# Multi-Objective Optimization for High-Speed Railway Network Based on “Demand–Supply–Management” Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Methodology

#### 3.1. Objectives for Optimizing the High-Speed Railway Network

#### 3.1.1. Micro-Objective for Traveler’s Demand

- Passenger travel time function of travelers: The travel time for high-speed railway passengers is defined as the total time between the starting point (origin, O) and the endpoint (destination, D). It includes the time from the travel point to the corresponding train station, the transfer times at different stations, the travel time of the train, and the time from the train station to the travel destination. Figure 1 shows a high-speed railway line.When a passenger travels between $OD$ (origin and destination are $O$ and $D$), they need to traverse a set of railway stations. The set of stations is denoted as $S=\left\{{S}_{1},{S}_{2},\cdots ,{S}_{n}\right\}$. For an $OD$ pair, their travel time is defined as:$${T}^{OD}={t}_{{o}_{1}}^{}+{\displaystyle \sum _{j=1}^{\left|n-1\right|}{t}_{i,i+1}^{}}+{t}_{nd}$$
- Passengers’ travel cost function: The travel cost, which is the total fare, includes all the expenses for high-speed railway travelling, and is the main factor for passengers in choosing a method of travel. This paper provides general fares, not the actual fare. The general fare is the comparable fare rate per kilometer per person on different routes, determined by the administrative department. The actual fare is the fare between the starting point and the ending point for each person, and it is uncertain due to seasonal fare discounts and other factors. We assume that the high-speed train journey takes place along an itinerary consisting of m sections along a route and that these are sequential. The lower the fare, the more attractive it is to passengers. Thus, the relative function of the cost per passenger can be described as:$$P={\displaystyle \sum _{i=1}^{m}\left({\alpha}_{i}\xb7{d}_{i}\right)}$$
- Micro-objective model for optimizing the high-speed railway network: The micro-objective is to minimize the generalized travel cost for passengers on high-speed railway travel: their travel expenses and travel time. It is the constraint condition that a high-speed train can go from 0 to 350 km/h in about 10 min, while it also takes about 10 min to decelerate from 350 km/h to 0. A high-speed train runs for at least 20 min, and 20 min is also the constraint condition (${t}_{\mathrm{min}}$). Therefore, the micro-objective function is constructed based on the two optimization objectives in Equations (1) and (2).$${F}_{1}=\mathrm{min}{F}^{\mathrm{micro}}={\gamma}_{1}{T}^{OD}+P$$

#### 3.1.2. Meso-Objective for Operator’s Supply

- The financial cost function of the operators: As the responsible person and transporter of the high-speed railway, operators bear not only the construction costs of the high-speed railway but also the responsibility for its operation and maintenance. This paper assumes that the service operator and infrastructure owner are the same (as in the case of China), and the cost structure is as follows:$${C}^{O}={C}_{1}+{C}_{2}+{C}_{3},$$$${C}_{1}={\displaystyle \sum _{i=1}^{m}\left({c}_{1i}+{c}_{2i}\right)},$$$${C}_{2}={\displaystyle \sum _{j=1}^{n}\left({c}_{1j}+{c}_{2j}\right)},$$$${C}_{3}={\displaystyle \sum _{k=1}^{l}({c}_{1k}+{c}_{2k})},$$
- Financial income function of the operator. High-speed railway operators must consider costs (operation and maintenance, etc.) and income (financial gain). The yield rate is the best indicator for measuring the operating efficiency of a high-speed railway. The relationship between the yield rates of the high-speed railway operators is as follows:$${I}^{O}={\displaystyle \sum _{i=1}^{m}{c}_{i1}}-{\displaystyle \sum _{i=1}^{m}{c}_{i2}}-{\displaystyle \sum _{i=1}^{m}{c}_{i3}},$$
- Meso-objective model for optimizing the high-speed railway network: The meso-objective aims to determine the operating requirements for the high-speed railway and to maximize its benefits. According to Chinese official documents, it costs ($\beta $), on average, USD 1.29 million per kilometer to build a high-speed railway track, which allows a train to travel at 350 km/h, while it costs, on average, USD 0.89 million per kilometer to build a track, which allows a train to travel at 250 km/h. F
_{2}is the constraint condition in high-speed railway enterprises minimizing costs and maximizing income. Therefore, the micro-objective function can be constructed based on two optimization objectives, as shown in Equations (4) and (8).$${F}_{2}=\mathrm{min}{F}^{meso}={C}^{O}-{I}^{O},$$

#### 3.1.3. Macro-Objective for Government Management

- Energy consumption function by the government: As a high-speed train is a fast vehicle that uses electrical power, energy consumption should be minimized. The relative function of energy consumption is:$$J={\displaystyle \sum _{i=1}^{m}{\displaystyle \sum _{k=1}^{z}\left[{q}_{ik}\xb7{\tau}_{k}({v}_{ik})\right]}}$$
- Efficiency function of the government: The line efficiency of a high-speed railway is the main indicator to comprehensively evaluate its operating state. The greater the line efficiency is, the better the operating state of the high-speed railway. The relative line efficiency function is:$$W=\frac{{\displaystyle \sum _{i=1}^{m}{\displaystyle \sum _{j=1}^{n-1}{q}_{i}^{j,j+1}}}}{{\displaystyle \sum _{i=1}^{m}{\displaystyle \sum _{j=1}^{n-1}{\delta}_{i}^{j,j+1}}}},$$
- Macro-objective model for optimizing the high-speed railway network: The macro-objective aims to optimize the efficiency of the high-speed railway line in terms of energy consumption, as line efficiency is an important indicator. Then, the macro-objective model is constructed based on the two optimization objectives in Equations (10) and (11).$${F}_{3}=\mathrm{min}{F}^{macro}={\gamma}_{2}J+{\gamma}_{3}W,$$

#### 3.2. Multi-Objective Optimization Model

#### 3.3. Solution Method

- Step 1.
- Initialize the contemporary population $G$

- Step 2.
- Fitness calculations

- Step 3.
- Cross-mutation operation

- Step 4.
- Search for non-inferior solutions

- Step 5.
- Population mixing

- Step 6.
- Obtain the elite population

## 4. Case Study

#### 4.1. Results of the Single Objective

#### 4.2. Result of the Multi-Objective Model

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Comparison of the performance indicators based on the micro-objective optimization model. (

**a**) Travel time (h); (

**b**) travel cost (CNY).

**Figure 4.**Comparison of performance indicators based on the meso-objective optimization model. (

**a**) Operation cost (million CNY); (

**b**) operation benefit rate (%).

**Figure 5.**Comparison of performance indicators based on the macro-objective optimization model. (

**a**) Energy consumption cost (million CNY); (

**b**) line efficiency (%).

**Figure 6.**The optimization process for the “four north–south and four east–west” railway lines. (

**a**) The 1st north–south railway line; (

**b**) the 2nd north–south railway line; (

**c**) the 3rd north–south railway line; (

**d**) the 4th east–west railway line; (

**e**) the 1st east–west railway line; (

**f**) The 3rd east–west railway line.

Citation | Year | Research Perspective | No. of Objectives | Solution Method |
---|---|---|---|---|

[8] | 2009 | Meso | Multi | Chaos self-adaptive evolutionary algorithm |

[9] | 2010 | Micro; meso | Multi | Genetic algorithm |

[10] | 2010 | Meso | Multi | Pareto optimal |

[11] | 2010 | Micro | Multi | Simulated annealing algorithm |

[12] | 2012 | Micro | Multi | Improved tolerance approach |

[13] | 2014 | Meso | Single | Distributed dynamic admission control |

[14] | 2014 | Meso | Multi | Genetic algorithm |

[15] | 2015 | Meso | Multi | Hybrid evolutionary algorithm |

[16] | 2015 | Meso | Single | Heuristic algorithm |

[17] | 2016 | Micro | Single | Time partitioning technique |

[18] | 2016 | Micro; meso | Multi | Genetic algorithm |

[19] | 2016 | Meso | Single | Lagrange multiplier |

[20] | 2017 | Meso | Single | Sequential quadratic programming |

[21] | 2018 | Micro; meso | Multi | Simulated annealing algorithm |

[22] | 2018 | Micro | Multi | Tabu search algorithm |

[23] | 2019 | Meso | Single | Ant colony optimization |

[24] | 2019 | Meso | Single | Simulated annealing algorithm |

[25] | 2019 | Meso | Single | Heuristic algorithm |

[26] | 2020 | Micro; meso | Multi | Heuristic algorithm |

High-Speed Railway Lines | Railway Station | ||
---|---|---|---|

Passenger Line | Average Speed [40] | ||

4 North–South | 1st North–South | 350 | Beijing–Tianjin–Jinan–Xuzhou–Bengbu–Nanjing–Shanghai |

2nd North–South | 350 | Beijing–Shijiazhuang–Zhengzhou–Wuhan–Changsha–Guangzhou–Shenzhen | |

3rd North–South | 350 | Beijing–Chengde–Shenyang–Tieling–Siping–Changchun–Harbin | |

4th North–South | 250,350 | Hangzhou–Ningbo–Taizhou–Wenzhou–Fuzhou–Xiamen–Shenzhen | |

4 East–West | 1st East–West | 250,350 | Xuzhou–Shangqiu–Zhengzhou–Luomen–Xi’an–Baoji–Lanzhou |

2nd East–West | 300,350 | Shanghai–Hangzhou–Nanchang–Changsha–Guiyang–Kunming | |

3rd East–West | 250 | Qingdao–Jinan–Dezhou–Shijiazhuang–Taiyuan | |

4th East–West | 200,250 | Shanghai–Nanjing–Hefei–Wuhan–Chongqing–Chengdu |

Optimization Target | Current Line | Optimized Line | Increase Ratio | |
---|---|---|---|---|

Micro-objective model | Total travel time | 3.57 h | 3.26 h | 8.68% |

Total travel cost | CNY 114.2296 | CNY 114.1572 | 0.063% | |

Meso-objective model | Operation costs | CNY 95,869 million | CNY 79,494 million | 17.08% |

Operation benefit rate | 1.0007% | 1.2208% | 21.99% | |

Macro-objective model | Energy consumption costs | CNY 29,008 million | CNY 19,598 million | 32.44% |

Line efficiency | 63.5164% | 69.9982% | 10.20% |

Highspeed Railway Lines | Optimized Speed (km/h) | Optimized Railway Station | |
---|---|---|---|

4 North–South | 1st North–South | 263.039 | Beijing–Tianjin–Jinan–Xuzhou–Nanjing–Shanghai |

2nd North–South | 250 | Beijing–Zhengzhou–Wuhan–Changsha–Guangzhou–Shenzhen | |

3rd North–South | 250.468 | Beijing–Chengde–Shenyang–Changchun–Harbin | |

4th North–South | 274.439 | Hangzhou–Ningbo–Wenzhou–Fuzhou–Xiamen–Shenzhen | |

4 East–West | 1st East–West | 251.371 | Xuzhou–Zhengzhou–Xi’an–Lanzhou |

2nd East–West | 314.710 | Shanghai–Hangzhou–Nanchang–Changsha–Guiyang–Kunming | |

3rd East–West | 250 | Qingdao–Jinan–Shijiazhuang–Taiyuan | |

4th East–West | 250 | Shanghai–Nanjing–Hefei–Wuhan–Chongqing |

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**MDPI and ACS Style**

Hu, Q.; Bian, L.; Lin, J.; Tan, M.
Multi-Objective Optimization for High-Speed Railway Network Based on “Demand–Supply–Management” Model. *Appl. Sci.* **2023**, *13*, 11406.
https://doi.org/10.3390/app132011406

**AMA Style**

Hu Q, Bian L, Lin J, Tan M.
Multi-Objective Optimization for High-Speed Railway Network Based on “Demand–Supply–Management” Model. *Applied Sciences*. 2023; 13(20):11406.
https://doi.org/10.3390/app132011406

**Chicago/Turabian Style**

Hu, Qizhou, Lishuang Bian, Juanjuan Lin, and Minjia Tan.
2023. "Multi-Objective Optimization for High-Speed Railway Network Based on “Demand–Supply–Management” Model" *Applied Sciences* 13, no. 20: 11406.
https://doi.org/10.3390/app132011406