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

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## 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

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**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 |

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**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