# Rescheduling Urban Rail Transit Trains to Serve Passengers from Uncertain Delayed High-Speed Railway Trains

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

- We present a rigorous mixed-integer programming model to solve the problem of robust rescheduling for capacitated URT trains, in which multiple scenarios are used to represent the uncertainty of arrival time at destinations of delayed HSR trains. The robust passenger assignment constraint is introduced to ensure the decision robustness of passenger assignment under different scenarios. Finally, the robust dispatching constraint is designed for the stable disrupting number of URT trains across different scenarios.
- We consider both objectives of delayed passengers and URT operators in the model. The resulting multi-objective optimization problem is to maximize the expected transported passengers and minimize the number of extra trains and operation-ending time of all extra trains.
- We design a novel iterative solution approach based on a revised version of the epsilon-constraint method combined with the weighted-sum method, which is designed for the computation of the multi-objective model.

## 3. Mathematical Formulation

#### 3.1. Problem Statement

- A High-Speed Railway (HSR) line and Urban Rail Transit (URT) lines

- 2.
- A set of extra trains of the URT system

- 3.
- A set of delayed HSR trains with uncertain delay information

- For each URT lines, we need to determine the number of extra trains that operate on it.
- For each extra URT train, we need to determine the departure time at the first station, the arrival time at the last station, the arrival and departure time at the intermediate stations, and the headway between two adjacent trains in the same line.
- For passengers of the delayed train, we need to determine which parts of it transfer to which extra train.

- We do not consider passenger transfer activities between URT trains.
- The rescheduling for the rolling stock plan of URT system is neglected.
- The passenger transfer walking time at the transfer station is known and fixed.
- In each station, passengers will always ride the first arriving train after they reach the platform to reduce their waiting time [40].
- We do not consider the train rescheduling of HSR system.

#### 3.2. Formulation of the Multi-Objective Model

#### 3.2.1. Objective Functions

#### 3.2.2. Constraints

- Start time constraints at the origin station:

- Departure time constraints:

- Arrival time constraints:

- Headway time constraints at origin station

- Minimum and maximum dwell time constraints at the intermediate station:

- Dwell time constraints of two adjacent trains:

- Transfer direction constraints:

- Passenger flow balance constraints:

- Train capacity constraints:

- Mapping constraints between passenger assignment and extra train passenger assignment constraints:

- Mapping constraints between transfer connection time and extra train connection constraints:

- Extra train constraints:

- Mapping constraints between passenger assignment and extra train operation constraints:

- Robust passenger assignment constraints:

- Robust train rescheduling constraints:

- Passenger assignment constraints:

#### 3.3. Solution Approach of the Multi-Objective Model

Algorithm 1 The solution approach for the multi-objective model |

Input: ${\theta}_{1},{\theta}_{2}$ |

Begin |

Step 1: Initialization, set $i=1$ |

1.1 max ${Z}_{P}$ subject to constraints (4) –(27), and set ${\alpha}_{1}={Z}_{P}^{*}$ |

1.2 min ${Z}_{U2}$ subject to constraints (4) –(27), and set ${\alpha}_{2}={Z}_{U2}^{*}$ |

1.3 min ${Z}_{U1}$ subject to constraints (4) –(27), and set ${\chi}_{1}={Z}_{U1}^{*}$, ${\chi}_{2}=\left|F\right|$. ${\chi}_{1}$ is the lower bound as well as ${\chi}_{2}$ is the upper bound of ${Z}_{U1}$ objective. We vary the $\chi $ from ${\chi}_{1}$ to ${\chi}_{2}$, and set the iteration interval as $\Theta $ |

1.4 set ${\lambda}_{1}={\theta}_{1}/{\alpha}_{1}$, ${\lambda}_{2}={\theta}_{2}/{\alpha}_{2}$, $Z={\lambda}_{1}\times {Z}_{P}-{\lambda}_{2}\times {Z}_{U2}$ |

1.5 max $Z$ subject to constraints (4) –(27), and set ${\beta}_{1}={Z}^{*}\left(i\right)$, ${\phi}_{2}={Z}_{U1}^{\prime \prime}\left(i\right)$ |

1.6 insert the pair $\left({Z}_{P}^{\prime},{Z}_{U1}^{\u2033},{Z}_{U2}^{\prime}\right)$ in the set of solution values $\Phi $ |

Step 2: while $\left({\phi}_{2}\ge {\chi}_{1}\right)$ do |

Begin |

2.1 Set $i++$ |

2.2 max $Z$ subjective to Constraints (4)–(28) plus constraint: ${Z}_{U1}\le \chi \left(i\right)$,$\chi \left(i\right)={\chi}_{2}-\left(i-1\right)\times \Theta $ |

2.3 set ${\beta}_{1}={Z}^{*}\left(i\right)$, ${\phi}_{2}={Z}_{U1}^{\u2033}\left(i\right)$ |

2.4 insert the pair $\left({Z}_{P}^{\prime},{Z}_{U1}^{\u2033},{Z}_{U2}^{\prime}\right)$ in the set of solution values $\Phi $ |

End |

Step 3: return the non-dominated pairs from the set of $\Phi $ |

End |

## 4. Numerical Experiments

#### 4.1. Test Case Description

- Network data:

- URT trains data:

- Delayed HSR trains data:

#### 4.2. Performance Evaluation of Single-Objective Model

#### 4.3. Search for Pareto-Optimal Solutions

#### 4.3.1. Parameter Analysis of the Weighted-Sum Method

#### 4.3.2. Results of the Pareto-Optimal Solutions

#### 4.3.3. Further Analysis of the Experimental Results of the Single-Objective Model and Multi-Objective Model

#### 4.4. Impact of Uncertainty towards Robustness

- WAR: Iterative approach with robustness consideration of both passenger assignment and number of extra trains.
- WPR: Iterative approach with robustness consideration of passenger assignment solutions and without robustness consideration of the number of extra trains.
- WER: Iterative approach with robustness consideration of the number of extra trains and without robustness consideration of passenger assignment solutions.
- NAR: Iterative approach without robustness consideration of both passenger assignment and number of extra trains.

#### Analysis of the Experimental Results of Four Cases

## 5. Conclusions and Future Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**Optimal solutions of three objectives. (

**a**) Expected transported passengers (${Z}_{P}^{*}$), (

**b**) Number of extra URT trains (${Z}_{U1}^{*}$ ), (

**c**) Operation-ending time of all extra trains (${Z}_{U2}^{*}$ ).

**Figure 9.**Results of the single-objective model and multi-objective model (${\theta}_{1}=0.5$), (

**a**) Objectives of expected transported passengers and number of extra URT trains, (

**b**) Objectives of operation-ending time and number of extra URT trains.

Symbol | Description |
---|---|

$l,{l}^{*}$ | Index of line; $l,{l}^{*}\in L$, $L$ is the set of lines in the network. |

$i,j$ | Index of station; $i,j\in S$, $S$ is the set of stations. |

$f,f\prime $ | Index of extra train of urban rail transit system, $f,f\prime \in F$, $F$ is the set of extra trains of urban rail transit system which are considered to schedule. |

${f}^{*}$ | Index of delayed trains,${f}^{*}\in {F}^{*}$, ${F}^{*}$ is the set of delayed trains. |

$w$ | Index of random scenario, $w\in W$,$W$ is the set of all scenarios. |

Parameters | Description |
---|---|

${F}_{l}$ | Set of extra trains could operate on line $l$, ${F}_{l}\subset F$. |

${S}_{f}$ | Set of stations extra train $f$ may use. |

${E}_{f}$ | Set of segments extra train $f$ may use. |

${F}_{{f}^{*}}$ | Set of extra trains of URT system which could connect the delayed train ${f}^{*}$. |

$\Delta $ | Set of the comprehensive hub stations, $\Delta \in S$. |

${p}_{{f}^{*}}$ | The volume of passengers on the delayed train ${f}^{*}$. |

$ca{p}_{f}$ | Available passenger carrying capacity of the extra train $f$ |

${o}_{f}$ | Origin node of extra train $f$. |

${d}_{f}$ | Destination node of extra train $f$. |

${\eta}_{f}$ | The first extra train which follows the extra train $f$ on the same line. |

$ES{T}_{f}$ | The predetermined earliest starting time of extra train $f$ at its origin node. |

$MIH$ | Minimum headway between two adjacent extra trains on the same line. |

${\xi}_{f,i,j}$ | The free-flow running time of extra train $f$ from station $i$ to $j$. |

${\delta}_{f,i}^{\mathit{max}}$ | The planned maximum dwell time for extra train $f$ at station $i$. |

${\delta}_{f,i}^{\mathit{min}}$ | The planned minimum dwell time for extra train $f$ at station $i$. |

${t}_{l,l\prime ,i}^{\mathit{walk}}$ | The passenger transfer walking time from line $l$ to line $l\prime $ at station $i$. |

${\nu}_{{f}^{*},f}$ | The probability of passengers from train ${f}^{*}$ that are assigned to train $f$. |

${\gamma}_{w}$ | Occurrence probability of scenario $w$. |

${t}_{\mathit{max}}^{C}$ | The maximum transfer connection time between connected train and feeder train at the transfer station. |

$\epsilon $ | A sufficiently small positive number. |

Decision Variables | Description |
---|---|

${y}_{f,{f}^{*},w}$ | Passenger assignment variables, the number of passengers from delayed train ${f}^{*}$ that is assigned to the extra connecting train $f$ under scenario $w$. |

$pextr{a}_{f,l,w}$ | 0–1 binary extra train passenger assignment variables, =1 if ${y}_{f,{f}^{*},w}$ is positive; =0, otherwise. |

${h}_{f,{f}^{\prime},l,w}$ | Headway time between departure time of train $f$ and train $f\prime $ on line $l$ at origin station under scenario $w$. |

${t}_{f,i,w}^{D}$ | The departure time of extra train $f$ at station $i$ under scenario $w$. |

${t}_{f,i,w}^{A}$ | The arrival time of extra train $f$ at station $i$ under scenario $w$. |

${t}_{f,i,w}^{\mathit{Dwl}}$ | The dwell time of extra train $f$ at station $s$ of line $l$ under scenario $w$. |

${t}_{f,{f}^{*},i,w}^{C}$ | Connection time of train ${f}^{*}$ and extra train $f$ at station $i$ under scenario $w$. |

$textr{a}_{f,l,w}$ | 0–1 binary extra train connection variables, =1 if ${t}_{{f}^{*},f,i,w}^{C}$ is positive; =0, otherwise. |

$extr{a}_{f,l,w}$ | 0–1 binary extra train operation variables, =1 if train $f$ operates on line $l$ under scenario $w$; =0, otherwise. |

Train Number | Planned Arrival Time at BSRS * (h:min) | Passenger Carrying Volume |
---|---|---|

G150 | 22:00 | 1015 |

G152 | 22:12 | 1015 |

G18 | 22:36 | 1152 |

G154 | 22:48 | 1015 |

G44 | 23:08 | 1015 |

G22 | 23:18 | 1152 |

G158 | 23:29 | 1015 |

Scenario | Delayed Time * (min) | Occurrence Probability ** |
---|---|---|

1 | 46 | 0.158 |

2 | 47 | 0.142 |

3 | 48 | 0.1268 |

4 | 49 | 0.113 |

5 | 50 | 0.1013 |

6 | 51 | 0.0896 |

7 | 52 | 0.0799 |

8 | 53 | 0.0708 |

9 | 54 | 0.0628 |

10 | 55 | 0.0558 |

**Table 6.**Number of variables, constraints, computational time, and optimal solutions of three objectives.

Number of URT Trains Considered | Number of Variables | Number of Constraints | Computational Time (s) | ${\mathit{Z}}_{\mathit{P}}^{*}$ | ${\mathit{Z}}_{\mathbf{U}1}^{\u2033}$ | ${\mathit{Z}}_{\mathbf{U}2}^{\u2033}$ |
---|---|---|---|---|---|---|

3 | 2471 | 4066 | 0.05 | 2030 | 2 | 2879.6 |

6 | 4751 | 8849 | 0.55 | 3178 | 5 | 7214.6 |

9 | 7031 | 13,572 | 0.65 | 5207 | 9 | 72,948.0 |

12 | 9311 | 18,295 | 0.47 | 7372 | 11 | 60,092.5 |

15 | 11,531 | 22,841 | 0.72 | 7369 | 14 | 50,512.8 |

18 | 13,751 | 27,387 | 1 | 7153 | 13 | 15,225.7 |

Case Study | Constraints (24) | Constraints (25) |
---|---|---|

WAR | on | on |

WPR | on | off |

WER | off | on |

NAR | off | off |

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## Share and Cite

**MDPI and ACS Style**

Wang, W.; Bao, Y.; Long, S.
Rescheduling Urban Rail Transit Trains to Serve Passengers from Uncertain Delayed High-Speed Railway Trains. *Sustainability* **2022**, *14*, 5718.
https://doi.org/10.3390/su14095718

**AMA Style**

Wang W, Bao Y, Long S.
Rescheduling Urban Rail Transit Trains to Serve Passengers from Uncertain Delayed High-Speed Railway Trains. *Sustainability*. 2022; 14(9):5718.
https://doi.org/10.3390/su14095718

**Chicago/Turabian Style**

Wang, Wanqi, Yun Bao, and Sihui Long.
2022. "Rescheduling Urban Rail Transit Trains to Serve Passengers from Uncertain Delayed High-Speed Railway Trains" *Sustainability* 14, no. 9: 5718.
https://doi.org/10.3390/su14095718