ramirez-Flores, M.D.; Lopez-Garcia, M.L.; Angulo, E. High-speed railway scheduling based on user preferences. Eur. J. Oper. Res.
**2015**, 246, 772–786. [Google Scholar] [CrossRef] - Cantarella, G.E.; Velona, P.; Watling, D.P. Day-to-day Dynamics & Equilibrium Stability in A Two-Mode Transport System with Responsive bus Operator Strategies. Netw. Spat. Econ.
**2015**, 15, 485–506. [Google Scholar] - Li, X.W.; Yang, H. Dynamics of modal choice of heterogeneous travelers with responsive transit services. Transp. Res. Pt. C-Emerg. Technol.
**2016**, 68, 333–349. [Google Scholar] [CrossRef] - Canca, D.; De-Los-Santos, A.; Laporte, G.; Mesa, J.A. A general rapid network design, line planning and fleet investment integrated model. Ann. Oper. Res.
**2016**, 246, 127–144. [Google Scholar] [CrossRef] - Li, X.W.; Liu, W.; Yang, H. Traffic dynamics in a bi-modal transportation network with information provision and adaptive transit services. Transp. Res. Pt. C-Emerg. Technol.
**2018**, 91, 77–98. [Google Scholar] [CrossRef] - Robenek, T.; Azadeh, S.S.; Maknoon, Y.; de Lapparent, M.; Bierlaire, M. Train timetable design under elastic passenger demand. Transp. Res. Pt. B-Methodol.
**2018**, 111, 19–38. [Google Scholar] [CrossRef] [Green Version] - Wen, J.; Chen, Y.X.; Nassir, N.; Zhao, J.H. Transit-oriented autonomous vehicle operation with integrated demand-supply interaction. Transp. Res. Pt. C-Emerg. Technol.
**2018**, 97, 216–234. [Google Scholar] [CrossRef] - Jeong, Y.; Saha, S.; Chatterjee, D.; Moon, I. Direct shipping service routes with an empty container management strategy. Transp. Res. Pt. e-Logist. Transp. Rev.
**2018**, 118, 123–142. [Google Scholar] [CrossRef]

Abbreviation | Full Definition |
---|---|

AH | Air and high-speed rail |

EMU | Electric multiple units |

HSR | High-speed rail |

LPP | Line planning problem |

MCP | Modal choice problem |

MC-TSPP | Modal choice and train stop planning problem |

MILP | Mixed integer linear programming |

MIP | Mixed integer programming |

MINLP | Mixed integer nonlinear programming |

OD | Origin and destination or origin–destination |

SP | Stated preference |

TR | Traditional rail |

TSPP | Train stop planning problem |

Study | Decision Variables | Objective Function | Solution Method |
---|---|---|---|

Chang et al. [4] | Stop-schedule, frequency, size of fleet, flow variables | Total operating cost, passenger’s total travel time loss | Fuzzy mathematical programming by LP software such as LINDO |

Goossens et al. [5] | Type of a station, flow variables | Total travel time | Lagrangian relaxation |

Goossens et al. [6] | Frequency, flow variables | Total operating cost | IBM ILOG Cplex |

Shi et al. [7] | Stop-schedule | Total operating cost, passenger’s total travel cost | Simulated annealing algorithm |

Deng et al. [8] | Stop-schedule, flow variables | Passenger’s total travel cost, total number of stops | Simulated annealing algorithms |

Ulusoy et al. [9] | Stop-schedule, frequency, | Total cost | Heuristic |

Wang et al. [10] | Route, stop-schedule, type of train, passenger assignment | Total operation cost and unserved passenger volume for the top layer, the served passenger volume and minimizing the total travel time for the bottom layer | Genetic algorithm for the top layer, IBM ILOG Cplex for the bottom layer |

Fu et al. [11] | Stop-schedule, passenger assignment | Total number of stops | Heuristic |

Huang and Peng [12] | Stop-schedule | Passengers’ traveling convenience, generalized cost | Heuristic with tabu search |

Jong et al. [13] | Stop-schedule | Total passenger in-vehicle time | Genetic algorithm |

Fu et al. [14] | Stop-schedule, passenger assignment | Total trains’ deadhead Kilometers, passengers’ generalized cost | Heuristic |

Li et al. [15] | Stop-schedule | Total number of stops | Heuristic |

Park et al. [16] | Stop pattern, frequency, passenger assignment | The sum of the total operating cost and total passenger travel time | Column generation-based heuristic |

Zhang et al. [17] | Stop-schedule, passenger assignment | Total seat-kilometers of unoccupied train-set seats | LINGO and heuristic |

Wang and Luo [18] | Stop-schedule | Total travel time, total operating cost | Hybrid of genetic algorithm and simulated annealing algorithm |

Lai et al. [19] | Stop-schedule, passenger assignment | Total passenger travel time | Heuristic |

Yang et al. [20] | Stop-schedule, train scheduling, train type | Total dwelling time at intermediate stations and total delay at origin station | GAMS with Cplex |

Yue et al. [21] | Stop-schedule, train scheduling | Total profit | Column-generation-based heuristic algorithm |

Qi et al. [22] | Stop-schedule, train scheduling, passenger assignment | Total travel time of all trains, total travel time of all passengers | Heuristic with GAMS and Cplex |

Study | Impact Factor | Type | Method |
---|---|---|---|

Arentze and Molin [30] | Travel time, cost | Modal split | Multinomial logit model |

Caseetta and Coppola [1] | Travel time, travel cost, and service frequency | Induced demand or generated demand | Trip frequency model |

Borjesson [2] | Travel time, cost | Modal split | A new non-linear model |

Cascetta and Carteni [31] | Stations’ architectural quality | Modal split | Binomial logit model |

Baek and Sohn [32] | Travel time, trip-related properties, individual-specific characteristics, latent factors | Modal split | Multinomial logit model |

Li and Sheng [33] | Total cost, en route travel time, connection time | Modal split | Multinomial logit model |

Sohoni et al. [34] | Waiting time, travel time, travel cost, transfer, discomfort level | Modal split | Multinomial logit model |

Carteni et al. [35] | Hedonic quality | Modal split | Binomial logit model |

Mattson et al. [36] | Individual, trip, and mode characteristics | Modal split | Mixed logit model |

Study | Integration or Interaction | Objective | Method |
---|---|---|---|

Cordone and Redaelli [3] | Timetable and modal choice | Demand | Heuristic |

Borndorfer [37] | Line planning (frequency), fare planning and modal choice | Revenue, profit, demand, welfare | GAMS and the NLP-solver snopt |

Espinosa-Aranda [38] | Timetable (departure time) and modal choice | Profit | Metaheuristic |

Cantarella et al. [39] and Li and Yang [40] | Transit service frequency and modal choice | Travel cost, profit | Investigate |

Li et al. [42] | Fare, frequency and transit demand | - | Static analysis |

Robenek et al. [43] | Timetable design with a multinomial logit-based passenger assignment | Revenue | Heuristic |

Wen et al. [44] | Service design (fleet size, vehicle capacity, fare policy, and hailing policy) and demand | - | Simulation |

Notation | Definition |
---|---|

$G=\left(S,E\right)$ | Physical network of high-speed railway consisting of stations $S$ and tracks $E$ between stations |

$S$ | Set of stations, indexed by $i,j\in S$ |

${S}_{r}$ | Set of stations on route $r$ including terminal stations, indexed by $p,q,k\in {S}_{r}$ |

${S}_{r}^{\prime}$ | Set of stations on route $r$ excluding terminal stations, indexed by ${k}^{\prime}\in {S}_{r}^{\prime}$ |

$E$ | Set of tracks, indexed by $e\in E$ |

${E}_{r}$ | Set of tracks on route $r$, indexed by ${e}^{\prime}\in {E}_{r}$ |

$R$ | Set of potential operated routes, indexed by $r\in R$ |

${\mathsf{\Omega}}_{r}$ | A set of ${L}_{r}$ train trips based on route $r$, indexed by $g\in {\mathsf{\Omega}}_{r}=\left\{1,\dots ,{L}_{r}\right\}$, where ${L}_{r}$ is the possible number of train trips on route $r$ |

${R}_{ij}$ | Set of routes that pass station $i$ and station $j$ simultaneously, indexed by ${r}^{\prime}\in {R}_{ij}$ |

${R}_{e}$ | Set of routes that go through track $e$, indexed by ${r}^{\u2033}\in {R}_{e}$ |

${R}_{i}$ | Set of routes that pass station $i$, indexed by $\widehat{r}\in {R}_{i}$ |

${A}_{ij}$ | Set of transportation modes that serve passengers between city $i$ and city $j$, indexed by $a\in {A}_{ij}$ |

${L}_{ij}^{a}$ | Set of services that serve passengers between city $i$ and city $j$ for transport mode $a\in {A}_{ij}$, indexed by $h\in {L}_{ij}^{a}$ |

Notation | Definition |
---|---|

${W}_{i}$ | Dwell time at station $i$ |

${o}_{r}$,${d}_{r}$ | Terminal stations of route $r$ |

${H}_{e}$ | Capacity of track e during the planning horizon |

${C}_{i}$ | Capacity of station i during the planning horizon |

${N}_{r}$ | The number of stations passed by route r including the origin and destination station |

${L}_{r}$ | The possible number of train trips on route $r$ |

$M$ | A large constant |

${\mu}_{rpqk}$ | Direction indicators indicating if the order of station $q$ is greater than the order of station $k$ and the order of station $p$ is less than or equal to the order of station $k$ on route $r$, where $p\ne q$ is required. |

${Q}_{rg}$ | Seating capacity for train trip $g$ on route $r$ |

${\rho}_{rpqk}$ | An indicator whose value takes one if station $k$ is located between station $p$ and station $q$ on route $r$ requiring $p\ne q,p\ne k$ and $q\ne k$, and zero otherwise |

${t}_{ij}^{a,h}$ | Travel time for service $h$ of mode $a$ between city $i$ and city $j$, $a\in {A}_{ij}$, $h\in {L}_{ij}^{a}$ |

${t}_{ij}^{a}$ | Average travel time of mode $a$ between city $i$ and city $j$, $a\in {A}_{ij}$ |

${\widehat{t}}_{ij}^{HSR}$ | HSR train running time between stations $i$ and $j$ |

${U}_{ij}^{a}$ | Utility of mode $a$ between city $i$ and city $j$ |

${V}_{ij}^{a}$ | The deterministic utility of mode $a$ between city $i$ and city $j$ |

${\epsilon}_{ij}^{a}$ | Random term for mode $a$ between city $i$ and city $j$ |

${P}_{ij}^{a}$ | Ticket price of alternative $a$ from $i$ to $j$ |

$\beta $,$\gamma $ | Unknown coefficients for ticket price and travel time, respectively |

${\mathsf{\Lambda}}_{ij}$ | Possible demand for all transport modes between $i$ and $j$ |

Notation | Definition |
---|---|

${x}_{gp}^{r}$ | Stop-schedule variable (1 if a train trip $g\in {\mathsf{\Omega}}_{r}$ on route $r\in R$ stops at station $p\in {S}_{r}$; 0 otherwise) |

${v}_{gpq}^{r}$ | Passenger flow variable specifying the number of passengers from station $p$ to station $q$ that is served by train trip $g$ on route $r$, which depends on the stop-schedule variable. |

${P}_{g{k}^{\prime}}^{r}$ | The number of passengers on board for train trip $g$ on route $r$ when the train trip stops at an intermediate station ${k}^{\prime}$, which results from the passenger flow variable and stop-schedule variable. |

${D}_{ij}^{HSR}$ | Travel demand for HSR from station $i$ to station $j$ during the planning horizon, which is a result of the demand forecasting method. |

Parameter | Value |
---|---|

Average dwell time at stations (including the acceleration time 1 min and deceleration time 2 min) (${W}_{i}$) | 5 min/station |

Train seating capacity (${Q}_{rg}$) | 1200 seats/train |

Tracks and stations capacity (${H}_{e}$,${C}_{i}$) | 200 train trips in one direction/day |

**Table 9.**Mixed integer programming (MIP) relative error with different time limit under the maximum demand.

Instance | Computation Time Limit (s) | MIP Relative Error |
---|---|---|

1 | 600 | 97.64% |

2 | 1200 | 84.24% |

3 | 1800 | 71.65% |

4 | 2400 | 59.86% |

5 | 3000 | 48.89% |

6 | 3600 | 38.72% |

7 | 4200 | 31.56% |

8 | 4800 | 24.97% |

9 | 5400 | 18.94% |

10 | 6000 | 13.48% |

11 | 6600 | 8.59% |

12 | 7200 | 4.26% |

13 | 7800 | 4.26% |

14 | 8400 | 4.26% |

15 | 9000 | 4.21% |

16 | 9600 | 4.21% |

17 | 10,200 | 4.21% |

18 | 10,800 | 4.21% |

19 | 11,400 | 4.21% |

20 | 12,000 | 4.21% |

21 | 12,600 | 4.21% |

22 | 13,200 | 4.21% |

23 | 13,800 | 4.21% |

24 | 14,400 | 4.21% |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jin, G.; He, S.; Li, J.; Li, Y.; Guo, X.; Xu, H.
An Integrated Model for Demand Forecasting and Train Stop Planning for High-Speed Rail. *Symmetry* **2019**, *11*, 720.
https://doi.org/10.3390/sym11050720

**AMA Style**

Jin G, He S, Li J, Li Y, Guo X, Xu H.
An Integrated Model for Demand Forecasting and Train Stop Planning for High-Speed Rail. *Symmetry*. 2019; 11(5):720.
https://doi.org/10.3390/sym11050720

**Chicago/Turabian Style**

Jin, Guowei, Shiwei He, Jiabin Li, Yubin Li, Xiaole Guo, and Hongfei Xu.
2019. "An Integrated Model for Demand Forecasting and Train Stop Planning for High-Speed Rail" *Symmetry* 11, no. 5: 720.
https://doi.org/10.3390/sym11050720