# Multiperiod Transfer Synchronization for Cross-Platform Transfer in an Urban Rail Transit System

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Transfer Optimization Model for Multiple Periods

#### 2.1. Assumptions

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

#### 2.2. Symbols and Variables

- $\mathrm{L}$
- the set of lines (with directions) in the network, $\mathrm{L}=\left\{\mathrm{l}|\mathrm{l}=1,2,\dots \dots ,\mathrm{M}\right\}$, where $\mathrm{M}$ is the total number of lines;
- ${\mathrm{S}}_{\mathrm{l}}$
- the set of stations on Line $\mathrm{l}$, ${\mathrm{S}}_{\mathrm{l}}=\left\{{\mathrm{s}}_{\mathrm{l}}{|\mathrm{s}}_{\mathrm{l}}=1,2,\dots \dots ,{\mathrm{N}}_{\mathrm{l}}\right\}$, where ${\mathrm{N}}_{\mathrm{l}}$ is the total number of stations online $\mathrm{l}$;
- $\mathrm{S}$
- the set of stations in the network, $\mathrm{S}=\left\{\mathrm{s}|\mathrm{s}=1,2,\dots \dots ,\mathrm{N}\right\}$, where $\mathrm{N}$ is the total number of stations;
- ${\mathrm{Q}}_{\mathrm{l}}$
- the set of trains on Line $\mathrm{l}$, ${\mathrm{Q}}_{\mathrm{l}}=\left\{{\mathrm{q}}_{\mathrm{l}}{|\mathrm{q}}_{\mathrm{l}}=1,2,\dots \dots ,{\mathrm{K}}_{\mathrm{l}}\right\}$, where ${\mathrm{K}}_{\mathrm{l}}$ is the total number of stations online $\mathrm{l}$;
- $\mathrm{Q}$
- the set of trains in the network, $\mathrm{Q}=\left\{\mathrm{q}|\mathrm{q}=1,2,\dots \dots ,\mathrm{K}\right\}$, where $\mathrm{K}$ is the total number of trains;
- $\mathrm{R}$
- the set of transfer arcs in the network, $\mathrm{R}=\left\{\mathrm{r}|\mathrm{r}=\left(\mathrm{l},\mathrm{s}\right)\to \left({\mathrm{l}}^{\prime},{\mathrm{s}}^{\prime}\right)\right\}$, where $\mathrm{r}$ donates a transfer in the network only if $\mathrm{s}$ and ${\mathrm{s}}^{\prime}$ are the same stations;
- ${\mathrm{h}}_{\mathrm{ls}}^{\mathrm{T},\mathrm{max}}$
- the maximum departure headway of Line $\mathrm{l}$ during time period $\mathrm{T}$;
- ${\mathrm{h}}_{\mathrm{ls}}^{\mathrm{T},\mathrm{min}}$
- the minimum departure headway of Line $\mathrm{l}$ during time period $\mathrm{T}$;
- ${\mathrm{dt}}_{\mathrm{lqs}}$
- the departure time of $\mathrm{q}$ on Line $\mathrm{l}$ at station$\text{}\mathrm{s}$;
- ${\mathrm{at}}_{\mathrm{lqs}}$
- the arrival time of $\mathrm{q}$ on Line $\mathrm{l}$ at station$\text{}\mathrm{s}$;
- ${\mathrm{rt}}_{\mathrm{lqs}}$
- the running time of $\mathrm{q}$ on Line $\mathrm{l}$ between station$\text{}\mathrm{s}$ and station$\text{}\mathrm{s}+1$;
- ${\mathrm{st}}_{\mathrm{lqs}}$
- the dwell time of $\mathrm{q}$ on Line $\mathrm{l}$ at station $\mathrm{s}$;
- ${\mathrm{TT}}_{\mathrm{lq}}$
- the trip time for train $\mathrm{q}$ on Line $\mathrm{l}$;
- ${\mathrm{CT}}_{\mathrm{ls}}$
- the time for trains on connecting Line $\mathrm{l}\text{}$to clear out the platform at station$\text{}\mathrm{s}$;
- ${\mathrm{ndt}}_{\mathrm{lq}}$
- the departure time for train $\mathrm{q}$ on Line $\mathrm{l}$ after turn-around at terminal station;
- ${\mathrm{ta}}_{\mathrm{ls}}^{\mathrm{min}}$
- the minimum time for turn-around at station s of Line $\mathrm{l}$;
- ${\mathrm{T}}_{{\mathrm{ll}}^{\prime}{\mathrm{sqq}}^{\prime}}$
- the transfer synchronization time from train $\mathrm{q}$ on Line$\text{}\mathrm{l}$ to train ${\mathrm{q}}^{\prime}$ on Line ${\mathrm{l}}^{\prime}$ at station $\mathrm{s}$;
- ${\mathrm{w}}_{{\mathrm{ll}}^{\prime}{\mathrm{sqq}}^{\prime}}$
- the transfer walking time from train $\mathrm{q}$ on Line $\mathrm{l}$ to train ${\mathrm{q}}^{\prime}$ on Line ${\mathrm{l}}^{\prime}$ at station $\mathrm{s}$;
- ${\mathrm{tw}}_{{\mathrm{ll}}^{\prime}{\mathrm{sqq}}^{\prime}}$
- the transfer waiting time from train$\text{}\mathrm{q}$ on Line$\mathrm{l}$ to train ${\mathrm{q}}^{\prime}$ on Line ${\mathrm{l}}^{\prime}$ at station $\mathrm{s}$;
- ${\mathrm{p}}_{{\mathrm{ll}}^{\prime}{\mathrm{sqq}}^{\prime}}$
- the number of transfer passengers from train$\text{}\mathrm{q}$ on Line$\text{}\mathrm{l}$ to train ${\mathrm{q}}^{\prime}$ on Line ${\mathrm{l}}^{\prime}$ at station $\mathrm{s}$;
- ${\mathrm{P}}_{{\mathrm{ll}}^{\prime}\mathrm{s}}$
- the total number of transfer passenger transfers from Line$\text{}\mathrm{l}$ to Line ${\mathrm{l}}^{\prime}$ at station $\mathrm{s}$.

- ${\mathrm{dt}}_{\mathrm{lq}}$
- the departure time of train q on the origin station of Line $\mathrm{l}$;
- ${\mathrm{ad}}_{\mathrm{lq}}$
- the adjustment of departure time of train q on the origin station of Line $\mathrm{l}$.

#### 2.3. Constraints

#### 2.4. Objective

## 3. Algorithm

- Chromosome Coding

- 2.
- Population/Individual Initialization

- 3.
- Fitness function

- 4.
- Selection and elitist preservation operator

- 5
- Recombination operator

- 6.
- Mutation

- 7.
- Elitist preservation operator

- 8.
- Termination Criterion

## 4. Case Study

#### 4.1. Case Description and Data Processing

#### 4.2. Results Analysis

## 5. Conclusions and Future Research

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Mohring, H.; Schroeter, J.; Wiboonchutikula, P. The Values of Waiting Time, Travel Time, and a Seat on a Bus. RAND J. Econ.
**1987**, 18, 40. [Google Scholar] [CrossRef] - Jansen, L.N.; Pedersen, M.B.; Nielsen, O.A. Minimizing Passenger Transfer Times in Public Transport Timetables. In Proceedings of the 7th Conference of the Hong Kong Society for Transportation Studies: Transportation in the Information Age, Hong Kong, China, 14 December 2002; pp. 229–239. [Google Scholar]
- Ye, Y.; Zhang, J.; Wang, Y. Transfer Coordination-Based Train Organization for Small-Size Metro Networks. Arab. J. Sci. Eng.
**2019**, 45, 3599–3610. [Google Scholar] [CrossRef] - Chakroborty, P. Genetic Algorithms for Optimal Urban Transit Network Design. Comput. Civ. Infrastruct. Eng.
**2003**, 18, 184–200. [Google Scholar] [CrossRef] - Yin, H.; Wu, J.; Sun, H.; Kang, L.; Liu, R. Optimizing last trains timetable in the urban rail network: Social welfare and synchronization. Transp. B Transp. Dyn.
**2018**, 7, 473–497. [Google Scholar] [CrossRef] - Teodorović, D.; Lucic, P. Schedule synchronization in public transit using the fuzzy ant system. Transp. Plan. Technol.
**2005**, 28, 47–76. [Google Scholar] [CrossRef] - Cevallos, F.; Zhao, F. Minimizing Transfer Times in Public Transit Network with Genetic Algorithm. In Transportation Research Record. J. Transp. Res. Board
**2006**, 1971, 74–79. [Google Scholar] [CrossRef] - Fang, X.; Zhou, L.; Xia, M. Research on Optimization of Urban Mass Transit Network Schedule Based on Coordination of Connecting Time Between Different Lines. In Proceedings of the 2010 Joint Rail Conference, Volume 1, ASME International, Urbana, IL, USA, 12–14 April 2010; pp. 465–477. [Google Scholar]
- Wong, R.C.W.; Yuen, T.W.Y.; Fung, K.W.; Leung, J.M.Y. Optimizing Timetable Synchronization for Rail Mass Transit. Transp. Sci.
**2008**, 42, 57–69. [Google Scholar] [CrossRef] - Shafahi, Y.; Khani, A. A practical model for transfer optimization in a transit network: Model formulations and solutions. Transp. Res. Part A Policy Pr.
**2010**, 44, 377–389. [Google Scholar] [CrossRef] - Wu, J.; Liu, M.; Sun, H.; Li, T.; Gao, Z.; Wang, D.Z. Equity-based timetable synchronization optimization in urban subway network. Transp. Res. Part C Emerg. Technol.
**2015**, 51, 1–18. [Google Scholar] [CrossRef] - Guo, X.; Sun, H.; Wu, J.; Jin, J.; Zhou, J.; Gao, Z. Multiperiod-based timetable optimization for metro transit networks. Transp. Res. Part B Methodol.
**2017**, 96, 46–67. [Google Scholar] [CrossRef] - Guo, X.; Wu, J.; Sun, H.; Liu, R.; Gao, Z. Timetable coordination of first trains in urban railway network: A case study of Beijing. Appl. Math. Model.
**2016**, 40, 8048–8066. [Google Scholar] [CrossRef][Green Version] - Kang, L.; Wu, J.; Sun, H.; Zhu, X.; Wang, B. A practical model for last train rescheduling with train delay in urban railway transit networks. Omega
**2015**, 50, 29–42. [Google Scholar] [CrossRef] - Zhou, W.; Deng, L.; Xie, M.; Yang, X. Coordination Optimization of the First and Last Trains’ Departure Time on Urban Rail Transit Network. Adv. Mech. Eng.
**2013**, 5, 848292. [Google Scholar] [CrossRef] - Kwan, C.M.; Chang, C. Timetable Synchronization of Mass Rapid Transit System Using Multiobjective Evolutionary Approach. IEEE Trans. Syst. Man, Cybern. Part C Applications Rev.
**2008**, 38, 636–648. [Google Scholar] [CrossRef] - Deb, K.; Agrawal, R.B. Simulated binary crossover for continuous search space. Complex Syst.
**1995**, 9, 115–148. [Google Scholar]

**Figure 6.**Comparison of operational performance between the original timetable and the optimized timetable for Transfer arc 1: (

**a**) weighted transfer waiting time; (

**b**) transfer waiting time for each transfer pair.

**Figure 7.**Comparison of operational performance between the original timetable and the optimized timetable for Transfer arc 2: (

**a**) weighted transfer waiting time; (

**b**) transfer waiting time for each transfer pair.

Headway (min) | First Train Period | Morning Peak Period | Morning Off-Peak | Afternoon Peak Period | Afternoon Off-Peak | Last Train Period |
---|---|---|---|---|---|---|

Line 1 | 10 | 5 | 8 | 5 | 8 | 10 |

Line 2 | 4.5 | 3 | 8 | 4 | 8 | 10 |

Parameter * | Parameter Value |

Transfer walking time (s) | 60 |

Trip time for down direction (s) | 1310 |

Trip time for up direction (s) | 1285 |

Train clear time of Line 1 (s) | 45 |

Train clear time of line 2 (s) | 45 |

Minimum turn-round time for down direction (s) | 120 |

Minimum turn-round time for up direction (s) | 120 |

Minimum technical headway (s) | 120 |

Maximum operational headway (s) | 900 |

Time of first train at initial station on down direction (-) | 05:30:00 |

Time of last train at initial station on down direction (-) | 22:55:00 |

Time of first train at initial station on up direction (-) | 05:05:00 |

Time of last train at initial station on up direction (-) | 22:30:00 |

Timetable | Just-Miss Transfer (-) | Average Transfer Waiting Time (s) | Average Weighted Transfer Waiting Time (s0 | |
---|---|---|---|---|

Original timetable | Transfer arc 1 | 23 | 273.08 | 212.35 |

Transfer arc 2 | 16 | 211.47 | 204.17 | |

Optimized timetable | Transfer arc 1 | 0 | 138.78 | 132.58 |

Transfer arc 2 | 0 | 138.78 | 132.58 |

$({\mathit{P}}_{\mathit{C}},{\mathit{P}}_{\mathit{m}})$ | Weighted Average Transfer Waiting Time (s) | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | Average | |

(0.5, 0.001) | 133.6046 | 131.5924 | 132.8749 | 132.8528 | 132.4779 | 132.6805 |

(0.5, 0.005) | 132.0606 | 132.1766 | 131.3146 | 131.9489 | 131.4251 | 131.7852 |

(0.5, 0.01) | 134.9172 | 135.2101 | 134.6301 | 135.4745 | 134.1522 | 134.8768 |

(0.5, 0.02) | 138.6853 | 139.9563 | 140.3819 | 140.4382 | 140.0707 | 139.9065 |

(0.5, 0.03) | 144.7713 | 144.2474 | 143.5480 | 145.8360 | 142.2524 | 144.1310 |

(0.6, 0.001) | 132.1175 | 133.2350 | 132.9381 | 132.3212 | 132.3879 | 132.5999 |

(0.6, 0.005) | 131.6824 | 131.6663 | 131.5824 | 131.5753 | 131.9280 | 131.6869 |

(0.6, 0.01) | 135.2351 | 134.5462 | 135.2084 | 135.4091 | 134.8768 | 135.0551 |

(0.6, 0.02) | 139.2688 | 141.1980 | 140.2803 | 139.2443 | 139.6086 | 139.9200 |

(0.7, 0.001) | 133.3183 | 132.9052 | 133.9585 | 132.8812 | 132.1245 | 133.0375 |

(0.7, 0.005) | 132.0622 | 131.2538 | 131.2728 | 131.6680 | 131.5487 | 131.5611 |

(0.7, 0.01) | 135.0258 | 134.8982 | 135.8670 | 135.9954 | 133.9305 | 135.1434 |

(0.7, 0.02) | 137.4536 | 139.6581 | 139.894 | 139.6751 | 140.3899 | 139.4141 |

(0.8, 0.001) | 134.7905 | 134.0476 | 133.3393 | 133.1477 | 132.7718 | 133.6194 |

(0.8, 0.005) | 131.6352 | 132.0475 | 132.2510 | 131.6513 | 131.9601 | 131.9090 |

(0.8, 0.01) | 135.8541 | 134.7069 | 134.7013 | 134.5917 | 135.6755 | 135.1059 |

© 2020 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**

Cao, N.; Tang, T.; Gao, C.
Multiperiod Transfer Synchronization for Cross-Platform Transfer in an Urban Rail Transit System. *Symmetry* **2020**, *12*, 1665.
https://doi.org/10.3390/sym12101665

**AMA Style**

Cao N, Tang T, Gao C.
Multiperiod Transfer Synchronization for Cross-Platform Transfer in an Urban Rail Transit System. *Symmetry*. 2020; 12(10):1665.
https://doi.org/10.3390/sym12101665

**Chicago/Turabian Style**

Cao, Nan, Tao Tang, and Chunhai Gao.
2020. "Multiperiod Transfer Synchronization for Cross-Platform Transfer in an Urban Rail Transit System" *Symmetry* 12, no. 10: 1665.
https://doi.org/10.3390/sym12101665