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

Abstract

This paper develops a multi-objective optimization model for national high-speed railway network planning. Three objectives are proposed from the macro, meso, and micro perspectives, and these three objectives are considered simultaneously. Using real data, case studies are conducted to optimize China’s “four east–west and four north–south railway lines” network, which includes Beijing, Shanghai, Wuhan, Guangzhou, and Zhengzhou as main hubs. The results show that, on the one hand, the optimization model reduces the overlap between long-distance passengers and short-distance passengers on the high-speed railway line, facilitates the travel of passengers, and improves the line service capability; on the other hand, optimization of the network shortens the travel time of the passengers, reduces the cost of the railway, and improves the operation efficiency of the high-speed railway line network. The results show that the total travel time of all high-speed railway passengers in the optimization model is reduced by 18.4%, while the benefit rate of the operator increased by 21.99%.

1. Introduction

As a symbol of modern technology and scientific development, the high-speed railway is considered the most competitive passenger transport technology [1]. It is the most complex engineering industry after the aerospace industry [2]. Although the high-speed railway derives from the traditional railway, it has broken from the concept of the traditional railway and developed into a new transport system compatible with the conventional railway network [3] High-speed railways in countries such as China, Japan, France, Germany, and Spain have developed rapidly on a large scale [4]. By the end of 2021, China’s high-speed railway had reached 40,000 km, ranking first in the world.

China’s high-speed railway network not only plays a critical role in meeting residents’ requirements for fast and comfortable travel but also drives the economic development of towns along the railway and the growth of the national economy [5,6]. The construction of the high-speed railway requires a large amount of investment, for example, the construction cost of the Beijing–Shanghai high-speed railway in China was USD 1.29 million per kilometer. Therefore, it is necessary to develop an optimal route structure for the railway network at the planning level. Nevertheless, the existing high-speed railway network is not fully optimized, due to a lack of knowledge on developing high-speed railways [7]. Therefore, the motivation of this study was to propose an optimization framework for high-speed railway network planning. The innovation of this study lies in the simultaneous consideration of macro, meso, and micro-objectives, which capture the interests of travelers, the benefits of operators, and the efficiency of managers. To be more specific, (1) the micro-objective was to minimize the generalized travel cost of the high-speed railway for travelers; (2) the meso-objective was to maximize the operators’ profit; (3) the macro-objective was to minimize the energy consumption of the high-speed railway system.

The rest of this paper is organized as follows. Section 2 discusses previous related works and establishes the contributions of this paper. Section 3 introduces the three objectives, the multi-objective optimization model, and the solution method. Section 4 conducts a case study using real data. Finally, Section 5 concludes this paper and provides directions for future research.

2. Literature Review

There have been a considerable number of studies on high-speed railway optimization in the scientific literature.

Table 1. A summary of high-speed railway optimization studies.

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

is a representative list of the most advanced approaches in chronological order, mainly focusing on three aspects, namely, the type of research perspective, the number of objectives, and the method applied to solve the proposed model.

In terms of research perspectives, railway optimization has been studied from the meso-level [8,9,10,13,16,18,20,23,24,26] and micro-level [9,11,12,17,18,21,22,26,27,28] perspectives. Studies from the micro-level perspective have attempted to capture the interests of travelers. For instance, Castillo et al. [17] proposed a time partitioning technique to optimize the schedule of a few trains. Rui et al. [11] evaluated the railway station location using accessibility and travel time consumption. Wang et al. [12] studied the timetable rescheduling problem based on the delay period and the number of trains impacted. Meanwhile, Xu et al. [28] formulated two mixed-integer programming models to simultaneously assign locomotives to trains and schedule trains so that delays were minimized. At the meso-level, Xu et al. [13] proposed using system utility to optimize resource allocation in high-speed railway wireless networks. Zheng and Liu [19] applied the total revenue of the train system to identify the optimal ticket fare for a single high-speed train. Wang et al. [20] employed optimum rail profiles to improve the stability of high-speed railway vehicles. Lin et al. [24] used the total costs to determine whether high-speed trains needed maintenance. Considering the application of the total turnaround time into maintenance, Wang et al. [25] studied high-speed train unit routing problems. Nevertheless, the above-mentioned studies suffer a shortcoming because they ignored the trade-off between the travelers’ costs and operators’ benefits. Chen et al. [18] overcame this disadvantage by integrating the interests of travelers and operators and applied a stop cost for railway corporations and the convenience of travelling for travelers to optimize the stop schedules. Both Zhang et al. [21] and Han and Ren [26], in line with Chen et al. [18], analyzed the optimization problem associated with the high-speed railway network and formed the micro- and meso-level perspectives. Zhang et al. [21] implemented a trade-off between a track occupation time and the total walking distance to solve the track allocation problem. Han and Ren [26] focused on the stop plan and ticket allocation problems that were significantly affected by passenger satisfaction and average seat occupancy rate. In addition, Binder et al. [29] focused on the railway timetable rescheduling problem in cases of large disruptions from a macroscopic point of view. Based on population coverage, budget constraints, and origin–destination flows, Blanco et al. [30] proposed a model for expanding transportation networks from a macro perspective.

However, the high-speed railway system is a multi-factor, multi-objective, and multi-function stochastic dynamic system. Therefore, in recent years, a few studies on optimizing high-speed railway problems have considered multi-objective models [8,15]. Sun et al. [14] proposed a multi-objective optimization model for train routing on a high-speed railway network, thereby providing an important reference for train planning to provide better services. Fernández-Rodríguez et al. [31] described the train scheduling problem as a dynamic multi-objective optimization model that utilizes accurate results provided by detailed real-time train simulations. Mateus et al. [32] described a multi-criteria decision analysis approach that identified the best alternative from a given set of possible alternatives. Based on the Beijing–Shanghai high-speed railway and the existing railway network, Ma et al. [33] established a multi-objective model for high-speed and medium-speed trains. At present, the model has been applied to the train diagram system of the Beijing–Shanghai high-speed railway. Nevertheless, the above studies only described the high-speed railway as a multi-objective model but did not optimize it.

As shown in the above review, existing models and tools are mainly concerned with high-speed railway management. Most of them use a single objective to compute the location of high-speed railway systems, and their output is an analysis of different indexes that reflect the characteristics of the systems. Some scholars have focused on optimizing the high-speed railway strictly according to the given operation management, which means that robustness characteristics are seldom considered. However, various disturbances can influence real-world high-speed railway operations. Therefore, if we use these results to evaluate high-speed railway operation management, we will obtain incorrect outcomes because of the neglected disturbances. To address the above issues, this paper describes the high-speed railway line problem as a dynamic multi-objective model and carries out optimization research to provide accurate results. Rather than just focusing on a single perspective, this paper proposes a more realistic model that takes into account multiple perspectives: macro, meso, and micro. An optimization model is used to solve the dimension problems of the different objective functions using the economic transformation coefficient, and the best operation scheme for the high-speed railway is obtained.

It is well known that there are many methods for solving optimization models. For highly complex and large-scale optimization models, an optimization algorithm is often used, such as the neural network algorithm [34,35,36], tabu search algorithm [22], genetic algorithm [9,14,18,25,36,37], or simulated annealing algorithm [11,21,24]. However, the process of improving the applicability of the algorithm to multi-objective optimization problems remains to be further studied.

In this paper, a multi-objective optimization model of the high-speed railway system is constructed by comprehensively considering the micro-, meso-, and macro-objectives. The Pareto optimization algorithms can offer a means to find a set of Pareto-optimal solutions in a single run [10,38,39]. Therefore, the main purpose of this paper was to introduce a novel Pareto frontier optimization algorithm to solve the tri-objective model and determine the optimal decision within the Pareto-optimal solution set range.

3. Methodology

3.1. Objectives for Optimizing the High-Speed Railway Network

The train operation scheme for the high-speed railway is an important part of the transportation organization problems, and the train stop scheme is the key link in the operation scheme. Therefore, this paper comprehensively considers the passenger transport cost, the passenger transport demand, and the energy consumption; establishes a multi-objective model to improve the network efficiency, and optimizes the train stop scheme, all while using ‘whether a railway station’ as the decision variable in this paper.

3.1.1. Micro-Objective for Traveler’s Demand

The proposed micro-objective is to minimize the generalized travel cost of passengers. Based on the interests of travelers, the optimization function for the high-speed railway line was constructed from two aspects: travel time and travel cost. To maximize the benefit for passengers, this section mainly builds on the optimization function at the micro-level. Based on the basic concepts and properties of the uncertainty theory, the proposed model is described as a quantitative linear model.

• 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:

  $$T^{OD}=t_{o_1}^{}+{\displaystyle \sum _{j=1}^{\left|n-1\right|}{t}_{i,i+1}^{}}+t_{nd}$$

  (1)

  where $t_{O_1}$ represents the travel time between the origin and the first railway station, $t_{i,i+1}$ represents the travel time between stations; $S_i$, $S_{i+1}$, and $t_{nd}$ represent the travel time between the last railway station and the destination.
• 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:

  $$P={\displaystyle \sum _{i=1}^m\left({\alpha }_i·d_i\right)}$$

  (2)

  where $P$ is the fare for a passenger traveling by high-speed railway; $d_i$ is the distance traveled by passengers on the i-th section of the high-speed railway; ${\alpha }_i$ is the fare rate per kilometer per person of the i-th section of the high-speed railway line, and it is determined by the operating speed $v_i$; $m$ is the number of high-speed railway lines.
• 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 ($t_{\min }$). Therefore, the micro-objective function is constructed based on the two optimization objectives in Equations (1) and (2).

  $$F_1=\min F^{\mathrm{micro}}={\gamma }_1{T}^{OD}+P$$

  (3)

  where $F_1$ is the micro-objective model for optimizing the high-speed railway network and ${\gamma }_1$ is the value of time in monetary terms used to solve the dimension problem of the different objective functions.

3.1.2. Meso-Objective for Operator’s Supply

The line optimization function of a high-speed railway is established to maximize the benefit to the operators, which includes both the financial cost and the income of the operators.

• 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:

  $$C^O=C_1+C_2+C_3,$$

  (4)

  where $C^O$ is the operating cost of the line, $C_1$ is the construction and maintenance costs of the high-speed railway lines, $C_2$ is the construction and maintenance costs of a high-speed station, and $C_3$ is the purchase and maintenance costs of a high-speed train. We can calculate $C_1$, $C_2$, and $C_3$, respectively, according to Equations (5)–(7).

  $$C_1={\displaystyle \sum _{i=1}^m\left(c_{1i}+c_{2i}\right)},$$

  (5)

  $$C_2={\displaystyle \sum _{j=1}^n\left(c_{1j}+c_{2j}\right)},$$

  (6)

  $$C_3={\displaystyle \sum _{k=1}^l(c_{1k}+c_{2k})},$$

  (7)

  where $c_{1i}$ is the annual amortization cost and $c_{2i}$ are the operating costs and the annual maintenance costs for passengers who traveled on the i-th high-speed railway line, respectively. $c_{1j}$ and $c_{2j}$ are the annual construction costs and the maintenance cost of the station for the j-th high-speed railway line, respectively, and $n$ is the number of stations on the high-speed railway line. $c_{1k}$ and $c_{2k}$ are the annual purchase cost and the annual maintenance cost of the k-th high-speed train, respectively, and $l$ is the number of high-speed trains.
• 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:

  $$I^O={\displaystyle \sum _{i=1}^m{c}_{i1}}-{\displaystyle \sum _{i=1}^m{c}_{i2}}-{\displaystyle \sum _{i=1}^m{c}_{i3}},$$

  (8)

  where $I^O$ is the annual operating income of the high-speed railway lines, and $c_{i1}$, $c_{i2}$, and $c_{i3}$ are the annual operating income (depending on the fare and the number of passengers), the annual staff expenses, and the other annual expenses for the i-th high-speed railway line traveled by passengers, respectively.
• 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. F₂ 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).

  $$F_2=\min F^{meso}=C^O-I^O,$$

  (9)

3.1.3. Macro-Objective for Government Management

Considering national interests, a macro-level line optimization function for a high-speed railway should include energy consumption and line efficiency. This paper focuses on low-carbon traffic optimization by the government at the macro level.

• 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:

  $$J={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{k=1}^z\left[q_{ik}·{\tau }_k(v_{ik})\right]}}$$

  (10)

  where $J$ denotes the energy consumption of a high-speed railway line; $q_{ik}^{}$, $v_{ik}^{}$, and ${\tau }_k\left(v_{ik}^{}\right)$ are the number, the operating speed, and the energy consumption per kilometer of the k-th train on the i-th high-speed railway line, respectively; $z$ is the number of high-speed trains in operation.
• 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:

  $$W=\frac{{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^{n-1}{q}_i^{j,j+1}}}}{{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^{n-1}{\delta }_i^{j,j+1}}}},$$

  (11)

  where $W$ is the line efficiency of the high-speed railway, the general requirement is that line efficiency should be more than 70%; $q_i^{j,j+1}$ and ${\delta }_i^{j,j+1}$ are the passenger flow and the passenger demand for the i-th high-speed railway line station $j$ to station $j+1$, respectively.
• 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).

  $$F_3=\min F^{macro}={\gamma }_{2}J+{\gamma }_{3}W,$$

  (12)

  where ${\gamma }_2$ and ${\gamma }_3$ are weight coefficients used to synthesize the line efficiency $W$ and the energy consumption $J$.

3.2. Multi-Objective Optimization Model

Multi-objective optimization is a vector optimization problem, which aims to find the design variables and optimize a vector of the objective function, subject to several constraints and bounds. The multi-objective optimization is often formulated as follows:

$$\min F\left(x\right)={\left\{f_1\left(x\right),f_2\left(x\right),\cdots ,f_n\left(x\right)\right\}}^T,$$

(13)

the optimization of the high-speed railway lines adopts a scientific method to optimize the operating line. The main purpose is to make the passenger flow distribution equal to the actual passenger flow and facilitate passengers to travel, improve the efficiency of enterprises, and promote the sustainable development of high-speed railways. The line optimization problem for high-speed railways is a management problem in the tri-objective system. The principle is that high-level management sends certain information to the lower-level management, and the low-level management responds according to their interests or preferences. Then, the high-level management makes decisions to meet their overall interests. When developing a tri-objective model to optimize high-speed railway lines, this paper considers the macro- (the country, i.e., the manager), meso- (enterprises, i.e., operators), and micro- (individuals, i.e., travelers) level aspects.

Since the tri-objective model for line optimization is a multi-objective non-linear programming problem with multiple targets, in practical application, the model should be simplified rationally and practically. Therefore, a tri-objective model for high-speed railway line optimization is constructed based on Equations (3), (9) and (12):

$$F\left(F_1,F_2,F_3\right)=\left\{\begin{array}{l}\min F^{micro}={\gamma }_1{T}^{OD}+P\\ \min F^{menso}=C^O-I^O\\ \min F^{macro}={\gamma }_{2}J+{\gamma }_{3}W\end{array}\right.,$$

(14)

$$s.t.\left\{\begin{array}{c}{\displaystyle \sum _{j=1}^{\left|n-1\right|}{t}_{i,i+1}^{}}\ge t_{\min }\\ p\ge \delta \\ C_1\le \beta ·x\\ W\ge W_{\min }\end{array}\right.$$

(15)

where $F\left(F_1,F_2,F_3\right)$ is a tri-objective model for high-speed railway line optimization; $t_{\min }$ is the minimum time for high-speed railway lines to operate between two stations. $\delta$ is the fare between the starting point and the ending point for each person; $x$ is the operating distance of a high-speed railway line; $\beta$ is the average cost per kilometer to build a high-speed railway; $W_{\min }$ is the minimum line efficiency for the high-speed railway.

3.3. Solution Method

The route planning of the high-speed railway is a vector optimization problem. Considering that the multi-objective optimization problem usually has multiple Pareto non-inferior solutions, the purpose of the multi-objective railway network planning method based on Pareto optimization is to search the complete and evenly distributed Pareto non-inferior solution set as the optimization scheme, and then, make the final decision based on certain principles and preferences in the optimization scheme [14].

This paper applies an improved PE algorithm to solve the tri-objective model. During the modeling, we considered the macro, meso, and micro disturbances, used various functions to describe the differences and applied an improved PE algorithm as the modeling tool. The flow chart of the improved PE algorithm for the tri-objective model is shown in Figure 2.

The specific process is as follows:

Step 1.

Initialize the contemporary population $G$

The elite population $G^{\prime }$ is generated and set empty. Then, the non-inferior solution of the contemporary population $G$ is found and stored in the elite population $G^{\prime }$.

Step 2.

Fitness calculations

The fitness of the elite population $G^{\prime }$ and contemporary population individuals $G$ is calculated, while the maximum number of iterations is set as $g_{\max }$.

Step 3.

Cross-mutation operation

A new population $Q$ is formed via cross-mutation operations, and the objective function values of the individuals in the new species group are calculated.

Step 4. 

Search for non-inferior solutions

The set of non-inferior solutions is solved, which is denoted as $Q^{\prime }$.

Step 5.

Population mixing

The repeated individuals in the population $Q$ and the elite population $G^{\prime }$ are eliminated, and the elite population $G^{\prime }$ is copied into the population $Q$ to form a new contemporary population $G$.

Step 6.

Obtain the elite population

If $g<g_{\max }$, the function returns to Step 3; otherwise, the algorithm is terminated.

4. Case Study

We selected the “four east–west and four north–south” railway network of China’s national high-speed railway as the research object to explain the rationality of the single-objective model. The high-speed railways incorporated into the “four east–west and four north–south” railway network connect almost all of the large cities in China. The details are shown in

Table 2. “Four east–west and four north–south” railway lines.

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

.

In this paper, eight high-speed railway lines in the “four east–west and four north–south” railway network are analyzed. From July to September 2015, our research team investigated and analyzed the above eight high-speed railways and obtained some valid and relevant data that were published by the Ministry of Transport of the People’s Republic of China. The operating status of these eight high-speed railways was also analyzed using the optimization model.

4.1. Results of the Single Objective

In this section, we used Equations (3), (9) and (12) to study the “four east–west and four north–south” railway networks of China’s National Railway. The values are defined as ${\gamma }_1=0.987$, ${\gamma }_2=0.503$, ${\gamma }_3=0.497$, $W_{\min }=0.7$, and $t_{\min }=20$. The optimized and unoptimized indicators are shown in Figure 3, Figure 4 and Figure 5.

As shown in Figure 3, Figure 4 and Figure 5, the total travel time of all high-speed railway passengers in the optimization model was reduced by 8.68%, and the benefit rate of the operator increased by 21.99%. This means that the proposed model can improve the operating ability of the high-speed railway, reduce passengers’ travel time, and improve the overall benefit of high-speed railways. The index comparison between the current and the optimized high-speed railway is shown in

Table 3. Index comparison between the current and optimized high-speed railway.

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%

.

4.2. Result of the Multi-Objective Model

As shown in Table 2, this network is composed of eight lines, and each line is optimized by PE. The optimization process for the “four east–west and four north–south” railway network is shown in Figure 6.

The optimized “four east–west and four north–south” railway network is shown in

Table 4. Optimized “four east–west and four north–south” railway network.

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

. The second east–west and the fourth east–west lines are unchanged, while the others (the first, second, third, and fourth north–south and the first and third east–west lines) are changed. Specifically, for the first north–south line, Bengbu station is not a stop; for the second north–south line, Shijiazhuang is not a stop; for the third north–south line, the Tieling and Siping stations are not stops; for the fourth north–south line, the Taizhou station is not a stop; for the first east–west line, the Shangqiu and Luomen stations are not stops; for the third east–west line, the Dezhou station is not a stop. Thus, the network becomes optimal after optimizing the transportation hubs of Beijing, Shanghai, Wuhan, Guangzhou, and Zhengzhou using the tri-objective model.

5. Conclusions

This paper comprehensively studied a high-speed railway operating problem by using the uncertain mathematics theory. Eight high-speed railway lines in the “four east–west and four north–south” railway network were selected for analysis. First, three single-objective models are proposed to optimize the railway network. Through optimization, the total travel time and travel cost for passengers are reduced by 18.04% and 0.063%, respectively. The operation costs and energy consumption costs of the high-speed railway are reduced by 17.08% and 21.99%, respectively. The line efficiency of the high-speed railway is increased by 10.20%. Second, based on the optimization function of the high-speed railway operation, a tri-objective model for line optimization is established. The improved Pareto efficient algorithm was selected to solve the tri-objective model. According to the optimization results, we suggest that the second east–west and the fourth east–west lines remain unchanged, while some stations in the first, second, third, and fourth north–south lines and the first and third east–west lines do not stop. The multi-objective optimization also provides the average speed for each line. Of course, the speed of some lines is quite different from their original speed, such as the first, second, third, and fourth north–south lines. The relevant results obtained by using the improved PE algorithm still need further verification, or even practical investigation, to prove their feasibility.

These results match the related national planning policies of the Beijing, Shanghai, Wuhan, Guangzhou, and Zhengzhou main hubs. The new PE railway line-optimizing method proposed in this work is especially suitable for optimization problems, although it can also be used as a general-purpose optimization method. This approach provides decision-makers with a Pareto set of choices. The solutions obtained through this method can reflect the entire Pareto optimal frontier. This approach also automatically captures the Pareto optimal frontier without any formal optimization process. It uses an approximation to guide the sampling process and does not require an accurate approximation model. Through testing and application, the PE method was found to be robust and efficient. This optimization model can be used in future research on high-speed railway network planning problems.