# A Multiobjective Integer Linear Programming Model for the Cross-Track Line Planning Problem in the Chinese High-Speed Railway Network

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## Abstract

**:**

## 1. Introduction

- (1)
- Due to the important role that the CTHST plays in China’s HSR system and the lack of an efficient computer method for cross-track line planning in China, a new process to generate a cross-track line plan in periodic context is proposed. The process is formed in two stages and we can obtain the cross-track line plan based on the individual-track line plans using this process. Then, both the cross-track line plans and individual-track line plans could produce a networked line plan adapted to Chinese travel habits and management rules.
- (2)
- A multiobjective integer linear programming model for cross-track line planning is developed by combining individual-track lines into cross-track lines, in the context of periodic operation. There are four goals in the objective function: periodicity of the line plan, train quantity, running mileage, and stop quantity. The constraints of the model include passenger demand and the number of individual-track lines available for combination. We first introduce the model formulation for a two-track case, and then expand it to the general formulation for generating the cross-track line plan for a multiple-track case.
- (3)
- We test the proposed model using China’s HSR cases. Optimal solutions are quickly obtained. In particular, we generate cross-track line plans for a large-scale HSR network, which was the first to be conducted in China to the best of our knowledge. We also explore the impacts of the goals’ weights and periodicity criteria parameter on the result and influencing factors of computation. Two comparisons with the existing classic methods and real-life line plans are also presented, respectively. These experiments verify the effectiveness and efficiency of our model.

## 2. Literature Review

## 3. Problem Statement

#### 3.1. A Two-Stage Line Planning Procedure

#### 3.2. Generating the Cross-Track Line Plan Based on Individual-Track Line Plans

- (1)
- Maximize the number of periodic CTHSTs.
- (2)
- Minimize the number of CTHSTs
- (3)
- Minimize the sum of running mileage of CTHSTs.
- (4)
- Minimize the sum of train stops of CTHSTs.

#### 3.2.1. Transformed Individual-Track Line Plan

#### 3.2.2. Range of Cross-Track Line’s Frequency

## 4. Line Planning Model

#### 4.1. Assumptions

- (1)
- The line plan of each individual track is fully periodic, i.e., the model is developed under a periodic context. However, the model can also be used in a nonperiodic pattern.
- (2)
- The length of a cycle is two hours as a common case. Therefore, the individual-track line plan used in this study is a 2-h line plan that could be repeated multiple times to obtain the line plan of the whole day.
- (3)
- Whether or not a CTHST is a periodic train is determined by its corresponding cross-track line’s frequency within a day. If a CTHST runs no less than a certain number of cycles a day, which is a parameter in the model, it is a periodic train; otherwise, it is a nonperiodic train.
- (4)
- To simplify the problem, there are only two types of train capacity among all tracks: 500 or 1000 seats per train.
- (5)
- We did not consider coupling and uncoupling work in the crossing station since they are not implemented in China. Therefore, only individual-track lines with the same train type can form a cross-track line.

#### 4.2. Model Formulation for a Two-Track Network

#### 4.3. Model Formulation for an N-Track Network

#### 4.4. Size of the Model

## 5. Case Study and Numerical Experiments

#### 5.1. Case Study for the Two-Track Network

#### 5.1.1. Input Data

#### 5.1.2. Results and Discussion

#### 5.2. Case Study for the N-Track Network

#### 5.2.1. Input Data

#### 5.2.2. Analysis of Weight Corresponding to Each Term in Objective Function

#### Periodicity Weight

#### Train Quantity Weight

#### Mileage Weight

#### Stop Quantity Weight

#### 5.2.3. Analysis of Periodicity Criteria Parameter

#### 5.3. Influence Factors of Computation

#### 5.3.1. Size of Cross-Track Line Pool

#### 5.3.2. Size of Cross-Track OD Pairs

#### 5.3.3. Physical Connectivity and Management Rules

#### 5.4. Comparison with Existing Approach

#### 5.5. Comparison with the Real-Life Plan

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The illustration of (

**a**) the individual-track passenger and cross-track passenger and (

**b**) the individual-track high-speed train (ITHST) and cross-track high-speed train (CTHST).

**Figure 2.**The high-speed railway (HSR) network for cross-track line planning among multiple tracks. The station names are presented in an abbreviated manner.

**Figure 3.**The illustration of a transfer pattern by individual-track trains (ITTs) and a direct pattern by CTHSTs.

**Figure 4.**Comparison of European train stop plan and Chinese train stop plan: (

**a**) A sample of a European IR train’s stop plan and (

**b**) a sample of possible Chinese trains’ stop plans of the same “departure station–terminal station” pair.

**Figure 6.**The illustration of combining individual-track lines of two individual tracks respectively into cross-track lines. The red boxes select individual-track lines of Track 1 whose destination stations are station B. The blue boxes select individual-track lines of Track 2 whose departure stations are station B.

**Figure 8.**Determining the maximum frequency of a CTHST within one day for the (

**a**) CTHST-AC case and (

**b**) CTHST-AD case. A–D are stations and a-c are trains; line ab represents a CTHST from A to C; line ac represents a CTHST from A to D; and the numbers below time horizon represent hours on 24-h clock.

**Figure 9.**The running results of the CTHSTs containing train a: (

**a**) only running CTHST-AC, (

**b**) only running CTHST-AD, and (

**c**) Running both CTHST-AC and CTHST-AD. A–D are stations and a-c are trains; line ab represents a CTHST from A to C; line ac represents a CTHST from A to D; and the numbers below time horizon represent hours on 24-h clock.

**Figure 10.**The original 2-h individual-track line plans of the two-track case: (

**a**) SCHSR’s line plan and (

**b**) CKHSR’s line plan. The top line without any train information is the station list. The big thick circles, big normal circles and small circles represent the big (terminal) stations, medium stations, and small stations, respectively. The number between two neighboring stations is the section distance in kilometers. The train no., frequency, and seat number are provided in front of each line.

**Figure 12.**The impact that the value of periodicity weight on the results: (

**a**) The impact on the number of CTHSTs and periodic CTHSTs and (

**b**) the impact on the average mileage of CTHSTs and average number of stops per CTHST. The abscissa axis labels different values of periodicity weight that we took for experiments.

**Figure 13.**The impact that the value of train quantity weight on the results: (

**a**) The impact on the number of CTHSTs and periodic CTHSTs and (

**b**) the impact on the average mileage of CTHSTs and average number of stops per CTHST. The abscissa axis labels different values of train quantity weight that we took for experiments.

**Figure 14.**The impact that the value of mileage weight on the results: (

**a**) The impact on the number of CTHSTs and periodic CTHSTs and (

**b**) the impact on the average mileage of CTHSTs and average number of stops per CTHST. The abscissa axis labels different values of mileage weight that we took for experiments.

**Figure 15.**The impact that the value of stop quantity weight on the results: (

**a**) The impact on the number of CTHSTs and periodic CTHSTs and (

**b**) the impact on the average mileage of CTHSTs and average number of stops per CTHST. The abscissa axis labels different values of stop quantity weight that we took for experiments.

**Table 1.**The comparison between studies of train stop plan optimization and stop planning in line pool generation stage.

Research | Background | Objective | Passenger Flow | Planning Stage |
---|---|---|---|---|

Freyss et al. [13] | Metro | Min passenger travel time; Min operation cost | Yes | Timetabling |

Abdelhafiez et al. [14] | Urban Rail | Min passenger travel time | Yes | Timetabling |

Jiang et al. [15] | Urban Rail | Min passenger waiting time | Yes | Timetabling |

Jamili and Aghaee [16] | Urban Rail | Max train speed | Yes | Timetabling |

Jiang et al. [17] | High-Speed Rail | Max the number of scheduled trains | No | Timetabling |

Yang et al. [18] | High-Speed Rail | Min total dwelling time; Min total deviation | No | Timetabling |

Yue et al. [19] | High-Speed Rail | Max profit: penalize stops | No | Timetabling |

Stop planning in line pool generation | High-Speed Rail | Generate a line pool under irregular skip–stop pattern | Yes | Line pool generation |

Set | Definition |

I | A transformed 2-h line plan of one individual track, $i\in I$ |

J | A transformed 2-h line plan of the other individual track, $j\in J$ |

E | Set of sections among two individual tracks, $e\in E$ |

D | Set of cross-track origin-destination (OD) pairs over two individual tracks, $d\in D$ |

Parameter | Definition |

${F}_{d}$ | Frequency requirement of the cross-track OD pair $d$ |

${P}_{d}$ | Passenger demand of the cross-track OD pair d |

${s}_{ij}^{d}$ | 0,1 parameter (1 if a cross-track line formed by i and j can serve the cross-track OD pair d and 0 otherwise) |

${t}_{ij}^{e}$ | 0,1 parameter (1 if a cross-track line formed by i and j passes through section e and 0 otherwise) |

${u}_{d}^{e}$ | 0,1 parameter (1 if a cross-track OD pair d covers the section e and 0 otherwise) |

${c}_{ij}$ | Seat capacity of a cross-track line formed by i and j |

$\theta $ | Periodicity criteria (i.e., the minimum frequency that a line needs to be operated in one day to be a periodic line) |

${K}_{ij}$ | The maximum frequency of a cross-track line formed by i and j in one day |

${K}_{i}$ | The maximum times that i can be used for forming a cross-track line in one day |

${K}_{j}$ | The maximum times that j can be used for forming a cross-track line in one day |

${L}_{ij}$ | Running mileage of a cross-track line formed by i and j |

${l}_{ij}$ | Relative running mileage of a cross-track line formed by $i\text{}\mathrm{and}\text{}j,\text{}{l}_{ij}={L}_{ij}/{L}_{ij}^{max}$, ${L}_{ij}^{max}$ is the maximum value of all ${L}_{ij}$ |

${h}_{ij}$ | The number of stops of a cross-track line formed by $i\text{}\mathrm{and}\text{}j,\text{}{h}_{ij}={h}_{i}+{h}_{j}+1$ |

${\lambda}_{1}$ | Weight of the periodicity objective |

${\lambda}_{2}$ | Weight of the objective for the total number of CTHSTs |

${\lambda}_{3}$ | Weight of the objective for the total mileage of CTHSTs |

${\lambda}_{4}$ | Weight of the objective for the sum of stops among all CTHSTs |

Decision Variable | Definition |

${x}_{ij}^{k}$ | 0,1 variable (1 if a cross-track line formed by i and j runs k cycles one day, i.e., its daily frequency is k, and 0 otherwise) |

Set | Definition |

${J}^{q}$ | A transformed 2-h line plan of individual track $q$, $q\in \left\{1,2,\dots ,N\right\}$$,\text{}N$ is the number of individual tracks in the network, ${j}^{q}\in {J}^{q}$ |

${M}_{n}$ | Set of all the combinations composed of $n$ individual-track lines belonging to $n$ individual tracks’ transformed 2-h line plans, respectively, $n\in \left\{2,3,\dots ,N\right\}$; each element is denoted as ${j}_{1}{j}_{2}\cdots {j}_{n}$; the element can also be expressed in more detail as ${j}_{1}\xb7{j}_{p}^{q}\cdots {j}_{n}$ $\mathrm{and}\text{}{j}_{p}^{q}$ $\mathrm{means}\text{}\mathrm{ITHST}\text{}{j}^{q}\in {J}^{q}$ is in the $p$ position of the combination |

Symbol | Definition |

$ob{j}_{m}$ | The objective value of cross-track line plan in the case of m-railway track combination, $m\in \left\{2,3,\dots ,N\right\}$; its mathematical expression can be imitated according to Equation (11) |

Parameter | Definition |

${K}_{{j}_{1}{j}_{2}\cdots {j}_{n}}$ | The maximum frequency of a cross-track line formed by ${j}_{1}$, ${j}_{2}$, …, ${j}_{n-1}$ $\mathrm{and}\text{}{j}_{n}$ in one day |

${K}_{n}^{{j}^{q}}$ | The maximum frequency of cross-track lines containing ${j}^{q}$ in the n-railway track combination case, $n\in \left\{2,3,\dots ,N\right\}$; ${K}_{n}^{{j}^{q}}=\mathrm{max}\left\{{K}_{{j}_{1}\cdots {j}_{p}^{q}\cdots {j}_{n}}\right\}$, ${j}_{1}\cdots {j}_{p}^{q}\cdots {j}_{n}\in {M}_{n}$ |

${s}_{{j}_{1}{j}_{2}\cdots {j}_{n}}^{d}$ | 0,1 parameter (1 if a cross-track line formed by ${j}_{1}$, ${j}_{2}$, …, ${j}_{n-1}$ $\mathrm{and}\text{}{j}_{n}$ can serve the cross-track OD pair $d$ and 0 otherwise) |

${t}_{{j}_{1}{j}_{2}\cdots {j}_{n}}^{e}$ | 0,1 parameter (1 if a cross-track line formed by ${j}_{1}$, ${j}_{2}$, …, ${j}_{n-1}$ $\mathrm{and}\text{}{j}_{n}$ passes through section $e$ and 0 otherwise) |

${c}_{{j}_{1}{j}_{2}\cdots {j}_{n}}$ | seat capacity of a cross-track line formed by ${j}_{1}$, ${j}_{2}$, …, ${j}_{n-1}$ $\mathrm{and}\text{}{j}_{n}$ |

Decision Variable | Definition |

${x}_{{j}_{1}{j}_{2}\cdots {j}_{n}}^{k}$ | 0,1 variable (1 if a cross-track line formed by ${j}_{1}$, ${j}_{2}$, …, ${j}_{n-1}$ $\mathrm{and}\text{}{j}_{n}$ $\mathrm{runs}\text{}k$ cycles one day, i.e., its daily frequency is $k$, and 0 otherwise) |

Number of Tracks | Number of Decision Variables | Number of Possible CTHSTs | Number of Lines of Individual-Track Line Plans | Number of Sections | Number of Cross-Track OD Pairs |
---|---|---|---|---|---|

1 | 0 | 0 | $n$ | $m$ | 0 |

2 | $k{n}^{2}$ | ${n}^{2}$ | 2$n$ | 2$m$ | ${m}^{2}$ |

3 | $k\left({C}_{3}^{2}{n}^{2}+{C}_{3}^{3}{n}^{3}\right)$ | ${C}_{3}^{2}{n}^{2}$ + ${C}_{3}^{3}{n}^{3}$ | 3$n$ | 3$m$ | ${C}_{3}^{2}{m}^{2}$ |

⋯ | ⋯ | ⋯ | ⋯ | ⋯ | ⋯ |

N | $k\left({C}_{N}^{2}{n}^{2}+\cdots +{C}_{N}^{N}{n}^{N}\right)$ | ${C}_{N}^{2}{n}^{2}+\cdots +{C}_{N}^{N}{n}^{N}$ | $Nn$ | $Nm$ | ${C}_{N}^{2}{m}^{2}$ |

**Table 5.**The cross-track passenger demand and the cross-track OD pair’s service frequency requirement.

OD No. | Original Station | Destination Station | Passenger’s Demand | OD Service Frequency Requirement |
---|---|---|---|---|

/person | /train | |||

1 | Fuzhou East | Anshun West | 1 | 1 |

2 | Fuzhou East | Guiding North | 1 | 1 |

3 | Fuzhou East | Guiyang North | 12 | 1 |

4 | Fuzhou East | Huaihua South | 7 | 1 |

⋯ | ⋯ | ⋯ | ⋯ | |

273 | Shanghai Hong Qiao | Guiyang North | 136 | 2 |

274 | Shanghai Hong Qiao | Huaihua South | 170 | 2 |

275 | Shanghai Hong Qiao | Kaili South | 57 | 1 |

276 | Shanghai Hong Qiao | Kunming South | 86 | 1 |

⋯ | ⋯ | ⋯ | ⋯ | |

403 | Zhuji | Xiangtan North | 5 | 1 |

404 | Zhuji | Xinhua South | 1 | 1 |

405 | Zhuji | Xinhuang West | 1 | 1 |

406 | Zhuji | Xupu South | 3 | 1 |

OD Group | Number of OD Pairs | Demand of Each OD Group | |
---|---|---|---|

Travel Demand/Person | Percentage of the Total | ||

All OD pairs | 406 | 5648 | 100.00% |

OD ≥ 10 pas. | 136 | 4582 | 81.13% |

OD ≥ 20 pas. | 69 | 3620 | 64.09% |

OD ≥ 30 pas. | 41 | 2946 | 52.16% |

OD ≥ 40 pas. | 31 | 2605 | 46.12% |

OD ≥ 50 pas. | 24 | 2286 | 40.47% |

OD Group (pas.) | Solving Time | Num of CTHSTs | Number of Periodic CTHSTs | Sum of Seats | Sum of Stops | Ave No. of Train Stops | Sum of Mil | Ave Train Mil | Ave Dis bet 2 Nei Stops |
---|---|---|---|---|---|---|---|---|---|

km | km | km | |||||||

All ODs | 0.54 s | 11 | 4 | 6500 | 204 | 18.55 | 17,071 | 1551.9 | 79.4 |

OD ≥ 10 | 0.44 s | 9 | 4 | 5400 | 135 | 15.00 | 13,334 | 1481.6 | 92.6 |

OD ≥ 20 | 0.51 s | 6 | 0 | 3750 | 88 | 14.67 | 9734 | 1622.3 | 103.6 |

OD ≥ 30 | 1.08 s | 5 | 0 | 3200 | 76 | 15.20 | 7641 | 1528.2 | 94.3 |

OD ≥ 40 | 0.54 s | 4 | 0 | 2650 | 59 | 14.75 | 6967 | 1741.8 | 110.6 |

OD ≥ 50 | 0.44 s | 4 | 0 | 2650 | 49 | 12.25 | 6226 | 1556.5 | 117.5 |

Tracks | Terminal Stations | Mileage (km) | No. of Stations | |
---|---|---|---|---|

Beijing–Shanghai HSR | BJS | SHHQ | 1318 | 23 |

Beijng–Shenzhen HSR | BJW | FT | 2409 | 42 |

Hangzhou–Shenzhen HSR | HZE | SZN | 1484 | 54 |

Qingdao–Jinan HSR | QD | JNW | 413 | 11 |

Xuzhou–Lanzhou HSR | XZE | BaJS | 1050 | 22 |

Nanjing–Chengdu HSR | NJS | CDE | 1674 | 30 |

Shanghai–Changsha HSR | SHHQ | CSS | 1083 | 28 |

Changsha–Kunming HSR | CSS | KMS | 1169 | 24 |

Weight | Solve Time (s) | Ave Train Mil (km) | Ave No. of Train Stops | Sum of Mil (km) | Sum of Stop-s | Sum of CTH-STs | Sum of Peri CTH-STs | Per Ratio (%) | |||
---|---|---|---|---|---|---|---|---|---|---|---|

Per | Qua | Mil | Stop | ||||||||

1 | 0 | 0 | 0 | 16.77 | 1119.8 | 7.39 | 304,580 | 2009 | 272 | 237 | 87.13 |

0 | 1 | 0 | 0 | 14.81 | 1325.6 | 10.51 | 136,533 | 1083 | 103 | 35 | 33.98 |

0 | 0 | 1 | 0 | 13.19 | 1163.7 | 8.72 | 119,863 | 898 | 103 | 22 | 21.36 |

0 | 0 | 0 | 1 | 13.99 | 1303.4 | 7.21 | 139,465 | 771 | 107 | 32 | 29.91 |

Per Criteria | Solve Time (s) | Ave Mil (km) | Ave No. of Train Stops | Sum of Mil (km) | Sum of Stops | Sum of CTH-STs | Ave Dis bet Two Nei Stops (km) | Per Ratio (%) |
---|---|---|---|---|---|---|---|---|

1 | 13.55 | 1273.8 | 7.59 | 131,204 | 782 | 103 | 148.3 | 100.0 |

2 | 13.94 | 1295.6 | 8.13 | 136,040 | 854 | 105 | 141.9 | 75.2 |

3 | 15.61 | 1326.3 | 7.92 | 139,261 | 832 | 105 | 148.6 | 65.7 |

4 | 12.63 | 1282.9 | 7.98 | 134,704 | 838 | 105 | 142.8 | 60.0 |

5 | 16.00 | 1194.7 | 8.24 | 127,828 | 882 | 107 | 129.2 | 55.1 |

6 | 19.00 | 1197.4 | 7.58 | 135,309 | 857 | 113 | 139.5 | 47.8 |

7 | 14.16 | 1273.8 | 7.59 | 131,204 | 782 | 103 | 148.3 | 0.0 |

Case No. | Num of Tracks | Num of Stations | Size of the Pool | Solving Time (s) | Num of CTHSTs | Periodicity Ratio |
---|---|---|---|---|---|---|

1 | 2 | 33 | 49 | 0.56 | 11 | 36.4% |

2 | 2 | 50 | 120 | 3.07 | 10 | 0.0% |

3 | 2 | 63 | 30 | 0.58 | 13 | 30.8% |

4 | 2 | 71 | 14 | 0.45 | 17 | 29.4% |

5 | 2 | 81 | 49 | 0.87 | 20 | 30.0% |

6 | 3 | 54 | 91 | 1.81 | 29 | 34.5% |

7 | 3 | 60 | 169 | 3.54 | 37 | 37.8% |

8 | 3 | 71 | 164 | 3.83 | 37 | 40.5% |

9 | 3 | 90 | 70 | 1.47 | 44 | 38.6% |

10 | 3 | 91 | 140 | 3.62 | 38 | 42.1% |

11 | 3 | 122 | 94 | 2.99 | 53 | 39.6% |

12 | 4 | 99 | 496 | 6.52 | 60 | 43.3% |

13 | 4 | 100 | 566 | 8.21 | 58 | 44.8% |

14 | 4 | 101 | 198 | 4.57 | 66 | 47.0% |

15 | 4 | 115 | 117 | 4.22 | 71 | 50.7% |

16 | 4 | 119 | 283 | 6.88 | 62 | 48.4% |

17 | 5 | 115 | 365 | 5.46 | 73 | 47.9% |

18 | 5 | 118 | 684 | 9.01 | 87 | 51.7% |

19 | 5 | 122 | 612 | 7.75 | 86 | 53.5% |

20 | 5 | 135 | 481 | 7.52 | 82 | 53.7% |

21 | 5 | 138 | 737 | 8.78 | 91 | 56.0% |

22 | 6 | 145 | 644 | 8.12 | 95 | 52.6% |

23 | 6 | 189 | 853 | 10.04 | 97 | 58.8% |

24 | 7 | 175 | 631 | 9.29 | 101 | 53.5% |

25 | 7 | 217 | 939 | 11.43 | 102 | 58.8% |

Case No. | Num of Tracks | Num of Stations | Size of the Pool | Num of OD Pairs | Solving Time (s) | Num of CTHSTs | Periodicity Ratio (%) |
---|---|---|---|---|---|---|---|

1 | 8 | 227 | 1028 | 899 | 12.63 | 105 | 60.0 |

2 | 8 | 227 | 2434 | 899 | 35.41 | 105 | 61.0 |

3 | 8 | 227 | 3185 | 899 | 43.32 | 105 | 61.9 |

4 | 8 | 227 | 5266 | 899 | 73.11 | 104 | 61.5 |

5 | 8 | 227 | 6471 | 899 | 87.03 | 104 | 61.5 |

6 | 8 | 227 | 8025 | 899 | 129.25 | 104 | 62.5 |

Case No. | OD Group | Proportion of Demand | Solving Time (s) | Ave Mil (km) | Ave No. of Train Stops | Sum of CTH-STs | Ave Dis bet Two Nei Stops (km) | Per Ratio (%) |
---|---|---|---|---|---|---|---|---|

1 | ≥500 pas. | 56.10% | 7.55 | 1017.6 | 6.5 | 58 | 135.7 | 77.6 |

2 | ≥400 pas. | 60.00% | 7.61 | 1071.1 | 6.35 | 62 | 145.6 | 79.0 |

3 | ≥300 pas. | 66.20% | 7.66 | 1073.3 | 6.19 | 72 | 149.2 | 75.0 |

4 | ≥250 pas. | 69.70% | 8.05 | 1076 | 6.22 | 76 | 148.9 | 73.7 |

5 | ≥200 pas. | 73.80% | 7.75 | 1058.9 | 6.41 | 79 | 143 | 74.7 |

6 | ≥150 pas. | 79.10% | 8.35 | 1139.6 | 6.46 | 83 | 152.8 | 77.1 |

7 | ≥100 pas. | 84.30% | 8.95 | 1214.4 | 6.91 | 86 | 153.6 | 80.2 |

8 | ≥50 pas. | 91.30% | 11.19 | 1156.4 | 6.52 | 95 | 153.9 | 69.5 |

9 | ≥30 pas. | 94.70% | 12.13 | 1170.2 | 6.92 | 99 | 147.8 | 69.7 |

10 | ≥20 pas. | 96.60% | 11.67 | 1233.2 | 7.28 | 100 | 148.9 | 67.0 |

11 | ≥10 pas. | 98.30% | 12.25 | 1205.9 | 7.26 | 105 | 146 | 61.9 |

12 | All | 100.00% | 12.63 | 1282.9 | 7.98 | 105 | 142.8 | 60.0 |

Network | Size of the Line Pool | Number of Stations | Time Range of the Plan | Solving Time |
---|---|---|---|---|

Hangzhou–Shenzhen HSR | 2880 | 54 | 2-h | 24 min 11 s |

Shanghai–Changsha HSR | 4617 | 28 | 2-h | 36 min 34 s |

Changsha–Kunming HSR | 2720 | 24 | 2-h | 18 min 41 s |

Qingdao–Jinan HSR | 366 | 11 | 2-h | 23 s |

Beijing–Shenzhen HSR | 5910 | 42 | 2-h | 1 h 27 min 3 s |

Beijing–Shanghai HSR | 4387 | 23 | 2-h | 49 min 21 s |

Nanjing–Chengdu HSR | 3518 | 30 | 2-h | 26 min 15 s |

Xuzhou–Lanzhou HSR | 3414 | 22 | 2-h | 21 min 57 s |

The Whole HSR Network | 21,165,743 | 227 | 1-day | out of memory |

Line Plan | Demand Served by Direct Service | Ave Milage of CTHSTs (km) | Ave Num of Stops per CTHST | Num of CTHSTs | Ave Distance between Two Neighboring Stops (km) | Periodicity Ratio |
---|---|---|---|---|---|---|

Experimental | 67.12% | 1282.90 | 7.98 | 105 | 142.85 | 60% |

Real-Life | 100% | 1277.62 | 14.41 | 123 | 82.91 | — |

© 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**

Han, P.; Nie, L.; Fu, H.; Gong, Y.; Wang, G.
A Multiobjective Integer Linear Programming Model for the Cross-Track Line Planning Problem in the Chinese High-Speed Railway Network. *Symmetry* **2019**, *11*, 670.
https://doi.org/10.3390/sym11050670

**AMA Style**

Han P, Nie L, Fu H, Gong Y, Wang G.
A Multiobjective Integer Linear Programming Model for the Cross-Track Line Planning Problem in the Chinese High-Speed Railway Network. *Symmetry*. 2019; 11(5):670.
https://doi.org/10.3390/sym11050670

**Chicago/Turabian Style**

Han, Peiwen, Lei Nie, Huiling Fu, Yantao Gong, and Gang Wang.
2019. "A Multiobjective Integer Linear Programming Model for the Cross-Track Line Planning Problem in the Chinese High-Speed Railway Network" *Symmetry* 11, no. 5: 670.
https://doi.org/10.3390/sym11050670