Optimization of Line Planning by Integrating Ticket Pricing and Seat Allocation Decisions for High-Speed Railway

Abstract

In the transportation organization optimization of high-speed railway (HSR), optimizations such as line planning, ticket pricing, and seat allocation are generally studied separately. However, in reality, when passengers choose trains, they need to consider multiple factors such as train routes, stop plans, seat prices, seat availability, and departure times. Therefore, there is an urgent need for an integrated optimization method to simultaneously make decisions regarding these multiple factors. This study constructs a nonlinear optimization model of line planning integrating differentiated pricing and seat allocation decisions for HSR under elastic demand. To efficiently solve the model, an improved heuristic algorithm based on the simulated annealing framework combined with a linear passenger flow allocation method is proposed. Finally, case analysis proves that the improved algorithm can effectively solve the model under the input conditions of an actual Y-shaped HSR network composed of 13 stations, with a potential for a 106.54% improvement from the initial solution to the final solution. The uniqueness of our study lies in the joint optimization of three critical HSR operations, which has not been comprehensively explored in prior studies and is of great significance for improving the level of HSR train operations and passenger services.

1. Introduction

High-speed railway (HSR) is an important transportation mode for passengers in today’s world, because of its high speed, high capacity, and high versatility [1]. With rapid urbanization and increasing mobility demands, the optimization of HSR operations has become a pressing issue for both researchers and practitioners. Efficient HSR operations not only enhance passenger experience but also contribute to the economic sustainability of railway enterprises.

If we regard an HSR train as a product, the line planning of HSR determines the train products and their attributes sold by railway companies, while ticket pricing decisions dictate the price levels of operating trains, and seat allocating decisions determine the seat utilization of operating trains. Train ticket prices and seat allocations define the marketing strategy for train products. Together, the line plan, ticket prices, and seat allocations impact passenger travel costs, service levels, and service capabilities, directly determining the demand for passenger travel and their choice of travel, thereby affecting the operational revenue of railway companies. Despite their interdependencies, these aspects are often studied separately, leading to inefficiencies in HSR operations. Given the increasing competition from other transportation modes, the integration of line planning, ticket pricing, and seat allocating has become more urgent than ever. A well-integrated approach can enhance service quality, optimize resource utilization, and improve overall profitability. Addressing this challenge, this study aims to provide a comprehensive optimization framework that captures the dynamic interplay among these three critical factors.

The optimization of line planning, ticket pricing, and seat allocation have long been focal points of study within the realm of railway transportation research. Currently, the optimization of line planning, ticket pricing, and seat allocation are often optimized in a step-by-step method. These topics have seen extensive and comprehensive research over the years. In a study on the optimization of line planning, Anthony [2] provided an early framework, and several studies [3,4,5] have reviewed the models and solution methods. From an objective perspective, line planning problems can be categorized into operator-oriented [6,7,8], passenger-oriented [9,10,11], and combined (both operator-oriented and passenger-oriented) [12,13,14,15]. In the literature on the optimization of ticket pricing, differentiated pricing is based on the varying sensitivities of different passengers compared to the attributes of different train products within the same O-D. Li and Wong [16] are two of the first scholars to apply differentiated pricing to the railway system, and differentiated pricing has since been gradually applied to HSR [17,18,19]. Seat allocation is a method of setting the number of tickets between train O-D pairs based on passenger demand, under certain train service capacity conditions, aimed at enhancing train load factors and operational revenue. Research on the optimization of seat allocating generally falls into two categories: allocation for a single train [20] and coordinated allocation across multiple trains [21,22].

Additionally, scholars have conducted certain studies on the integrated optimization of these three aspects. The interrelated issues of line plans and ticket prices significantly influence passenger travel choices. To more accurately describe these choices, a few studies have combined these two aspects for integrated optimization. Wang [23] developed a bi-level programming model for line planning and differentiated pricing based on user equilibrium, which was solved using a combination of an enhanced particle swarm optimization and the Frank–Wolfe algorithm. Gao [24] based his study on a price response function to explore the integrated optimization of line planning and ticket pricing, implementing section-based pricing and solving the established model with an improved genetic algorithm. Zhou et al. [22] achieved the integrated optimization of HSR line planning and differentiated pricing based on time-dependent elastic demand. Their model comprehensively considers the train route, train frequency, train stops, train departure and arrival times, and differentiated pricing between O-D pairs. The optimization process effectively integrates train operation choices with passenger travel choices, and an efficient solution algorithm was designed based on simulated annealing and linear passenger flow allocation to solve the integrated optimization model.

The level of ticket pricing influences passenger demand, and seat allocating is based on this demand, making ticket pricing and seat allocation interrelated issues. Recent studies have explored their integrated optimization. Wu et al. [25] combined the seat allocation decision with the HSR dynamic pricing for optimization and used a two-stage algorithm to solve it. Hu et al. [17] explored an integrated optimization method for ticket pricing and seat allocation that incorporated differentiated pricing and multi-stage pricing strategies, proposing a quasi-Newton algorithm framework capable of solving large-scale integrated optimization problems. Deng et al. [26] researched a bi-level programming integrated optimization method for ticket pricing and seat allocation, designing a divide-and-conquer optimization approach to solve the model. Meng et al. [27] proposed a chance constrained stochastic nonlinear programming method to jointly optimize dynamic pricing and ticket allocation and designed a combined algorithm of particle swarm optimization and mixed-integer linear programming to solve the model. Xu et al. [28] proposed a mixed-integer linear programming approach for the integrated optimization of train ticket pricing and seat allocation, with case study results demonstrating the method’s ability to find the global optimal.

Table 1. Comparison of relevant publications.

Decisions						Transit Mode	Demand	Solution Method	Publication
Line Planning				Differential Pricing	O-D Seat Allocating
Line	Frequency	Stop	Time
Y	Y	N	N	N	N	Bus and tram	Daily demand	Column generation algorithm	[12]
Y	Y	Y	Y	N	N	Railway	Time-dependent demand	Simulated annealing algorithm	[13]
Y	Y	Y	Y	Y	N	Railway	Time-dependent elastic demand	Simulated annealing algorithm	[14]
N	N	N	N	Y	Y	Railway	Elastic demand	Quasi-Newton algorithm	[17]
N	N	N	N	Y	Y	Railway	Elastic demand	Adopting the range reduction method	[28]
Y	Y	Y	Y	Y	Y	Railway	Time-dependent elastic demand	Simulated annealing algorithm + linear passenger flow allocation	This paper

Symbol Y means to consider the corresponding decision; symbol N means not to consider the corresponding decision.

lists some comparative studies, showing the differences in models, demand types, and solution methods of the relevant literature. By comparison, it can be found that the current step-by-step optimization method cannot simultaneously reflect the impact of train products, ticket prices, and seat availability on passenger travel choices, but can only consider some of these influencing factors in each step of optimization. This results in an inability to accurately depict passenger travel choices. Therefore, the current step-by-step optimization method struggles to produce train operation products, ticket prices, and seat availability that match passenger demand. Only through integrating ticket pricing and seat allocation decisions can models accurately describe passenger demand and their travel choices, achieve an optimized line plan of HSR that matches travel demand, and enhance the level of train service and operational efficiency.

Moreover, the optimization of existing line planning is based on passenger volume, and passenger flow allocation is combined in the process of optimizing the line plan. The resulting passenger flow distribution is a static outcome that does not consider the distribution of seat allocation on the train from a revenue management perspective, nor does it dynamically consider the comprehensive utilization process of seat allocation. Therefore, only by integrating ticket pricing and seat allocation decisions into the optimization of line planning can the influence of price level and seat allocation be dynamically considered during the optimization process, and the line planning matching with the passenger demand can be optimized to realize the improvement of train service level and operating benefits.

This study addressed the passenger travel choice under the elastic demand, constructed an optimization model of line planning integrating ticket pricing and seat allocation decisions for HSR, designed an effective solution algorithm, and realized the overall optimization of the number of trains, the origin and destination stations of each train, the train route, the running time period, the stop plan, the train price level, and the train seat allocation. In order to improve the operation efficiency of HSR trains and the quality of passenger travel service and improve the matching degree of train products and passenger demands, we aimed to provide the following three primary contributions:

• The model in this paper was constructed based on the elastic demand that comprehensively considers the HSR travel demand, train service level, and train service capacity. It also integrates passenger travel choices at the time–space network level, achieving the first-time optimization of line planning integrating differentiated ticket pricing and O-D seat allocation decisions. This joint optimization enables simultaneous decision making for the number of high-speed trains, the origin and destination stations of each train, the train route, the train running time period, the train stop plan, the O-D ticket price level, and the O-D ticket allocation.
• The algorithm in this paper improves on the one designed by Zhou et al. [14], proposing a heuristic method based on the simulated annealing framework combined with the linear passenger flow allocation method. This algorithm provides the process for generating the initial solution and introduces seven neighborhood structures combining line plans, ticket prices, and seat allocations. The algorithm effectively solves the complex and challenging joint optimization model.
• The case study analysis in this paper demonstrates the correctness of the constructed model. It also proves that the improved algorithm designed in this study can effectively solve a real Y-shaped railway network composed of 13 stations, indirectly showing that the joint optimization model and algorithm designed in this study can enhance the operational efficiency of HSR trains and the quality of passenger travel service.

The remainder of this paper is organized as follows: Section 2 describes passenger travel choices at the time–space network level and constructs an elastic demand that comprehensively considers HSR travel demand, train service level, and train service capacity. Section 3 constructs an optimization model for line planning, integrating differentiated ticket pricing and O-D seat allocating decisions. Section 4 presents a heuristic method based on the simulated annealing framework combined with the linear passenger flow allocation method. Section 5 provides case studies based on a virtual railway network and a real HSR network.

2. Problem Statement

This section first gives the decision variables and assumptions, then constructs a candidate travel network that considers line planning integrating ticket pricing and seat allocating decisions, then describes passenger travel choice behavior under elastic demand, and finally summarizes the mutual influence of the decision variables.

2.1. Decision Variables and Assumptions

Line planning is based on the nature, characteristics, and patterns of passenger flow, and it scientifically and reasonably arranges train elements such as the departure and terminal stations, the number of trains, the routes taken, the train formation, the stop plan, and the utilization of passenger capacity. The ticket pricing decision determines the price levels for each train, while differential pricing is designed to scientifically and reasonably set the price for each train between each serviceable O-D pair. The seat allocation decision determines the utilization of train seats, meaning that the total number of seats on the train is scientifically and reasonably allocated based on the serviceable O-D pairs.

If we think of the train as a product, the ticket pricing decision and seat allocation decision determine the marketing strategy of the train products sold by the railway company. These decisions directly impact the operational effectiveness and revenue of the train products offered by the railway company, as well as the travel costs and train choices of passengers. The line planning, ticket pricing decision, and seat allocation decision collectively influence passengers’ travel costs, service levels, and service capacities, directly determining the demand for passenger travel and their travel choices and also affecting the operational revenue of the railway company.

Therefore, the optimization of line planning integrating ticket pricing and seat allocation decisions is a complex combinatorial decision problem involving multiple objectives, various constraints, and multiple decisionmakers, encompassing the interests of both the railway company and traveling passengers.

Table 2. Definitions of decision variables.

Objects	Symbols	Definitions
Line planning	$x_f$	0-1 decision variable: If train $f\in F$ is an operating train, then $x_f=1$; otherwise, it is 0
	$y_f\left(k\right)$	0-1 decision variable: If train $f\in F$ is an operating train and stops at station $k\in S_f$, then $y_f\left(k\right)=1$; otherwise, it is 0
	$a_f\left(k\right)$	Arrival time of train $f\in F$ at station $k\in S_f$. If train $f\in F$ is not selected as an operating train, its value is 0
	$d_f\left(k\right)$	Departure time of train $f\in F$ at station $k\in S_f$. If train $f\in F$ is not selected as an operating train, its value is 0
Ticket pricing	$p_{r,s}^f$	Ticket price for train $f\in F$ for the O-D pair from station $r$ to station $s$
Seat allocation	$e_{r,s}^f$	Seat quota for train $f\in F$ for the O-D pair from station $r$ to station $s$
Passenger	${\rho }_{r,s}^h\left(i,j\right)$	Passenger flow on the directed arc $\left(i,j\right)$ from station $r$ to station $s$ during passenger flow period $h\in H_r$
	$z\left(i,j\right)$	0-1 auxiliary variable: If the transfer arc $\left(i,j\right)\in A_{\mathrm{H}}$ is valid under the minimum transfer time constraint, then $z\left(i,j\right)=1$; otherwise, it is 0
	$w_h\left(i,j\right)$	0-1 auxiliary variable: If the boarding arc $\left(i,j\right)$ allows passengers from the departure period $h$ to board, then $w_h\left(i,j\right)=1$; otherwise, it is 0

provides the definitions of the relevant decision variables.

Before providing a detailed description, some assumptions are given as follows:

(A1) An optimal line plan of the optimization problem can be chosen from a set of pre-determined candidate trains that satisfies the transport capacity of the physical rail section. This is widely used in some existing studies [12,14,29].

(A2) The elastic demand for a given O-D pair in each time period decreases with the average generalized travel cost of passengers, which includes time and price costs [26,28].

(A3) The seat allocations for each train between different O-D pairs only consider one class of seat; specifically, for HSR trains, only the second-class seat category is considered. This assumption is consistent with the approach adopted in many studies on train seat allocations [17,28].

2.2. Candidate Travel Network

The method for constructing the candidate travel network continues the approach used in our previous research. In Reference [14], we constructed a candidate travel network using a space–time network approach, aimed at the combined optimization problem of train line planning and differential pricing. This paper addresses the optimization problem of line planning that integrates ticket pricing and ticket allocation decisions. The interrelationships among the decision variables are more complex, as passengers need to consider ticket price costs, travel time costs, and the constraints of seat allocation during the travel choice process.

The travel network constructed in this paper effectively integrates the optimization problem-related decisions with passenger travel choices, consisting of three types of node sets denoted as $N$ and six types of directed arc sets denoted as $\tilde{A}$. Targeting railway stations $k\in S$ and candidate trains $f\in F$, the three types of nodes are station nodes $n_k\in N_S\in N$ corresponding to station $k$, arrival nodes $u_f^k$, and departure nodes $v_f^k$, corresponding to train $f$ passing through station $k\in S_f$. Moreover, $i$ and $j$ denote any type of nodes, and ${\widehat{A}}_i$ and ${\check{A}}_i$ represent sets of directed arcs flowing out of and into node $i$ in the candidate travel network, respectively, with $\left(i,j\right)$ being any type of directed arc within.

In targeting each candidate train, boarding arcs $\left(n_k,v_f^k\right)\in A_{\mathrm{B}}$, running from each station node to the corresponding departure node of the station, are differentiated by origin station, destination station, and passenger flow time period, reflecting the fare cost for different passenger flows and their waiting time cost at the origin station. Furthermore, the total passenger flow carried on each boarding arc is restricted by the seat allocation for each train between each O-D pair. The travel network’s boarding arcs are restricted by the completion of the passenger flow’s start and end and the passenger flow time period, with the decision-making seat allocation limit for trains between each serviceable O-D pair being restricted by the total passenger flow carried on the related boarding arcs. Additionally, the upper limit of the passenger flow carried on other types of arcs is constrained by the train’s capacity. Travel arcs $\left(v_f^k,u_f^{k^{\prime }}\right)\in A_{\mathrm{T}}$, running from each departure node to the corresponding adjacent station’s arrival node, are differentiated by the destination station and passenger flow time period, reflecting the different passenger flows’ interval travel time cost and the train’s interval travel time cost. Dwell arcs $\left(u_f^k,v_f^k\right)\in A_{\mathrm{D}}$, running from each arrival node to the corresponding departure node within the same station, need to differentiate the destination station and passenger flow time period, reflecting different passenger flows’ stopover time cost and the train’s stopover time cost. Transfer arcs $\left(u_f^k,v_{f^{\prime }}^k\right)\in A_{\mathrm{H}}$, running from the arrival node of the previous train to the departure node of the following train at the same station, need to differentiate the destination stations and passenger flow time period, reflecting different passenger flows’ transfer time cost at the transfer station and the fare cost for transferring to a second train. Additionally, the seat allocation for each train between each O-D pair restricts the passenger flow volume transferring to the second train. Getting-off arcs $\left(u_f^k,n_k\right)\in A_{\mathrm{O}}$, running from each arrival node to the corresponding station node, need to differentiate the destination stations and passenger flow time period and do not bear any costs. Virtual arcs $\left(n_r,n_s\right)\in A_{\mathrm{V}}$, running from the starting station node of each O-D pair to the destination station node, are used to carry the retained passenger flow within the travel network.

2.3. Description of Elastic Passenger Flow and Passengers’ Travel Choices

Due to the close interrelationship between ticket pricing, seat allocation, and passenger flow demand for each serviceable O-D pair of the train, this study adopted the formulation of an elastic demand function to reflect the elastic relationship between passenger flow demand and the level of passenger travel services. It constructed an elastic demand function based on the passenger-weighted average generalized travel cost. The generalized travel cost for passengers considered in this paper includes the integrated impact of the train fare cost, travel time cost, and seat allocation. It is assumed that for each time period, the level of passenger travel services is better for an O-D pair when the average fare and travel time cost for passengers are lower, and the total allocated seat volume for the trains serving that O-D pair is relatively higher.

Let $e_{r,s}^f$ and $p_{r,s}^f$, respectively, denote the O-D seat allocation (unit: piece) and O-D fare (unit: RMB Yuan) for train $f$ from station $r\in S_f\backslash \left\{d_f\right\}$ to station $s\ne r\in S_f\backslash \left\{o_f\right\}$, and $⟨\left(r,s\right)|h|f⟩$ represents the train product offered by train $f$ from station $r$ to station $s$ within time period $h$. Considering that the passenger travel cost for any train product between any O-D pair is influenced by the train fare and travel time, the generalized travel cost $C_{r,s}^{h,f}$ for each train product $⟨\left(r,s\right)|h|f⟩$ can be expressed as follows, where ${\alpha }_1$ is the time cost parameter (unit: RMB Yuan/minute):

$$C_{r,s}^{h,f}=p_{r,s}^f+\left(a_f\left(s\right)-{\mathcal{R}}_h\right)\times {\alpha }_1$$

(1)

Next, based on the generalized travel costs $C_{r,s}^{h,f}$ of each train product, the weighted average generalized travel cost $C_{r,s}^h$ for passengers traveling from station $r$ to station $s$ within time period $h$ can be determined, which is affected by the O-D average fare cost and travel time cost.

$$C_{r,s}^h=\left[\sum _{f\in \mathit{F}}{C}_{r,s}^{h,f}\times x_f\times y_f\left(r\right)\times y_f\left(s\right)\right]/\sum _{f\in \mathit{F}}{x}_f\times y_f\left(r\right)\times y_f\left(s\right)$$

(2)

Now, the elastic demand function can be constructed based on the weighted average generalized travel costs for passenger travel between each O-D pair during each time period, where demand increases when the level of travel service is better than the initial level, and vice versa. Here, $q_{r,s}^h$ represents the total demand for the O-D pair $\left(r,s\right)$ during time period $h$, $q_{r,s}^{h0}\left(C_{r,s}^{h0}\right)$ represents the initial total demand for the O-D pair $\left(r,s\right)$ during time period $h$, corresponding to the initial weighted average generalized travel cost $C_{r,s}^{h0}$, and ${\eta }_{r,s}^h>0$ is the elasticity coefficient for the O-D pair $\left(r,s\right)$ during time period $h$:

$$q_{r,s}^h\left(C_{r,s}^h\right)=q_{r,s}^{h0}\left(C_{r,s}^{h0}\right)\times exp\left[-{\eta }_{r,s}^h\left({\displaystyle \frac{C_{r,s}^h}{C_{r,s}^{h0}}}-1\right)\right]$$

(3)

As shown in Figure 1, a small example is presented to illustrate the constructed candidate travel network and the travel choices of passengers within the network. The figure includes three stations, two trains, and one virtual train, along with the corresponding display of three types of nodes and six types of directed arcs. When passengers make choices on directed arcs in the travel network, which corresponds to selecting trains, they must bear the O-D ticket price costs from boarding arcs or transfer arcs, as well as the time costs from travel arcs and dwell arcs. Additionally, passengers are constrained by the seat availability between the O-D on the boarding arcs or transfer arcs. The decision-making seat allocation limit for each serviceable O-D pair of a train is restricted by the total passenger flow carried on the related boarding arcs or transfer arcs. In the meantime, the upper limit of the passenger flow carried on other types of arcs is constrained by the train’s capacity. For example, when a passenger chooses to board the first train at station A to station C in the candidate travel network, the passenger must first satisfy the constraints related to seat allocation $e_{A,C}^1$ on boarding arc $\left(n_1,v_1^1\right)$. Once this seat constraint is met, the passenger is allowed to board the first train. However, this passenger also needs to bear the departure waiting time cost on boarding arc $\left(n_1,v_1^1\right)$, O-D fare cost $p_{A,C}^1$, pure travel time cost on travel arc $\left(v_1^A,u_1^{\mathrm{B}}\right)$, stop time cost on dwell arc $\left(u_1^B,v_1^B\right)$, and pure travel time cost on travel arc $\left(v_1^B,u_1^{\mathrm{C}}\right)$. Of course, if the total travel cost of this option is higher than that of other options, the passenger can automatically choose a more favorable travel plan. However, sometimes there may be no available seats in the more optimal travel plans, forcing the passenger to select a less favorable option. Similarly, the process of passengers choosing to board a train via the transfer arc is the same as the process described above for boarding a train via the boarding arc.

2.4. Description of Joint Decision-Making Relationship

The decision-making process and interrelationships among line planning, ticket pricing, and seat allocation in this study are illustrated in Figure 2. The mutual influence relationship in Figure 2 reflects the elastic changes in demand. Seat allocation represents a limited capacity resource of HSR trains. By reasonably distributing train capacity, seat allocation aims to improve train load factors and determines the upper limit of travel demand for each serviceable O-D pair on a train. The revenue generated from ticket sales is jointly determined by ticket pricing levels and seat allocation schemes. Meanwhile, train line planning, ticket pricing, and seat allocation together define the quality of train services. Consequently, these three factors collectively influence the total operating costs and total revenue of railway transportation enterprises, as well as the total travel costs of passengers.

Therefore, this paper focuses on optimizing HSR line planning by integrating ticket pricing and seat allocation decisions. By incorporating passenger travel choices, our approach comprehensively determines train operation details, ticket pricing levels, and seat allocation schemes. This ensures that the optimized train products and their corresponding sales strategies effectively adapt to actual transportation market demands, thereby enhancing railway enterprises’ revenue while considering both operational costs and passenger travel expenses.

3. The Optimization Model

This paper takes into account the objectives of both railway companies and passengers to establish an integrated model for optimizing line planning with ticket pricing and seat allocation decisions. The goal of the model is to maximize the total revenue of the railway companies minus the total travel cost of passengers, taking into consideration decisions and constraints related to line planning, ticket pricing, and seat allocation. Next, based on the description of the candidate travel network, this section organizes and summarizes the decision variables related to line planning, ticket pricing, seat allocation, and travel demand.

3.1. Constraints

The model must satisfy constraints related to the line planning and ticket pricing on one hand, and on the other hand, it must integrate flow balance constraints as well as capacity constraints for various types of arcs to set seat allocation-related constraints. This will ensure that the constraints related to line planning, ticket pricing, and seat allocation are comprehensively addressed.

3.1.1. Line Planning-Related Constraints

$$t_f^s\times x_f\le d_f\left(o_f\right)\le t_f^e\times x_f,\forall f\in F$$

(4)

$$a_f\left(k^{\prime }\right)-d_f\left(k\right)={\tau }_f^{k,k^{\prime }}\times x_f,\forall f\in F,\forall \left(k,k^{\prime }\right)\in E_f$$

(5)

$$d_f\left(k\right)-a_f\left(k\right)=w_f^k\times y_f\left(k\right),\forall f\in F,\forall k\in S_f\backslash \left\{o_f,d_f\right\}$$

(6)

$$y_f\left(o_f\right)+y_f\left(d_f\right)=x_f\times 2,\forall f\in F$$

(7)

$$\sum _{k\in S_f}{y}_f\left(k\right)\le x_f\times {\mathcal{L}}_f,\forall f\in F$$

(8)

Constraint (4) ensures that each train departs from its originating station within a given departure time range and is consistent with the train operation choice. Constraint (5) ensures that the travel time between intervals for each train is consistent with the train operation choice. Constraint (6) ensures that the dwell time at stops for each train aligns with the train stopping decision. Constraint (7) guarantees that each operating train must stop at its originating and terminating stations. Constraint (8) ensures that the number of stops made by each operating train does not exceed the total number of stations ${\mathcal{L}}_f$.

3.1.2. Ticket Pricing-Related Constraints

$$0\le p_{r,s}^f\le y_f\left(r\right)\times y_f\left(s\right)\times M,\forall f\in F,\forall r\in S_f\backslash \{d_f\},\forall s\ne r\in S_f\backslash \{o_f\}$$

(9)

$${\check{p}}_{r,s}^f\le p_{r,s}^f\le {\widehat{p}}_{r,s}^f,\forall f\in F,\forall r\in S_f\backslash \{d_f\},\forall s\ne r\in S_f\backslash \{o_f\}$$

(10)

$$p_{r,k}^f+p_{k,s}^f\ge p_{r,s}^f,\forall f\in F,\forall r\le k\in S_f\backslash \{d_f\},\forall s\ge k\in S_f\backslash \{o_f\}$$

(11)

Constraint (9) ensures that the ticket prices set for each operational train between any serviceable O-D pair are consistent with the stopping decisions for those trains. Constraint (10) guarantees that the ticket prices for each operational train between any serviceable O-D pair are bounded, meaning there are predefined minimum and maximum limits for ticket prices. Constraint (11) ensures that the sum of segment prices for the same O-D pair is higher than the price for a direct ticket for the entire journey, thereby preventing the occurrence of fare inversion.

3.1.3. Seat Allocation-Related Constraints

Seat allocation for each train is manifested through the boarding and transfer arcs in the passenger travel network, meaning that the seat allocation constraints for each train between any O-D pair are enforced through these arcs. This approach allows for precise control over ticket allocation because each boarding and transfer arc can distinguish between passengers based on their starting station, their destination station, and the time period of their travel. This precision enables seat allocation to be closely aligned with passenger travel choices within the travel network.

$$\sum _{h\in H_r}\sum _r\sum _s\left({\rho }_{r,s}^h\left(n_r,v_f^r\right)+{\rho }_{r,s}^h\left({u_{f^{\prime }}^r,v}_f^r\right)\right)\le y_f\left(r\right)\times y_f\left(s\right)\times e_{r,s}^f,\forall f\in F,\forall r\in S_f\backslash \left\{d_f\right\},\forall s\ne r\in S$$

(12)

Constraint (12) ensures that the total passenger flow on boarding and transfer arcs, which carry passengers boarding at station $r$ and alighting at station $s$, is limited by the seat allocation for trains operating between the O-D pair from station $r$ to station $s$.

$$\sum _{h\in H_r}\sum _s{\rho }_{r,s}^h\left(v_f^k{,u}_f^{k^{\prime }}\right)\le x_f\times \sum _{s\in S_f\backslash \{o_f\}}{e}_{r,s}^f,\forall f\in F,\forall \left(k,k^{\prime }\right)\in E_f$$

(13)

$$\sum _{h\in H_r}\sum _s{\rho }_{r,s}^h\left({u_f^k,v}_f^k\right)\le y_f\left(k\right)\times \sum _{s\in S_f\backslash \{o_f\}}{e}_{r,s}^f,\forall f\in F,\forall k\in S_f\backslash \left\{o_f,d_f\right\}$$

(14)

Constraint (13) ensures that the total passenger flow on travel arcs ending at station $s$ in the travel network satisfies the seat allocation limit for the O-D pairs ending at station $s$. Constraint (14) ensures that the total passenger flow on dwell arcs ending at station $s$ in the travel network also adheres to the seat allocation limit for O-D pairs ending at station $s$.

$$\sum _{h\in H_r}\sum _k\sum _s{\rho }_{k,s}^h\left({u_f^k,v}_{f^{\prime }}^k\right)\le z\left({u_f^k,v}_{f^{\prime }}^k\right)\times \sum _{s\in S_{f^{\prime }}\backslash \{o_{f^{\prime }}\}}{e}_{r,s}^{f^{\prime }},\forall f^{\prime }\ne f\in F, \forall k\in S_f\cap S_{f^{\prime }},\forall s\ne k\in S$$

(15)

$$z\left({u_f^k,v}_{f^{\prime }}^k\right)\le \left[y_f\left(k\right)+y_{f^{\prime }}\left(k\right)\right]/2,\forall f^{\prime }\ne f\in F,\forall k\in S_f\cap S_{f^{\prime }}$$

(16)

$$d_{f^{\prime }}\left(k\right)-a_f\left(k\right)\ge {{\rm Y}}_k-\left[1-z\left({u_f^k,v}_{f^{\prime }}^k\right)\right]\times M,\forall f^{\prime }\ne f\in F,\forall k\in S_f\cap S_{f^{\prime }}$$

(17)

Constraint (15) ensures that the total passenger flow on transfer arcs, which represent passengers transferring at station $k$ and alighting at station $s$, does not exceed the seat allocation limit for the subsequent train $f^{\prime }$ in the transfer relationship destined for station $s$. Constraint (16) ensures the consistency between the transfer arc validity variables and the stopping decisions of trains. Constraint (17) ensures that the transfer arc validity variables are consistent with the minimum transfer time at the station and that the interval between the arrival and departure times of the two trains involved in a transfer meets the minimum transfer time requirements.

$$\sum _{h\in H_r}\sum _{s\in S_f\backslash \left\{o_f\right\}}{\rho }_{k,s}^h\left(u_f^s,n_s\right)+\sum _{h\in H_r}\sum _k\sum _{k^{\prime }}{\rho }_{k,k^{\prime }}^h\left({u_f^s,v}_{f^{\prime }}^s\right)\le y_f\left(k\right)\times \sum _{s\in S_f\backslash \{o_f\}}{e}_{r,s}^f,\forall f\in F,\forall k\in S_f\backslash \left\{d_f\right\},\forall k^{\prime }\ne k\in S$$

(18)

Constraint (18) ensures that the total passenger flow on getting-off arcs and transfer arcs, which carry passengers alighting at station $s$, does not exceed the seat allocation limit for trains destined for station $s$ in terms of O-D pairs.

$$\sum _{r\in {\mathit{S}}_{\mathit{f}}\backslash \left\{d_f\right\}}\sum _{s\ne r\in \mathit{S}}{e}_{r,s}^f\times y_f\left(r\right)\times y_f\left(s\right)={Cap}_f,\forall f\in F$$

(19)

$$e_{r,s}^f\in N^{+},\forall f\in F,\forall r\in S_f\backslash \left\{d_f\right\}, \forall s\ne r\in S$$

(20)

Constraint (19) ensures that the total seat allocation for each train across all serviceable O-D pairs does not exceed the train’s service capacity limit. Constraint (20) ensures that the seat allocation for each train across all serviceable O-D pairs is a non-negative integer.

3.1.4. Passenger Flow Related Constraints

$$0\le \sum _{h\in H_r}\sum _r{\rho }_{r,s}^h\left(n_r,v_f^r\right)\le w_h\left(n_r,v_f^r\right)\times {Cap}_f,\forall f\in F,\forall s\ne r\in S$$

(21)

$$d_f\left(k\right)\ge w_h\left(n_r,v_f^r\right)\times {\mathcal{R}}_h,\forall f\in F,\forall h\in H_r,\forall s\ne r\in S$$

(22)

Constraints (21) and (22) ensure that boarding arcs within the travel network can only carry passengers for specific time periods if the departure times of the trains at the stations occur later than those time periods. Essentially, these constraints regulate the flow of passengers onto trains based on the scheduled departure times, aligning passenger boarding with the operational timetable of the train services.

$$\sum _{\left(n_r,j\right)\in {\widehat{A}}_{n_r}}{\rho }_{r,s}^h\left(n_r,j\right)+\sum _{\left(n_r,n_s\right)\in A_V}{\rho }_{r,s}^h\left(n_r,n_s\right)=q_{r,s}^h,\forall h\in H_r,\forall r\in S_f\backslash \{d_f\},\forall s\ne r\in S$$

(23)

$$\sum _{\left(i,j\right)\in {\check{A}}_j}{\rho }_{r,s}^h\left(i,j\right)=\sum _{\left(j,i^{\prime }\right)\in {\widehat{A}}_j}{\rho }_{r,s}^h\left(j,i^{\prime }\right),\forall h\in H_r,\forall r\ne s\in S,\forall s\ne r\in S$$

(24)

$$\sum _{\left(i,n_s\right)\in {\check{A}}_{n_s}}{\rho }_{k,s}^h\left(i,n_s\right)+\sum _{\left(i,n_s\right)\in A_V}{\rho }_{k,s}^h\left(i,n_s\right)=q_{r,s}^h,\forall h\in H_r,\forall s\ne r\in S$$

(25)

These three constraints are fundamental to ensuring flow balance for each node within the travel network, aligning closely with the overall demand for travel between any given O-D pair during specific time periods. Constraint (23) ensures that the total volume of passengers boarding trains at stations via boarding arcs and virtual arcs matches the total actual demand for each specific travel requirement $q_{r,s}^h$. Constraint (25) ensures that the total volume of passengers alighting at stations through getting-off arcs and virtual arcs is consistent with the total actual demand for each specific travel requirement $q_{r,s}^h$. Constraint (24) ensures that the total inflow of passengers into each node within the travel network matches the total outflow from that node, maintaining flow balance across the system.

3.2. Objective

First, analyze the total ticket revenue ${\tilde{R}}_{\mathrm{P}}$ and the total operational cost $H$ of the railway company. The total ticket revenue ${\tilde{R}}_{\mathrm{P}}$ can be represented as

$${\tilde{R}}_P=\sum _f\sum _{h\in H_r}\sum _r\sum _{s\ne r}{p}_{r,s}^f\times w_{r,s}^f\times x_f$$

(26)

$$w_{r,s}^f=\left(\sum _{h\in H_r}{\rho }_{r,s}^h\left(n_r,v_f^r\right)+\sum _{h\in H_r}{\rho }_{r,s}^h\left(u_{f^{\prime }}^r,v_f^r\right)\right)\times y_f\left(r\right)\times y_f\left(s\right)$$

(27)

where $w_{r,s}^f$ represents the total passenger flow on boarding and transfer arcs corresponding to train $f$ between each O-D pair in the travel network, that is, the total actual seat allocation. Since the total actual seat allocation for these two sets of corresponding arcs is constrained by constraint (12), it must satisfy the seat allocation limit for train $f$ between each O-D pair.

The total operational cost $H$ of the railway company consists of the total organizational cost $C_{\mathrm{F}}$ and the total train travel cost $C_{\mathrm{V}}$:

$$H=C_F+C_V$$

(28)

$$H=\sum _f\left\{x_f\times {\phi }_f+\left(\sum _{\left(k,k^{\prime }\right)\in {\mathit{E}}_{\mathit{f}}}{x}_f\times {\tau }_f^{k,k^{\prime }}+\sum _{k\in {\mathit{S}}_{\mathit{f}}}{y}_f\left(k\right)\times w_f^k\right)\times \alpha \right\}$$

(29)

where ${\phi }_f$ represents the organizational cost per train (unit: RMB Yuan), and $\alpha$ is the time cost parameter (unit: RMB Yuan/minute). Furthermore, the total revenue $\tilde{R}$ of the railway company can be described as the total ticket revenue minus the total operational costs:

$$\tilde{R}={\tilde{R}}_P-H$$

(30)

Furthermore, the total travel cost $\Pi$ for passengers is specifically reflected in the travel network through the directed arcs that passengers traverse to complete their journey. These costs include the total ticket price cost $C_{\mathrm{P}}$, the total departure waiting time cost $C_{\mathrm{W}}$, the total pure travel time cost $C_{\mathrm{T}}$, the total dwell time cost $C_{\mathrm{D}}$, the total transfer time cost $C_{\mathrm{H}}$, and the total delay cost $C_{\mathrm{E}}$. Below are the specific calculation equations:

$$C_P=\sum _f\sum _{h\in H_r}\sum _r\sum _{s\ne r}{p}_{r,s}^f\times w_{r,s}^f\times x_f$$

(31)

$$C_W=\sum _f\sum _{h\in H_r}\sum _r\sum _{s\ne r}\sum _{\left(n_k,v_f^k\right)\in A_B}(d_f\left(k\right)-R_h)\times {\rho }_{r,s}^h\left(n_k,v_f^k\right)\times x_f\times \alpha$$

(32)

$$C_T=\sum _f\sum _{h\in H_r}\sum _s\sum _{\left(v_f^k,u_f^{k^{\prime }}\right)\in A_T}{\tau }_f^{k,k^{\prime }}\times {\rho }_{k,s}^h\left(v_f^k,u_f^{k^{\prime }}\right)\times x_f\times \alpha$$

(33)

$$C_D=\sum _f\sum _{h\in H_r}\sum _s\sum _{\left(u_f^k,v_f^k\right)\in A_D}{w}_f^k\times {\rho }_{k,s}^h\left(u_f^k,v_f^k\right)\times y_f\left(k\right)\times \alpha$$

(34)

$$C_H=\sum _{f^{\prime }}\sum _{h\in H_r}\sum _s\sum _{\left(u_f^k,v_{f^{\prime }}^k\right)\in A_H}\left(d_{f^{\prime }}\left(k\right)-a_f\left(k\right)\right)\times {\rho }_{k,s}^h\left(u_f^k,v_{f^{\prime }}^k\right)\times y_f\left(k\right)\times y_{f^{\prime }}\left(k\right)\times \alpha$$

(35)

$$C_E=\sum _{h\in H_r}\sum _{\left(n_k,n_{k^{\prime }}\right)\in A_V}{\rho }_{k,k^{\prime }}^h\left(n_k,n_{k^{\prime }}\right)\times M_V$$

(36)

$$\Pi =C_P+C_W+C_T+C_D+C_H+C_E$$

(37)

Now, based on the total revenue of the railway company and the total travel cost for passengers, the objective function of the model considered is constructed by combining the total revenue of the railway company and the total travel cost for passengers using a weighting coefficient. The objective is to maximize the following equation:

$$\mathit{max}\tilde{Z}=\omega \times \tilde{R}-\left(1-\omega \right)\times \Pi$$

(38)

Here, $\omega$ is the weighting coefficient, with a value in the interval (0,1]. The value of $\omega$ influences the weight of the two components in the optimization result.

3.3. Construction of the Optimization Model

Building on the analysis and descriptions from Section 3.1 and Section 3.2, this paper constructs a mathematical model based on the objective function, constraints, and elastic demand function. Observations and analyses of the relevant content reveal that the mathematical model constructed in this section is a nonlinear model.

The objective is to maximize the following equation:

$$m\mathit{ax}\tilde{Z}=\omega \times \tilde{R}-\left(1-\omega \right)\times \Pi$$

(39)

Here, $\tilde{R}$ is calculated using Equations (26)–(30), and $\Pi$ is derived from Equations (31)–(37), subject to the following:

Elastic demand functions (1) to (3);

Constraints (4) to (25).

4. Solution Algorithm

This section presents the design of a solution algorithm for the optimization model of line planning that integrates ticket pricing and seat allocation decisions. The overall approach of the algorithm is designed based on the simulated annealing framework and linear passenger flow allocation.

4.1. Algorithm Framework

The optimization model formulated in this study is inherently nonlinear due to the integration of line planning, ticket pricing, and seat allocation decisions under elastic demand. The nonlinearity arises primarily from the interactions between ticket prices, passenger choices, and capacity constraints, which result in complex interdependencies that cannot be easily represented in a linear form. While convex relaxation techniques, such as second-order cone programming and semidefinite programming, are commonly used to improve the tractability of nonlinear problems; applying them directly in this context poses significant challenges. First, the discrete nature of seat allocation and train operation decisions makes it difficult to construct a convex formulation without sacrificing key operational constraints. Second, transforming the pricing and demand elasticity relationships into a convex structure may lead to infeasible or unrealistic solutions that do not align with real-world railway operations. Given these limitations, we adopted a heuristic approach based on simulated annealing, which effectively handles the nonlinearities while ensuring practical feasibility.

This paper uses a heuristic method that combines the basic framework of simulated annealing with linear passenger flow allocation methods to solve the constructed model. The algorithm implementation involves eight processes, repeatedly iterating processes 5 to processes 7 to continuously optimize the model’s objective function value until reaching the algorithm’s termination condition:

• Generation of candidate train set and ticket price boundaries: Based on the physical railway network, current time-dependent demand data, and existing ticket price levels, generate a given candidate train set; then, based on the candidate train set, generate upper and lower bounds for ticket prices for each candidate train’s serviceable O-D pairs.
• Initial seat allocation generation: Based on the model and solving algorithm in Reference [14] and corresponding input data, solve for the final travel demand volume for each train between each O-D pair, and then set this demand volume as an initial seat allocation set for each candidate train.
• Generation of the initial solution: Based on the candidate train set, ticket price boundaries, and initial seat allocating, generate an initial solution.
• Evaluation of the initial solution: Calculate the weighted average generalized travel cost for each O-D pair within each passenger flow period based on the initial solution, update the time-dependent actual demand using the elastic demand function, obtain specific passenger flow information for each arc in the travel network using the linear passenger flow allocation method, i.e., obtain the passenger flow distribution results for each operating train, calculate the objective function value corresponding to the initial solution, and evaluate it.
• Generation of neighborhood solutions: Obtain the neighborhood solution of the current solution based on the constructed neighborhood structure; update the travel network according to the generated neighborhood solution.
• Evaluation of neighborhood solutions: Calculate the weighted average generalized travel cost for each O-D pair within each passenger flow period based on the current neighborhood solution, update the time-dependent elastic demand using the elastic demand function, obtain specific passenger flow information for each arc in the travel network using the linear passenger flow allocation method, calculate the objective function value corresponding to the initial solution, and evaluate the neighborhood solution.
• Solution update: Determine whether to accept the neighborhood solution based on its evaluation.
• Algorithm termination condition.

4.2. Initial Solution Generation

The generation of the initial solution mainly includes the generation of the initial line plan, ticket pricing plan, and seat allocation plan.

Step 0: Generate a candidate train set based on the generation method for candidate trains described in Reference [14].

Step 1: Set all trains in the candidate train set as operating trains, set all stations passed by each train as initial stops, and set the lower bound of the departure time range at the originating station for each candidate train as the initial departure time of the train at its originating station to obtain the initial line plan.

Step 2: Set the initial differentiated ticket prices for each train between each serviceable O-D pair to the current real fixed ticket prices to obtain the initial differentiated ticket pricing plan.

Step 3: Based on the model and solving algorithm in Reference [14] and corresponding input data, solve for the final travel demand volume for each train between each O-D pair, and then set this demand volume as the initial seat allocation plan in the initial solution.

Step 4: Based on the initial line plan, ticket pricing plan, and seat allocation plan, use the elastic demand function to calculate the actual demand between each O-D pair within each passenger flow period, and then use the linear passenger flow allocation method and the GUROBI solver to solve for the passenger flow volume and information on each type of arc in the travel network.

Step 5: Based on the sum of passenger flows for each train at each stop (boarding arcs, getting-off arcs, transfer arcs), randomly select stations where the total number of passengers for each train is below a certain value to be designated as stations where the train does not stop. This value can be determined based on the scale of the case study.

Step 6: Randomly set some stations as non-stop stations among all the operating trains’ stop stations and update the train operation plan; simultaneously update the ticket pricing scheme and ticket allocation plan based on the updates to the stops; afterward, an initial solution is obtained, and this initial solution is set as the current and final solution.

4.3. Neighborhood Structure Construction

Seven types of neighborhood structures were constructed. These include strategies for suspending train services, adding train services, removing stops, adding stops, adjusting train departure times, adjusting train ticket prices, and adjusting train seat allocations. The first six strategies continue to use the approaches discussed in Reference [14]. The difference lies in how the adjustment processes for each strategy further incorporate the impact of seat allocation on problem solving. Adjustments related to train operation strategies (the first four types) will be synchronized with price adjustments and seat allocation adjustments. The fifth type, related to train operation, will involve seat allocation adjustments. The sixth type, related to ticket price adjustments, will involve adjustments to the line plan and seat allocations. The seventh strategy, related to seat allocation adjustments, will affect ticket pricing adjustments, indicating that the adjustments of line plans, ticket pricing, and seat allocations are interconnected, dynamic, and mutually influencing processes.

• Suspending train services strategy: This strategy primarily considers the ticket revenue of each operating train. On one hand, the ticket revenue calculations include the impact of each train’s seat allocation on the final passenger load, influencing the final ticket revenue. On the other hand, the final decision to suspend services will involve removing the ticket pricing and seat allocating for that train in the neighborhood solution.
• Adding new train services strategy: For trains that are ultimately added under this strategy, an initial ticket pricing and seat allocation based on the train stopping plan will be added to the neighborhood solution.
• Removing stops strategy: This strategy is based on the total ticket revenue and passenger volumes boarding and alighting at each station. It considers the impact of each train’s seat allocation on the final passenger load when calculating ticket revenue and boarding volumes at each stop, influencing the final ticket revenue and boarding volumes. Stops ultimately removed will have their related ticket information and seat allocation information deleted in the neighborhood solution.
• Adding stops strategy: This strategy is based on the average ticket revenue of other trains stopping at stations where the train does not currently stop. This includes the impact of seat allocations on the final passenger load, influencing the average ticket revenue. Stops ultimately added will have related seat allocation information added to the neighborhood solution, setting up initial seat allocations accordingly.
• Adjusting train departure times strategy: This strategy is primarily based on the average waiting time for passengers at the originating station, where different seat allocations influence the final passenger distribution and thereby affect passengers’ path choices within the travel network, ultimately affecting the different waiting times associated with various trains.
• Adjusting train ticket prices strategy: The strategy is primarily based on the ticket revenue for each O-D pair per train service. The ticket revenue is influenced by the ticket prices set for that O-D pair and significantly by the seat allocations assigned to that O-D pair, making the adjustment process for ticket prices and seat allocations interdependent and dynamic, where both aspects mutually influence and complement each other.
• Adjusting seat allocations strategy: This strategy determines seat allocation adjustments based on the difference between the seat allocation and actual passenger flow for each train across serviceable O-D pairs. The actual situation for each train and each O-D pair is handled in three scenarios:
  • When a train has both remaining seat allocations in some O-D pairs and completely used allocations in others, the remaining ticket allocations for those O-D pairs will be adjusted based on ticket revenue to the O-D pairs where allocations are exhausted.
  • When all seat allocations for a train across all O-D pairs are used up, it is assumed that the current seat allocation scheme is relatively good. The current neighborhood search process will not adjust its seat allocation scheme, but the train may undergo seat allocation adjustments in subsequent neighborhood searches due to the influence of ticket price adjustments.
  • When all seat allocations for a train across all O-D pairs are not exhausted, indicating that the train’s operation is not very efficient, its seat allocation scheme will not be adjusted. However, it will be influenced by neighborhood search processes involving suspending train services, removing stops, adjusting departure times, and adjusting ticket prices. Its seat allocation scheme may be continuously optimized in subsequent neighborhood search iterations to improve the utilization rate of ticket allocations.
    • Specific adjustment steps are as follows:
    • First, generate a random number ${\lambda }_6$ within the interval $\left[0,{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}6}\right]$, where ${\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}6}\le 1$ is the maximum adjustment ratio for the number of operating trains in the operating train set $F_{\mathrm{O}}$. As the simulated annealing iterations increase, this maximum adjustment ratio gradually decreases. Therefore, the actual number of trains whose seat allocation schemes are adjusted under the current strategy can be calculated as ${\mathsf{\Omega }}_6=\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{l}\left(F_{\mathrm{O}}\right)\times {\lambda }_6$.
    • Second, calculate the actual passenger flow for each serviceable O-D pair for each operating train and compute the difference between the allocated seat volume and the actual passenger flow for each O-D pair. Sort all these differences from smallest to largest, denoted as the set ${TA}_{r,s}$, and record the largest value in this set as ${TA}_{r,s}^{\mathrm{m}\mathrm{a}\mathrm{x}}$. Calculate the ticket revenue for each O-D pair, sort all O-D pair ticket revenues from smallest to largest, denoted as the set ${\overline{R}}_{r,s}$, and record the maximum value in this set as ${\overline{R}}_{r,s}^{\mathrm{m}\mathrm{a}\mathrm{x}}$.
    • Third, for the first scenario: calculate the probability $P_{{TA}_{r,s}}^f$ of reducing seat allocations for all O-D pairs with seat allocation differences greater than zero as $\left({TA}_{r,s}^{\mathrm{m}\mathrm{a}\mathrm{x}}-{TA}_{r,s}^f\right)/\sum _{f^{\prime }\in F_{\mathrm{O}}}\left({TA}_{r,s}^{\mathrm{m}\mathrm{a}\mathrm{x}}-{TA}_{r,s}^{f^{\prime }}\right)$. Calculate the probability $P_{{\overline{R}}_{r,s}}^f$ of increasing seat allocations for all O-D pairs with seat allocation differences of zero as $\left({\overline{R}}_{r,s}^{\mathrm{m}\mathrm{a}\mathrm{x}}-{\overline{R}}_{r,s}^f\right)/\sum _{f^{\prime }\in F_{\mathrm{O}}}\left({\overline{R}}_{r,s}^{\mathrm{m}\mathrm{a}\mathrm{x}}-{\overline{R}}_{r,s}^{f^{\prime }}\right)$. Then, based on these two probabilities, use the roulette wheel selection method to choose to increase seat allocations from O-D pairs with remaining seat allocations to those without remaining seat allocations. For the second and third scenarios, temporarily abstain from making seat allocation adjustments.

The seven constructed neighborhood structures can generate neighborhood solutions aligned with the model’s objectives or the characteristics of the problem. The generation process of neighborhood solutions will be based on all neighborhood structures or randomly based on some neighborhood structures. This ensures that the generated neighborhood solutions are well directed by the model’s objectives and randomly perturbed to expand the solution search space, avoiding falling into local optima.

4.4. Algorithm Steps

The detailed solution steps of the algorithm based on the simulated annealing framework and linear passenger flow allocation are given in Algorithm 1.

Algorithm 1. Simulated annealing algorithm

Input: Sets: Station set $S$, section set $E$, candidate train set $F$, passenger flow time period set $H_r$, upper and lower bounds of differentiated ticket prices ${\widehat{p}}_{r,s}^f$ and ${\check{p}}_{r,s}^f$, seat allocation scheme $e_{r,s}^f$. Model Parameters: Weight coefficient $\omega$, time parameters $\alpha$ and ${\phi }_f$, elasticity coefficient ${\eta }_{r,s}^h$. Algorithm Parameters: Initial temperature $T_0$, final temperature $T_{\mathrm{e}\mathrm{n}\mathrm{d}}$, temperature decay rate $\vartheta$, maximum iterations at the current temperature $I_{\mathrm{m}\mathrm{a}\mathrm{x}}$, upper limit of iterations where the objective function value remains unchanged $Z_{\mathrm{e}\mathrm{n}\mathrm{d}}$. Elastic Demand: Initial demand $q_{r,s}^h$ for each passenger flow period in each O-D pair, elastic demand function $q_{r,s}^h\left(C_{r,s}^h\right)$. Output: Train line planning, differentiated ticket pricing, and seat allocation plan $X^{*}=\left\{x_f,y_f\left(k\right),a_f\left(k\right),d_f\left(k\right),p_{r,s}^f,e_{r,s}^f\right\}$, objective function value ${\tilde{Z}}^{*}$, passenger flow volume on each arc after linear passenger flow allocation ${{\rho }_{r,s}^h\left(i,j\right)}^{*}$. 0. Step 0: Initialization Set initial temperature $T_0$, final temperature $T_{\mathrm{e}\mathrm{n}\mathrm{d}}$, temperature decay rate $\vartheta$. Let current temperature $T\leftarrow T_0$, iteration count $n\leftarrow 1$. Proceed to Step 1. 1. Step 1: Generate initial train line planning, ticket pricing, and seat allocation Generate initial line planning, differentiated ticket pricing, and seat allocation $X_0$. Set current solution $X\leftarrow X_0$, final solution $X^{*}\leftarrow X_0$; calculate passenger weighted average generalized travel cost $C_{r,s}^h$ using the elastic demand function $q_{r,s}^h\left(C_{r,s}^h\right)$ to obtain passenger demand for each period and O-D pair. Proceed to Step 2. 2. Step 2: Calculate objective function for initial solution using linear passenger flow allocation Use linear passenger flow allocation to obtain passenger flow volume ${\rho }_{r,s}^h\left(i,j\right)$ on each arc; calculate objective function value $\tilde{Z}\left(X_0\right)$; set $\tilde{Z}\leftarrow \tilde{Z}\left(X_0\right)$, ${\tilde{Z}}^{*}\leftarrow \tilde{Z}\left(X_0\right)$. Proceed to Step 3. 3. Step 3: Generate neighborhood solution for initial plan Produce a neighborhood solution $X_n$ for the current solution $X$ using the neighborhood structure; calculate passenger weighted average generalized travel cost and demand using the elastic demand function. Proceed to Step 4. 4. Step 4: Calculate objective function for neighborhood solution using linear passenger flow allocation Use linear passenger flow allocation to obtain passenger flow volume on each arc; calculate objective function value $\tilde{Z}\left(X_n\right)$. Proceed to Step 5. 5. Step 5: Update current and final solutions Calculate $∆\tilde{Z}\leftarrow \tilde{Z}\left(X\right)-\tilde{Z}\left(X_n\right)$; If $∆\tilde{Z}\le 0$, then set $X\leftarrow X_n$, $\tilde{Z}\leftarrow \tilde{Z}\left(X_n\right)$, $X^{*}\leftarrow X_n$, ${\tilde{Z}}^{*}\leftarrow \tilde{Z}\left(X_n\right)$; Else if $\mathrm{e}\mathrm{x}\mathrm{p}\left[-∆\tilde{Z}/T\right]>Random\left(\mathrm{0,1}\right)$, then set $X\leftarrow X_n$, $\tilde{Z}\leftarrow \tilde{Z}\left(X_n\right)$. Set $n=n+1$. Proceed to Step 6. 6. Step 6: Iteration count check If $n\ge I_{\mathrm{m}\mathrm{a}\mathrm{x}}$, then $T\leftarrow \vartheta \times T$; otherwise, proceed to Step 3. 7. Step 7: Algorithm termination check If $T<T_{\mathrm{e}\mathrm{n}\mathrm{d}}$, or if the objective function value $\tilde{Z}$ has not changed in ${\tilde{Z}}_{\mathrm{e}\mathrm{n}\mathrm{d}}$ iterations, proceed to Step 8; otherwise, set $n\leftarrow 1$, proceed to Step 3. 8. Step 8: Output results Output the final solution $X^{*}$ and its objective value ${\tilde{Z}}^{*}$.

5. Case Study

In this study, a case study was conducted using both a virtual railway network and a real HSR network to validate the effectiveness of the model. Additionally, the efficiency of the heuristic algorithm based on the simulated annealing framework and linear passenger flow allocation method was tested. Small-scale case studies based on the virtual railway network were initially performed with sensitivity analysis on some key parameters. Then, based on the real high-speed rail network, practical examples were analyzed. All case analyses utilized MATLAB R2019b, GUROBI Optimizer version 10.0.0, and the YALMIP toolbox version R20210331. The small-scale cases were run on a personal computer with an Intel Core i5-12400 2.50 GHz CPU and 16 GB of memory, while the large-scale cases were run on a workstation with an Intel Xeon W-2145 3.70 GHz CPU and 128 GB of memory.

5.1. Small-Scale Case Study

The small-scale case study begins with a virtual railway network, as depicted in Figure 3. This network consists of two railway lines, seven stations, and six sections. The pure travel time required for trains to run through each section is given below the network diagram. This network setup was used to facilitate the comparison of solution results, thereby highlighting the differences between optimization models.

Table 3. Candidate train information.

Train Number	Sequence of Stations	Initial Stopping Scheme	Departure Time Range at Origin Station (min)
1	1-2-3-4-5	1-2-3-4-5	[0, 60]
2	1-2-3-6-7	1-2-3-6-7	[0, 60]
3	1-2-3	1-2-3	[0, 60]
4	3-4-5	3-4-5	[0, 60]
5	3-6-7	3-6-7	[0, 60]
6	1-2-3-4-5	1-2-3-4-5	[0, 60]
7	1-2-3-6-7	1-2-3-6-7	[0, 60]
8	1-2-3-4-5	1-2-3-4-5	[60, 120]
9	1-2-3-6-7	1-2-3-6-7	[60, 120]
10	1-2-3	1-2-3	[60, 120]
11	3-4-5	3-4-5	[60, 120]
12	3-6-7	3-6-7	[60, 120]
13	1-2-3-4-5	1-2-3-4-5	[60, 120]
14	1-2-3-6-7	1-2-3-6-7	[60, 120]
15	1-2-3-4-5	1-2-3-4-5	[120, 180]
16	1-2-3-6-7	1-2-3-6-7	[120, 180]
17	1-2-3	1-2-3	[120, 180]
18	3-4-5	3-4-5	[120, 180]
19	3-6-7	3-6-7	[120, 180]
20	1-2-3-4-5	1-2-3-4-5	[120, 180]
21	1-2-3-6-7	1-2-3-6-7	[120, 180]
22	1-2-3-6-7	1-2-3-6-7	[120, 180]

presents a candidate train set consisting of 22 trains, along with each train’s sequence of stations, initial stopping scheme, and departure time range at the origin station. Each candidate train’s passenger service capacity was set to 600 seats per train. The initial differentiated ticket prices for each candidate train were generated based on the initial set prices for each O-D pair. The objective function’s weight coefficient $\omega$ was set at 0.8. The operational cost per train ${\phi }_f$ was set at RMB 5000 per train. The dwell time $w_f^k$ at each stop for each train was 5 min, the minimum transfer time at stations ${{\rm Y}}_k$ was 20 min, the elasticity coefficient for each O-D pair in each passenger flow period ${\eta }_{r,s}^h$ was set to 1, and the time cost parameter $\alpha$ was 0.5.

Additionally, the initial seat allocation for the candidate trains was randomly assigned based on the passenger flow volumes between O-D pairs during different time periods.

Table 4. Initial seat allocation and ticket price of two candidate trains.

Candidate Train 1			Candidate Train 4
O-D	Initial Seat Allocation	Initial Ticket Price (RMB Yuan)	O-D	Initial Seat Allocation	Initial Ticket Price (RMB Yuan)
1–2	40	30	3–6	150	20
1–3	100	65	3–7	300	46
1–4	40	82	6–7	150	30
1–5	150	105	—	—	—
2–3	40	40	—	—	—
2–4	40	57	—	—	—
2–5	120	82	—	—	—
3–4	20	20	—	—	—
3–5	30	48	—	—	—
4–5	20	30	—	—	—

lists the initial seat allocation and ticket price for two of the candidate trains. The ticket prices for each train were constrained with upper and lower limits, while the seat allocations were governed by the passenger service capacity of each train. Specifically, in the small-scale case study, during each iteration process, the total seat allocations for each train across all serviceable O-D pairs were set to match the train’s service capacity.

Based on the input data of the small-scale case study, the convergence curve of the model’s objective function value is shown in Figure 4. The algorithm reaches convergence after 598 s of CPU time and 331 iterations, with no increase in the objective function value after the 192nd iteration. The curve illustrates that the objective function value converges from an initial loss of RMB −88,510.00 to a final profit of RMB 46,912.70, indicating an improvement of RMB 135,422.7 (153.00% increase). This demonstrates the suitability of the designed simulated annealing framework and linear passenger flow allocation algorithm for the model, showing its ability to converge to a final solution rapidly and improve significantly from the initial solution.

In the final solution of the comprehensive optimization strategy, seven trains were selected for operation.

Table 5. The revenue information for the seven trains that are ultimately operated.

Train Number	Initial Stopping Plan	Final Stopping Plan	Final Revenue (RMB Yuan)
9	1-2-3-6-7	1-2-6-7	14,441
12	3-6-7	3-6-7	8803
13	1-2-3-4-5	1-2-3-4-5	41,609.5
14	1-2-3-6-7	1-2-3-6-7	15,663.5
15	1-2-3-4-5	1-2-3-4-5	18,237.5
21	1-2-3-6-7	1-2-3-6-7	24,980.5
22	1-2-3-6-7	1-2-3-6-7	22,418.5

presents the revenue information for the seven trains that were ultimately selected for operation, and all these trains could achieve profitability. Compared to the initial stopping scheme, Train 9 ultimately did not stop at the third intermediate station. This indicates that the model and the corresponding solution method can effectively determine train operations and achieve profitability after integrating ticket pricing and ticket allocation decisions.

Table 6. Initial ticket price scheme and final ticket price scheme of two operating trains.

Operational Train 9				Operational Train 15
O-D	Initial Ticket Price (RMB Yuan)	Final Ticket Price (RMB Yuan)	Difference Value	O-D	Initial Ticket Price (RMB Yuan)	Final Ticket Price (RMB Yuan)	Difference Value
1-2	30	26	−4	1-2	30	28	−2
1-3	65	/	/	1-3	65	89	24
1-6	90	87	−13	1-4	82	112	30
1-7	105	104	−1	1-5	105	105	0
2-3	40	/	/	2-3	40	45	5
2-6	48	66	18	2-4	57	50	−7
2-7	65	89	24	2-5	82	75	−7
3-6	30	/	/	3-4	20	19	−1
3-7	48	/	/	3-5	48	45	−3
6-7	20	18	−2	4-5	30	33	3

presents the initial and final OD ticket prices for Train 9 and Train 15. Overall, some ticket prices between certain O-Ds decreased, some remained unchanged, and some increased. For example, the ticket price for Train 9 from Station 2 to Station 6 increased by RMB 18 due to strong demand in this O-D section, leading to an increase of 108 tickets in the allocation for this section. This indicates that throughout the optimization process, the optimization of train operations, ticket price schemes, and ticket allocation schemes is integrated and mutually influential, and that comprehensive optimization can yield better results.

Table 7. Initial seat allocation scheme and final seat allocation scheme of two operating trains.

Operational Train 9				Operational Train 13
O-D	Initial Seat Allocation	Final Seat Allocation	Difference Value	O-D	Initial Seat Allocation	Final Seat Allocation	Difference Value
1-2	50	72	22	1-2	50	70	20
1-3	60	/	−60	1-3	80	1	−79
1-6	70	72	2	1-4	40	3	−37
1-7	80	77	−3	1-5	120	109	−11
2-3	60	/	−60	2-3	50	2	−48
2-6	50	158	108	2-4	50	64	14
2-7	60	154	94	2-5	100	90	−10
3-6	60	/	−60	3-4	30	71	41
3-7	50	/	−50	3-5	50	92	42
6-7	60	67	7	4-5	30	98	68

presents the initial and final seat allocations for two of these operating trains. Changes in the train stopping schemes compared to the initial solution led to adjustments in the seat allocations among the serviceable O-D pairs. For instance, operational Train 9 reduced the number of serviceable O-D pairs related to Station 3, reallocating some of its seats from reduced O-D pairs to those with higher demand and profitability. Operational Train 13, in contrast, showed no changes in its stopping scheme between the initial and final solutions, but seats increased for 5 out of 10 serviceable O-D pairs. This demonstrates how passenger choices in the travel network influence the distribution of passenger loads across the operating trains and highlights how the neighborhood search strategy effectively manages and allocates seat quotas on trains.

Next, using the same data inputs and based on the HSR line planning and differentiated pricing model in Reference [14] and its designed solving algorithm, the corresponding objective function convergence curve was obtained. This curve was then compared synchronously with the convergence curve shown in Figure 4 and displayed in Figure 5. This comparison visually illustrates the differences and gaps between the model results with and without seat allocation integration. As shown in Figure 5, the graph displays a comparison of the convergence curves for the model from Reference [14] and model from this paper. In the graph, the blue line represents the convergence curve of the comprehensive optimization model under a small-scale case study from this study, while the gray line represents the convergence curve of the optimization model without seat allocation under the same small-scale conditions. Under the same input data conditions, the final objective function value of the blue line is RMB 46,912.70, and that of the gray line is RMB 38,532.60, with an increased gap in objective function values between the two models of RMB 8380.10, or 21.75%. This indicates that under the current case study, the optimization effect of the optimization of line planning integrating ticket pricing and seat allocation decisions is superior to that of the optimization of line planning and ticket pricing, as the integrated model achieves better optimization results and obtains a larger objective function value. Additionally, the objective function value of the gray line does not change after the 44th iteration, whereas for the blue line, it remains unchanged after the 192nd iteration. This suggests that incorporating seat allocation decisions and their constraints into the model makes the solution and neighborhood search processes more complex, requiring more iterations and CPU time for the model to converge.

5.2. Sensitivity Analysis

5.2.1. Impact Analysis of $\omega$

The value of the weight coefficient $\omega$ in the objective function influences the balance between the total revenue $R$ of the railway company and the total travel cost $\Psi$ of the passengers. By keeping other data inputs and parameters constant and continuously changing the value of $\omega$ in the objective function, the trend of changes in the objective values under different weight coefficients can be observed. Figure 6 displays the trend of the objective function values as $\omega$ varies from 0.65 to 1. It is evident from the graph that both the objective function value and the total travel cost of passengers increase as the weight coefficient increases. However, the total revenue of the railway company first increases and then decreases with an increase in the weight coefficient.

When the weight coefficient is set to 0.65, there is an unexpected negative value in the objective function, which does not meet the target expectations, suggesting that the weight coefficient should not be set lower than 0.65. When $\omega$ equals 0.8, the objective function value reaches a relatively high level, maximizing the total revenue of the railway company while maintaining the total travel cost of passengers at a moderate level. When $\omega$ equals 1, the objective function value equals the total revenue of the railway company, but correspondingly, the total travel cost for passengers is at its maximum. In practical optimization processes, to consider the interests of both the railway company and the passengers, it is recommended that $\omega$ be set around 0.8. This setting allows for a balanced consideration of maximizing company revenue while controlling passenger costs to a reasonable level, thus achieving an optimal balance in the model’s objectives.

5.2.2. Impact Analysis of Passenger Service Capacity

The passenger service capacity of candidate trains determines the total seat allocation for each operating train across all serviceable O-D pairs, which significantly influences the level of service provided by the train and passengers’ travel choices. Figure 7 illustrates the trend in the objective function value as the train passenger service capacity varies from 600 seats per train to 1200 seats per train in increments of 100. The objective function value increases with the increase in train service capacity when the capacity ranges between 400 and 800 seats per train. However, when the train service capacity exceeds 900 seats per train, the increase in the objective function value nearly stagnates with further increases in service capacity.

Within a certain range, as shown in the graph where the train service capacity is below 900 seats per train, the increase in available seats enriches passenger travel choices. An increase in train capacity can lead to fewer operating trains carrying more passengers, thereby reducing both the total operating costs for the railway company and the overall travel costs for passengers. Once the train service capacity exceeds a certain threshold, due to constraints imposed by ticket price boundaries and passenger travel preferences, the objective function value ceases to change.

5.3. Case Study Based on the Beijing–Zhengzhou and Beijing–Taiyuan HSR

5.3.1. Data Preparation

This large-scale case study considers a real HSR network consisting of two lines connecting Beijing to Zhengzhou and Beijing to Taiyuan. This network is shown in Figure 8, shaped like a ‘Y’, includes two HSR lines, 13 stations, and 12 segments, with five stations shared by both lines. The data input focuses solely on the southbound direction from Beijing West to Zhengzhou East and from Beijing West to Taiyuan South. The Beijing West and Shijiazhuang stations can serve as departure stations, while Shijiazhuang, Taiyuan South, Handan East, and Zhengzhou East can serve as terminal stations.

Moreover, the objective function’s weight coefficient $\omega$ was set at 0.8. Each train’s passenger service capacity was 1200 seats, and the operational cost per train ${\phi }_f$ was set at RMB 30,000 per train. The dwell time $w_f^k$ at each stop for each train was 5 min, the minimum transfer time at stations ${{\rm Y}}_k$ was 30 min, the elasticity coefficient for each O-D pair in each passenger flow period ${\eta }_{r,s}^h$ was set to 1, and the time cost parameter $\alpha$ was 0.5. Based on the spatial and temporal distribution of demand within the physical railway system and the actual requirements of the lines, a candidate set consisting of 61 trains was predefined. Figure 9 shows the initial demand data of the example. Each column shows the total number of passengers at each station during the corresponding passenger departure period. In terms of passenger flow period, a day was divided into five periods (one passenger flow departure period every three hours, for example, 6:00–8:59). The input data included 66 O-D pairs, corresponding to a total of 15,040 passengers.

5.3.2. Convergence Analysis

As illustrated in Figure 10, the objective function value experiences a pronounced and continuous improvement over 331 iterations and 32.78 h of CPU time. Initially, the solution remains below zero (indicating a net loss), but by the 40th iteration, the approach surpasses the break-even threshold, reaching RMB 396,700.00. Following this swift ascent, the convergence progresses more gradually, with an apparent plateau emerging around the 182nd iteration. Ultimately, the objective function settles at RMB 819,346.30, representing a 106.54% increase compared to its value at the 40th iteration.

This convergence profile suggests a two-phase optimization process. In the early phase, the algorithm rapidly overcomes suboptimal configurations, exploiting promising neighborhoods to eliminate loss-making solutions. In the later phase, more subtle refinements in train schedules, ticket prices, and seat allocations persist, culminating in a high-profit, near-stationary solution. Such a turnaround—from an initial deficit to substantial profitability—underscores the robustness of the simulated annealing framework coupled with linear passenger flow allocation. More importantly, it affirms the practical viability of our integrated approach, offering railway operators a powerful method to navigate the complex interplay of scheduling, pricing, and demand elasticity in large-scale high-speed rail systems.

5.3.3. Model Performance

Based on the final solution of the current case study, a train operation plan diagram is depicted in Figure 11. In the final solution, 21 trains from the candidate train set were selected to operate, with 17 serving the first line and 4 serving the second line. The operational trains are evenly distributed across the train operation plan diagram, with most trains originating from Beijing West, including 19 from Beijing West and 2 from Shijiazhuang. The majority of trains terminate at Zhengzhou East and Taiyuan South, including 15 at Zhengzhou East, 4 at Taiyuan South, 1 at Shijiazhuang, and 1 at Handan East. This distribution underscores the model’s capacity to prioritize profitable routes and effectively manage railway resources.

Table 8. Some information about the 21 trains that were ultimately operated.

Train Number	Initial Stopping Plan	Final Stopping Plan	InitialDeparture Time	FinalDeparture Time	Final Revenue (RMB Yuan)
1	1-3-5-7-8-9-10-11	1-3-5-9-10-11	07:30	06:00	127,025
2	1-3-5-6-7-8-9-11	1-3-7-8-9-11	07:30	06:16	259,818
3	1-2-3-4-5-6-7	1-2-3-4-5-6-7	07:30	06:22	40,588
4	5-6-7-8-9-10-11	5-6-7-8-9-10-11	07:30	06:33	9401
5	1-2-3-4-5-12-13	1-2-3-4-5-12-13	07:30	06:44	164,790
9	1-2-5-6-7-10-11	1-2-5-6-7-10-11	07:30	07:28	76,463
10	1-3-5-7-8-9-10-11	1-3-5-7-8-9-10-11	07:30	07:39	39,136
11	5-6-7-8-9-10-11	5-6-8-9-10-11	07:30	07:50	62,119
12	1-2-3-5-7-9-11	1-2-3-5-7-9-11	07:30	08:01	71,170
13	1-3-4-5-7-9-11	1-3-4-5-7-9-11	10:30	09:00	107,904
14	1-2-3-5-6-7-8-10-11	1-2-3-5-6-7-8-10-11	10:30	09:10	150,924
15	1-3-5-7-8-10-11	1-3-7-8-10-11	10:30	09:33	456,536
16	1-3-5-6-7-8-10-11	1-3-6-7-8-10-11	10:30	09:42	60,982
19	1-2-3-4-5-12-13	1-2-3-4-5-12-13	10:30	10:24	229,937
21	1-2-3-5-6-7-10-11	1-2-3-5-6-7-10-11	10:30	10:52	−504
25	1-2-3-4-5-12-13	1-2-3-4-5-12-13	10:30	11:43	−11,450
26	1-2-3-4-5	1-4-5	10:30	12:00	2453
28	1-2-3-5-7-8-11	1-2-5-7-8-11	13:30	12:09	41,800
31	1-3-4-5-12-13	1-4-12-13	13:30	12:52	912
41	1-3-5-6-7-9-11	1-3-5-6-7-9-11	13:30	13:44	74,753
57	1-5-7-8-9-11	1-5-7-8-9-11	15:30	14:16	9625

displays the initial stopping plan, final stopping plan, initial departure time, final departure time, and the final revenue information for the 21 trains that were ultimately selected for operation. From the changes in the stopping scheme of the trains, it can be observed that five trains had less intermediate stops in their final stopping scheme compared to the initial ones. This indicates that during the optimization process, trains can reasonably adjust their stopping schemes based on the revenue situation between O-D pairs. From the departure times at the origin stations, it can be seen that each train is able to choose an appropriate departure time within the given range based on passenger train choices. This also indicates that the model can better match time-dependent demand, allowing for train departure times to be arranged according to passenger choices during different time periods. From the final revenue of the trains, 19 trains achieved profitability, indicating that the majority of the operational trains in the final solution had good service quality. However, two trains incurred losses, which was due to the need to deploy an additional train during a certain time period to better meet passenger demand, ensuring that passengers could complete their journeys.

Table 9. Initial and final seat allocations for two operating trains.

Operational Train 9				Operational Train 31
O-D	Initial Seat Allocation	Final Seat Allocation	Difference Value	O-D	Initial Seat Allocation	Final Seat Allocation	Difference Value
1-2	50	22	−28	1-3	70	/	−70
1-5	300	23	−277	1-4	50	5	−45
1-6	10	10	0	1-5	300	/	−300
1-7	200	25	−175	1-12	70	294	224
1-10	10	10	0	1-13	407	629	222
1-11	200	29	−171	3-4	10	/	−10
2-5	10	7	−3	3-5	50	/	−50
2-6	10	10	0	3-12	5	/	−5
2-7	10	10	0	3-13	10	/	−10
2-10	10	7	−3	4-5	50	/	−50
2-11	10	7	−3	4-12	3	5	2
5-6	70	167	97	4-13	5	5	0
5-7	100	389	289	5-12	40	/	−40
5-10	10	28	18	5-13	90	/	−90
5-11	10	22	12	12-13	40	262	222
6-7	10	64	54	—	—	—	—
6-10	10	10	0	—	—	—	—
6-11	10	61	51	—	—	—	—
7-10	10	17	7	—	—	—	—
7-11	50	222	172	—	—	—	—
10-11	100	60	−40	—	—	—	—

lists the initial and final seat allocations for two of these 21 operating trains. For operational Train 9, there was no change in the stopping plan between the initial and final solutions. Among the 21 serviceable O-D pairs, 8 experienced an increase in seat allocations, 8 saw a decrease, and 5 remained unchanged. Operational Train 31, compared to the initial solution, reduced stops at stations 3 and 5, which resulted in the reallocation of seat allocations from these stations to other serviceable O-D pairs. Changes in seat allocations among these pairs included both increases and decreases, with the reduced allocations being redistributed to O-D pairs with higher demand and profitability. This indicates that the neighborhood search strategy effectively manages and reallocate seat allocations on trains.

From a railway operations management perspective, these findings highlight the strategic value of integrating line planning, ticket pricing, and seat allocation within a single optimization framework. First, operators can pinpoint underperforming stations and adjust train stopping patterns accordingly, reducing wasteful capacity. Second, the ability to modify departure times ensures that services coincide with peak demand windows, thus maximizing ridership. Third, dynamic seat reallocation prevents seat underutilization or oversupply in lower-demand segments, leading to improved revenue yields and higher passenger satisfaction. Overall, this joint optimization approach constitutes a powerful decision support tool, aiding high-speed railway enterprises in achieving balanced objectives of profitability, service quality, and resource efficiency.

5.3.4. Stop Frequency at Station

Figure 12 compares the total passenger flow volume (ranging from 150 to 8361) and the number of train stops (ranging from 4 to 19) at each station in the final solution. This broad disparity underscores the network’s heterogeneous travel patterns: high-demand nodes, such as Beijing West (8361 passengers), correspond to more frequent stops (up to 19), while lower-demand locations (e.g., Yangquan North, with 150 passengers) receive fewer stops. The alignment between ridership levels and stop frequencies highlights a demand-driven allocation of train resources.

Notably, intermediate stations (e.g., Shijiazhuang, Xintai East, Handan East) exhibit fluctuating passenger volumes, prompting context-sensitive adjustments to stop frequencies. High-demand segments benefit from additional services, improving load factors and overall revenue, while stations with fewer passengers experience fewer stops to minimize idle capacity. Through this selective scheduling, the model efficiently directs operational focus to nodes with the greatest revenue potential or passenger uptake.

Overall, these results demonstrate the integrated framework’s capacity to optimize train services at a granular station level. By synchronizing stops with actual passenger flow, the model ensures that high-demand stations receive adequate coverage, thereby reducing wait times and bolstering profitability. From a practical operations viewpoint, such adaptability allows railway operators to respond swiftly to evolving demand patterns, balancing service quality and resource efficiency.

6. Results

This paper comprehensively considers the impact of train O-D seat allocation on the problems of high-speed train line planning and ticket pricing. Starting from the perspective of improving the operational level of railway companies and the service quality of travel demand, a line planning optimization model that integrates ticket pricing and seat allocation decisions was developed. The model takes into account the impact of train capacity, ticket prices, and operating time periods on train operation choices and passenger travel choices. A heuristic algorithm based on the simulated annealing framework and linear passenger flow allocation method was designed to solve the optimization model, taking into account the interrelationships between train operations, train stops, initial departure times of trains, train ticket prices, and seat allocations. Seven effective neighborhood structures were proposed to effectively search for neighborhood solutions. Case studies were conducted using a virtual railway network and a real high-speed railway network composed of the Beijing West–Zhengzhou East and Beijing West–Taiyuan South lines.

Integrated with ticket pricing and seat allocation decisions, our heuristic algorithm demonstrated strong adaptability by increasing the objective function value by 106.54%, thereby refining railway operations under time-dependent passenger demand. In the final solution, 21 trains were operated, 19 of which were profitable, while certain trains had less intermediate stops to better align with revenue potential. Concurrently, seat allocations were redistributed from low-demand segments to higher-profit O-D pairs, reflecting the model’s capacity to dynamically adapt line plans. Stations experiencing greater passenger flow received more frequent stops, whereas stations with lower demand were served less often, thus enhancing overall resource utilization. Collectively, these outcomes confirm that the integrated method can boost total revenue for railway operators and improve travel service quality for passengers, underscoring the practical effectiveness of this approach in high-speed railway networks.

For the future, there remain many issues worth further research and exploration. (1) Operating high-speed trains have various formations and multiple seat classes, and there is passenger flow transfer between different seats on the train, such as some passengers opting for first-class seats when second-class seats are sold out. However, this paper only considers a single-train formation and a single seat class. Further research could consider the optimization for multiple-train formations and multiple seat classes. (2) When making travel choices, passengers consider various factors such as travel time and ticket prices. At the same time, other modes of transportation, such as aviation and road, also compete with high-speed rail to some extent. Future studies could consider optimizing the problem under competitive multi-transport models. (3) Future research could explore hybrid approaches that incorporate convex relaxation techniques to approximate certain nonlinear components before applying heuristic optimization, potentially improving computational efficiency without compromising solution feasibility.