This study investigates the impact of flagship trains on high-speed railway capacity utilization and develops a brand value-oriented optimization framework that balances service quality enhancement with operational efficiency.
A mathematical optimization model based on integer programming is developed, incorporating flagship train constraints into capacity optimization. Case studies compare scenarios with and without flagship train considerations using the Beijing–Shanghai High-Speed Railway data across 20 experimental groups.
Operating flagship trains with hourly departure constraints results in an average decrease of 0.9 trains and an 8.4% reduction in capacity utilization rate. When scheduling 2 flagship trains within a 2-h timeframe, capacity utilization decreases from 86.43% to 83.73%, quantifying the trade-off between brand positioning and operational capacity.
This research provides the first quantitative framework for brand value-oriented railway capacity optimization, establishing clear definitions for flagship trains and mathematical foundations for evaluating service quality versus efficiency trade-offs. The findings offer practical decision support for railway operators balancing competitive positioning with capacity maximization.
1. Introduction
The rapid development of high-speed railway systems worldwide has fundamentally transformed intercity transportation, offering unprecedented travel speeds and service quality. As competition intensifies among different transportation modes and passenger expectations continue to rise, railway operators face the dual challenge of maximizing infrastructure capacity utilization while enhancing service quality and brand value. This challenge is particularly pronounced in densely operated high-speed railway corridors, where the strategic scheduling of premium services can significantly impact both operational efficiency and market competitiveness. The concept of brand value in railway operations has emerged as a critical factor, influencing not only passenger choice behavior but also the fundamental approaches to capacity planning and timetable design. This research addresses the intricate relationship between brand-oriented service design and railway capacity utilization, proposing a novel framework that balances operational efficiency with service quality enhancement in high-speed railway systems.
1.1 Research background
Railway transportation optimization is a comprehensive optimization problem encompassing a series of optimization processes, including passenger flow demand, network planning, train operation scheme optimization, timetable optimization, rolling stock utilization optimization, and crew scheduling optimization. In railway operations, high-speed trains with high travel speeds, fewer stops, and serving intercity routes with large passenger volumes are commonly referred to as “flagship trains” by field practitioners. During the timetable compilation phase, China Railway Corporation currently adopts a phased approach, first scheduling same-line and cross-line flagship trains, followed by other trains. Flagship trains have gradually become premium services in high-speed railway transportation organization, possessing significant brand value. However, there is currently no clear definition for existing flagship trains.
To further quantify the impact of flagship trains on timetable compilation and capacity utilization, this study proposes the concept of “flagship trains” and investigates the issue of carrying capacity utilization from a brand value-oriented perspective. The operation of flagship trains is regarded as key to enhancing brand value. This study characterizes the number of train paths consumed by flagship trains under different transportation organization schemes, and further explores the impact of operating clock-face flagship trains—a service enhancement measure—on the actual capacity “deduction.” This research establishes non-value-oriented carrying capacity utilization as the foundation, and further develops a brand value-oriented high-speed railway carrying capacity utilization model. By comparing capacity optimization results under different train stopping pattern alternatives, case studies are conducted on the Beijing South-Jinan West section and the entire Beijing-Shanghai High-Speed Railway line. The results reveal the average changes in capacity indicators such as train numbers and capacity utilization rates under the brand value-oriented constraint of fixed clock-face departures for flagship trains.
1.2 Literature review
Due to their characteristics of fewer stops, higher average speed, superior service quality, operation on high-speed grade lines, and departure times at regular hourly intervals that are convenient for passengers to remember and recognize, flagship trains have become the preferred travel option for some passengers. With their higher occupancy rates, they should possess significant operational value. However, their scheduling requires sacrificing the number of operating train paths, which also has a notable impact on capacity utilization.
Lu (2019) analyzed the number of non-stop trains deducted (deduction coefficient) on high-speed railway parallel operation diagrams from the perspective of flagship train overtaking characteristics, and based on the analysis results, proposed relevant suggestions for the Beijing-Shanghai high-speed railway timetable planning. Zhang, Fan, Wang, and Guo (2022) considered the periodic train operation structure and utilized the close relationship between section passing capacity and train stops to optimize the number of flagship trains and their travel speeds. Xia, Li, Zhang, and Zhang (2024) constructed an integer linear programming model based on Event Activity Network (EAN) with the objectives of OD service balance and capacity utilization. The model considered basic timetabling constraints such as flagship train section running time and connection time, combined with special constraints including flagship train departure time constraints and grouping constraints, providing reference for high-speed railway flagship train timetable compilation and optimization.
Qu, Shi, and Xu (2022) explored the increase in the number of high-speed railway flagship trains, adopting an hourly balance-centralized scheduling approach to explore improvements in flagship train service quality. When analyzing the factors affecting the passing capacity of corridor-type high-speed railways in existing train timetables, they identified the pattern that flagship trains generally depart at regular hourly intervals to improve service quality. Fan, Shen, Liu, Li, and Liang (2024) analyzed from a systematic perspective, simplifying train path efficiency as the ratio of technical indicators to invested transportation resources to objectively and effectively evaluate the efficiency of various types of train paths. Through DEA projection analysis, they provided targeted suggestions for improving train path efficiency. The results showed that the comprehensive efficiency of trains adopting high-speed grade operating standards was higher than other categories of trains in the entire timetable.
Mei, Lai, Zhang, Pan, and Cui (2024) examined flagship trains' operational characteristics through comprehensive evaluation metrics. Their analysis using the improved TOPSIS method revealed that flagship trains such as G4, G3, and G8 ranked among the top performers on the Beijing-Shanghai HSR, characterized by high speed, high grade, high mileage rates, and fewer stops. Sun, Jing, Li, Tian, and Tian (2024) proposed a method for predicting passenger flow distribution in high-speed railway train riding sections based on an interpretable machine learning framework to explain the impact of passenger service product characteristics on passenger flow distribution prediction. They focused on analyzing the relationships between features such as benchmark trains, ticket prices, travel time, date, day of the week, train number, departure time, and passenger flow. The results showed that date and benchmark trains exhibited a nonlinear positive correlation with passenger flow distribution prediction, while travel time showed a nonlinear negative correlation. Benchmark trains with low travel time, high ticket prices, and departure times at regular hourly intervals had a positive impact on passenger flow distribution prediction, with benchmark trains having a significant influence on both passenger flow and revenue. Wu, Shan, and Zhao (2025) identified that flagship trains exhibit relatively low stop ratios in their optimization model for the Beijing-Shanghai high-speed railway, as these trains primarily serve longer travel distances and high-level stations. Their analysis showed that a small number of flagship trains maintain stop ratios below 0.6, distinguishing them from other train types that make more frequent stops at county-level stations to serve local passenger demand.
In other studies regarding timetables, capacity, and service elements, Jensen, Schmidt, and Nielsen (2020) addressed railway infrastructure capacity determination for strategic long-term planning without requiring fixed timetables. They formulated capacity determination as an optimization problem to find the maximal train set that minimizes average waiting time between trains of the same type, testing feasibility through train sequence sampling rather than fixed timetable generation. Zhong, Zang, Yang, Sheng, and Xu (2024) investigated the impact of timetable elements on railway line capacity, establishing the relationship between consumed capacity and timetable elements as the signed sum along a “critical path.” They demonstrated that changing timetable elements impacts capacity with neighboring trains in a combinatorial manner, analyzing stop plans and extending findings to capacity factors including average speed and heterogeneity. Pouryousef and Lautala (2015) investigated applying European timetable management methodologies to the less structured U.S. rail system through a hybrid simulation approach combining RTC and RailSys tools. They demonstrated that optimizing dwell time constraints can significantly improve level of service parameters while managing the trade-off with capacity utilization, showing the potential benefits of structured timetable management in traditionally unstructured rail networks. Gu, Wei, and Xia (2025) transformed the train timetable scheduling problem into a node routing problem where nodes represent train paths and directed arcs depict sequential connections between paths. They constructed an optimization model maximizing the number of trains under the constraint that total arc weight does not exceed available time-space resources, revealing how train path sequences impact section carrying capacity and proposing optimization strategies for improving capacity utilization of high-speed railways. Xu et al. (2023) addressed railway capacity optimization through quantum computing by developing a space-time network-based binary integer programming model for train timetabling. They demonstrated that quantum simulators, particularly the coherent Ising machine (CIM), can effectively solve capacity-constrained timetabling problems, offering a novel computational approach for maximizing line capacity utilization in complex high-speed rail networks where traditional methods face computational limitations. Gao, Chen, Xu, Liu, and Chen (2025) developed a bi-objective optimization model for high-speed train rescheduling under disruptions that enhances capacity recovery and service quality by minimizing both timetable deviation and impacts on transferring passengers. Their computational intelligence framework achieved significant service improvements by preserving line capacity and service levels during disruptions. D'Ariano, Pacciarelli, and Pranzo (2008) introduced flexible timetables as an alternative to traditional time reserves for improving railway punctuality without reducing line capacity. Their approach plans less rigidity in baseline schedules while empowering real-time management to resolve inter-train conflicts dynamically. Jensen, Landex, Nielsen, Kroon, and Schmidt (2017) developed a strategic planning framework for calculating railway infrastructure occupation and capacity consumption independent of specific timetables. Their model generates and assesses different train sequences to achieve timetable independence, employing stochastic delay simulation to determine capacity consumption.
Existing research has not clearly defined the quantitative relationships between service levels for different types of trains, including flagship trains, cross-line trains, and other mainline trains. Research on flagship train timetable compilation methods is relatively limited overall. Existing studies indicate that operating flagship trains is an effective way to improve service levels, with their establishment being driven by both the necessity and public welfare nature of macro policies on one hand, and the convenience and comfort of passenger travel on the other. However, in existing research, there is limited analysis of the impact of flagship trains on capacity, making it difficult to provide effective analysis for capacity utilization.
2. Definition and classification of flagship trains
2.1 Definition and characteristics of flagship trains
Before discussing flagship train classification, this study first explores the characteristics and connotations of flagship trains. In railway operations, high-speed trains with high travel speeds, fewer stops, and serving high-traffic intercity routes are commonly referred to as flagship trains. Train timetabling involving flagship trains represents a special category with distinct objectives and constraints compared to conventional timetabling methods.
Flagship trains feature fast travel speeds, fewer stops, operation at maximum speed standards, stops mainly at key provincial capitals, and typically whole-hour departures. Taking the Beijing-Shanghai high-speed line in Q4 2022 as an example, five of the 28 trains (G1-G28) depart at non-whole-hour times. Through analyzing current flagship train data, existing research has found it difficult to identify common characteristics. This paper identifies the following key issues:
Consideration of whole-hour departures
While most whole-hour trains on the Beijing-Shanghai line are flagship trains, it remains unclear whether non-whole-hour departure trains can be flagship trains and whether whole-hour departure necessarily defines a flagship train.
Consideration of stops
Flagship trains on the Beijing-Shanghai line stop at 2–5 stations, but the specific threshold for “few stops” among 23 stations remains undefined. Whether trains stopping at 6–7 stations qualify as flagship trains is debatable. While flagship trains primarily serve provincial capitals, some also stop at high-traffic stations like Xuzhou East, making stopping standards difficult to define.
Consideration of travel speed
Whether “flagship trains” should exist among lower-grade trains operating below maximum speed, and whether lower-grade trains with whole-hour departures and few stops qualify as flagship trains requires further discussion.
Given these conceptual uncertainties, this study proposes the concept of flagship trains and examines their impact on capacity utilization.
This study defines flagship trains as a special category of flagship trains that: depart at whole hours, operate at maximum speed grade, stop at no more than one-quarter of total line stations, and serve major prefecture-level cities with high passenger flow. Key characteristics include short travel times and balanced whole-hour departures, which are crucial for optimizing capacity utilization in high-speed railway systems.
Short travel time
All flagship trains operate at 350 km/h with limited stops (≤25% of line stations), enabling higher speeds and reduced stopping time, thereby improving passenger travel efficiency and satisfaction.
Balanced whole-hour departures
Whole-hour departure times provide regularity and predictability, facilitating passenger planning while enhancing reliability and optimizing transportation organization efficiency.
Flagship trains are primarily characterized as flagship trains—high-quality trains with strict timetabling requirements that serve as reference standards for timetable optimization.
2.2 Classification and value types of flagship trains
Based on operating characteristics, flagship trains can be classified as:
Different stopping patterns
Flagship trains can be divided into direct and limited-stop services. Different stopping patterns help evaluate operational performance by comparing travel time, station spacing, and passenger demand satisfaction to select optimal stopping schemes.
Different time periods
Peak and off-peak flagship trains may have different timetables and stopping schemes to accommodate varying passenger demand and traffic flow.
This classification helps planners evaluate different train schemes to improve railway efficiency and service quality.
As reference standards in high-speed railway systems, flagship trains possess significant value:
Brand value
Flagship trains provide high-quality transportation services that meet rapid travel demands, promote economic exchange, and drive regional integration. They attract passenger flow from other modes and serve as the flagship brand of high-speed rail products.
Economic value
Short travel times reduce time costs and promote business/tourism development. Whole-hour departures improve reliability and capacity utilization while attracting more passengers, increasing revenue. Flagship trains enhance economic benefits through improved efficiency and optimized organization.
Structural value
In timetables, flagship trains exhibit hourly pseudo-periodic patterns that provide structural value for overall timetable layout and compilation, significantly impacting capacity optimization.
The value of flagship trains encompasses brand significance, economic benefits, and structural contributions, providing efficient travel options while serving as references for system planning and management to improve timetable efficiency and capacity utilization.
3. Throughput capacity utilization optimization model with flagship train considerations
3.1 Problem description and assumptions
This chapter addresses the throughput capacity utilization problem for high-speed railways considering the operation of flagship trains. Flagship trains, characterized by their fixed departure times at round hours and premium service levels, represent a critical component of brand value in railway operations. The model optimizes the selection and scheduling of trains from a candidate set while ensuring flagship trains operate at their designated hourly departure times.
The following assumptions are made for the capacity utilization model:
Known candidate train set: The candidate train set is predetermined and serves as input to the optimization problem. The set includes trains of different speed grades (300km/h and 350km/h) with various stopping patterns, where flagship trains represent a premium subset with mandatory whole-hour departures and limited stops. Each candidate train is defined by its origin-destination pair, stopping scheme, and service characteristics.
Sufficient rolling stock and crew resources: The model assumes adequate availability of train units and crew, without explicitly optimizing their allocation.
Train-kilometer conversion coefficient as objective: The optimization uses weighted train-kilometer conversion coefficients instead of simple train counts, allowing proper consideration of short-haul services converted to long-haul equivalent capacity.
3.2 Mathematical formulation
As shown in Figure 1, train has the longest running distance. Without considering other factors, the train-kilometer coefficient of train should be 1, while the train-kilometer coefficients of train and train are converted to 0.4 and 0.6 respectively.
The time-space diagram shows the movement of three trains across three stations. The vertical axis is labeled “Stations,” showing “Station i” at the bottom, “Station i plus 1” in the middle, and “Station i plus 2” at the top. The distance between “Station i” and “Station i plus 1” is marked “6 kilometers.” The distance between “Station i plus 1” and “Station i plus 2” is marked “4 kilometers.” The horizontal axis is labeled “Time.” A green rectangular grid forms the plot background, with vertical lines and a horizontal green line dividing the area. Three trains labeled “Train a,” “Train b,” and “Train c” are shown as diagonal lines moving upward and to the right. “Train a” travels from “Station i plus 1” to “Station i plus 2,” covering 4 kilometers. “Train b” passes between “Station i” and “Station i plus 2,” covering 10 kilometers. “Train c” moves from “Station i” to “Station i plus 1,” covering 6 kilometers.Illustration of the train-kilometer conversion coefficient. Source: Author’s own work
The time-space diagram shows the movement of three trains across three stations. The vertical axis is labeled “Stations,” showing “Station i” at the bottom, “Station i plus 1” in the middle, and “Station i plus 2” at the top. The distance between “Station i” and “Station i plus 1” is marked “6 kilometers.” The distance between “Station i plus 1” and “Station i plus 2” is marked “4 kilometers.” The horizontal axis is labeled “Time.” A green rectangular grid forms the plot background, with vertical lines and a horizontal green line dividing the area. Three trains labeled “Train a,” “Train b,” and “Train c” are shown as diagonal lines moving upward and to the right. “Train a” travels from “Station i plus 1” to “Station i plus 2,” covering 4 kilometers. “Train b” passes between “Station i” and “Station i plus 2,” covering 10 kilometers. “Train c” moves from “Station i” to “Station i plus 1,” covering 6 kilometers.Illustration of the train-kilometer conversion coefficient. Source: Author’s own work
The sets, indices, parameters, and decision variables used in the model are shown in Table 1 and Table 2.
Set, indexes, parameters of the model
| Symbols | Definitions |
|---|---|
| Set of stations on the line | |
| Set of trains | |
| Set of low-speed grade trains (300km/h) | |
| Set of high-speed grade trains (350km/h) | |
| Set of flagship trains, | |
| Station index, | |
| Total number of trains running on the line | |
| Train index, | |
| Set of stations commonly passed by trains and , | |
| Origin station of train | |
| Terminal station of train | |
| Stopping status of train at station , 0 means passing without stopping, 1 means stopping | |
| Minimum number of stops required at station within the specified time | |
| Pure running time of train in section | |
| Departure time of train , generally taken as whole hours, i.e. 420min, 480min, etc., | |
| Starting additional time of train at station | |
| Stopping additional time of train at station | |
| Minimum dwell time of trains at stations | |
| Maximum dwell time of trains at stations | |
| Upper limit of operation time window at station | |
| Lower limit of operation time window at station | |
| Minimum train departure interval | |
| Maximum train passing interval | |
| Minimum passing-to-departure interval | |
| Minimum arrival-to-arrival interval | |
| Minimum arrival-to-passing interval | |
| Minimum passing-to-arrival interval | |
| Minimum passing-to-passing interval | |
| A large positive number | |
| Train-kilometer conversion coefficient for train | |
| Set of whole hour times |
Variables of the model
| Symbols | Definitions |
|---|---|
| Binary variable, indicating whether train is operated. If operated, ; otherwise | |
| Integer variable, the arrival time of train at station | |
| Integer variable, the departure time of train at station | |
| Binary variable, indicating the running order of trains and in section . If , train is the preceding train of train ; if , train is the following train of train |
The capacity utilization optimization model considering flagship trains proposed in this section is an integer programming model based on transportation resource request conflicts. The model is constructed using the Big M method, which has certain similarities with the capacity calculation model based on train diagram optimization graphical methods. The optimization objective is adjusted from the number of trains to maximizing the weighted train-kilometer conversion coefficient, that is, establishing the optimization objective function by maximizing the sum of weighted train-kilometer conversion coefficients of selected trains to be operated.
The model includes the following constraints:
Station Service Frequency Constraint
To ensure passengers' basic travel needs are met, it is necessary to limit the total number of trains stops at each station. The minimum service frequency at each station should be set based on passenger demand, using either passenger boarding/alighting numbers or passenger travel demand as approximation methods.
Train Dwell Time Constraint
To ensure smooth passenger boarding/alighting and necessary technical operations, minimum dwell times must be established for trains at each station. To minimize negative impacts on passenger experience, train dwell times should also not be excessive. Therefore, both minimum and maximum time limits are necessary. Since train 's operation is uncertain, both sides of the equation are multiplied by .
Due to the binary decision variable for train operation, the constraints are linearized using the Big-M method:
Inter-station Running Time Constraint
The inter-station running time includes the travel time between adjacent stations. For train in section , the running time equals the pure running time plus the starting additional time at station and the stopping additional time at station .
Similarly, linearized using the Big-M method:
Train Departure Interval Constraint
To ensure safe operating distances between trains, adjacent trains must maintain specified minimum departure intervals. These intervals are affected by the stopping strategies of adjacent trains at stations. Four types of intervals must be considered: minimum stop-to-stop departure interval (preceding train stops, following train stops); minimum stop-to-pass departure interval (preceding train stops, following train passes); minimum pass-to-stop departure interval (preceding train passes, following train stops); minimum pass-to-pass departure interval (both trains pass). The specific interval type depends on both trains' stopping decisions at the station and their sequence order in the section.
When both trains and operate and stop at station : if (train precedes in section ), equation (13) applies; if (train follows ), equation (14) applies.
When train stops at station while passes:
When both trains pass station :
Train Arrival Interval Constraint
Similar to departure intervals, adjacent trains must maintain specified arrival intervals at stations. Four types of arrival intervals are considered: , , , and .
Time Window Constraint
To ensure safe train operations on high-speed railways, time window constraints such as maintenance windows must be established. This study uses parallelogram (diamond) shaped time windows, where each train's arrival and departure times at each station must fall within the time window.
Section Running Order Constraint
This constraint defines the running order of any two trains in a specific section. On the same track section, one train must be the preceding train while the other follows.
Flagship Train Hourly Departure Constraint
Flagship trains provide new approaches for brand value-oriented capacity optimization. All flagship trains belong to high-grade train categories and must depart at the top of the hour.
Decision Variable Constraints
4. Capacity optimization case study considering flagship trains
4.1 Case parameter settings
The Beijing-Shanghai High-Speed Railway is one of the busiest high-speed rail lines in China, comprising 23 stations in total. This case study focuses on the southbound Beijing-Jinan section (from Beijing South Station to Jinan West Station), which includes 6 stations. The time range selected is a two-hour daytime period from 7:30 to 9:30.
The minimum dwell time for trains at intermediate stations is set at 2 minutes, with a maximum of 10 minutes. The departure headway between trains at each station is 5 minutes, the arrival headway is 4 minutes, the passing headway is 3 minutes, the passing-departure headway is 3 minutes, the departure-passing headway is 5 minutes, the arrival-passing headway is 4 minutes, and the passing-arrival headway is 3 minutes.
Two speed categories of trains operate on this line: high-speed trains (350 km/h) and low-speed trains (300 km/h). The running times of trains with different speed levels in each section are shown in Table 3.
Jingji section line and parameter information
| Station | Mileage (km) | Section pure running time (min) | Additional time for train starting/topping (min) | Min service frequency | ||
|---|---|---|---|---|---|---|
| 350km/h | 300km/h | Starting | Stopping | |||
| BJN | 0 | – | – | 2 | 3 | 10 |
| LF | 59 | 15 | 16 | 2 | 3 | 1 |
| TJN | 131 | 11 | 13 | 2 | 3 | 5 |
| CZX | 219 | 16 | 18 | 2 | 3 | 3 |
| DZD | 327 | 18 | 20 | 2 | 3 | 5 |
| JNX | 419 | 17 | 19 | 2 | 3 | 10 |
The models proposed in this paper are implemented using Python 3.11 with the Branch-and-Cut algorithm. The computing environment consists of a personal computer with an Intel Core i7-11700 CPU @ 2.50 GHz and 32 GB of RAM. The construction of candidate train sets follows the approach of Zhang et al. (2022), whose research was based on a quarterly timetable from 2022. After providing a basic set of candidate trains, the candidate stopping patterns were continuously supplemented, using a 2-h time range as the research example. This case study establishes 20 different candidate train sets, each maintaining a fixed number of 22 candidate trains without further increases. Among these, 3 are high-speed train candidates and 19 are low-speed train candidates. The main difference between each candidate train set lies in the low-speed train candidate sets, which are configured based on different schemes derived from actual operational timetables. The specific generation rules are shown in Figure 2, with low-speed train stopping patterns detailed in Appendix A.
The train stopping scheme diagram visually represents the stopping patterns of trains at various stations along a route, divided into two main sections. The left section is labeled “Low-grade actual train stopping scheme.” It lists “G 101,” “G 103,” “G 105,” “G 107,” and “G 901” with horizontal lines representing their routes. Each line includes several circular stops. The middle of each line has a small open circle, and each route extends horizontally from left to right. The right section is labeled “Form 20 sets of alternative experimental schemes.” It shows two sections. Each section contains train routes labeled with stations “B J N,” “L F,” “T J N,” “C Z X,” “D Z D,” and “J N X” along the top. Horizontal lines with circular nodes represent different train stopping patterns. Inside both sections, these schemes are divided into two: “Low grade” and “High grade.” High-grade points are placed inside the red box, and low-grade points are placed inside the green box. The top section is labeled “Train stopping alternative set Scheme 1.” High grade at top, arrows pointing from left to right: 3, B J N, J N X. Low grade, starting from top to bottom, and arrows pointing from left to right: 1, B J N, L F, T J N, D Z D, J N X. 1, B J N, L F, T J N, J N X. 1, B J N, L F, C Z X, J N X. 1, B J N, T J N, D Z D, J N X. 3, B J N, C Z X, D Z D, J N X. 3, B J N, C Z X, J N X. 2, B J N, D Z D, J N X. 1, B J N, L F, T J N, C Z X, D Z D, J N X. 2, B J N, D Z D. 2, B J N, T J N, D Z D. 1, B J N, C Z X, D Z D. 1, T J N, C Z X, D Z D, J N X. The bottom section is labeled “Train stopping alternative set Scheme 20.” High grade at top, arrows pointing from left to right: 3, B J N, J N X. Low grade, starting from top to bottom, and arrows pointing from left to right: 2, B J N, L F, C Z X, J N X. 2, B J N, T J N, C Z X, J N X. 2, B J N, T J N, D Z D, J N X. 2, B J N, C Z X, D Z D, J N X. 2, B J N, C Z X, J N X. 2, B J N, D Z D, J N X. 2, B J N, J N X. 1, B J N, C Z X, D Z D. 1, B J N, D Z D. 1, T J N, C Z X, D Z D, J N X. A blue arrow is labeled “Select 19 trains from low-grade actual train,” and an orange arrow is labeled “Select another 19 trains from low-grade actual train.” Four blue arrows are pointing from the left section to the right section's top, labeled “Train stopping alternative set Scheme 1.” Four orange arrows are pointing from the left section to the right section's bottom, labeled “Train stopping alternative set Scheme 20.”Candidate train set generation rules. Source: Author’s own work
The train stopping scheme diagram visually represents the stopping patterns of trains at various stations along a route, divided into two main sections. The left section is labeled “Low-grade actual train stopping scheme.” It lists “G 101,” “G 103,” “G 105,” “G 107,” and “G 901” with horizontal lines representing their routes. Each line includes several circular stops. The middle of each line has a small open circle, and each route extends horizontally from left to right. The right section is labeled “Form 20 sets of alternative experimental schemes.” It shows two sections. Each section contains train routes labeled with stations “B J N,” “L F,” “T J N,” “C Z X,” “D Z D,” and “J N X” along the top. Horizontal lines with circular nodes represent different train stopping patterns. Inside both sections, these schemes are divided into two: “Low grade” and “High grade.” High-grade points are placed inside the red box, and low-grade points are placed inside the green box. The top section is labeled “Train stopping alternative set Scheme 1.” High grade at top, arrows pointing from left to right: 3, B J N, J N X. Low grade, starting from top to bottom, and arrows pointing from left to right: 1, B J N, L F, T J N, D Z D, J N X. 1, B J N, L F, T J N, J N X. 1, B J N, L F, C Z X, J N X. 1, B J N, T J N, D Z D, J N X. 3, B J N, C Z X, D Z D, J N X. 3, B J N, C Z X, J N X. 2, B J N, D Z D, J N X. 1, B J N, L F, T J N, C Z X, D Z D, J N X. 2, B J N, D Z D. 2, B J N, T J N, D Z D. 1, B J N, C Z X, D Z D. 1, T J N, C Z X, D Z D, J N X. The bottom section is labeled “Train stopping alternative set Scheme 20.” High grade at top, arrows pointing from left to right: 3, B J N, J N X. Low grade, starting from top to bottom, and arrows pointing from left to right: 2, B J N, L F, C Z X, J N X. 2, B J N, T J N, C Z X, J N X. 2, B J N, T J N, D Z D, J N X. 2, B J N, C Z X, D Z D, J N X. 2, B J N, C Z X, J N X. 2, B J N, D Z D, J N X. 2, B J N, J N X. 1, B J N, C Z X, D Z D. 1, B J N, D Z D. 1, T J N, C Z X, D Z D, J N X. A blue arrow is labeled “Select 19 trains from low-grade actual train,” and an orange arrow is labeled “Select another 19 trains from low-grade actual train.” Four blue arrows are pointing from the left section to the right section's top, labeled “Train stopping alternative set Scheme 1.” Four orange arrows are pointing from the left section to the right section's bottom, labeled “Train stopping alternative set Scheme 20.”Candidate train set generation rules. Source: Author’s own work
4.2 Numerical experiments
This study examines 3 high-speed trains, with 2 designated as flagship trains scheduled to depart on the hour. A capacity optimization case considering flagship trains (FO) is designed, with 20 experimental groups named FO1, FO2, ..., FO20, to further analyze the capacity results.
Under the condition of scheduling 2 flagship trains, the optimization results are shown in Figure 3. The figure illustrates that two high-speed trains operate as flagship trains departing at 8:00 and 9:00 sharp. In subfigures 3(1), (3), and (6), the third high-speed train is not selected for operation. The specific solution results for case FO are presented in Table 4.
Six graphs are arranged in a 3 by 2 grid. The graphs labeled from top left “(1) Result of case F O 1,” “(2) Result of case F O 2,” “(3) Result of case F O 3,” “(4) Result of case F O 4,” “(5) Result of case F O 5,” “(6) Result of case F O 6,” Each row contains two graphs. In all graphs, the vertical axis is labeled with categories from bottom to top as follows: “J N X,” “D Z D,” “C Z N,” “L F N,” “L F,” and “B J N,” and the horizontal axis ranges from “7:00” to “12:00,” with increments of 1:00 hour. In all cases, multiple pink and blue lines slope diagonally downward from the upper left to the lower center and lower right sides. The blue lines are inside and overlapping the pink lines. In graphs (1), (3), and (6), 2 blue lines overlap the pink lines. In graphs (2), (4), and (5), 3 blue lines overlap the pink lines. Each graph begins with lines at the top in the range of “7:30” to “9:30” and ends with lines toward the bottom in the range of “9:10” to “11:10” across all six plots. Note: All numerical data values are approximated.Timetable example of the Case FO. Source: Author’s own work
Six graphs are arranged in a 3 by 2 grid. The graphs labeled from top left “(1) Result of case F O 1,” “(2) Result of case F O 2,” “(3) Result of case F O 3,” “(4) Result of case F O 4,” “(5) Result of case F O 5,” “(6) Result of case F O 6,” Each row contains two graphs. In all graphs, the vertical axis is labeled with categories from bottom to top as follows: “J N X,” “D Z D,” “C Z N,” “L F N,” “L F,” and “B J N,” and the horizontal axis ranges from “7:00” to “12:00,” with increments of 1:00 hour. In all cases, multiple pink and blue lines slope diagonally downward from the upper left to the lower center and lower right sides. The blue lines are inside and overlapping the pink lines. In graphs (1), (3), and (6), 2 blue lines overlap the pink lines. In graphs (2), (4), and (5), 3 blue lines overlap the pink lines. Each graph begins with lines at the top in the range of “7:30” to “9:30” and ends with lines toward the bottom in the range of “9:10” to “11:10” across all six plots. Note: All numerical data values are approximated.Timetable example of the Case FO. Source: Author’s own work
Analysis of the results of the case FO
| Case | Number of trains | Number of high-priority trains | Number of low-priority trains | Number of constraints | Number of variables | Objective function | Solution time (s) |
|---|---|---|---|---|---|---|---|
| FO1 | 20/22 | 2/3 | 18/19 | 10,550 | 2,582 | 18.71 | 212.24 |
| FO2 | 19/22 | 3/3 | 16/19 | 9,641 | 2,572 | 17.94 | 246.67 |
| FO3 | 20/22 | 2/3 | 18/19 | 10,734 | 2,584 | 19.17 | 218.19 |
| FO4 | 18/22 | 3/3 | 15/19 | 10,763 | 2,584 | 17.77 | 193.42 |
| FO5 | 20/22 | 3/3 | 17/19 | 10,367 | 2,580 | 18.78 | 186.38 |
| FO6 | 19/22 | 2/3 | 17/19 | 11,529 | 2,592 | 18.7 | 238.88 |
| FO7 | 20/22 | 3/3 | 17/19 | 9,641 | 2,572 | 18.71 | 208.99 |
| FO8 | 19/22 | 3/3 | 16/19 | 9,997 | 2,576 | 18.24 | 204.35 |
| FO9 | 19/22 | 3/3 | 16/19 | 10,734 | 2,584 | 18.47 | 262.39 |
| FO10 | 20/22 | 2/3 | 18/19 | 10,546 | 2,582 | 18.94 | 250.49 |
| FO11 | 19/22 | 3/3 | 16/19 | 10,589 | 2,582 | 18.24 | 226.73 |
| FO12 | 19/22 | 3/3 | 16/19 | 11,126 | 2,588 | 18.47 | 250.7 |
| FO13 | 19/22 | 3/3 | 16/19 | 10,176 | 2,578 | 18.4 | 256.71 |
| FO14 | 19/22 | 2/3 | 17/19 | 11,137 | 2,588 | 18.4 | 215.31 |
| FO15 | 18/22 | 2/3 | 16/19 | 11,348 | 2,590 | 18 | 252.84 |
| FO16 | 19/22 | 3/3 | 16/19 | 10,546 | 2,582 | 18.47 | 248.43 |
| FO17 | 18/22 | 3/3 | 15/19 | 10,589 | 2,582 | 17.47 | 189.02 |
| FO18 | 19/22 | 3/3 | 16/19 | 10,738 | 2,584 | 18.47 | 242.74 |
| FO19 | 19/22 | 3/3 | 16/19 | 10,561 | 2,582 | 18.7 | 279.33 |
| FO20 | 19/22 | 3/3 | 16/19 | 10,176 | 2,578 | 18.4 | 254.33 |
To compare the optimization results with those considering flagship trains, a capacity utilization optimization model without flagship train considerations (NFO) is designed using the same candidate train sets. Twenty experimental groups are conducted, named NFO1, NFO2, ..., NFO20, to further analyze the capacity results. The optimization outcomes are shown in Figure 4.
Six graphs are arranged in a 3 by 2 grid. The graphs are labeled, from the top left, “(1) Result of case N F O 1,” “(2) Result of case N F O 1,” “(3) Result of case N F O 1,” “(4) Result of case N F O 4,” “(5) Result of case N F O 5,” “(6) Result of case N F O 6” two graphs in each row. In all graphs, the vertical axis is labeled with categories from bottom to top as follows: “J N X,” “D Z D,” “C Z N,” “L F N,” “L F,” and “B I N,” and the horizontal axis plots time from “7:00” to “12:00,” with increments of 1:00 hour. Across all graphs, multiple pink and blue diagonal lines slope downward from left to right. The blue lines are positioned overlapping the pink lines. In graphs (1), (5), and (6), 2 blue lines overlap the pink lines. In graphs (2) and (3), 3 blue lines overlap the pink lines. In graph (4), 1 blue line overlaps the pink lines. Each graph begins with lines at the top in the range of “7:30” to “9:30” and ends with lines toward the bottom in the range of “9:10” to “11:10” across all six plots. Note: All numerical data values are approximated.Timetable example of the Case NFO. Source: Author’s own work
Six graphs are arranged in a 3 by 2 grid. The graphs are labeled, from the top left, “(1) Result of case N F O 1,” “(2) Result of case N F O 1,” “(3) Result of case N F O 1,” “(4) Result of case N F O 4,” “(5) Result of case N F O 5,” “(6) Result of case N F O 6” two graphs in each row. In all graphs, the vertical axis is labeled with categories from bottom to top as follows: “J N X,” “D Z D,” “C Z N,” “L F N,” “L F,” and “B I N,” and the horizontal axis plots time from “7:00” to “12:00,” with increments of 1:00 hour. Across all graphs, multiple pink and blue diagonal lines slope downward from left to right. The blue lines are positioned overlapping the pink lines. In graphs (1), (5), and (6), 2 blue lines overlap the pink lines. In graphs (2) and (3), 3 blue lines overlap the pink lines. In graph (4), 1 blue line overlaps the pink lines. Each graph begins with lines at the top in the range of “7:30” to “9:30” and ends with lines toward the bottom in the range of “9:10” to “11:10” across all six plots. Note: All numerical data values are approximated.Timetable example of the Case NFO. Source: Author’s own work
In the capacity utilization optimization case NFO without flagship train considerations, trains are scheduled within the parallelogram time window (2-h pink area) with the objective of maximizing the weighted train-kilometer conversion coefficient. Analysis reveals that in Figure 4(b) and (c), high-speed trains appear in “paired” tracking patterns, which is a common approach demonstrating reasonable capacity utilization. In Figure 4(a), (d), (e), and (f), high-speed and low-speed trains make reasonable use of spatial-temporal resources through stopping patterns and overtaking arrangements. These cases typically feature more low-speed trains with high-speed trains appearing in a dispersed manner. Detailed calculation results are shown in Table 5.
Analysis of the results of the case NFO
| Case | Number of trains | Number of high-priority trains | Number of low-priority trains | Number of constraints | Number of variables | Objective function | Solution time (s) |
|---|---|---|---|---|---|---|---|
| NFO1 | 20/22 | 2/3 | 18/19 | 10,546 | 2,582 | 18.94 | 232.43 |
| NFO2 | 20/22 | 2/3 | 18/19 | 9,637 | 2,572 | 18.41 | 342.7 |
| NFO3 | 20/22 | 2/3 | 18/19 | 10,730 | 2,584 | 19.17 | 225.25 |
| NFO4 | 20/22 | 2/3 | 18/19 | 10,759 | 2,584 | 19.01 | 201.2 |
| NFO5 | 20/22 | 3/3 | 17/19 | 10,363 | 2,580 | 19.01 | 266.67 |
| NFO6 | 19/22 | 1/3 | 18/19 | 11,525 | 2,592 | 18.7 | 183.19 |
| NFO7 | 22/22 | 3/3 | 19/19 | 9,637 | 2,572 | 19.88 | 89.52 |
| NFO8 | 20/22 | 2/3 | 18/19 | 9,993 | 2,576 | 18.71 | 216.92 |
| NFO9 | 20/22 | 2/3 | 18/19 | 10,730 | 2,584 | 19.17 | 211.38 |
| NFO10 | 20/22 | 2/3 | 18/19 | 10,542 | 2,582 | 19.01 | 223.35 |
| NFO11 | 20/22 | 3/3 | 17/19 | 10,585 | 2,582 | 19.01 | 240.16 |
| NFO12 | 19/22 | 2/3 | 17/19 | 11,122 | 2,588 | 18.7 | 187.4 |
| NFO13 | 20/22 | 2/3 | 18/19 | 10,172 | 2,578 | 18.64 | 252.18 |
| NFO14 | 19/22 | 0/3 | 19/19 | 11,133 | 2,588 | 18.4 | 276.3 |
| NFO15 | 20/22 | 1/3 | 19/19 | 11,344 | 2,590 | 19.31 | 224.44 |
| NFO16 | 20/22 | 3/3 | 17/19 | 10,542 | 2,582 | 19.17 | 245.33 |
| NFO17 | 20/22 | 1/3 | 19/19 | 10,585 | 2,582 | 18.55 | 257.22 |
| NFO18 | 20/22 | 2/3 | 18/19 | 10,734 | 2,584 | 19.17 | 228.65 |
| NFO19 | 20/22 | 3/3 | 17/19 | 10,557 | 2,582 | 19.4 | 261.31 |
| NFO20 | 21/22 | 3/3 | 18/19 | 10,172 | 2,578 | 19.94 | 219.08 |
In the headers of Tables 4 and 5, the numbers to the left of the slash in the fields for train count, high-speed train count, and low-speed train count represent the number of trains selected by the model optimization, while the numbers to the right represent the provided candidate train count. Comparative analysis between case FO (considering flagship trains) and case NFO (without flagship train considerations) reveals that cases with flagship train operating constraints select fewer trains on average. The following sections provide a comparative analysis of the two cases based on train count and capacity utilization rate indicators.
4.3 Results discussion and analysis
Changes in Train Numbers
As shown in Figure 5, the average number of trains in case FO is lower than that in case NFO. The average number of trains in case NFO is 20, while case FO has an average of 19.1 trains, representing a decrease of 0.9 trains on average. In experiments 1, 3, 5, 6, 10, 12, and 14, the number of trains remains equal, indicating that under identical stopping pattern alternatives, both models select the same number of trains. Research analysis reveals that under a given set of alternative train options, operating flagship trains may reduce the total number of trains. In the 20 experimental groups, under the given stopping pattern alternatives and other case parameters, when two flagship trains with hourly departures are operated within a 2-h period, the number of trains decreases by an average of 0.9.
The vertical axis is labeled “Train Number,” ranging from 15 to 23 in increments of 2. The horizontal axis is labeled “Cases,” ranging from 1 to 20 in increments of 1. The “Case F O” curve, a solid line with circular markers, starts at (1, 20), shows a steady pattern until (5, 20), falls slightly to (6, 19), rises sharply to a peak at (7, 22), then drops again to (8, 20). It remains nearly stable with minor fluctuations between (11, 20) and (15, 20), maintains a steady trend until (19, 20), and ends with a small rise near (20, 21). The “Case N F O” curve, a solid line in a different color with circular markers, begins at (1, 20), falls gradually to (2, 19), rises again around (3, 20), and then shows several minor rises and falls between (4, 18) and (18, 19) before ending near (20, 19). Note: All numerical data values are approximated.Train Number Change of the Case FO and NFO. Source: Author’s own work
The vertical axis is labeled “Train Number,” ranging from 15 to 23 in increments of 2. The horizontal axis is labeled “Cases,” ranging from 1 to 20 in increments of 1. The “Case F O” curve, a solid line with circular markers, starts at (1, 20), shows a steady pattern until (5, 20), falls slightly to (6, 19), rises sharply to a peak at (7, 22), then drops again to (8, 20). It remains nearly stable with minor fluctuations between (11, 20) and (15, 20), maintains a steady trend until (19, 20), and ends with a small rise near (20, 21). The “Case N F O” curve, a solid line in a different color with circular markers, begins at (1, 20), falls gradually to (2, 19), rises again around (3, 20), and then shows several minor rises and falls between (4, 18) and (18, 19) before ending near (20, 19). Note: All numerical data values are approximated.Train Number Change of the Case FO and NFO. Source: Author’s own work
Changes in Capacity Utilization Rate
The capacity utilization rate here is calculated as the current weighted train-kilometer conversion coefficient divided by the total number of alternative stopping trains, which equals the model's optimization objective value divided by the upper limit of alternative stopping trains. As shown in Figure 6, the capacity utilization rate of the optimization model considering flagship trains is lower on average than that of the capacity utilization model without value distinction. The average capacity utilization rate for case NFO is 86.43%, while case FO achieves 83.73%, showing an average decrease of 8.4% in capacity utilization rate when flagship trains are considered. In experimental cases FO3, FO6, and FO14, the capacity utilization rates remain identical, indicating that under these experimental parameters, both models produce consistent train operation optimization schemes. The capacity utilization rate calculated using train-kilometer conversion coefficients can reflect capacity changes in long and short routes that cannot be distinguished by train numbers alone. Research analysis indicates that under a given set of alternative trains, operating flagship trains as a brand value behavior may reduce capacity utilization rate by 8.4%.
The vertical axis is labeled “Train Number,” ranging from 78 percent to 92 percent in increments of 2 percent. The horizontal axis is labeled “Cases,” ranging from 1 to 20 in increments of 1. The “Case F O” curve, a solid line with circular markers, starts at (1, 86), falls slightly to (2, 83.941), rises to (3, 87.113), and dips marginally to (6, 85.013), then rises sharply to a peak at (7, 90.356). After a short fall to (8, 85.241), it fluctuates between (9, 87.102) and (15, 88.572), dips again at (17, 84.304), and climbs gradually to end near (20, 91.032). The “Case N F O” curve, also a solid line in a different color with circular markers, begins at (1, 85.012), declines to (2, 81.876), and rises again near (3, 87.274). It falls steadily to (4, 80.745), recovers to (5, 85.112), and fluctuates slightly through (6, 84.926) and (7, 84.702). The curve dips to (8, 83.112), rises again to (9, 86.045), and continues alternating between minor peaks and troughs until it reaches (17, 79.043), the lowest point. It then rises modestly and ends at (20, 83.646). Note: All numerical data values are approximated.Capacity utilization of the case FO and NFO. Source: Author’s own work
The vertical axis is labeled “Train Number,” ranging from 78 percent to 92 percent in increments of 2 percent. The horizontal axis is labeled “Cases,” ranging from 1 to 20 in increments of 1. The “Case F O” curve, a solid line with circular markers, starts at (1, 86), falls slightly to (2, 83.941), rises to (3, 87.113), and dips marginally to (6, 85.013), then rises sharply to a peak at (7, 90.356). After a short fall to (8, 85.241), it fluctuates between (9, 87.102) and (15, 88.572), dips again at (17, 84.304), and climbs gradually to end near (20, 91.032). The “Case N F O” curve, also a solid line in a different color with circular markers, begins at (1, 85.012), declines to (2, 81.876), and rises again near (3, 87.274). It falls steadily to (4, 80.745), recovers to (5, 85.112), and fluctuates slightly through (6, 84.926) and (7, 84.702). The curve dips to (8, 83.112), rises again to (9, 86.045), and continues alternating between minor peaks and troughs until it reaches (17, 79.043), the lowest point. It then rises modestly and ends at (20, 83.646). Note: All numerical data values are approximated.Capacity utilization of the case FO and NFO. Source: Author’s own work
5. Conclusions
This paper focuses on solving the capacity utilization optimization problem under the condition of flagship train hourly departure scheduling, supporting research on capacity utilization with brand value considerations. The study identifies brand value with the operation of “flagship trains” as high-value popular services, and based on the operational characteristics of flagship trains, treats the scheduling of fast hourly trains as a means of brand value enhancement. A brand value-oriented capacity utilization optimization model for flagship train operation is constructed and compared with high-speed rail capacity utilization models that do not consider flagship trains as the benchmark for value-oriented capacity utilization optimization results. Analysis of cases NFO and FO demonstrates that under given alternative train sets, operating 2 flagship trains within a 2-h timeframe results in an average capacity utilization rate decrease of 8.4%. This finding provides decision support for brand value-oriented high-speed rail capacity utilization research.
The supplementary material for this article can be found online.
