# A Multi-Objective Optimization Model for the Intercity Railway Train Operation Plan: The Case of Beijing-Xiong’an ICR

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Railway Train Operation Plan

#### 2.2. Research Ideas

## 3. Mathematical Model

#### 3.1. Problem Description

_{0}), big-station-stop (m

_{1}), selected-station-stop (m

_{2}) and all-stop (m

_{3}) modes, as shown in Figure 1. There is only one stop plan for the modes of no-stop, big-station stop and all-stop, while the mode of select-station-stop has a combination of multiple stop plans.

#### 3.2. Model Assumptions

- (1)
- Closed assumption

- (2)
- Deterministic assumption

- (3)
- No transfer assumption

- (4)
- Capability assumption

- (5)
- Passenger time value assumption

#### 3.3. Model Notations

#### 3.4. Mathematical Model

#### 3.4.1. Objective Functions

- (1)
- Minimize the operating cost of intercity railway enterprises

_{0}can be expressed as Equation (1).

_{1}can be expressed as Equation (2).

_{2}can be expressed as Equation (3).

- (2)
- Minimize the travel time of passengers

_{3}can be expressed as Equation (5).

_{4}can be expressed as Equation (6).

#### 3.4.2. Constraint Conditions

#### 3.5. Processing of Multi-Objective

## 4. Methods

#### 4.1. Genetic Algorithm Design

- (1)
- Chromosome Coding

^{m}stops at station s. If ${x}_{s}^{{k}^{m}}=1$, it means stop; if ${x}_{s}^{{k}^{m}}=0$, it means no stop.

^{m}; ${f}_{k}^{m}$ is a non-negative integer. Choose a binary code with a code string length of 8 bits to represent the departure frequency of intercity trains. When encoding, it is necessary to convert the departure frequency of the train from decimal to binary. Considering the actual passing capacity of China’s ICRs, the maximum departure frequency that can be represented by this code is 256 trains, which can fully cover the possible situation of the current departure frequency of intercity trains. The two codes are combined to obtain a gene fragment of the chromosome, as shown in Figure 2.

- (2)
- Fitness Evaluation

- (3)
- Crossover and Mutation Operation

#### 4.2. Computing Procedure

_{c}, the mutation probability is P

_{m}, the maximum number of iterations of evolution is T, and the current iteration number t = 0.

## 5. Case Study

#### 5.1. Data

_{1}–S

_{6}, and the OD passenger flow data is shown in Table 4.

#### 5.2. Results

- (1)
- Scenario 1 optimization plan

- (2)
- Scenario 2 optimization plan

#### 5.3. Discussion

#### 5.3.1. Number of Train Stops

#### 5.3.2. Station Service Frequency

#### 5.3.3. Comparison of Important Indicators

- (1)
- Down direction

- (2)
- Up direction

## 6. Conclusions

- The influence of the change in passenger travel time value on the train operation plan is considered. A multi-objective optimization model aiming at the minimum operating cost of the enterprise and the minimum consumption time of passengers is constructed.
- According to the characteristics of the model, a genetic algorithm is designed to solve the model. The algorithm is calculated on the MATLAB platform, and the optimal solution can be obtained quickly.
- Taking BXICR as a research case, two types of intercity railway train operation plan under different travel time values of passengers are obtained. Scenario 1 runs 14 pairs of trains, which saves a lot of operating costs for the company; Scenario 2 runs 15 pairs of trains, which saves a lot of travel time for passengers.
- Comparing and analyzing different train operation plans, the results show that both optimization plans are better than the original plan. In the down direction, operating costs in Scenario 1 and Scenario 2 decreased by 7.3% and 1.1%, total time consumption decreased by 1.1% and 1.7%, and the number of stops per train for two scenarios decreased by 8.8% and 14.9%, respectively. In the up direction, operating costs in Scenario 1 and Scenario 2 decreased by 7.7% and 1.6%, total time consumption decreased by 0.7% and 1.5%, and the number of stops per train for two scenarios decreased by 12.6% and 18.2%, respectively. The optimized plan reduces the operating cost of the enterprise, attracts more passenger flow, and realizes the sustainable development of the intercity railway enterprise.
- The model established in this paper is a general model suitable for the optimization of the train operation plan of the ICR, taking the Beijing-Xiong‘an Intercity Railway as a case study. ICRs similar to BXICR include Beijing-Tianjin ICR, Nanchang-Jiujiang ICR, Guangzhou-Zhuhai ICR and Wuhan-Huangshi ICR, etc. After investigation, the length and average daily passenger flow of these lines are at the same level as the BXICR. The optimization model established in this paper can be applied to the above-mentioned similar lines, and similar conclusions can be obtained, including the reduction in the total operating costs, the reduction in the number of stops per train, and the reduction in the total time consumption.
- Due to the limited data collection capacity, this paper studies the ICR passenger flow under the condition of fixed demand. In the actual operation process, the passenger flow has strong uncertainty, and the change in the passenger flow law in different periods will also affect the preparation of the train operation plan. In future research, it is necessary to combine passenger flow forecasting and other information, and consider factors other than the train operation plan as comprehensively as possible, so as to strengthen the applicability of the model.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Notations | Definition |
---|---|

$G$ | Intercity railway line, G = (S, E) |

$S$ | Intercity railway station, s, i, j ∈ S |

$E$ | Train running sections |

$L$ | Line length |

${l}_{ij}$ | Distances between i and j |

m | Train stop mode, m ∈ M = {m_{0}, m_{1}, m_{2}, m_{3}} |

${k}^{m}$ | Stop plan for m stop mode, k^{m} ∈ K^{m} |

${C}_{0}$ | The fixed cost of all trains |

${C}_{1}$ | The stop cost of all trains |

${C}_{2}$ | The running cost of all trains |

${C}_{3}$ | The time consumption with train stops |

${C}_{4}$ | The time consumption with train running |

$D$ | Train capacity |

${t}_{s}$ | The stop time of station s, s ∈ S |

${R}_{s}$ | The stop cost of station s, s ∈ S |

$h$ | Running cost, CNY per train per kilometer |

$p$ | Fixed cost of an intercity train |

${v}_{ij}^{{k}^{m}}$ | Average running speed between i and j by train with the stop plan k^{m} |

${q}_{ij}^{{k}^{m}}$ | The passenger flow transported by train with the stop plan k^{m} |

${Q}_{ij}$ | The total passenger flow that needs to be transported between i and j |

$\delta $ | Average passenger time value |

$\overline{\theta}$ | Maximum occupancy rate |

$\underset{\_}{\theta}$ | Minimum occupancy rate |

$\overline{{\partial}_{}^{m}}$ | Maximum number of trains of m stop mode |

$\underset{\_}{{\partial}_{}^{m}}$ | Minimum number of trains of m stop mode |

$\overline{{\phi}_{}^{m}}$ | Maximum number of trains stops of m stop mode |

$\underset{\_}{{\phi}_{}^{m}}$ | Minimum number of trains stops of m stop mode |

$N$ | Maximum starting capacity |

Decision Variables | Definition |
---|---|

${x}_{s}^{{k}^{m}}$ | 0–1 variable; 1 means train running with stop plan k^{m} stops at station s; 0 means otherwise |

${f}_{}^{{k}^{m}}$ | The frequency of train running with stop plan k^{m} |

Parameter | Value |
---|---|

Train type | CR400AF EMU |

Starting/terminal station | Beijingxi/Xiong’an |

Running speed | 250~350 km/h |

Stop time | 3 min (Including start and stop additional time) |

Train formation | 8 carriages |

Seating capacity | 576 seats |

$\mathrm{VVT}\text{}({\delta}^{low}$$/{\delta}^{high}$) | 0.5/0.75 ¥/min |

Fixed cost | 8000 ¥ |

Stop cost | 500 ¥ |

Running cost | 60 ¥/km |

Departure capacity | 100 trans/day |

Stations | S_{1} | S_{2} | S_{3} | S_{4} | S_{5} | S_{6} |
---|---|---|---|---|---|---|

S_{1} | — | 15 | 1696 | 430 | 1404 | 3351 |

S_{2} | 50 | — | 12 | 8 | 24 | 29 |

S_{3} | 1411 | 31 | — | 77 | 216 | 767 |

S_{4} | 573 | 16 | 124 | — | 7 | 28 |

S_{5} | 1302 | 16 | 266 | 11 | — | 40 |

S_{6} | 3178 | 77 | 833 | 41 | 38 | — |

Direction | Scenarios | Number of Train Stops | Total Frequency | Average Stop Times (per Train) | ||||
---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | ||||

Up direction | Original/freq | 0 | 2 | 8 | 4 | 1 | 15 | 2.27 |

Scenario 1/freq | 1 | 0 | 11 | 1 | 1 | 14 | 2.07 | |

Scenario 2/freq | 1 | 1 | 12 | 0 | 1 | 15 | 1.93 | |

Down direction | Original/freq | 0 | 1 | 1 | 8 | 3 | 15 | 2.53 |

Scenario 1/freq | 1 | 4 | 1 | 7 | 1 | 14 | 2.21 | |

Scenario 2/freq | 1 | 5 | 1 | 8 | 0 | 15 | 2.07 |

Direction | Scenarios | Stations | Total | |||||
---|---|---|---|---|---|---|---|---|

Beijingxi | Beijing Daxing | Daxing Airport | Gu’an dong | Bazhou bei | Xiong’an | |||

Up direction | Original | 15 | 3 | 15 | 7 | 9 | 15 | 64 |

Scenario 1 | 14 | 1 | 13 | 6 | 9 | 14 | 57 | |

Scenario 2 | 15 | 1 | 13 | 6 | 9 | 15 | 59 | |

Down direction | Original | 15 | 6 | 15 | 8 | 9 | 15 | 68 |

Scenario 1 | 14 | 3 | 13 | 7 | 8 | 14 | 59 | |

Scenario 2 | 15 | 3 | 12 | 8 | 8 | 15 | 61 |

Indicators | Original | Scenario 1 | Scenario 2 |
---|---|---|---|

Fixed cost/¥ | 120,000 | 112,000 | 120,000 |

Running cost/¥ | 81,900 | 76,440 | 81,900 |

Stop cost/¥ | 17,000 | 14,500 | 14,500 |

Total cost/¥ | 218,900 | 202,940 | 216,400 |

Stop time/min | 37,421 | 32,612 | 30,224 |

Running time/min | 393,477 | 393,477 | 393,477 |

Total time/min | 430,898 | 426,089 | 423,701 |

Number of stops at intermediate stations | 34 | 29 | 29 |

Station service frequency | 64 | 57 | 59 |

OD service frequency | 106 | 92 | 91 |

Indicators | Original | Scenario 1 | Scenario 2 |
---|---|---|---|

Fixed cost/¥ | 120,000 | 112,000 | 120,000 |

Running cost/¥ | 81,900 | 76,440 | 81,900 |

Stop cost/¥ | 19,000 | 15,500 | 15,500 |

Total cost/¥ | 220,900 | 203,940 | 217,400 |

Stop time/min | 39,512 | 36,637 | 33,389 |

Running time/min | 370,199 | 370,199 | 370,199 |

Total time/min | 409,711 | 406,836 | 403,588 |

Number of stops at intermediate stations | 38 | 31 | 31 |

Station service frequency | 68 | 59 | 61 |

OD service frequency | 132 | 104 | 102 |

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**MDPI and ACS Style**

Fan, Z.; Liu, D.; Rong, W.; Li, C.
A Multi-Objective Optimization Model for the Intercity Railway Train Operation Plan: The Case of Beijing-Xiong’an ICR. *Sustainability* **2022**, *14*, 8557.
https://doi.org/10.3390/su14148557

**AMA Style**

Fan Z, Liu D, Rong W, Li C.
A Multi-Objective Optimization Model for the Intercity Railway Train Operation Plan: The Case of Beijing-Xiong’an ICR. *Sustainability*. 2022; 14(14):8557.
https://doi.org/10.3390/su14148557

**Chicago/Turabian Style**

Fan, Zilong, Di Liu, Wenyu Rong, and Chengrui Li.
2022. "A Multi-Objective Optimization Model for the Intercity Railway Train Operation Plan: The Case of Beijing-Xiong’an ICR" *Sustainability* 14, no. 14: 8557.
https://doi.org/10.3390/su14148557