Train Planning for Through Operation Between Intercity and High-Speed Railways: Enhancing Sustainability Through Integrated Transport Solutions

Abstract

In order to advocate for green and environmentally friendly travel modes, enhance the attractiveness of rail transit, and promote the sustainable development of rail transport, we focus on the transportation organization problem under limited-resource conditions. This paper studies the formulation of a train plan under the condition of through operation between intercity and high-speed railway, constructing a multi-objective nonlinear optimization model with train frequency, a stop plan, and turn-back station locations as decision variables. Given the high dimensionality of model variables and complex constraints, an improved multi-population genetic algorithm (IMGA) is designed. Through an actual case study of the through operation between the Chengdu–Mianyang–Leshan Intercity Railway and the Chengdu–Chongqing High-Speed Railway, a staged solution method is adopted for analysis. The results indicate that the through-operation mode can save operational costs for enterprises and travel costs for passengers, while also better adapting to changes in passenger flow. Additionally, the IMGA demonstrates better solution quality and higher efficiency compared to the classical genetic algorithm. The main contribution of this paper is to propose a novel approach to solve the train plan problem. It also contributes to creating a high-quality, high-efficiency, and high-comfort integrated transportation service network, promoting the sustainable development of rail transit.

1. Introduction

Currently, China ranks among the world’s leaders in total rail transit volume. Various regions have developed extensive rail transit networks of different types, with high-speed railways and urban rail transit systems leading globally in terms of operational mileage. Additionally, the construction and operational mileage, as well as the passenger volumes, of suburban and intercity railways, are also steadily increasing year by year. By the end of 2023, the total operating mileage of railways nationwide reached 159,000 km, with high-speed rail accounting for 45,000 km. The national railway network density stood at 161.1 km per 10,000 km², and the passenger volume of national railways reached 3.68 billion people. In the same year, excluding Hong Kong, Macau, and Taiwan, a total of 59 cities opened and operated 338 urban rail transit lines, with a total length of 11,224.54 km, completing an annual passenger volume of 29.466 billion trips. With the rapid growth of urban rail transit operational mileage and the densification of the network, the total passenger volume continues to climb. The summary of urban rail transit passenger statistics for various cities in 2023 is shown in

Table 1. The summary of urban rail transit passenger statistics for various cities in 2023.

City	Passenger Volume [×10,000]	Entry Volume [×10,000]	Maximum Daily Passenger Volume of the Line [×10,000]	Maximum Daily Passenger Volume of the Station [×10,000]
Beijing	345,294.26	190,296.03	183.54	45.76
Shanghai	366,906.65	202,888.93	144.60	68.80
Tianjin	57,132.93	35,661.96	55.10	33.03
Chongqing	132,582.81	83,377.33	93.67	37.95
Guangzhou	313,413.28	171,501.20	236.65	77.74
Shenzhen	271,112.14	158,890.28	139.37	60.33
Wuhan	135,163.39	85,652.12	138.29	45.23
Nanjing	100,998.93	61,028.79	120.66	74.32
Chengdu	212,191.00	120,161.00	106.00	52.58
Hangzhou	138,357.17	82,857.29	124.56	36.49

.

In the operation of rail transit, it is common for multiple modes to operate simultaneously. With the continuous emergence of new technologies and new models, rail transit is exhibiting a trend of multi-layered and multi-modal diversification. Against the backdrop of a large-scale network and the flourishing development of various rail transit modes, the drawbacks of the conventional single-mode independent operation model are becoming increasingly apparent. The shortcomings of the existing operation models and the urgent issues that need to be addressed are summarized as follows:

• The problem of transportation resource sharing under the operation mode of “dedicated line for dedicated trains”

At present, although some densely populated and economically developed areas have formed a situation of the synchronous operation of different rail transit systems, the overall degree of coordination is not high, and the interconnection in the actual sense has not been realized in China. Most rail transit systems still use the “dedicated line for dedicated trains” operational model. While this model is simple and practical, without impacting on each other, it also has drawbacks such as asynchronous information, low resource sharing rates, and poor alignment between capacity and transportation demand. These issues can limit service quality and, in the event of emergencies, result in an over-reliance on the line-specific rescue resources. This leads to inadequate integrated scheduling and command, limited coordinated transportation interaction, and overall low emergency response capability.

Figure 1 shows the peak-hour maximum sectional passenger flow of Beijing rail transit in 2022–2023, from which it can be seen that there is a large gap between the passenger flow of different line sections. For example, Line 6, with its densely located transfer stations, sees peak-hour passenger flows reaching up to 61,100, with some segments experiencing a peak load factor of 170%. In contrast, the Capital Airport Line, Daxing Airport Line, and Yanfang Line have peak-hour passenger flows of less than 5000, with daily load factors below 8%. This indicates a severe imbalance in passenger flow distribution over time and space. In segments with lower passenger volumes, there is a considerable wastage of capacity, while transportation pressure is high at key nodes.

• Limited improvement in the quality and efficiency of rail transit services

Achieving point-to-point passenger transport with a single-rail transit mode is challenging. For instance, some passengers may need to take the subway in City A to reach a certain station, then transfer to a regional or intercity train to arrive at their destination in City B. In City B, they might need to take the subway again to reach their final destination. As the travel distance increases, the likelihood of transfers and travel time also increase, resulting in poor travel convenience. The inability to achieve direct transportation affects passengers’ perception of transportation service quality and travel efficiency.

Taking the Beijing subway as an example, a total of 490 stations are currently in operation, including 83 transfer stations. As can be seen from Figure 2, the service pressure of Beijing rail transit transfer station during the morning (evening) peak hour is huge, with a high proportion of transfer passenger flow, making passenger flow management challenging. Some transfer stations have long transfer corridors, with some taking as long as 6–7 min to walk through, causing a significant inconvenience to passengers. Additionally, due to crowded platforms at transfer stations, it is common for passengers to miss the first-arriving train. For those who manage to board, the high passenger volume during peak hours makes it difficult to ensure a comfortable ride. The transfer problem has increased the negative factors of passengers’ travel experience and seriously affected the travel efficiency of passengers and the attractiveness of subway travel modes. These challenges also place immense pressure on maintaining safe, stable, and efficient operational management.

The independent operation mode of a single-rail transit system cannot provide multi-level travel services. Therefore, multi-network integration and through operation has become an inevitable trend.

Through operation means that trains cross from one line to another, sharing a section with the existing trains on the new line. Through operation can be either unidirectional or bidirectional. As shown in Figure 3, if only the A-line trains can enter the B-line and share resources with B-line trains, while B-line trains only run within the scope of their own line, this mode is called unidirectional through operation. If the two lines have different operators, it is essential to clarify the division of line and facility usage rights, as well as the allocation of fares and operating costs. In such cases, the line or train rental model is usually adopted, where the A-line operator pays the B-line operator for the use of their line, or the A-line operator charges the B-line operator for train rental fees.

Bidirectional through operation refers to a mode where trains from both lines can cross over to the other line, sharing transportation resources with the existing trains on that line, as shown in Figure 4. In this mode, if the A-line and B-line have different operators, the operating income and expenditure costs are typically borne separately by the operators of each line.

Generally speaking, no matter what kind of rail transit system, the process from planning and construction to operation involves completing a comprehensive transportation organization process. This starts with network design (strategic level), followed by the formulation of train plans and train schedules (tactical level), and finally the compilation of rolling stock usage and crew scheduling plans (operational level) [1]. This paper focuses on the tactical-level issue of train plan formulation, which serves as an intermediate operational plan, playing a crucial role in bridging the preceding and following stages of the transportation organization process. Train plans typically require strong robustness to suit fixed infrastructure (such as lines and stations), but they usually need to be recompiled or adjusted when there are significant changes in passenger flow or connections to new lines and stations. The main purpose of formulating a train plan is to determine the train grade and type, service frequency and composition, optimal route, and stop plan [2,3]. Many studies focus on the individual sub-problems mentioned above, such as the stop plan [4], the determination of service frequency [5,6], optimal route planning [7,8,9], and train compositions [10,11,12]. Additionally, numerous scholars have researched the combination of these sub-problems [13,14].

It is important to note that the formulation of a train plan is not the same as the development of a train timetable. In practical engineering contexts, the train plan is usually prepared first, followed by the formulation of the train timetable. For urban rail transit trains, due to the limited variety of train types, fixed operation routes, and the fact that most stopping patterns adopt the “stop at each station” mode, determining the train service frequency typically means determining the interval between train departures. In this case, the train plan and the train timetable can be considered equivalent. However, for high-speed or intercity trains, the variety of train types, stopping plans, operation routes, and train formations is much wider. Therefore, the train plan must first be developed based on passenger flow during a specific period (monthly, weekly, or daily). The timetable is then formulated by considering passenger flow fluctuations during specific time periods as well as technical constraints.

Research on train plans has always been a focus of attention, both for networked lines and for through-operation conditions within a single-mode rail transit system. Tang et al. classified network nodes into inflow and outflow types and designed a train plan formulation method based on the service network concept [15]. Shi et al. proposed a service level-oriented method for formulating a high-speed rail train plan [16]. Meng et al. transformed the train plan into a train service network with small-world and complex network characteristics and defined related evaluation indicators [17]. Mao et al. analyzed the physical environment and basic characteristics of passenger flow in networked operations and reviewed the existing theories and practices of urban rail transit networked operations. After summarizing the technical characteristics of different methods and their impacts on travelers, they concluded the applicability of various methods [18]. Huang et al. proposed an optimization model for a networked train plan under through-operation conditions within an urban rail system, using a passenger flow distribution algorithm considering rational passengers and an improved neighborhood search algorithm to solve the model [19]. Sun incorporated passenger arrival characteristics and choice behavior into the optimization process of networked train plans [20]. Lópezramos et al. utilized the operation of express and local trains in the road network to improve the spatial imbalance of station loads [21]. Yang et al. demonstrated through a case study of Changping Line that the introduction of cross-line express trains significantly reduces the number of transfers and passenger travel time and improves operational efficiency [22]. Huang et al. took Xi’an’s rail transit network as an example and constructed a bi-level planning model for a multi-line connected train plan with energy-saving goals [23]. Zeng addressed the issue of unbalanced cross-line operation capacity, calculating the mean-square deviation of the load factors of cross-line and non-cross-line trains to determine the multi-composition cross-line train plan for Y-shaped routes [24].

In recent years, a few scholars have begun to focus on the scheduling of metro and suburban railways through operation. Feng et al. explored the relationship among passenger flow, the location of through-service terminal stations, and the total time saved by passengers [25]. The study by Li et al. indicated that the through-operation mode of metro and suburban railways can save passenger travel time and increase operational revenue. They found that with an increase in through-passenger flow and transfer time, as well as a decrease in through-service transportation costs, the benefits of through operation also increase [26]. Lin et al. conducted an in-depth study on the train plan under the operation mode of line sharing between metro and suburban railways, aiming to reduce the number of passenger transfers and save travel time [27].

The aforementioned studies have provided valuable insights and references for the research approach of this paper. Overall, current research on train plans is mostly based on single-rail transit systems, such as urban rail transit or national railway mainlines. Few scholars have focused on the through operation of different rail transit systems, and the existing literature is limited to the through operation of metro and suburban railways, without considering connections with other systems. Distinguished from earlier studies, this paper explores train plans under the through-operation mode of intercity and high-speed railways. A multi-objective comprehensive optimization model is established and solved using an improved genetic algorithm.

The key contributions of this paper are as follows.

(1)

The concept of through operation is explored from the perspective of train planning. A new model is developed to plan the train operations for through operation between intercity and high-speed railways. Furthermore, a detailed description of the stop plan under through-operation mode is provided.

(2)

An enhanced multi-population genetic algorithm is proposed to solve the train plan problem. It is an innovation in the application of the algorithm.

(3)

The method proposed in this paper can meet passenger travel demand under limited-resource conditions, reduce operating costs for the enterprise, and contribute to the long-term and stable development of rail transit.

The remainder of this paper is organized as follows. Section 2 presents a mathematical description of the train plan problem. Section 3 introduces the improved multi-population genetic algorithm (IMPGA) designed to address this problem. In Section 4, numerical experiments and comparative analyses are conducted to validate the effectiveness of the proposed approach. Finally, Section 5 concludes the paper and outlines potential directions for future research.

2. Methodologies

2.1. Problem Description

As shown in Figure 5, the intercity line and the high-speed line are connected through station $S_b$. Let $S=\left\{S_1,\dots ,S_N\right\}$ be the set of stations, and $N$ be the total number of stations. The section from station $S_1$ to $S_b$ is the operation section for intercity trains without through operation (referred to as intercity trains). The section from station $S_1$ to $S_c$ is the operation section for intercity trains running through both the intercity and high-speed lines (referred to as through-operation intercity trains). The section from station $S_b$ to $S_N$ is the operation section for high-speed trains without through operation (referred to as high-speed trains). The section from station $S_a$ to $S_N$ is the operation section for high-speed trains running through both the high-speed and intercity lines (referred to as through-operation high-speed trains). Using these starting and terminal stations, the line is divided into four travel sections (Section 1, Section 2, Section 3 and Section 4) to facilitate model description. Specifically, the section from station $S_1$ to $S_a$ is Section 1, the section from station $S_a$ to $S_b$ is Section 2, the section from station $S_b$ to $S_c$ is Section 3, and the section from station $S_c$ to $S_N$ is Section 4. Based on the above research scenario, we explore how to schedule trains to meet the total passenger demand between stations $S_1$ and $S_N$. The meanings of all parameters and variables involved in the train plan model are listed in Appendix A.

2.2. Assumptions

The establishment of the model in this paper is based on the following idealized assumptions.

(1)

Transfer Assumption: Passengers prefer direct travel and will only transfer when there are no direct trains available between two stations. Transfers are allowed only once during the journey.

(2)

Unidirectional Assumption: Since high-speed and intercity trains usually operate in pairs for both directions, this study focuses on the train plan in only one direction (downward direction).

(3)

Through-Operation Assumption: The two lines are capable of through operation, and some intermediate stations can accommodate turn-back trains. The time for turning back at each station is constant.

(4)

Operating Speed Assumption: To maintain line capacity and considering the design speed and the use of EMUs (Electric Multiple Units), high-speed trains should run at the same speed as intercity trains when running on intercity lines. In other words, high-speed trains need to reduce their speed when operating on intercity lines. Due to technical constraints, through-operation intercity trains should maintain their original speed when running on high-speed lines.

(5)

Assumption of Train Types: It is assumed that through-operation trains and non- through-operation trains use the same train types. For the sake of differentiation, different symbols and icons are used in the following text, but the actual train types are the same.

2.3. Formulation of the Objective Function

2.3.1. Minimizing Operating Costs for Enterprises

The operating costs of an enterprise are mainly composed of two parts: the expenses incurred in the operation process and the stopping costs. Since the fixed cost investment in the early stage has been completed and has little relevance to the preparation of the train plan, it is not considered. The costs incurred in the operation process include equipment and facility usage fees, maintenance fees, wear and depreciation costs, as well as staff wages, all of which can be expressed as the product of the cost per vehicle-kilometer, the number of operating kilometers, and the number of vehicles. The train stopping costs are the sum of the costs incurred for each stop made by each train. Since the cost of a single stop varies for different types of trains, the calculation needs to be performed separately for each type of train.

$$\min W_{\mathrm{com}}=W_{\mathrm{run}}+W_{\mathrm{stop}}$$

(1)

$$W_{\mathrm{run}}={\displaystyle \sum _{V_u\subseteq V}{\displaystyle \sum _{u\in V_u}cos{t}_{\mathrm{run}}^{V_u}\cdot m^{V_u}\cdot f^u\cdot d^{V_u}}}$$

(2)

$$\begin{array}{l}{W}_{\mathrm{stop}}={\displaystyle \sum _{i=2}^{b-1}{\displaystyle \sum _{u\in V_{\mathrm{int}}}{x}_{S_i}^u\cdot f^u\cdot cos{t}_{\mathrm{stop}}^{V_{\mathrm{int}}}}}+{\displaystyle \sum _{c=b+1}^N{y}_c\cdot {\displaystyle \sum _{i=2}^{c-1}{\displaystyle \sum _{u\in V_{\mathrm{int}}^{\mathrm{cros}}}{x}_{S_i}^u\cdot f^u\cdot cos{t}_{\mathrm{stop}}^{V_{\mathrm{int}}^{\mathrm{cros}}}}}+}\\ {\displaystyle \sum _{i=b+1}^{N-1}{\displaystyle \sum _{u\in V_{\mathrm{hig}}}{x}_{S_i}^u\cdot f^u\cdot cos{t}_{\mathrm{stop}}^{V_{\mathrm{hig}}}}}+{\displaystyle \sum _{a=1}^{b-1}{y}_a}\cdot {\displaystyle \sum _{i=a+1}^{N-1}{\displaystyle \sum _{u\in V_{\mathrm{hig}}^{\mathrm{cros}}}{x}_{S_i}^u\cdot f^u\cdot cos{t}_{\mathrm{stop}}^{V_{\mathrm{hig}}^{\mathrm{cros}}}}}\end{array}$$

(3)

$d^{V_u}$ represents the total operating mileage for $V_u$-type trains. For $V_{\mathrm{int}}$-type trains, the operating mileage $d^{V_{\mathrm{int}}}$ is the distance between the starting station $S_1$ and the final station $S_b$, that is, $d_{S_1{S}_b}$. Similarly, $d^{V_{\mathrm{int}}^{\mathrm{cros}}}=d_{S_1{S}_c}$, $d^{V_{\mathrm{int}}^{\mathrm{cros}}}=d_{S_1{S}_c}$, and $d^{V_{\mathrm{hig}}^{\mathrm{cros}}}=d_{S_a{S}_N}$.

2.3.2. Minimizing Travel Costs for Passengers

The travel cost for passengers includes both the time cost and the ticket cost. The ticket cost can be represented by the product of the (per passenger-kilometer) fare rate, the passenger volume (number of passengers), and the train operating mileage. Since fare rates, operating mileage, and allocated passenger flow vary for different types of trains, the calculation needs to be performed by category and then summed.

$$\min W_{\mathrm{pas}}=W_{\mathrm{time}}+W_{\mathrm{tic}}$$

(4)

$$\begin{array}{l}{W}_{\mathrm{tic}}=({\displaystyle \sum _{1\le o\le a}{\displaystyle \sum _{o<d\le c}\cup }}{\displaystyle \sum _{a\le o\le N}{\displaystyle \sum _{o<d\le N}){\displaystyle \sum _{V_u\subseteq V}{\displaystyle \sum _{u\in V_u}cos{t}_{\mathrm{tic}}^u\cdot d_{S_o{S}_d}\cdot q_{S_o{S}_d}^u}}}}\cdot x_{S_o}^u\cdot x_{S_d}^u+\\ {\displaystyle \sum _{1\le o<a}{\displaystyle \sum _{c<d\le N}{\displaystyle \sum _{a\le k\le c}{\displaystyle \sum _{u\in V_{\mathrm{int}}\cup V_{\mathrm{int}}^{\mathrm{cros}}}{\displaystyle \sum _{v\in V_{\mathrm{hig}}\cup V_{\mathrm{hig}}^{\mathrm{cros}}}\left(cos{t}_{\mathrm{tic}}^u\cdot d_{S_o{S}_k}+cos{t}_{\mathrm{tic}}^v\cdot d_{S_k{S}_d}\right)}}}}}\cdot q_{S_o{S}_d}^{u-v}\cdot x_{S_o}^u\cdot x_{S_k}^u\cdot x_{S_k}^v\cdot x_{S_d}^v\end{array}$$

(5)

where $x_{S_o}^u$, $x_{S_d}^u$, and $x_{S_k}^u$ are auxiliary 0–1 variables indicating whether the train stops at certain stations. If train $u$ stops at stations $S_o$, $S_d$, or $S_k$, the corresponding variable takes the value of 1; otherwise, it is 0. $x_{S_k}^v$ and $x_{S_d}^v$ are auxiliary 0–1 variables indicating whether train $v$ stops at certain stations. If the train stops at stations $S_k$ or $S_d$, the corresponding variable takes the value of 1; otherwise, it is 0.

According to the division of travel sections shown in Figure 5, the total travel time for passengers can be calculated separately for each travel section. The time can then be converted into costs by using the product of travel time and the value coefficient of non-working time.

$$W_{\mathrm{time}}=\gamma \cdot {\displaystyle \sum _{g=1}^4{\displaystyle \sum _{l=g}^4{T}_{gl}}}$$

(6)

The travel time for passengers mainly consists of four parts: pre-departure waiting time, in-vehicle travel time, station stop waiting time, and transfer time. For suburban railway or metro systems, train departure intervals are relatively short, and passengers arrive at the station relatively evenly within fixed periods, following a first-come, first-served principle. Therefore, the pre-departure waiting time is closely related to the train frequency (departure interval), and it can generally be approximated as half of the departure interval. In contrast, passengers traveling by high-speed rail typically do not arrive at the station evenly to wait for their train. Instead, they decide when to go to the station based on factors such as the distance from the station, the convenience of reaching the station, and personal preferences. This waiting time is not directly related to the departure interval, train frequency, or stop-schedule plan. Some passengers prefer to arrive at the station well in advance, while others are accustomed to arriving just before the check-in process begins. In summary, the pre-departure waiting time is not considered in this model, and when calculating $T_{gl}$, only in-vehicle travel time, station stop waiting time, and transfer time are taken into account.

According to assumption (1), only passengers whose travel route starts in Section 1 and ends in Section 4 need to transfer during their journey, while all other types of passenger flows can reach their destinations directly. The expressions for $T_{11}$, $T_{12}$, and $T_{13}$ are as follows.

$$T_{11}={\displaystyle \sum _{a=1}^{b-1}{y}_a}\cdot {\displaystyle \sum _{1\le o<a}{\displaystyle \sum _{o<d\le a}{\displaystyle \sum _{u\in V_{\mathrm{int}}\cup V_{\mathrm{int}}^{\mathrm{cros}}}{q}_{S_o{S}_d}^u\cdot x_{S_o}^u\cdot x_{S_d}^u\cdot \left(\frac{d_{S_o{S}_d}}{spee{d}^{V_{\mathrm{int}}}}+{\displaystyle \sum _{o<m<d}{x}_{S_m}^u\cdot t_{S_m}^u}\right)}}}$$

(7)

$$T_{12}={\displaystyle \sum _{a=1}^{b-1}{y}_a}\cdot {\displaystyle \sum _{1\le o<a}{\displaystyle \sum _{a<d\le b}{\displaystyle \sum _{u\in V_{\mathrm{int}}\cup V_{\mathrm{int}}^{\mathrm{cros}}}{q}_{S_o{S}_d}^u\cdot x_{S_o}^u\cdot x_{S_d}^u\cdot \left(\frac{d_{S_o{S}_d}}{spee{d}^{V_{\mathrm{int}}}}+{\displaystyle \sum _{o<m<d}{x}_{S_m}^u\cdot t_{S_m}^u}\right)}}}$$

(8)

$$T_{13}={\displaystyle \sum _{a=1}^{b-1}{\displaystyle \sum _{c=b+1}^N{y}_a}}\cdot y_c\cdot {\displaystyle \sum _{1\le o<a}{\displaystyle \sum _{b<d\le c}{\displaystyle \sum _{u\in V_{\mathrm{int}}^{\mathrm{cros}}}{q}_{S_o{S}_d}^u\cdot x_{S_o}^u\cdot x_{S_d}^u\cdot \left(\frac{d_{S_o{S}_d}}{spee{d}^{V_{\mathrm{int}}}}+{\displaystyle \sum _{o<m<d}{x}_{S_m}^u\cdot t_{S_m}^u}\right)}}}$$

(9)

where $x_{S_m}^u$ is a 0–1 variable indicating whether train $u$ stops at station $S_m$; if train $u$ stops at station $S_m$, the corresponding variable takes the value of 1, otherwise it is 0. $spee{d}^{V_{\mathrm{int}}}$ represents the operating speed of intercity trains (including through-operation intercity trains), which is also the speed at which high-speed trains run on intercity lines.

For the portion of passengers requiring a transfer, the origin station $S_o$ is located in Section 1, while the destination station $S_d$ is in Section 4. Assuming passengers take train $u$ to station $S_k$ and then transfer to train $v$, based on assumption (1), which states that transfers are only made when necessary and at most once, the chosen transfer station $S_k$ must satisfy $a\le k\le c$.

According to Figure 6, if passengers choose station $S_k$ for transfer and it is located at or before station $S_b$ in the upward direction, they can purchase either an intercity train ticket or a through-operation intercity train ticket at the departure station, and then transfer to a through-operation high-speed train to reach their destination. The through-operation high-speed train runs at a reduced speed on intercity lines; therefore, all types of trains that passengers may transfer to run at speed $spee{d}^{V_{\mathrm{int}}}$ before reaching station $S_b$, and at speed $spee{d}^{V_{\mathrm{hig}}}$ after $S_b$. If the chosen transfer station $S_k$ is located downward from station $S_b$, passengers can only take the through-operation intercity train to station $S_k$, and then transfer to a high-speed train or a through-operation high-speed train. As a result, the train runs at speed $spee{d}^{V_{\mathrm{int}}}$ before reaching station $S_k$, and after transferring to a high-speed train (or a through-operation high-speed train) at station $S_k$, the speed of the transferred train becomes $spee{d}^{V_{\mathrm{hig}}}$. Based on the above analysis, the total travel time for passengers can be obtained by adding the in-vehicle travel time, the stop time at intermediate stations, and the transfer time at the transfer station. Calculations for other types of passengers follow a similar process and are not described in detail.

$$T_{14}=\left\{\begin{array}{l}{\displaystyle \sum _{a=1}^{b-1}{\displaystyle \sum _{c=b+1}^N{y}_a}}\cdot y_c\cdot {\displaystyle \sum _{1\le o<a}{\displaystyle \sum _{c<d\le N}{\displaystyle \sum _{u\in V_{\mathrm{int}}\cup V_{\mathrm{int}}^{\mathrm{cros}}}{\displaystyle \sum _{v\in V_{\mathrm{hig}}^{\mathrm{cros}}}{q}_{S_o{S}_d}^{u-v}\cdot \left[x_{S_o}^u\cdot x_{S_k}^u\cdot \left(\frac{d_{S_o{S}_b}}{spee{d}^{V_{\mathrm{int}}}}\right.\right.+}}}}\\ \left.\left.{\displaystyle \sum _{o<m_1<k}{x}_{S_{m_1}}^u\cdot t_{S_{m_1}}^u}\right)+x_{S_k}^v\cdot x_{S_d}^v\cdot \left(t_{\mathrm{trans}}+{\displaystyle \frac{d_{S_b{S}_d}}{spee{d}^{V_{\mathrm{hig}}}}}+{\displaystyle \sum _{k<m_2<d}{x}_{S_{m_2}}^v\cdot t_{S_{m_2}}^v}\right)\right], a\le k\le b\\ {\displaystyle \sum _{a=1}^{b-1}{\displaystyle \sum _{c=b+1}^N{y}_a}}\cdot y_c\cdot {\displaystyle \sum _{1\le o<a}{\displaystyle \sum _{c<d\le N}{\displaystyle \sum _{u\in V_{\mathrm{int}}^{\mathrm{cros}}}{\displaystyle \sum _{v\in V_{\mathrm{hig}}\cup V_{\mathrm{hig}}^{\mathrm{cros}}}{q}_{S_o{S}_d}^{u-v}\cdot \left[x_{S_o}^u\cdot x_{S_k}^u\cdot \left(\frac{d_{S_o{S}_k}}{spee{d}^{V_{\mathrm{int}}}}+\right.\right.}}}}\\ \left.\left.{\displaystyle \sum _{o<m_1<k}{x}_{S_{m_1}}^u\cdot t_{S_{m_1}}^u}\right)+x_{S_k}^v\cdot x_{S_d}^v\cdot \left(t_{\mathrm{trans}}+{\displaystyle \frac{d_{S_k{S}_d}}{spee{d}^{V_{\mathrm{hig}}}}}+{\displaystyle \sum _{k<m_2<d}{x}_{S_{m_2}}^v\cdot t_{S_{m_2}}^v}\right)\right], b<k\le c\end{array}\right.$$

(10)

where $x_{S_{m_1}}^u$ is a 0–1 variable indicating whether train $u$ stops at station $S_{m_1}$; if train $u$ stops at station $S_{m_1}$, the corresponding variable takes the value of 1; otherwise, it is 0. $x_{S_{m_2}}^v$ is a 0–1 variable indicating whether train $v$ stops at station $S_{m_2}$; if train $v$ stops at station $S_{m_2}$, the corresponding variable takes the value of 1; otherwise, it is 0.

$$T_{22}={\displaystyle \sum _{a=1}^{b-1}{y}_a}\cdot {\displaystyle \sum _{a\le o<b}{\displaystyle \sum _{o<d\le b}{\displaystyle \sum _{u\in V_{\mathrm{int}}\cup V_{\mathrm{int}}^{\mathrm{cros}}\cup V_{\mathrm{hig}}^{\mathrm{cros}}}{q}_{S_o{S}_d}^u\cdot x_{S_o}^u\cdot x_{S_d}^u\cdot \left(\frac{d_{S_o{S}_d}}{spee{d}^{V_{\mathrm{int}}}}+{\displaystyle \sum _{o<m<d}{x}_{S_m}^u\cdot t_{S_m}^u}\right)}}}$$

(11)

$$T_{23}=\left\{\begin{array}{l}{\displaystyle \sum _{a=1}^{b-1}{\displaystyle \sum _{c=b+1}^N{y}_a}}\cdot y_c\cdot {\displaystyle \sum _{a\le o<b}{\displaystyle \sum _{b<d\le c}{\displaystyle \sum _{u\in V_{\mathrm{int}}^{\mathrm{cros}}}{q}_{S_o{S}_d}^u\cdot x_{S_o}^u\cdot x_{S_d}^u\cdot \left(\frac{d_{S_o{S}_d}}{spee{d}^{V_{\mathrm{int}}}}+{\displaystyle \sum _{o<m<d}{x}_{S_m}^u\cdot t_{S_m}^u}\right),\forall u\in V_{\mathrm{int}}^{\mathrm{cros}}}}}\\ {\displaystyle \sum _{a=1}^{b-1}{\displaystyle \sum _{c=b+1}^N{y}_a}}\cdot y_c\cdot {\displaystyle \sum _{a\le o<b}{\displaystyle \sum _{b<d\le c}{\displaystyle \sum _{u\in V_{\mathrm{hig}}^{\mathrm{cros}}}{q}_{S_o{S}_d}^u\cdot x_{S_o}^u\cdot x_{S_d}^u\cdot \left(\frac{d_{S_o{S}_b}}{spee{d}^{V_{\mathrm{int}}}}+\frac{d_{S_b{S}_d}}{spee{d}^{V_{\mathrm{hig}}}}+{\displaystyle \sum _{o<m<d}{x}_{S_m}^u\cdot t_{S_m}^u}\right),\forall u\in V_{\mathrm{hig}}^{\mathrm{cros}}}}}\end{array}\right.$$

(12)

$$T_{24}={\displaystyle \sum _{a=1}^{b-1}{\displaystyle \sum _{c=b+1}^N{y}_a}}\cdot y_c\cdot {\displaystyle \sum _{a\le o<b}{\displaystyle \sum _{c<d\le N}{\displaystyle \sum _{u\in V_{\mathrm{hig}}^{\mathrm{cros}}}{q}_{S_o{S}_d}^u\cdot x_{S_o}^u\cdot x_{S_d}^u\cdot \left(\frac{d_{S_o{S}_b}}{spee{d}^{V_{\mathrm{int}}}}+\frac{d_{S_b{S}_d}}{spee{d}^{V_{\mathrm{hig}}}}+{\displaystyle \sum _{o<m<d}{x}_{S_m}^u\cdot t_{S_m}^u}\right)}}}$$

(13)

$$T_{33}=\left\{\begin{array}{l}{\displaystyle \sum _{a=1}^{b-1}{\displaystyle \sum _{c=b+1}^N{y}_a}}\cdot y_c\cdot {\displaystyle \sum _{b\le o<c}{\displaystyle \sum _{o<d\le c}{\displaystyle \sum _{u\in V_{\mathrm{int}}^{\mathrm{cros}}}{q}_{S_o{S}_d}^u\cdot x_{S_o}^u\cdot x_{S_d}^u\cdot \left(\frac{d_{S_o{S}_d}}{spee{d}^{V_{\mathrm{int}}}}+{\displaystyle \sum _{o<m<d}{x}_{S_m}^u\cdot t_{S_m}^u}\right),\forall u\in V_{\mathrm{int}}^{\mathrm{cros}}}}}\\ {\displaystyle \sum _{a=1}^{b-1}{\displaystyle \sum _{c=b+1}^N{y}_a}}\cdot y_c\cdot {\displaystyle \sum _{b\le o<c}{\displaystyle \sum _{o<d\le c}{\displaystyle \sum _{u\in V_{\mathrm{hig}}\cup V_{\mathrm{hig}}^{\mathrm{cros}}}{q}_{S_o{S}_d}^u\cdot x_{S_o}^u\cdot x_{S_d}^u\cdot \left(\frac{d_{S_o{S}_d}}{spee{d}^{V_{\mathrm{hig}}}}+{\displaystyle \sum _{o<m<d}{x}_{S_m}^u\cdot t_{S_m}^u}\right),\forall u\in V_{\mathrm{hig}}\cup V_{\mathrm{hig}}^{\mathrm{cros}}}}}\end{array}\right.$$

(14)

$$T_{34}={\displaystyle \sum _{c=b+1}^N{y}_c}\cdot {\displaystyle \sum _{b\le o<c}{\displaystyle \sum _{c<d\le N}{\displaystyle \sum _{u\in V_{\mathrm{hig}}\cup V_{\mathrm{hig}}^{\mathrm{cros}}}{q}_{S_o{S}_d}^u\cdot x_{S_o}^u\cdot x_{S_d}^u\cdot \left(\frac{d_{S_o{S}_d}}{spee{d}^{V_{\mathrm{hig}}}}+{\displaystyle \sum _{o<m<d}{x}_{S_m}^u\cdot t_{S_m}^u}\right)}}}$$

(15)

$$T_{44}={\displaystyle \sum _{c=b+1}^N{y}_c}\cdot {\displaystyle \sum _{c\le o<N}{\displaystyle \sum _{o<d\le N}{\displaystyle \sum _{u\in V_{\mathrm{hig}}\cup V_{\mathrm{hig}}^{\mathrm{cros}}}{q}_{S_o{S}_d}^u\cdot x_{S_o}^u\cdot x_{S_d}^u\cdot \left(\frac{d_{S_o{S}_d}}{spee{d}^{V_{\mathrm{hig}}}}+{\displaystyle \sum _{o<m<d}{x}_{S_m}^u\cdot t_{S_m}^u}\right)}}}$$

(16)

2.3.3. Combination of Objective Functions

Given the complexity and difficulty of solving multi-objective models, we adopt the linear weighted sum method to assign different weights to the objective functions, thereby converting the multi-objective optimization model into a single-objective one. The coefficients ${\xi }_1$ and ${\xi }_2$ are weights ranging between 0 and 1, with ${\xi }_1+{\xi }_2=1$. To unify the magnitudes of the operating costs for enterprises and the travel costs for passengers, the maximum value $W_{\mathrm{com}}^{\max }$ and the minimum value $W_{\mathrm{com}}^{\min }$ of Objective (1), as well as the maximum value $W_{\mathrm{pas}}^{\max }$ and the minimum value $W_{\mathrm{pas}}^{\min }$ of Objective (2), are obtained in advance and used for normalization.

$$W={\xi }_1\cdot {\displaystyle \frac{W_{\mathrm{com}}-W_{\mathrm{com}}^{\min }}{W_{\mathrm{com}}^{\max }-W_{\mathrm{com}}^{\min }}}+{\xi }_2\cdot {\displaystyle \frac{W_{\mathrm{pas}}-W_{\mathrm{pas}}^{\min }}{W_{\mathrm{pas}}^{\max }-W_{\mathrm{pas}}^{\min }}}$$

(17)

2.4. Constraints of the Train Plan Model

1.

Passenger demand satisfaction constraint

During the process of developing the train plan, the principle of “operating trains according to passenger flow” must be followed. Therefore, the carrying capacity of all trains to be run between each OD pair $(S_o,S_d)$ must meet the passenger demand for that pair.

$${\displaystyle \sum _{V_u\subseteq V}{\displaystyle \sum _{u\in V_u}{f}^u\cdot A^u\cdot x_{S_o}^u\cdot x_{S_d}^u\cdot \overline{\eta }\ge q_{S_o{S}_d}} \forall S_o,S_d\in S}$$

(18)

Formula (18) indicates that the passenger capacity of trains operating between station $S_o$ and $S_d$ should be greater than the passenger flow demand between these two stations. The left-hand side of the inequality represents the train’s passenger capacity, expressed as the sum of the product of the train service frequency, train capacity, and capacity utilization.

2.

Available vehicle number constraint

The number of vehicles used must not exceed the available vehicles for any train type.

$${\displaystyle \sum _{u\in V_{\mathrm{int}}\cup V_{\mathrm{int}}^{\mathrm{cros}}}⌈\frac{f^u\cdot m^{V_u}}{T_{\mathrm{ope}}/[2(d^{V_u}/spee{d}^{V_u}+{\displaystyle \sum _{S_{\mathrm{start}}^{V_u}<m<S_{\mathrm{end}}^{V_u}}{x}_{S_m}^u\cdot t_{S_m}^u})+t_{\mathrm{turn}}^{V_u}+t_{\mathrm{pre}}^{V_u}]}⌉}\le M^{V_{\mathrm{int}}}$$

(19)

$$\begin{array}{l}{\displaystyle \sum _{u\in V_{\mathrm{hig}}^{\mathrm{cros}}}⌈\frac{f^u\cdot m^{V_u}}{T_{\mathrm{ope}}/[2(\frac{d_{S_a{S}_b}}{spee{d}^{V_{\mathrm{int}}}}+\frac{d_{S_b{S}_N}}{spee{d}^{V_{\mathrm{hig}}}}+{\displaystyle \sum _{S_{\mathrm{start}}^{V_u}<m<S_{\mathrm{end}}^{V_u}}{x}_{S_m}^u\cdot t_{S_m}^u})+t_{\mathrm{turn}}^{V_u}+t_{\mathrm{pre}}^{V_u}]}⌉}\\ +{\displaystyle \sum _{u\in V_{\mathrm{hig}}}⌈\frac{f^u\cdot m^{V_u}}{T_{\mathrm{ope}}/[2(\frac{d_{S_b{S}_N}}{spee{d}^{V_{\mathrm{hig}}}}+{\displaystyle \sum _{S_{\mathrm{start}}^{V_u}<m<S_{\mathrm{end}}^{V_u}}{x}_{S_m}^u\cdot t_{S_m}^u})+t_{\mathrm{turn}}^{V_u}+t_{\mathrm{pre}}^{V_u}]}⌉\le M^{V_{\mathrm{hig}}}}\end{array}$$

(20)

Here, ⌈⌉ represents the ceiling function. $S_{\mathrm{start}}^{V_u}$ is the origin and $S_{\mathrm{end}}^{V_u}$ is the destination for $V_u$-type train routes. For example, when $V_u=V_{\mathrm{int}}$, $S_{\mathrm{start}}^{V_u}=S_1$ and $S_{\mathrm{end}}^{V_u}=S_b$. When $V_u=V_{\mathrm{int}}^{\mathrm{cros}}$, $S_{\mathrm{start}}^{V_u}=S_1$ and $S_{\mathrm{end}}^{V_u}=S_c$. When $V_u=V_{\mathrm{hig}}$, $S_{\mathrm{start}}^{V_u}=S_b$ and $S_{\mathrm{end}}^{V_u}=S_N$. When $V_u=V_{\mathrm{hig}}^{\mathrm{cros}}$, $S_{\mathrm{start}}^{V_u}=S_a$ and $S_{\mathrm{end}}^{V_u}=S_N$. Taking Formula (19) as an example, the numerator on the left-hand side of the inequality represents the required vehicle number of intercity trains (i.e., the product of the train service frequency and the marshaling number), while the denominator represents the number of times that the corresponding vehicle can be turned back within the daily operating period (calculated as the total operating time divided by the time for a round trip). The ratio of these two values should be less than or equal to the total available number of intercity trains on the right-hand side of the inequality.

3.

Line capacity constraint

The number of trains operated must not exceed the line capacity limit.

$${\displaystyle \sum _{u\in V_{\mathrm{int}}\cup V_{\mathrm{int}}^{\mathrm{cros}}\cup V_{\mathrm{hig}}^{\mathrm{cros}}}{f}^u}\le C_{\max }^{\mathrm{int}}$$

(21)

$${\displaystyle \sum _{u\in V_{\mathrm{hig}}\cup V_{\mathrm{int}}^{\mathrm{cros}}\cup V_{\mathrm{hig}}^{\mathrm{cros}}}{f}^u}\le C_{\max }^{\mathrm{hig}}$$

(22)

4.

Station service frequency constraint

Within the study period, the total number of trains served by any station node on the line must not exceed the capacity limit of the station. Additionally, to attract more passengers, meet diverse travel needs, and balance transportation service resources, a lower-bound constraint must also be applied to station service frequency.

$$F_{S_i}^{\mathrm{low}}\le {\displaystyle \sum _{u\in V_{\mathrm{int}}\cup V_{\mathrm{int}}^{\mathrm{cros}}}{f}^u}\cdot x_{S_i}^u\le F_{S_i}^{\mathrm{up}},\forall i\in \left\{1,2,\dots ,a-1\right\}$$

(23)

$$F_{S_i}^{\mathrm{low}}\le {\displaystyle \sum _{u\in V_{\mathrm{int}}\cup V_{\mathrm{int}}^{\mathrm{cros}}\cup V_{\mathrm{hig}}^{\mathrm{cros}}}{f}^u}\cdot x_{S_i}^u\le F_{S_i}^{\mathrm{up}},\forall i\in \left\{a,a+1,\dots ,b-1\right\}$$

(24)

$$F_{S_i}^{\mathrm{low}}\le {\displaystyle \sum _{u\in V_{\mathrm{int}}^{\mathrm{cros}}\cup V_{\mathrm{hig}}\cup V_{\mathrm{hig}}^{\mathrm{cros}}}{f}^u}\cdot x_{S_i}^u\le F_{S_i}^{\mathrm{up}},\forall i\in \left\{b,b+1,\dots ,c-1\right\}$$

(25)

$$F_{S_i}^{\mathrm{low}}\le {\displaystyle \sum _{u\in V_{\mathrm{hig}}\cup V_{\mathrm{hig}}^{\mathrm{cros}}}{f}^u}\cdot x_{S_i}^u\le F_{S_i}^{\mathrm{up}},\forall i\in \left\{c,c+1,\dots ,N\right\}$$

(26)

5.

Position constraint for the through-operation segment

The location of the terminal stations in the through-operation section is unique and must be set at stations with turnaround capability.

$${\displaystyle \sum _{c=b+1}^N{y}_c}=1,S_c\in S_{\mathrm{turn}}$$

(27)

$${\displaystyle \sum _{a=1}^{b-1}{y}_a}=1,S_a\in S_{\mathrm{turn}}$$

(28)

6.

Train stop constraint

All trains must stop at their origin and destination stations.

$$x_{S_1}^u=x_{S_b}^u=1,\forall u\in V_{\mathrm{int}}$$

(29)

$$x_{S_1}^u=x_{S_c}^u=1,\forall u\in V_{\mathrm{int}}^{\mathrm{cros}}$$

(30)

$$x_{S_b}^u=x_{S_N}^u=1,\forall u\in V_{\mathrm{hig}}$$

(31)

$$x_{S_a}^u=x_{S_N}^u=1,\forall u\in V_{\mathrm{hig}}^{\mathrm{cros}}$$

(32)

7.

Decision variable value constraint

The train service frequency is a natural number, while the decision variables for train stop selection and the terminal station position of the through-operation route are 0–1 variables.

$$f^u\in {\rm N}\forall u\in V_u$$

(33)

$$x_{S_i}^u\in \{0,1\}\forall u\in V_u,S_i\in S$$

(34)

$$y_c\in \left\{0,1\right\}\forall c\in \left\{b+1,\dots ,N\right\}$$

(35)

$$y_a\in \left\{0,1\right\}\forall a\in \left\{1,\dots ,b-1\right\}$$

(36)

3. Algorithm Design

Since the above model is nonlinear and characterized by high-dimensional decision variables and complex constraints, compared to other train plan models, it contains a large number of 0–1 variables related to stop plans. Therefore, the encoding complexity and solution space of the model are significantly larger. Genetic algorithms are well suited for dealing with discrete and integer variable problems, and through proper encoding strategies, such as binary encoding, discrete decisions can be transformed into a form that genetic algorithms can handle [28,29]. Moreover, the multi-population genetic algorithm has strong global search capabilities, is easy to parallelize, and has good robustness, making it suitable for the computation. First, the enumeration method is used to determine the turnaround station locations for through operations. Then, the improved multi-population genetic algorithm is employed to calculate the optimal service frequency and stop plans for each turnaround station option, storing the solutions in the candidate set. Finally, a global optimal solution is selected and organized into a complete train plan.

3.1. Genetic Algorithm Fundamentals

In 1975, J.H. Holland proposed the genetic algorithm (GA). When using GA to solve a problem, the solution is first encoded into chromosomes, and an initial population is generated randomly. The algorithm then evolves this population through operations such as crossover and mutation to select chromosomes that are better adapted to the environment. During this process, those with higher fitness values have a greater chance of being selected and passed on. After multiple iterations, the result converges toward the optimal solution.

Solving with a classic genetic algorithm mainly involves the following core steps.

• Algorithm initialization

Define specific optimization parameters for the algorithm, such as the search range of the solution space, population size, probabilities of chromosome crossover and mutation, termination conditions, and the maximum number of iterations. An initial population is then randomly generated within the solution space.

A key issue in constructing the initial population is the encoding strategy. Binary encoding is the most widely used method. Additionally, symbolic encoding, gray coding, dynamic encoding, and other encoding methods can also be used. The most suitable approach should be determined based on the specific problem at hand.

2.

Fitness evaluation

The fitness value of a chromosome reflects the quality of the solution it represents and serves as the basis for population evolution and survival of the fittest. For different problems to be solved, various methods can be used for fitness evaluation, such as employing the objective function or a standard fitness function.

3.

Selection operation

Selection is the process of survival of the fittest. After calculating the fitness of each individual, those with better fitness values are selected. The most common selection method is roulette wheel selection, where the probability of selecting an individual is determined by its proportion to the fitness function. For a maximization problem, the probability of any individual being selected is as follows:

$$p_i={\displaystyle \frac{fi{t}_i}{{\displaystyle \sum _{i=1}^{npop}fi{t}_i}}}$$

(37)

where $fi{t}_i$ represents the fitness value of the $i$-th individual in the population, and $npop$ is the population size.

4.

Crossover operation

Crossover refers to the process of recombining genes from two parents. The design of replacement recombination is closely related to the problem being studied and should typically be considered alongside the encoding method of the individuals. Common crossover operations include uniform crossover, simulated binary crossover, single-point crossover, and two-point crossover.

5.

Mutation operation

Mutation operation involves making slight adjustments to certain gene loci of individuals, thereby introducing new genes into the chromosome population. Common mutation methods include uniform mutation, non-uniform mutation, and polynomial mutation.

6.

Termination condition

Evaluate the fitness of individuals in the new population and determine whether the termination condition of the algorithm is met. If the termination condition is satisfied, the loop ends; otherwise, return to Step (3) Selection Operation and continue.

3.2. Design of the Improved Algorithm

Based on the computational process and design principles of the classic genetic algorithm, this section presents improvements to the multi-population genetic algorithm in terms of the encoding strategy, generation of the initial solution, enhancements to crossover and mutation operations, and the introduction of a migration operator.

• Encoding strategy

Based on the characteristics of the model, this paper adopts binary encoding to design the chromosome structure. The service frequency $f={\displaystyle \sum _{u\in V_u}{f}^u}$ is determined according to the maximum passenger flow density. As shown in Figure 7, the entire through-operation line, including the origin station, terminal station, and intermediate stations, has $N$ stations. Thus, each chromosome designed in this paper consists of $N+2$ gene loci, with all the genes collectively forming an $f\times (N+2)$ chromosome matrix.

For each chromosome encoding, the first to the $N$-th gene loci correspond to the stop plan for stations $S_1$ to $S_N$, where a value of 1 indicates that the train either departs, terminates, or stops at that station, and 0 indicates it does not depart, terminate, or stop at that station. For intercity trains, the route starts at station $S_1$ and ends at station $S_b$. Therefore, the values of the first and $b$-th gene locus in the chromosome must be 1, and the values from $b+1$ to $N$ must be 0, since the route does not extend beyond station $S_b$. The values of the other intermediate loci are determined by the train’s stop plan. Similarly, for through-operation intercity trains, the values of the first and $c$-th gene locus must be 1, and the values from $c+1$ to $N$ must be 0, with the other intermediate loci values determined by the stop plan. For high-speed trains, the values of the $b$-th and $N$-th gene locus must be 1, and the values from the first to $(b-1)$-th loci must be 0, with the other intermediate loci values determined by the stop plan. For through-operation high-speed trains, the values of the $a$-th and $N$-th gene loci must be 1, and the values from the first to $(a-1)$-th loci must be 0, with the other intermediate loci values determined by the stop plan.

The $(N+1)$-th and $(N+2)$-th gene loci of each chromosome encode the train type. Four types of trains are involved in the model: ‘00’ represents intercity trains, ‘01’ represents through-operation intercity trains, ‘10’ represents high-speed trains, and ‘11’ represents through-operation high-speed trains. Summing the trains with the same stopping plan and type in the chromosome yields the operating frequency $f^u$ for trains of type $u$.

2.

Generation of the initial solution

The population size is determined according to the literature [30] and extensive preliminary data experiments. First, randomly generate three populations, A, B, and C, each containing $npop$ train plans that satisfy constraint (18) and follow the above encoding strategy. Each population thus consists of $npop$ chromosomes, where each chromosome is an $f\times (N+2)$ matrix encompassing all types of trains. Next, calculate the train frequencies corresponding to different stopping schemes based on the randomly generated chromosome matrices. If the train frequencies and stop plans meet the remaining constraints, the train plan is included in the initial solution. Otherwise, a new chromosome matrix is generated until all constraints are met.

3.

Improvements in crossover and mutation operations

The chromosomes in populations A, B, and C can undergo crossover operations simultaneously, demonstrating the parallelism and efficiency of multi-population genetic algorithms. First, two chromosome matrices are randomly selected from the population for pairing. Then, based on the crossover operations in classical genetic algorithms and considering the characteristics of the model and encoding in this paper, any two rows of chromosomes from the matrices are selected for exchange based on a probability $p_{\mathrm{crossover}}$, completing the algorithm’s crossover.

Then, a mutation operation is performed with a probability of $p_{\mathrm{mutation}}$, allowing chromosomes from different populations to mutate simultaneously. This procedure uses multi-point mutation, which starts with a few genetic sites in a randomly selected chromosomal matrix. According to the chromosome encoding rules described in this paper, for individuals with the last two bits of the chromosome encoded as 00 (intercity trains), the mutation can occur at any gene point from the 1st to the $(b-1)$-th position. If the mutation occurs at the 1st position, the train is deleted. Similarly, for individuals with the last two bits of the chromosome encoded as 01 (intercity trains with through operation), the mutation can occur at any gene point from the 1st to the $(c-1)$-th position. If the mutation occurs at the 1st position, it means the train must be deleted. For individuals with the last two bits of the chromosome encoded as 10 (high-speed trains), the mutation can occur at any gene point from the $b$-th to the $(N-1)$-th position. If the mutation occurs at the $b$-th position, the train is deleted. For individuals with the last two bits of the chromosome encoded as 11 (high-speed trains with through operation), the mutation can occur at any gene point from the $a$-th to the $(N-1)$-th position. If the mutation occurs at the $a$-th position, the train is deleted.

Unlike the fixed parameter mode in classical genetic algorithms, this paper adopts an adaptive approach for the probability and the number of mutation points. The average fitness value $fi{t}_{avg}$ of all solutions in the population is calculated. Based on the relationship between the smaller fitness value of the paired parent chromosomes and $fi{t}_{avg}$, different crossover probabilities are then generated using Formula (38). Similarly, based on the relationship between the selected mutation individual’s fitness value and $fi{t}_{avg}$, different mutation probabilities and mutation point counts are generated using Formulas (39) and (40), respectively. The core idea is to assign lower probabilities $p_{\mathrm{crossover}}$ and $p_{\mathrm{mutation}}$ to individuals whose fitness is better than the average, making them more likely to be preserved. In contrast, higher probabilities $p_{\mathrm{crossover}}$ and $p_{\mathrm{mutation}}$ are assigned to individuals whose fitness is worse than the average, making them more likely to evolve.

$$p_{\mathrm{crossover}}=\left\{\begin{array}{ll}\hfill p_{\mathrm{crossover}}^{\mathrm{initial}}\hfill & fit\left(X\right)>fi{t}_{\mathrm{avg}}\\ {\displaystyle \frac{p_{\mathrm{crossover}}^{\mathrm{initial}}\cdot \left[fit\left(X\right)-fi{t}_{\min }\right]}{fi{t}_{\mathrm{avg}}-fi{t}_{\min }}}& fit\left(X\right)\le fi{t}_{\mathrm{avg}}\end{array}\right.$$

(38)

$$p_{\mathrm{mutation}}=\left\{\begin{array}{ll}\hfill p_{\mathrm{mutation}}^{\mathrm{initial}}\hfill & fit\left(Y\right)>fi{t}_{\mathrm{avg}}\\ {\displaystyle \frac{p_{\mathrm{mutation}}^{\mathrm{initial}}\cdot \left[fit\left(Y\right)-fi{t}_{\min }\right]}{fi{t}_{\mathrm{avg}}-fi{t}_{\min }}}& fit\left(Y\right)\le fi{t}_{\mathrm{avg}}\end{array}\right.$$

(39)

$$num=\left\{\begin{array}{ll}\hfill nu{m}^{\mathrm{initial}}\hfill & fit\left(Y\right)>fi{t}_{\mathrm{avg}}\\ {\displaystyle \frac{nu{m}^{\mathrm{initial}}\cdot \left[fit\left(Y\right)-fi{t}_{\min }\right]}{fi{t}_{\mathrm{avg}}-fi{t}_{\min }}}& fit\left(Y\right)\le fi{t}_{\mathrm{avg}}\end{array}\right.$$

(40)

In this context, $p_{\mathrm{crossover}}^{\mathrm{initial}}$ is the baseline crossover probability, which refers to the crossover probability when the smaller fitness value in the paired parent chromosomes is still greater than the average fitness value of the population. $p_{\mathrm{mutation}}^{\mathrm{initial}}$ is the baseline mutation probability, which refers to the mutation probability when an individual’s fitness value exceeds the average fitness value of the population. $fit\left(X\right)$ represents the smaller fitness value in the paired parent chromosomes, while $fit\left(Y\right)$ denotes the fitness value of the mutated individual. $fi{t}_{\min }^{\mathrm{I}-\mathrm{H}}$ is the minimum fitness value within the population. $nu{m}^{\mathrm{initial}}$ refers to the baseline number of mutation points, which indicates the number of mutated gene points when the fitness value of the mutated individual is greater than the average fitness value of the population.

The chromosomes generated through the above-mentioned crossover and mutation operations need to be modified by using model constraints. Gene loci or chromosome individuals that fail to meet the constraints are randomly adjusted to comply with the requirements.

4.

Introduction of the migration operator

In addition, based on the characteristics of the model and the encoding scheme of the solutions, this paper introduces a migration operator to improve the algorithm. The migration operator is an operation used to exchange individuals or information between different populations. Its purpose is to facilitate the flow of information between populations, thereby enhancing the search performance of the algorithm and promoting global exploration. A migration operation is performed every $I_{\mathrm{migration}}$ generations. With a probability of $p_{\mathrm{migration}}$, the stop scheme of through-operation trains on the original route section is transferred from the chromosome matrix of the population with the best fitness (i.e., the lowest fitness value) to the matrix of another population with the worst fitness (i.e., the highest fitness value), replacing the stop scheme of non-through-operation trains of the same type.

3.3. Solving the Process of the Improved Algorithm

This paper transforms the constraints into penalty functions to construct the fitness function, the basic form of which is shown in Equation (41). Here, $fit$ represents the fitness function, $W\left(X\right)$ is the original objective function, ${\Psi }_1$ is the amplification factor, ${\Psi }_2$ is the penalty factor, and $pe{n}^i\left(X\right)$ is the penalty term for violating the $i$-th constraint.

$$\min fit={\Psi }_1\cdot W\left(X\right)+{\Psi }_2\cdot {\displaystyle \sum _i\max \left(0,pe{n}^i\left(X\right)\right)}$$

(41)

Since the model in this paper aims to solve a minimization problem, while the selection strategy of classical genetic algorithms is designed for maximization problems, an improved roulette wheel selection strategy is employed for the selection process.

Based on the calculation of chromosome fitness values, the probability of selecting the $i$-th chromosome from the population is given by the following:

$$p_{\mathrm{select},i}={\displaystyle \frac{1/fi{t}_i}{{\displaystyle \sum _{i=1}^{npop}1/fi{t}_i}}}$$

(42)

The following steps illustrate the improved multi-population genetic algorithm, with a clearer visual depiction available in Figure 8.

Step 1:

Import the basic data and set the parameter values of the algorithm. Set the current iteration count to $iter=1$.

Step 2:

Enumerate all possible through-operation route schemes $\left\{1,2,\dots ,\chi \right\}$. Let the initial through-operation route scheme be $plan=1$. Randomly generated $3\cdot npop$ chromosome matrices.

Step 3:

Check whether the chromosome matrix satisfies all the constraints.

Step 4:

Divide the chromosomes that satisfy all constraints into three populations: A, B, and C. Use Formula (42) to calculate the probability of each chromosome being selected.

Step 5:

Randomly pair the selected individuals in pairs. Perform the crossover operation on the paired chromosomes.

Step 6:

Perform mutation operations on the selected chromosomes, and the newly generated individuals are screened and corrected.

Step 7:

Let $iter=iter+1$, and check whether the iteration count $iter$ reaches $\varpi \cdot I_{\mathrm{migration}}$ ($\varpi =1,2,3,\dots \cdot$). If $iter=\varpi \cdot I_{\mathrm{migration}}$, a migration operator is introduced for interpopulation learning.

Step 8:

If $iter\ge Cycl{e}_{\max }$, terminate the current iteration and output the optimal solution for the through-operation route, placing it into the candidate set $\Im$. If $iter<Cycl{e}_{\max }$, proceed to Step 4.

Step 9:

Let $plan=plan+1$. If $plan\le \chi$, proceed to Step 3; otherwise, output the optimal train plan from the candidate set $\Im$ as the global optimal solution.

4. Computation Case and Results Analysis

This section takes the Chengdu–Mianyang–Leshan Intercity Railway and the Chengdu–Chongqing High-Speed Railway, which are connected through Chengdu East Station, as case studies. Based on the description of the problem presented earlier, the stations and lines are simplified, and the train plan is developed using the model and algorithm proposed in this paper so as to verify the effectiveness of the method.

4.1. Basic Data Settings

The schematic diagram of the study line is shown in Figure 9a, which can be abstracted into the form shown in Figure 9b. Based on the station location information, the mileage of each section can be obtained: $d_{S_1{S}_2}=14\mathrm{km}$, $d_{S_2{S}_3}=25\mathrm{km}$, $d_{S_3{S}_4}=26\mathrm{km}$, $d_{S_4{S}_5}=21\mathrm{km}$, $d_{S_5{S}_6}=20\mathrm{km}$, $d_{S_6{S}_7}=14\mathrm{km}$, $d_{S_7{S}_8}=11\mathrm{km}$, $d_{S_8{S}_9}=21\mathrm{km}$, $d_{S_9{S}_{10}}=53\mathrm{km}$, $d_{S_{10}{S}_{11}}=30\mathrm{km}$, $d_{S_{11}{S}_{12}}=41\mathrm{km}$, $d_{S_{12}{S}_{13}}=29\mathrm{km}$, $d_{S_{13}{S}_{14}}=26\mathrm{km}$, $d_{S_{14}{S}_{15}}=34\mathrm{km}$, $d_{S_{15}{S}_{16}}=10\mathrm{km}$, $d_{S_{16}{S}_{17}}=22\mathrm{km}$, $d_{S_{17}{S}_{18}}=30\mathrm{km}$, and $d_{S_{18}{S}_{19}}=24\mathrm{km}$. As can be seen from Figure 9, $d^{V_{\mathrm{int}}}=d_{S_1{S}_9}=152\mathrm{km}$, $d^{V_{\mathrm{int}}^{\mathrm{cros}}}=d_{S_1{S}_c}=152\mathrm{km}+d_{S_9{S}_c}$, $d^{V_{\mathrm{hig}}}=299\mathrm{km}$, and $d^{V_{\mathrm{hig}}^{\mathrm{cros}}}=d_{S_a{S}_N}=299\mathrm{km}+d_{S_a{S}_9}$.

The detailed passenger flow is provided in Appendix B. For the sake of calculation, the Chengdu–Mianyang–Leshan Intercity Railway uniformly adopts the CRH2A model train, which is suitable for railway lines with speeds ranging from 200 to 250 km/h. The train primarily consists of an eight-formation train configuration (four powered units and four trailing units), with a seating capacity of 610 passengers. The Chengdu–Chongqing High-Speed Railway uniformly adopts the CRH380A electric multiple unit (EMU), which has a commercial operational speed of 350 km/h. This EMU primarily consists of an eight-formation train configuration (six powered units and two trailing units), with a seating capacity of 494 passengers.

Through investigation and survey, it is found that the average load factor of the EMUs in the region remains around 70–80%. Therefore, this section assigns a value of 75% to the load factor. Additionally, $S_{\mathrm{turn}}=\left\{S_1,S_3,S_5,S_7,S_9,S_{11},S_{12},S_{13},S_{15},S_{17},S_{19}\right\}$ refers to the set of stations with turning-back capabilities; for these stations, the departure capacity is $F_{S_i}^{\mathrm{up}}=200$ trains/day, and $S_i\in S_{\mathrm{turn}}$, while for other intermediate stations, $F_{S_i}^{\mathrm{up}}=144$ trains/day, and $S_i\in S\backslash S_{\mathrm{turn}}$. To ensure that all stations are served by trains, $F_{S_i}^{\mathrm{low}}=5$ trains/day, and $S_i\in S$ is applied.

Referring to actual operational data and values from existing studies [30], the values for other parameters involved in the model are listed in

Table 2. Parameter setting of the model.

Parameter	Value	Measurement Unit
$cos{t}_{\mathrm{run}}^{V_{\mathrm{int}}}$, $cos{t}_{\mathrm{run}}^{V_{\mathrm{int}}^{\mathrm{cros}}}$, $cos{t}_{\mathrm{run}}^{V_{\mathrm{hig}}}$, $cos{t}_{\mathrm{run}}^{V_{\mathrm{hig}}^{\mathrm{cros}}}$	75, 75, 80, 80	CNY/vehicle·km
$cos{t}_{\mathrm{stop}}^{V_{\mathrm{int}}}$, $cos{t}_{\mathrm{stop}}^{V_{\mathrm{int}}^{\mathrm{cros}}}$, $cos{t}_{\mathrm{stop}}^{V_{\mathrm{hig}}}$, $cos{t}_{\mathrm{stop}}^{V_{\mathrm{hig}}^{\mathrm{cros}}}$	450, 450, 500, 500	CNY/train·station
$m^{V_{\mathrm{int}}}$, $m^{V_{\mathrm{int}}^{\mathrm{cros}}}$, $m^{V_{\mathrm{hig}}}$, $m^{V_{\mathrm{hig}}^{\mathrm{cros}}}$	8, 8, 8, 8	Vehicle
$cos{t}_{\mathrm{tic}}^{V_{\mathrm{int}}}$, $cos{t}_{\mathrm{tic}}^{V_{\mathrm{int}}^{\mathrm{cros}}}$, $cos{t}_{\mathrm{tic}}^{V_{\mathrm{hig}}}$, $cos{t}_{\mathrm{tic}}^{V_{\mathrm{hig}}^{\mathrm{cros}}}$	0.4, 0.4, 0.5, 0.5	CNY/person·km
$\gamma$	25	CNY/h
$spee{d}^{V_{\mathrm{int}}}$, $spee{d}^{V_{\mathrm{hig}}}$	250, 300	Km/h
$t_{\mathrm{trans}}$	0.25	h
$t_{S_m}^u$, $t_{S_{m_1}}^u$, $t_{S_{m_2}}^v$	0.05, 0.05, 0.05	h
$A^{V_{\mathrm{int}}}$, $A^{V_{\mathrm{int}}^{\mathrm{cros}}}$, $A^{V_{\mathrm{hig}}}$, $A^{V_{\mathrm{hig}}^{\mathrm{cros}}}$	610, 610, 494, 494	Person
$\overline{\eta }$	0.75	-
$t_{\mathrm{turn}}^{V_{\mathrm{int}}}$, $t_{\mathrm{turn}}^{V_{\mathrm{int}}^{\mathrm{cros}}}$, $t_{\mathrm{turn}}^{V_{\mathrm{hig}}}$, $t_{\mathrm{turn}}^{V_{\mathrm{hig}}^{\mathrm{cros}}}$	0.17, 0.25, 0.33, 0.42	h
$t_{\mathrm{pre}}^{V_{\mathrm{int}}}$, $t_{\mathrm{pre}}^{V_{\mathrm{int}}^{\mathrm{cros}}}$, $t_{\mathrm{pre}}^{V_{\mathrm{hig}}}$, $t_{\mathrm{pre}}^{V_{\mathrm{hig}}^{\mathrm{cros}}}$	0.5, 0.5, 0.75, 0.75	h
$T_{\mathrm{ope}}$	16	h
$M^{V_{\mathrm{int}}}$, $M^{V_{\mathrm{hig}}}$	480, 620	Vehicle
$C_{\max }^{\mathrm{int}}$, $C_{\max }^{\mathrm{hig}}$	144, 144	Trains/day

.

Based on the parameter values from the existing literature [28,29,30,31], combined with extensive numerical experiments and summarization, the algorithm parameter values used in this paper are listed in

Table 3. Parameter setting of the algorithm.

Parameter	Definition	Value
${\Psi }_1$	Amplification factor	1000
${\Psi }_2$	Penalty factor	1000
$npop$	Population size	50
$p_{\mathrm{crossover}}^{\mathrm{initial}}$	Baseline crossover probability	0.8
$p_{\mathrm{mutation}}^{\mathrm{initial}}$	Baseline mutation probability	0.1
$nu{m}^{\mathrm{initial}}$	Baseline number of mutation points	6
$I_{\mathrm{migration}}$	Migration interval	10
$p_{\mathrm{migration}}$	Migration probability	0.3
$Cycl{e}_{\max }$	Maximum number of iterations	3500

.

4.2. Analysis of Optimal Solution

The improved algorithm is implemented using Python to balance the interests of both parties. Under the condition where the objective function weights ${\xi }_1$ and ${\xi }_2$ are both set to 0.5, the optimal solution encoding is compiled. Based on the calculation results, the train plan under the through-operation conditions of the intercity and high-speed railways can be abstracted as shown in Figure 10. At this point, $S_a$ is located at Mianyang Station, and $S_c$ is located at Zizhongbei Station. The operating cost for the company is CNY 500,149,230, the passenger travel cost is CNY 18,943,237.99, and the total objective function value is 0.316196. A total of 31 intercity trains operating, with 19 through-operation intercity trains; 39 high-speed trains operating, with 38 through-operation high-speed trains.

4.2.1. Sensitivity Analysis of Objective Function Weights

For analyzing the impact of weight coefficients on the results, all possible weight coefficients of two objective functions (with 0.1 as the smallest unit) are enumerated. The calculated results, after statistical collation, are presented in

Table 4. The impact of objective function weights on calculation results.

$\left({\mathit{\xi }}_1,{\mathit{\xi }}_2\right)$	${\mathit{S}}_{\mathit{a}}$	${\mathit{S}}_{\mathit{c}}$	${\mathit{f}}^{{\mathit{V}}_{\mathbf{int}}}$	${\mathit{f}}^{{\mathit{V}}_{\mathbf{int}}^{\mathbf{cros}}}$	${\mathit{f}}^{{\mathit{V}}_{\mathbf{hig}}}$	${\mathit{f}}^{{\mathit{V}}_{\mathbf{hig}}^{\mathbf{cros}}}$	${\mathit{W}}_{\mathbf{com}}$	${\mathit{W}}_{\mathbf{pas}}$	$\mathit{W}$
(0.1, 0.9)	$S_3$	$S_{11}$	38	19	25	54	536,482,170	13,028,074.87	0.270526
(0.2, 0.8)	$S_3$	$S_{11}$	29	27	40	39	520,094,820	13,242,495.94	0.275129
(0.3, 0.7)	$S_5$	$S_{11}$	38	24	40	42	518,215,950	13,629,969.87	0.285876
(0.4, 0.6)	$S_7$	$S_{12}$	32	36	30	43	505,613,220	18,636,557.94	0.334503
(0.5, 0.5)	$S_3$	$S_{12}$	31	19	39	38	500,149,230	18,943,237.99	0.316196
(0.6, 0.4)	$S_5$	$S_{12}$	20	33	33	42	498,663,480	19,361,447.88	0.302926
(0.7, 0.3)	$S_5$	$S_{12}$	30	30	26	45	494,401,110	19,422,301.39	0.276259
(0.8, 0.2)	$S_7$	$S_{13}$	24	40	28	39	492,552,690	24,154,043.66	0.273703
(0.9, 0.1)	$S_7$	$S_{13}$	30	33	28	37	468,172,110	24,400,748.18	0.155434

. As shown in Table 4, with the increase in the weight coefficient of a specific optimization objective, the value of the corresponding objective function generally exhibits a decreasing trend. In real-world scenarios, operating companies can decide whether to assign more weight to the enterprise operating costs or to passenger experience based on their decision-making preferences.

To provide a more intuitive representation of the relationship between each objective function value and its corresponding weight coefficient, the relationship curves of objective values with varying weights are plotted, as shown in Figure 11. $W_{\mathrm{com}}$, $W_{\mathrm{pas}}$, and $W$ are jointly determined by train service frequency, stop plans, and the turnaround station setting of through-operation sections. These variables are influenced by the allocation proportions of the objective function within a certain range. As the length of the through-operation interval and the total number of trains operated increase, this typically leads to higher enterprise operating costs and lower passenger travel costs. Therefore, different operating companies should choose different weight assignments based on the actual situation when making decisions in order to better balance the interests of both the company and the passengers.

4.2.2. Comparison with Non-Through-Operation Scheme

• Comparison of optimal schemes

To demonstrate the advantages of the through-operation mode, the optimal train plan under the non-through-operation mode is used as the control group. Similarly, the impact of the operational mode is analyzed by fixing the objective function weights. To balance the interests of both the enterprises and passengers, the weights are set as ${\xi }_1={\xi }_2=0.5$. A comparison of the optimal train plans under the two modes is presented in

Table 5. Impact of different operation modes on the optimal train plan.

$\mathit{a}$	$\mathit{c}$	${\mathit{f}}^{{\mathit{V}}_{\mathbf{int}}}$	${\mathit{f}}^{{\mathit{V}}_{\mathbf{int}}^{\mathbf{cros}}}$	${\mathit{f}}^{{\mathit{V}}_{\mathbf{hig}}}$	${\mathit{f}}^{{\mathit{V}}_{\mathbf{hig}}^{\mathbf{cros}}}$	${\mathit{W}}_{\mathbf{com}}$	${\mathit{W}}_{\mathbf{pas}}$	$\mathit{W}$
3	12	31	19	39	38	500,149,230	189,43,237.99	0.316196
-	-	64	-	73	-	6,035,83,050	36,156,921.48	0.821191

. The results indicate that adopting the through-operation mode reduced the operating costs of the enterprises by 17.14%, decreased passenger travel costs by 47.61%, and lowered the total number of trains in operation by 7.30%.

When passenger flow fluctuates within the range of −60% to 60% based on the values provided in Appendix B, the changes in enterprise operating costs, passenger travel costs, and the total number of trains operating under and without the through-operation mode are shown in Figure 12. As illustrated in the figure, the through-operation mode can reduce passenger travel time, enhance enterprise operational efficiency, and demonstrate better adaptability to variations in passenger flow.

2.

Comparison with classical genetic algorithm

To demonstrate the advantages of the algorithm designed in this paper, the classical genetic algorithm before improvement is selected as the comparative algorithm. The parameter settings of the comparative algorithm are consistent with the benchmark parameters used in this study, and the problem is implemented in Python on the same computer. The comparative analysis of the two algorithms is presented in

Table 6. Comparative analysis of the two algorithms.

Comparison Items	The Improved Algorithm	The Classical Genetic Algorithm
Iteration count	1453	1790
$W$	0.316196	0.342535
$W_{\mathrm{com}}$	500,149,230	514,139,430
$W_{\mathrm{pas}}$	18,943,237.99	18,679,335.90
Operation sections of through- operation intercity trains	$[S_1,S_{12}]$	$[S_1,S_{12}]$
Operation sections of through- operation high-speed trains	$[S_3,S_{19}]$	$[S_7,S_{19}]$
$f^{V_{\mathrm{int}}}$	31	31
$f^{V_{\mathrm{int}}^{\mathrm{cros}}}$	19	40
$f^{V_{\mathrm{hig}}}$	39	38
$f^{V_{\mathrm{hig}}^{\mathrm{cros}}}$	38	35

, and the iteration process comparison is shown in Figure 13.

As shown in Table 6 and Figure 13, compared with the classical genetic algorithm, the improved algorithm designed in this paper achieves an 18.83% increase in convergence speed and a 7.69% improvement in solution quality. Consequently, the improved genetic algorithm not only converges faster but also delivers superior solution performance, making it better suited to the characteristics of the proposed model.

5. Conclusions

This paper focuses on the train plan problem in the context of sustainability. A multi-objective optimization model is developed to determine train service frequency, stop plans, and the location of turnaround stations within the through-operation sections. The model balances the interests of both enterprises and passengers and is characterized by high-dimensional decision variables, nonlinearity, and complex constraints. Therefore, the solution process is divided into two stages. First, the enumeration method is employed to determine the locations of the turnaround stations in the through-operation sections, and the chromosome structure as well as the initial solution generation strategy are designed based on the model’s features. Second, adaptive crossover and mutation strategies are applied, along with the introduction of a migration operator to enhance the optimization capability of the algorithm. Finally, the optimal solution for each type of through-operation route is added to a candidate set, and the globally optimal train plan is selected from this set.

The proposed method is conducive to saving transportation resources, reducing the number of transfers, and improving the efficiency and satisfaction of rail transit as an eco-friendly mode of travel. The research findings provide theoretical support and valuable references for the integration of multiple networks, the optimization of transportation resource allocation, and the construction of a green and environmentally friendly travel transportation system. However, this study is based on a series of idealized assumptions, such as fixed passenger OD demand and the allowance of only one transfer during passenger travel. And we use the daily average passenger flow as the basis for making the train plan. There are indeed variations in passenger flow during different time slots of the day, which results in varying numbers of trains being scheduled for each time slot. To reduce unnecessary adjustments and feedback, the train plan and timetable can also be integrated and optimized collaboratively. In addition, future research could incorporate dynamic passenger flows within large-scale rail networks to further expand the application scenarios and network scale.