# Column Generation Accelerated Algorithm and Optimisation for a High-Speed Railway Train Timetabling Problem

^{*}

## Abstract

**:**

Graphical Abstract

## 1. Introduction

## 2. Problem Formulation

#### 2.1. Problem Description

#### 2.2. Notation

#### 2.3. Mathematical Model

## 3. Column Generation-Based Algorithm

#### 3.1. Column Generation Algorithm

#### 3.2. Branching Strategies

- Step 1
- Construct a space-time network. Generate an initial path set ${P}_{0}$ as the root node of the branch tree and add it to $A$. Set $UB=+\infty $ and $LB=-\infty $.
- Step 2
- Judge whether $A$ is an empty set. If it is, the algorithm stops; otherwise, go to Step 3.
- Step 3
- Judge whether $\left(UB-LB\right)/UB\text{\hspace{0.17em}}\u2a7d\text{\hspace{0.17em}}\epsilon $ is true or not. If it is true, the algorithm stops; otherwise, go to Step 4.
- Step 4
- Judge whether the minimum objective function value of the nodes in $A$ is greater than $LB$. If it is, update $LB$. Sort $A$ according to the depth-first strategy, select the first node $w$, and remove it from $A$. Judge whether node $w$ is the root node. If it is, go to Step 6; otherwise, go to Step 5.
- Step 5
- Judge whether ${B}_{w}$ of node $w$ is empty. If it is, go to Step 2; otherwise, the first variable ${k}_{0}$ to be branched is selected from ${B}_{w}$ and added to ${C}_{w}$. Each branch variable $k$ in ${C}_{w}$ is circulated, and ${x}_{k}=1$ is added to the model as the constraint. Remove ${k}_{0}$ from ${B}_{w}$, and go to Step 6.
- Step 6
- The column generation algorithm is used to solve the linear relaxation problem of node $w$.
- Step 6.1
- The initial feasible solution of the RMP is generated. Suppose that ${G}^{\mathrm{min}}=\varnothing $.
- Step 6.2
- The dual variables are obtained by solving the RMP. If an integer solution is obtained and is less than $UB$, update $UB$.
- Step 6.3
- The dual variables are used to update the weights of the space-time arcs and calculate the shortest path of each sub-problem. The path satisfying the requirement of reduced cost ${\xi}_{p}$ is added to set ${G}^{\mathrm{min}}$.
- Step 6.4
- Judge whether ${G}^{\mathrm{min}}$ is empty. If it is, go to Step 7; otherwise, add the paths in ${G}^{\mathrm{min}}$ to the RMP. Remove the paths from ${G}^{\mathrm{min}}$, and go to Step 6.2.

- Step 7
- Judge whether the objective function of $w$ is less than $UB$. If it is, the new active node ${w}^{\prime}$ is added to $A$, ${C}_{w}$ is added to ${C}_{{w}^{\prime}}$, the fractional variables in the results are sorted according to certain rules, and a certain number of variables are selected and added to ${B}_{{w}^{\prime}}$ according to the beam search algorithm. Delete the variable $k$ from ${B}_{w}$, and go to Step 5. Otherwise, pruning is conducted, and the variable ${k}_{0}$ is deleted from ${C}_{w}$. Then, go to Step 5.

## 4. Acceleration Strategies of the Column Generation-Based Algorithm

#### 4.1. Preprocessing Stage

#### 4.2. RMP Stage

#### 4.2.1. Column Management Strategy

#### 4.2.2. Initial Solution Iteration Strategy

#### 4.3. PP Stage

#### 4.3.1. Partial Pricing Strategy

#### 4.3.2. Multiple Paths Strategy

#### 4.3.3. Delayed Constraint Strategy

#### 4.4. Branch-and-Bound Stage

#### 4.4.1. Early Branching Strategy

#### 4.4.2. Column Replacement Strategy

#### 4.5. Postprocessing Stage

## 5. Case Study

- Case 1
- A train timetable calculated using the original column generation-based algorithm.
- Case 2
- A train timetable calculated using the improved column generation-based algorithm with the acceleration strategies of the column management strategy and initial solution iteration strategy.
- Case 3
- A train timetable calculated using the improved column generation-based algorithm with the acceleration strategies of the early branching strategy and column replacement strategy.
- Case 4
- A train timetable calculated using the improved column generation-based algorithm with the acceleration strategies of the delayed constraint strategy and neighbourhood search strategy.
- Case 5
- A train timetable calculated using the improved column generation-based algorithm with all nine acceleration strategies.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Relationships among the train timetable, railway lines and space-time network. The upper part of Figure 1 represents a train timetable, the middle part represents a railway line and the lower part represents a space-time network diagram.

**Figure 5.**Variation in the objective function value of the root node over time: (

**a**) the objective function curve of Model M1; and (

**b**) the objective function curve of Model M2.

Symbol | Definition |
---|---|

Subscripts | |

$t$ | Index of times, $t\in T$, where $T$ is the set of time intervals, $T=\left\{0,\text{\hspace{0.17em}}1,\cdots ,\text{\hspace{0.17em}}1440\right\}$; |

$i,\text{\hspace{0.17em}}{i}^{\prime}$ | Index of stations, $i,\text{\hspace{0.17em}}{i}^{\prime}\in S:\text{\hspace{0.17em}}i\ne {i}^{\prime}$, where $S$ is the set of stations; |

$j$ | Index of trains, $j\in N$, where $N$ is the set of trains, $\left|N\right|=l$; |

$p,\text{\hspace{0.17em}}{p}^{\prime}$ | Index of paths, $p,\text{\hspace{0.17em}}{p}^{\prime}\in P:p\ne {p}^{\prime}$, where $P$ is the set of paths; |

$e$ | Index of sections, $e\in E$, where $E$ is the set of sections; |

Sets | |

${P}_{j}$ | Space-time path set belonging to train $j$; |

${P}_{e}$ | Space-time path set passing section $e$; |

Input parameters | |

${c}_{p}^{1},\text{\hspace{0.17em}}{c}_{p}^{2}$ | Cost of a space-time path for different mathematical models; |

${t}_{p,e}^{\mathrm{dep}},\text{\hspace{0.17em}}{t}_{p,e}^{\mathrm{arr}}$ | Time when a train enters and leaves section $e$; |

${t}_{e}^{\mathrm{dep}},\text{\hspace{0.17em}}{t}_{e}^{\mathrm{arr}}$ | Minimum departure headway and minimum arrival headway of section $e$; |

${t}_{p,i}^{\mathrm{beg}},\text{\hspace{0.17em}}{t}_{p,i}^{\mathrm{end}}$ | Arrival time and departure time of station $i$ in path $p$; |

${t}_{i}^{\mathrm{min}},\text{\hspace{0.17em}}{t}_{i}^{\mathrm{max}}$ | Minimum and maximum dwell time requirements for station $i$; |

${n}_{i}$ | Number of side tracks at station $i$; |

${\mu}_{p}^{i}$ | Stop index, which is equal to 1 if path $p$ stops at station $i$ and is equal to 0 otherwise; |

${\tau}_{p}^{i,\text{\hspace{0.17em}}{i}^{\prime}}$ | Coupled-stop index, which is equal to 1 if path $p$ stops at both station $i$ and station ${i}^{\prime}$ and is equal to 0 otherwise, ${\tau}_{p}^{i,\text{\hspace{0.17em}}{i}^{\prime}}={\mu}_{p}^{i}\cdot {\mu}_{p}^{{i}^{\prime}}$; |

${\beta}_{1}$ | Penalty coefficient of the number of train stops, the unit is 1/num; |

${\beta}_{2}$ | Penalty coefficient of the stopping time, the unit is 1/min; |

${v}_{i,\text{\hspace{0.17em}}{i}^{\prime}}$ | Minimum number of direct trains serving the passenger flow from station $i$ to station ${i}^{\prime}$; |

${Q}_{i,\text{\hspace{0.17em}}{i}^{\prime}}$ | Average passenger flow OD matrix from station $i$ to station ${i}^{\prime}$ over a period of time counted by the passenger ticket department, the unit is one person; |

$\phi $ | Passenger capacity of a train, the unit is one person; |

${w}_{i,\text{\hspace{0.17em}}{i}^{\prime}}$ | Average attendance rate of the trains from station $i$ to station ${i}^{\prime}$. |

Station Name | Station ID | Distance (km) | Number of Tracks | ||
---|---|---|---|---|---|

Downward Direction | Upward Direction | All | |||

Beijing South | 1 | 0 | 20 | 20 | 24 |

Langfang | 2 | 59 | 6 | 6 | 12 |

Tianjin South | 3 | 131 | 4 | 4 | 6 |

Cangzhou West | 4 | 219 | 6 | 6 | 8 |

Dezhou East | 5 | 327 | 8 | 8 | 12 |

Jinan West | 6 | 419 | 10 | 11 | 17 |

Taian | 7 | 462 | 6 | 6 | 8 |

Qufu East | 8 | 533 | 6 | 6 | 8 |

Tengzhou East | 9 | 589 | 3 | 3 | 4 |

Zaozhuang | 10 | 625 | 6 | 6 | 8 |

Xuzhou East | 11 | 688 | 9 | 10 | 15 |

Suzhou East | 12 | 767 | 6 | 6 | 8 |

Bengbu South | 13 | 844 | 10 | 11 | 14 |

Dingyuan | 14 | 897 | 3 | 3 | 4 |

Chuzhou | 15 | 959 | 4 | 4 | 6 |

Nanjing South | 16 | 1018 | 6 | 6 | 10 |

Zhenjiang South | 17 | 1087 | 6 | 6 | 8 |

Danyang North | 18 | 1112 | 3 | 3 | 4 |

Changzhou North | 19 | 1144 | 6 | 6 | 8 |

Wuxi East | 20 | 1201 | 6 | 6 | 8 |

Suzhou North | 21 | 1227 | 6 | 6 | 8 |

Kunshan South | 22 | 1259 | 4 | 4 | 6 |

Shanghai Hongqiao | 23 | 1302 | 10 | 11 | 19 |

Adjacent Station Pair | Section Travel Time (min) | |
---|---|---|

Grade G | Grade D | |

1–2 | 19 | 22 |

2–3 | 13 | 16 |

3–4 | 18 | 22 |

4–5 | 22 | 26 |

5–6 | 20 | 23 |

6–7 | 13 | 15 |

7–8 | 15 | 18 |

8–9 | 12 | 14 |

9–10 | 8 | 9 |

10–11 | 14 | 16 |

11–12 | 14 | 17 |

12–13 | 18 | 22 |

13–14 | 12 | 14 |

14–15 | 13 | 16 |

15–16 | 16 | 19 |

16–17 | 17 | 20 |

17–18 | 6 | 7 |

18–19 | 7 | 8 |

19–20 | 12 | 14 |

20–21 | 6 | 7 |

21–22 | 7 | 8 |

22–23 | 15 | 17 |

Station | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | — | 8 | 11 | 8 | 8 | 14 | 8 | 8 | 8 | 8 | 11 | 8 | 11 | 2 | 8 | 14 | 9 | 5 | 8 | 8 | 10 | 8 | 14 |

2 | — | 5 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 5 | 2 | 5 | 1 | 2 | 8 | 1 | 2 | 2 | 2 | 4 | 2 | 8 | |

3 | — | 5 | 5 | 11 | 5 | 5 | 5 | 5 | 8 | 3 | 8 | 0 | 4 | 11 | 1 | 2 | 5 | 3 | 7 | 3 | 11 | ||

4 | — | 2 | 8 | 2 | 2 | 2 | 2 | 5 | 2 | 5 | 2 | 2 | 8 | 3 | 2 | 2 | 2 | 4 | 2 | 8 | |||

5 | — | 8 | 2 | 2 | 2 | 2 | 5 | 2 | 5 | 2 | 2 | 8 | 3 | 1 | 2 | 2 | 4 | 2 | 8 | ||||

6 | — | 8 | 8 | 8 | 8 | 11 | 8 | 11 | 3 | 8 | 14 | 9 | 7 | 8 | 8 | 10 | 8 | 14 | |||||

7 | — | 2 | 2 | 2 | 5 | 2 | 5 | 1 | 2 | 8 | 3 | 1 | 2 | 2 | 4 | 2 | 8 | ||||||

8 | — | 2 | 2 | 5 | 2 | 5 | 0 | 2 | 8 | 3 | 2 | 2 | 2 | 4 | 2 | 8 | |||||||

9 | — | 2 | 5 | 2 | 5 | 2 | 2 | 8 | 3 | 2 | 2 | 2 | 4 | 2 | 8 | ||||||||

10 | — | 5 | 2 | 5 | 1 | 2 | 8 | 3 | 2 | 2 | 2 | 4 | 2 | 8 | |||||||||

11 | — | 5 | 8 | 5 | 5 | 11 | 6 | 5 | 5 | 5 | 7 | 5 | 11 | ||||||||||

12 | — | 5 | 2 | 2 | 8 | 3 | 2 | 2 | 2 | 4 | 2 | 8 | |||||||||||

13 | — | 3 | 5 | 11 | 6 | 5 | 5 | 5 | 7 | 5 | 11 | ||||||||||||

14 | — | 2 | 6 | 2 | 2 | 2 | 2 | 4 | 2 | 5 | |||||||||||||

15 | — | 8 | 3 | 2 | 2 | 2 | 4 | 2 | 8 | ||||||||||||||

16 | — | 9 | 8 | 8 | 8 | 10 | 8 | 14 | |||||||||||||||

17 | — | 3 | 3 | 3 | 5 | 3 | 9 | ||||||||||||||||

18 | — | 2 | 2 | 4 | 2 | 8 | |||||||||||||||||

19 | — | 2 | 4 | 2 | 8 | ||||||||||||||||||

20 | — | 4 | 2 | 8 | |||||||||||||||||||

21 | — | 4 | 10 | ||||||||||||||||||||

22 | — | 8 | |||||||||||||||||||||

23 | — |

Parameters | Actual Case | M1 | M2 |
---|---|---|---|

Number of trains | 142 | 142 | 200 |

Average travel speed (km/h) | 216.1 | 238.2 | 213.2 |

Average number of stops | 5.2 | 2.8 | 5.2 |

Average dwell time (min) | 16.2 | 6.8 | 15.8 |

Overtaking | 124 | 38 | 281 |

Max dwell time of Grade G (min) | 12 | 7 | 12 |

Max dwell time of Grade D (min) | 34 | 25 | 28 |

Station | Instance | Station | Instance | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

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

1 | N | N | N | N | N | 13 | N | Y | Y | N | Y |

2 | N | Y | Y | N | Y | 14 | Y | N | Y | N | Y |

3 | N | Y | Y | N | Y | 15 | N | Y | Y | N | Y |

4 | N | Y | Y | N | Y | 16 | Y | N | Y | N | Y |

5 | Y | N | Y | N | Y | 17 | N | Y | Y | N | Y |

6 | N | Y | Y | N | Y | 18 | N | Y | Y | N | Y |

7 | N | Y | Y | N | Y | 19 | N | Y | Y | N | Y |

8 | Y | N | Y | N | Y | 20 | N | Y | Y | N | Y |

9 | N | Y | Y | N | Y | 21 | N | Y | Y | N | Y |

10 | N | Y | Y | N | Y | 22 | N | Y | Y | N | Y |

11 | N | Y | Y | N | Y | 23 | N | N | N | N | N |

12 | N | Y | Y | N | Y | — | — | — | — | — | — |

Instance | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | |||||
---|---|---|---|---|---|---|---|---|---|---|

Time | Gap | Time | Gap | Time | Gap | Time | Gap | Time | Gap | |

1 | 2564 | 0.302 | 1847 | 0.282 | 1215 | 0.287 | 780 | 0.125 | 223 | 0.109 |

2 | 2441 | 0.63 | 2187 | 0.522 | 1015 | 0.522 | 2301 | 0.515 | 178 | 0.162 |

3 | 4019 | 0.61 | 1945 | 0.600 | 2164 | 0.600 | 1364 | 0.184 | 325 | 0.154 |

4 | 2028 | 0.455 | 1327 | 0.455 | 907 | 0.460 | 544 | 0.345 | 106 | 0.363 |

5 | 2575 | 0.463 | 2319 | 0.463 | 1520 | 0.470 | 867 | 0.325 | 317 | 0.288 |

Average | 2725 | 0.492 | 1925 | 0.464 | 1364 | 0.468 | 1171 | 0.299 | 230 | 0.215 |

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## Share and Cite

**MDPI and ACS Style**

Wang, J.; Zhou, L.; Yue, Y.
Column Generation Accelerated Algorithm and Optimisation for a High-Speed Railway Train Timetabling Problem. *Symmetry* **2019**, *11*, 983.
https://doi.org/10.3390/sym11080983

**AMA Style**

Wang J, Zhou L, Yue Y.
Column Generation Accelerated Algorithm and Optimisation for a High-Speed Railway Train Timetabling Problem. *Symmetry*. 2019; 11(8):983.
https://doi.org/10.3390/sym11080983

**Chicago/Turabian Style**

Wang, Jin, Leishan Zhou, and Yixiang Yue.
2019. "Column Generation Accelerated Algorithm and Optimisation for a High-Speed Railway Train Timetabling Problem" *Symmetry* 11, no. 8: 983.
https://doi.org/10.3390/sym11080983