Collaborative Optimal Train Carriage Flexible Release Strategy and Passenger Flow Control Strategy for the Metro System

Abstract

In the oversaturated metro system, the mismatch between supply and demand leads to unequal allocation of train capacity at different stations, resulting in a transportation inequity issue. This paper proposes a collaborative optimization method to use train carriage flexible release strategy and passenger flow control strategy, which is described as a mixed‐integer nonlinear programming (MINLP) model considering the trade‐off between equity and efficiency. To solve this model, it is reformulated into a mixed‐integer linear programming (MILP) model, which is solved by the GUROBI solver. An efficient variable neighborhood search algorithm is then proposed to find a high‐quality solution to the proposed problem. Finally, two sets of numerical experiments, including a small‐scale case and a real‐world case of Chengdu metro system, are conducted to verify the proposed model. The experimental results show that the train release scheme and passenger flow control scheme generated by our proposed method can perform well on the trade‐off between equity and efficiency.

1. Introduction

The rapid development of the metro system can be attributed to its numerous advantages, such as large capacity, high speed, eco-friendliness, and enhanced security, making it vital in alleviating severe traffic congestion in megacities [1]. However, with the acceleration of urbanization, the passenger demand for the metro system increases rapidly, resulting in a spatial and temporal mismatch between supply and demand, especially during rush hours. In response to the significant demand pressure, further operation schemes should be carried out by the operation and management departments of the metro system to improve service quality and safety.

From the supply perspective, a wide range of studies have focused on designing train capacity allocation strategies for the metro system to mitigate passenger congestion. A practical approach to enhance service quality is to optimize the supply-oriented timetable. By adjusting the arrival and departure times of trains, a well-designed timetable alleviates passenger congestion at busy stations. For example, focusing on reducing the number of stranded passengers, Niu and Zhou [2] proposed a train scheduling method based on the time-varying OD passenger flow data. Sun et al. [3] developed optimization models to reduce passenger traveling time. Beyond timetable studies, other researchers have explored various flexible train capacity allocation techniques, such as skip-stop program [4], rolling stock circulation planning [5], and express/local train strategy [6]. These methods provide guidance for practical operation, but it is difficult to implement them accurately because a large number of staff and equipment in different stations need to cooperate. Recently, a flexible train capacity allocation measure involving reserving and releasing carriages has been proposed to balance the spatial and temporal distribution between train supply and passenger demand [7, 8]. Before the train is put into operation at the departure station, some reserved carriages are closed, and the reserved carriages are gradually released according to the passenger demand during the train’s travel from the upstream station to the downstream station. In this way, the train capacity can be ensured through an equitable allocation of train capacity across the entire line [8]. Although this strategy has been improved, it only allocates the train capacity from the perspective of supply, ignoring the demand side.

From the demand perspective, imposing passenger flow control is another widely used strategy to alleviate passenger congestion issues in the metro system. The essence of the passenger flow control strategy is to limit the number of passengers entering the platform from the entry hall to wait for the train, allowing only a part of them to access the platform through the entry gates. This strategy can effectively improve the efficiency of train operations, such as reducing passenger travel time or reducing the number of stranded passengers at the station. For instance, to reduce passenger waiting time or the number of stranded passengers at stations, some researchers have established corresponding optimization models by considering the passenger flow control strategy [1, 9, 10]. Besides, some researchers have focused on minimizing the passenger accumulation risks at all the involved stations and the total passenger waiting time with passenger flow control [11]. It is worth mentioning that the implementation of passenger flow control strategy depends not only on the distribution of demand but also on the capacity of trains. Therefore, to effectively tackle the supply–demand mismatch problem, it becomes crucial to collaboratively optimize both train capacity allocation and the passenger flow control strategy from both supply and demand perspectives. However, previous studies have primarily focused on optimizing transportation efficiency, often overlooking transportation equity. Notably, in the oversaturated metro system, empty trains departing from the departure station cannot accommodate passengers from downstream stations. The inequity arises from the overwhelming number of passengers boarding at upstream stations, which rapidly fills the train’s capacity [12]. In this case, there is an unequal allocation of train capacity between upstream and downstream stations. Consequently, a substantial disparity emerges between the number of stranded passengers at upstream and downstream stations, leading to varying travel priorities for passengers across different stations. This disparity significantly diminishes the metro system’s service level and obstructs its sustainable development.

To address these issues, this paper proposes a collaborative optimization approach that dynamically allocates train capacity and implements passenger flow control strategy from supply and demand perspectives. A MINLP model is developed to enhance transportation equity in the metro system while maintaining efficiency. The main contributions of this paper are as follows:

• (1)

  This paper proposes a collaborative optimization method to rationally allocate the train capacity resources by using train carriage flexible release strategy and passenger flow control strategy. A MINLP model is formulated to describe the passenger loading dynamics with train carriage flexible release constraints and passenger flow control constraints. The objective is to minimize the maximum number of stranded passengers at the station while also ensuring the total number of stranded passengers at all stations, thus ensuring a trade-off between equity and efficiency.

• (2)

  To solve the complex MINLP model with nonlinear constraints, this paper first reformulates it into a MILP model, which can be effectively solved by a commercial solver (i.e., GUROBI). However, due to the limitation of commercial solvers on the scale of the model, large-scale examples cannot get the optimal solution in an effective time. Therefore, a VNS meta-heuristic algorithm is designed to solve the nonlinear programming model efficiently, and the good solutions of the model can be obtained in a short time. Through the above two solving methods, we can flexibly and efficiently meet the needs of different solving efficiencies in various scale application scenarios.

• (3)

  To verify the effectiveness of the strategy proposed in this paper, we apply the collaborative optimization strategy to a small-scale case and a real-world case of the Chengdu metro system. It is verified that the collaborative optimization model proposed in this paper can perform well on the trade-off between equity and efficiency. Furthermore, by comparing the calculation time and effect of the GUROBI solver and VNS algorithm for different scale cases, it is verified that VNS algorithm can quickly find the good solutions of large-scale case. The collaborative optimization model is applied to Chengdu Metro Line 2 again. The results show that the carriage release scheme and passenger flow control scheme obtained after implementing the collaborative optimization strategy can effectively improve the balanced distribution of train capacity resources in Chengdu Metro Line 2 and enhance transportation equity.

The rest of this paper is arranged as follows: The Literature Review part reviews the related literature. The Mathematical Formulations part introduces the formulation process and methodologies of the proposed model in detail. The Solution Approach part designs the solution methods of the model. The Numerical Experiments part conducts two sets of numerical experiments with a small-scale case and real-world operation data from the Chengdu metro system to verify the proposed model. The Conclusions part concludes this paper and provides an outlook for future research.

2. Literature Review

The inefficiency and inequity of passengers derive from the mismatch between supply and demand in the oversaturated metro system. A variety of studies have been proposed to tackle these issues. This subsection will review the related studies about train capacity allocation optimization, passenger flow control optimization, and transportation equity optimization.

2.1. Train Capacity Allocation Optimization

Train capacity allocation is a systematic process, which uses optimization methods to effectively allocate limited train resources (such as seats). It allocates these resources to specific transportation needs, such as passenger flow between stations. To better allocate train resources, many scholars have put forward optimization strategies, such as the rolling stock circulation planning [5] and the train timetable (Niu and Zhou [2]; Sun et al. [3]). Currently, these strategies primarily aim to tackle three crucial issues: enhancing transportation efficiency, optimizing corporate operating costs, and mitigating the risk of passenger accumulation.

To improve transportation efficiency, Hassannayebi et al. [13] introduced a path-indexed nonlinear formulation to address the train timetable problem and minimize the average waiting time per passenger. Cadarso and Marín [14] took a different approach by integrating train timetable planning and rolling stock allocation, leading to the development of the RT&RS model, which aimed to minimize operating and leasing costs. In a similar vein, Niu et al. [4] developed a uniform quadratic integer programming model to optimize train timetable with given time-varying start-to-destination passenger demand data and train skip-stop patterns, aiming at minimizing the total passenger waiting time.

In pursuit of cost reduction, Yang et al. [15] proposed a cooperative scheduling method that synchronizes the acceleration and braking times of consecutive trains so that the energy generated by braking can be recovered and developed a cooperative scheduling model to maximize the overlap time. Su et al. [16] proposed an integrated train operation method by jointly optimizing the train timetable and driving strategy and developed an integrated energy-saving optimization model to reduce the metro system’s net energy consumption.

Moreover, several researchers have sought to optimize transportation efficiency while mitigating the risk of passenger congregation at stations. For instance, Shi et al. [8]; Guo et al. [7] introduced a flexible carriage reservation strategy and formulated a nonlinear integer programming model. They aimed to minimize the total passenger waiting time and the cumulative passenger risk at each station. By adopting this approach, they enhanced the overall performance and safety of the metro system during periods of high demand.

2.2. Passenger Flow Control Optimization

To alleviate congestion on station platforms, stabilize passenger flow, and improve service quality, recent studies have shown interest in designing a passenger flow control strategy. For example, Meng et al. [17] proposed a robust passenger flow control strategy based on random dynamic passenger flow and developed an integer linear programming model to minimize the passenger’s expected waiting time. Yuan et al. [18] proposed a passenger flow control model based on the network-level system to reduce passenger traveling time and total waiting time. Jiang et al. [9] adopted the passenger flow control strategy based on a reinforcement learning approach to minimize the safety risk for passengers at metro stations. Lu et al. [19] developed a two-stage distribution robust optimization (DRO) model based on the consideration of a mean conditional value-at-risk (CvaR) criterion to optimize the expected number of passengers waiting for the train and the congestion risk at the stations. Shi et al. [11] proposed a collaborative passenger flow control optimization approach under a specific metro network and developed a biobjective integer linear programming model to minimize the total passenger waiting time and the risk of passenger aggregation at all stations. Yoo et al. [20] developed a linear dwell time model that considered the passenger flow control strategy to reduce the delay time of scheduled train dwell and passenger traveling time.

The above studies mainly optimize from the demand perspective and ignore the supply perspective. Therefore, to make up for this deficiency, some researchers have proposed optimization methods by combining supply and demand. First, combining train timetable and the passenger flow control strategy, Shi et al. [10] formulated a rigorous integrated integer linear programming model to minimize the total passenger waiting time at all of the involved stations; Li et al. [21] proposed a dynamic optimization problem to minimize the schedule and headway deviation of metro lines; Yuan et al. [22] developed a MILP model to minimize the total number of waiting passengers. Some researchers also considered the train skip-stop strategy. Jiang et al. [23] developed an optimization model by incorporating the passenger flow control strategy and the train skip-stop strategy to minimize the number of stranded passengers on the whole line. Hu et al. [24] investigated the co-optimization problem of the metro line schedule, passenger flow control, and skip-stop mode. A MINLP model is proposed, which requires a performance trade-off between service levels and operating costs. In addition, there are also some researchers considering train timetable. Liu et al. [1] proposed a collaborative optimization method for metro train timetable and train connection and developed a MINLP model in combination with passenger flow control strategy to minimize the number of stranded passengers. Zhang et al. [25] developed a multiobjective MINLP model by collaboratively optimizing train timetable, passenger control, and vehicle schedules to minimize the waiting time, the difference of bidirectional platform passenger density, and the use cost of trains. Zhou et al. [5] proposed a joint optimization method for train timetable and vehicle circulation planning, considering the passenger flow control strategy on tidal oversaturated metro lines, and developed a MINLP model to simultaneously minimize passenger waiting time and the operating cost of the metro system.

However, most studies focus on optimizing passenger efficiency or operational cost while neglecting passenger equity, a crucial factor in enhancing service quality.

2.3. Transportation Equity Optimization

Recently, transportation equity has garnered significant attention from researchers concerned with passenger interests. Transportation equity refers to striving for equal opportunities in passengers’ participation in socioeconomic activities, especially when social transportation resources are limited. However, several studies have revealed significant inequalities concerning workplace reachability, commuting time via vehicles and public transport, and disparities among various locations in metropolitan areas (Kawabata and Shen [26]). Particularly in the oversaturated metro system, transportation inequity is evident.

Therefore, to improve transportation equity, some researchers have adopted corresponding optimization strategies. For instance, Liang et al. [27] developed real-time passenger flow control policies to manage the inflow of crowds at each station to optimize the total load carried or revenue earned (efficiency) and to ensure that adequate service is provided to passengers on each o-d pair (equity), as much as possible. Wu et al. [28] proposed a train timetable optimization model to reduce the waiting time and the equity of waiting time at all transfer stations in the metro network. Li et al. [29] analyzed two equity criteria, min-max equity and α-equity in the metro system, and they developed a MILP model to optimize the train timetable based on the trade-off between efficiency and equity. Shang et al. [12] optimized the train skip-stop pattern to achieve an equitable distribution of all passengers. Wu et al. [30] mitigated the inequality of time-varying passenger waiting time by optimizing the train skip-stop patterns and train timetable. Han et al. [31] developed a multiobjective MINLP model to find the optimal train skip-stop plan and train timetable to optimize train inequality and total travel time. Huan et al. [32] developed mathematical programming with equilibrium constraints (MPEC) approach to optimize a demand-responsive passenger flow control strategy, highlighting the importance of balancing the operational efficiency of the metro system with service equity perceived by passengers. Gong et al. [33] combined the train timetable and the passenger flow control strategy to optimize the equity of the passenger service. Lu et al. [34] studied the joint optimization of the robust passenger flow control strategy and train timetable and developed deterministic models of train timetables and passenger flow control at each station to trade-off between operational efficiency and service equity.

2.4. Comparison With Relevant Studies

Passenger flow control and train capacity allocation are valid strategies to improve the service level of the metro system. This paper aims to develop an equity-oriented collaborative optimization model of capacity dynamic allocation and passenger flow control strategy for the metro system. Table 1 summarizes the related works in terms of their problems, objectives, model characteristics, and solution methods. From the perspective of optimization strategy, compared with most early literature, this paper considers a novel train resource allocation method proposed by Shi et al. [8]: train carriage flexible release strategy. But that literature only considers the supply perspective. This paper integrates the passenger flow control strategy on the basis of this strategy. The advantage of collaborative optimization is that both supply and demand perspectives have promoted the balanced distribution of train capacity. From the perspective of research objectives, this paper focuses on the game relationship between efficiency and equity. In this paper, equity or efficiency is not calculated as a penalty term, but weight coefficients are set for efficiency and equity in the objective function of the model. Whether it is more concerned with equity or efficiency, the model established in this paper can be better adapted. In actual operation, it can be flexibly adjusted according to the needs of metro operation organizations. In summary, this paper proposes a model that takes into account both train carriage flexible release strategy and passenger flow control strategy, encompassing both efficiency and equity aspects.

Table 1 The comparison of different publications of related models and methods.

Literature	Problems	Objectives	Consideration of equity	Model	Solution method
Shi et al. [8]	Train carriage flexible release	Total passenger waiting time	N	INLP	VNS
Cumulative risk value for passengers at each station
Cost of reserving carriages
Shi et al. [11]	Passenger flow control	Passenger waiting time	N	ILP	CPLEX
Risk level of passenger accumulation
Yuan et al. [18]	Train timetable	Train utilization rate	N	MINP	LR
Train circulation plan	Number of passengers controlled
Passenger flow control	Number of passengers waiting on the platform
Liang et al. [27]	Passenger flow control	Ex ante fill rate target for every O-D pair	Y	DP	OPFCwF
Li et al. [29]	Train timetable	The maximum average waiting time	Y	MINLP	SA + NS
Summation of logarithm of average waiting time dividing the in-vehicle times for all OD pairs
Yang et al. [35]	Train timetable	Total number of passengers waiting for trains	N	MILP	CPLEX
Passenger flow control
This paper	Passenger flow control	Total number of stranded passengers at stations	Y	MINLP	GUROBI VNS
Train carriage flexible release	Number of stranded passengers at the most congested station

3. Mathematical Formulations

3.1. Problem Statement

In the studied problem, we consider a single-directional metro line with a total number of |K| stations. The stations are numbered K = {1, 2, …, i, …|K|} consecutively, in which K denotes the set of stations and k denotes the index of stations. The trains are numbered as I = {1, 2, …, i, …|I|} consecutively, in which I denotes the set of trains, and i denotes the index of stations. The carriages are numbered as J = {1, 2, …, j, …|J|} consecutively, in which J denotes the set of carriages. For the overcrowded metro lines, passenger oversaturation usually happens during the morning peak hours (i.e., about 7:00 a.m.–9:00 a.m.) and evening peak hours (i.e., about 5:30 p.m.–7:30 p.m.). In this study, we simplify the analysis by considering passenger flow and service trains in one direction. A similar strategy can be applied in the opposite direction.

To illustrate the phenomenon of transportation inequity, a simple example is used to introduce transportation inequity, as shown in Figure 1. In this case, there are 3 trains and 4 stations, and the capacity of each train is set to two passengers. The passenger flow of each station is as follows: Three passengers arrive at Station 1, numbered ①, ②, and ③ according to the arrival time, and their destinations are Station 4, Station 3, and Station 2, respectively. Four passengers arrive at Station 2, numbered ④, ⑤, ⑥, and ⑦ according to the arrival time, and the first two passengers’ destination is Station 4, and the others’ destination is Station 3. Three passengers arrive at Station 3, numbered ⑧, ⑨, and ⑩ according to the arrival time, and their destinations are Station 4. It should be noted that the arrival time of all passengers is earlier than the time when the first train arrives at the departure station, and the principle of first come, first served is followed.

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Without optimization strategy, the passenger loading process is shown in Figure 1. In this case, the total number of stranded passengers at three stations is 1, 6, and 4, respectively, with a variance of 4.2, which is an important index to measure the transportation inequity. Because Station 1 occupies most or all of the train capacity, there are more stranded passengers at Station 2 and Station 3 than at Station 1. To improve the above situation, we take Strategy 1. At Station 1, we restrict that only one passenger can take the train. After the implementation of Strategy 1, the passenger loading process is shown in Figure 2. It can be seen that passenger ④ is stranded at Station 2 due to the capacity limitation of Train 1 without strategy. However, after taking Strategy 1, passenger ④ got the opportunity to take Train 1. After all train services, the number of stranded passengers at three stations is 4, 5, and 5, respectively. Compared with that without strategy, the number of stranded passengers at Station 1 increases by 3, that at Station 2 decreases by 1, and that at Station 3 increases by 1. The variance has been reduced from 4.2 to 0.2, and the inequity improves by 95%.

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From the above introduction, it can be concluded that it is necessary and meaningful to adopt optimization strategy to improve transportation equity when there is a mismatch between passenger demand and train supply. In addition, a problem can be found in the above case: although the equity increases by 95%, the total number of stranded passengers at all stations increases from 11 to 14. The total number of stranded passengers at all stations is an important index to express the efficiency of train operation. Therefore, the transportation efficiency before and after the implementation of Strategy 1 decreases by 27%. The decrease in efficiency mainly depends on the restriction of the train resources at the upstream station, which restricts the passengers at the upstream stations from boarding.

Obviously, we cannot just pursue the improvement of equity and ignore efficiency. Therefore, in view of this case, we take Strategy 2 which considers both equity and efficiency. We restrict only one passenger can take Train 1 at Station 1, but Train 2 and Train 3 do not restrict passengers from boarding. After the implementation of Strategy 2, the passenger loading process is shown in Figure 3. After the service, the number of stranded passengers at three stations is 2, 5, and 5, respectively. The comparison results of the strategies are shown in Table 2. Compared with that without strategy, the equity is increased by 52%, and the efficiency is decreased by 9%. Compared with strategy 1, the equity is decreased, but the efficiency is obviously improved.

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Table 2 Data comparison with or without strategies.

	The number of stranded passengers	Variance	Equity	Efficiency
Station 1	Station 2	Station 3	Sum
Without strategy	1	6	4	11	4.2	—	—
Strategy 1	4	5	5	14	0.2	95%	−27%
Strategy 2	2	5	5	12	2.0	+52%	−9%

Through the introduction of relevant theories, it can be further clarified that there is an inequity phenomenon in the metro system. Therefore, to solve the problem of transportation inequity, this paper comprehensively considers two perspectives of supply and demand. The collaborative optimization process of train carriage flexible release strategy and passenger flow control strategy is shown in Figure 4. The optimization objective of this strategy is a trade-off between equity and efficiency from the perspective of the system through the reservation of carriages and the control of passenger flow on the whole line. From the supply perspective, when the total train capacity is determined, this paper adopts train carriage flexible release to allocate the train capacity dynamically according to the passenger demand. Because the carriages in the train are connected, it is necessary to install an isolation door between any two adjacent carriages, develop an automatic door control function, and embed it into ATO. According to the given carriage condition, the door is automatically controlled. Meanwhile, the side gates on the carriages and platform screen doors should be controlled synchronously to ensure that the carriage is completely closed or opened (Shi et al. [8]). From the demand perspective, under the implementation of passenger flow control strategy, the number of passengers entering the platform from the entrance can be limited according to the remaining capacity of the carriage. Among them, the calculation dimension of train carriage flexible release strategy is based on the number of carriages, while the calculation dimension of passenger flow control strategy is based on a single passenger, which can achieve the purpose of more accurate reservation of train capacity. This improves the balance of the number of passengers served at upstream stations and downstream stations, thus enhancing equity.

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3.2. Assumptions

Before modeling, to simplify the problem, the following reasonable assumptions are made:

$$

Statement

Assumption 1.

Passenger demands are predetermined and given, and passengers do not change their travel plans under the train carriage flexible release strategy and the passenger flow control strategy.

$$

Statement

Assumption 2.

On a metro line, the train stops at each station.

$$

Statement

Assumption 3.

All arriving passengers on the line need to wait in the queue outside the platform first and then enter the queue waiting inside the platform.

3.3. Notations and Decision Variables

In our study, there are two sets of decision variables. The first is the intensity of passenger flow control c_i(k), denoted by the number of passengers at station k who are restricted from entering the platform to wait for the train i. The second is the state of the train carriage release $c{o}_i^j\left(k\right)$, using the binary variable to indicate whether carriage j is released when train i arrives at station k. Moreover, the related indices, parameters, and variables are first listed in Table 3. Based on the defined parameters and variables, the detailed optimization model will be given in the following subsections.

Table 3 Indices, parameters, and variables used in this section.

Notation	Definition
i	Index of trains
j	Index of carriages
k	Index of stations
I	{1, 2, …, I}, the set of involved trains
J	{1, 2, …, J}, the set of involved carriages
K	{1, 2, …, K}, the set of involved stations
cpi(k)	Capacity of train i at station k
cc	Train capacity of one carriage
wi(k)	The number of arriving passengers between the departure of train i − 1 and the arrival of train i at station k.
γi(k)	Passenger alighting rate of train i at station k
pwi(k)	Number of stranded passengers at station k after train i departs station k
pji(k)	In the period between the departure of train i at station k and the departure of train i at station k, the number of passengers waiting at the platform at station i under the application of the passenger flow control strategy
pei(k)	Number of passengers boarding train i at station k
psi(k)	Number of alighting passengers of train i at station k
pi(k)	Number of passengers aboard the train i at station k
pyi(k)	Remaining capacity aboard the train when train i arrives at station k
cmax	The maximum intensity of passenger flow control
$c_{\max }^{imp}$	The maximum intensity of passenger flow control at important station
comin	Minimum number of released carriages
η	Maximum number of stranded passengers at a station
Decision variable	Definition
ci(k)	Number of passengers who are controlled to enter the platform of station k and wait for train i
$c{o}_i^j\left(k\right)$	If the carriage of train i is released at station k, $c{o}_i^j\left(k\right)=1$; otherwise, $c{o}_i^j\left(k\right)=0$

3.4. Train Carriage Flexible Release Constraints

The capacity of the train employing the train carriage flexible release strategy can be determined by the number of released carriages, as the capacity of each carriage remains fixed. Therefore, the train capacity cp_i(k) can be expressed as 1 $\begin{array}{}c{p}_i\left(k\right)={\displaystyle \sum }_{j=1}^{J}c{o}_i^j\left(k\right)cc,\forall i\in I,k\in K,j\in J,\end{array}$ where $c{o}_i^j\left(k\right)$ represents the release status of carriage j when train i arrives at station k. If carriage j is released, $c{o}_i^j\left(k\right)=1$; otherwise, $c{o}_i^j\left(k\right)=0$. cc represents the train capacity of one carriage.

During train service, to avoid the situation that train i stops at station k without releasing the carriages (this situation may cause passenger dissatisfaction and reduce the quality of train service), the number of train carriages released at the departure station should be larger than the minimum number of carriages to be released co_min: 2 $\begin{array}{}{\displaystyle \sum }_{j=1}^{J}c{o}_i^j\left(1\right)\ge c{o}_{\min },\forall i\in I.\end{array}$

Moreover, the carriages should not be closed after release. Therefore, it is necessary to consider the constraint that carriages cannot be closed after they are released: 3 $\begin{array}{}c{o}_i^j\left(k\right)\le c{o}_i^j\left(k+1\right),\forall i\in I,k\in K\backslash \left\{\left|K\right|\right\}.\end{array}$

3.5. Passenger Flow Control Constraints

We need to calculate the number of waiting passengers at the platform. Two cases are considered here. First, for the departing train (i = 1), the number of waiting passengers at the platform is determined by the number of passenger arrivals between the departure of train i − 1 from station k and the arrival of train i at station k and the number of passengers outside the platform who are controlled to enter the platform. For the following trains (i > 1), after the previous train leaves the station, the number of passengers waiting at the station is divided into two parts: the passengers stranded pw_i−1(k) and the newly arrived passengers w_i(k). Besides the above two parts, the number of waiting passengers at the platform also depends on the number of passengers outside the platform who are controlled from entering the platform c_i(k). Therefore, the number of passengers waiting at the platform pj_i(k) can be expressed as 4 $\begin{array}{}p{j}_i\left(k\right)=\left\{\begin{array}{cc}{w}_i\left(k\right)-c_i\left(k\right)& \text{if\hspace{0.17em}}i=1\\ w_i\left(k\right)-c_i\left(k\right)+p{w}_{i-1}\left(k\right)& \text{if\hspace{0.17em}}i\in I\backslash \left\{1\right\}\end{array}\right.,\forall k\in K,\end{array}$ where pw_i−1(k) represents the number of stranded passengers at station k after train i − 1 leaves the station.

To implement the passenger flow control strategy more effectively, it is necessary to consider the constraints related to passenger flow control in this model. First, the number of passengers entering the station cannot be controlled without a limit. Therefore, it is necessary to limit the intensity of passenger flow control to less than the maximum value c_max: 5 $\begin{array}{}{c}_i\left(k\right)\le c_{\max },\forall i\in I,k\in K.\end{array}$

In addition, the metro system is closely related to neighborhood planning during the development process. There are a few important stations in a metro line. When adopting the passenger flow control, strategy may cause inefficiency at important stations. Therefore, to avoid this situation, the control intensity of important stations should be less than a certain value $c_{\max }^{\text{imp}}$: 6 $\begin{array}{}{c}_i\left(k_{\text{imp}}\right)\le c_{\max }^{\text{imp}},\forall i\in I,\end{array}$ where k_imp denotes the designated important station.

3.6. Passenger Loading Dynamics

The passenger dynamic loading process refers to Liu et al. [1]. First, we need to calculate the remaining capacity of trains arriving at the station since it directly influences the number of boarding passengers. The remaining passenger capacity when train i arrives at station k depends on the total capacity of the train, the number of passengers aboard the train, and the number of alighting passengers at station k. Therefore, the remaining capacity aboard the train py_i(k) can be formulated as follows: 7 $\begin{array}{}p{y}_i\left(k\right)=c{p}_i\left(k\right)-p_i\left(k\right)+p{s}_i\left(k\right),\forall i\in I,k\in K,\end{array}$ where p_i(k) represents the number of passengers on board when train i arrives at station k, and ps_i(k) denotes the number of passengers who get off at station k on train i.

Specifically, given the alighting rate, the number of alighting passengers ps_i(k) of train i at station k is equal to the product of the number of passengers aboard the train p_i(k) and the alighting rate γ_i(k), calculated as below: 8 $\begin{array}{}p{s}_i\left(k\right)={\gamma }_i\left(k\right)p_i\left(k\right),\forall i\in I,k\in K.\end{array}$

There are two kinds of stations where the train arrives, which are the departure station and other stations. When the train arrives at the departure station (k = 1), there are no passengers on the train, namely p_i(k) = 0. When the train arrives at other stations (k > 1), the number of passengers on board depends on the number of passengers aboard the train at station k − 1 and the number of individuals boarding and alighting the train at station k − 1. Mathematically, the number of passengers aboard the train p_i(k) can be expressed as 9 $\begin{array}{}{p}_i\left(k\right)=\left\{\begin{array}{cc}0& \text{if\hspace{0.17em}}k=1,\\ p_i\left(k-1\right)-p{s}_i\left(k-1\right)+p{e}_i\left(k-1\right)& \text{if\hspace{0.17em}}k\in K\backslash \left\{1\right\}\end{array}\right.,\forall i\in I\end{array}$ where pe_i(k − 1) represents the number of passengers ready to board train i at station k − 1.

In reality, considering the constraints of train capacity, only a limited number of passengers can successfully board the train. Equation (10) gives the relationships among the number of boarding passengers pe_i(k), the train’s remaining capacity py_i(k), and the number of passengers waiting on the platform of station k for train ipj_i(k). When pj_i(k) is less than py_i(k), pe_i(k) is equal to pj_i(k); otherwise, it is equal to py_i(k). Intuitively, pe_i(k) equals the smaller value between py_i(k) and pj_i(k): 10 $\begin{array}{}p{e}_i\left(k\right)=\min \left(p{j}_i\left(k\right),p{y}_i\left(k\right)\right),\forall i\in I,k\in K.\end{array}$

When the first train arrives at the departure station, the number of passengers arriving at the station at this time is w₁(1). The number of stranded passengers can be calculated by subtracting the number of boarding passengers from the number of waiting passengers. When the following train arrives at the departure station, the number of passengers waiting at this station is the number of passengers stranded by the previous train plus the number of newly arriving passengers at this station. After passengers get on and off, the number of stranded passengers is updated to pw_i(k) + w_i(k) − pe_i(k). Therefore, the number of stranded passengers pw_i(k) at the station can be expressed as 11 $\begin{array}{}p{w}_i\left(k\right)=\left\{\begin{array}{cc}{w}_i\left(k\right)-p{e}_i\left(k\right)& \text{if\hspace{0.17em}}i=1\\ p{w}_{i-1}\left(k\right)+w_i\left(k\right)-p{e}_i\left(k\right)& \text{if}i\in I\backslash \left\{1\right\}\end{array}\right.,\forall k\in K\end{array}$

3.7. Objective Function

The optimization objective of this paper is to improve transportation equity while ensuring transportation efficiency.

3.7.1. Efficiency

The number of stranded passengers at a station is an important criterion for the efficiency of train operation. Therefore, operational efficiency can be ensured by minimizing the total number of stranded passengers. The number of stranded passengers at all stations can be described as 12 $\begin{array}{}{T}_1={\displaystyle \sum }_{i=1}^{\left|I\right|}{\displaystyle \sum }_{k=1}^{\left|K\right|}p{w}_i\left(k\right).\end{array}$

3.7.2. Equity

This paper focuses on equity in view of the great difference in the number of stranded passengers at different stations. To improve passenger equity in the metro system, this paper minimizes the maximum number of stranded passengers at the station to alleviate inequity between different stations. The maximum number of stranded passengers at the station can be described as 13 $\begin{array}{}{T}_2=\max \left({\displaystyle \sum }_{i=1}^{\left|I\right|}p{w}_i\left(k\right),\forall k\in K\right).\end{array}$

Therefore, the objective function is set to minimize the weighted sum of the number of stranded passengers at all stations and the maximum number of stranded passengers at the station: 14 $\begin{array}{}\text{mi}\text{n}Z={\omega }_1{T}_1+{\omega }_2{T}_2,\end{array}$ where ω₁ and ω₂ are weights for efficiency and equity. The weighting factor ω₁ and ω₂ is determined by the practical requirements of efficiency and equity in train operation. If the aim is to improve operational efficiency, ω₁ will be larger than ω₂. In contrast, if equity is the primary consideration, ω₁ should be smaller than ω₂. In general, both parameters can be flexibly adjusted according to the optimization purpose or actual operational requirements.

3.8. Complexity of Formulation

The model comprises two types of decision variables. The first type is an integer variable representing the number of passengers restricted from entering the station, and the second type is a binary variable indicating the release state of train carriages. In addition, equations (10) and (13) are nonlinear constraints. That is to say, the proposed optimization model belongs to a MINLP model, which cannot be effectively solved by commercial solvers (e.g., GUROBI). In this paper, two algorithms are designed to solve the MINLP model. The first one is solved using a solver, which requires the model to be linearized first. Therefore, it is necessary to transform the model into a linear one. The second is using the VNS heuristic algorithm to solve the model. In the following subsection, we will introduce the details of the approach.

4. Solution Approach

4.1. Model Reformulation Into Mixed-Integer Linear Programming

To solve the proposed optimization model by commercial solvers, the nonlinear constraint (10) and objective function (13) must be linearized. We use the theories of propositional logical transformations in linear inequalities proposed by Bemporad and Morari [36], considering the following properties of nonlinear constraints.

First, we consider the statement f(x) ≤ 0, where $f:{\mathbb{R}}^n⟶\mathbb{R}$ linear, assume that x ∈ χ, where χ is a given bounded set, and define 15 $\begin{array}{c}{f}_{\max }=\underset{x\in \chi }{\max }f\left(x\right),f_{\min }=\underset{x\in \chi }{\min }f\left(x\right).\end{array}$

Then, we introduce logic variable μ ∈ {0, 1}, and it is easy to verify that 16 $\begin{array}{}\left[f\left(x\right)\le 0\right]⟷\left[\mu =1\right]\text{\hspace{0.17em}is\hspace{0.17em}true\hspace{0.17em}if}\left\{\begin{array}{c}f\left(x\right)\le f_{\max }\left(1-\mu \right),\\ f\left(x\right)\ge \epsilon +\left(f_{\min }-\epsilon \right)\mu ,\end{array}\right.\end{array}$ where ε is a small tolerance (typically the machine precision) beyond which the constraint is regarded as violated.

Second, the term μf(x), where $f:{\mathbb{R}}^n⟶\mathbb{R}$ and μ ∈ {0, 1} can be replaced by auxiliary real variable y = μf(x), which satisfies [μ = 0]⟶[y = 0], [μ = 1]⟶[y = f(x)]. Therefore, defining f_max, f_min as in equation (15), y = μf(x) is equivalent to 17 $\begin{array}{}\left\{\begin{array}{c}y\le f_{\max }\mu ,\\ y\ge f_{\min }\mu ,\\ y\le f\left(x\right)-f_{\min }\left(1-\mu \right),\\ y\ge f\left(x\right)-f_{\max }\left(1-\mu \right).\end{array}\right.\end{array}$

With the above properties, we then linearize constraint (10). Let 18 $\begin{array}{}f=p{y}_i\left(k\right)-p{j}_i\left(k\right),\end{array}$ and define 19 $\begin{array}{}\mu =\left\{\begin{array}{cc}1,& f\le 0,\\ 0,& f\ge 0.\end{array}\right.\end{array}$

Then, we have 20 $\begin{array}{}\min \left(p{j}_i\left(k\right),p{y}_i\left(k\right)\right)=p{j}_i\left(k\right)+\left(p{y}_i\left(k\right)-p{j}_i\left(k\right)\right)\mu =p{j}_i\left(k\right)+f\mu .\end{array}$

Then, let 21 $\begin{array}{}z=f\mu .\end{array}$

With the above transformation, constraint (21) is equivalent to 22 $\begin{array}{}\left\{\begin{array}{c}f\le f_{\max }\left(1-\mu \right),\\ f\ge \epsilon +\left(f_{\min }-\epsilon \right)\mu ,\\ z\le f_{\max }\mu ,\\ z\ge f_{\min }\mu ,\\ z\le f-f_{\min }\left(1-\mu \right),\\ z\ge f-f_{\max }\left(1-\mu \right),\end{array}\right.\end{array}$ where f_max = cc·|J| and f_min = −pj_max. The meaning of pj_max is platform capacity. This paper does not consider the platform capacity to simplify the model, so the platform capacity is set to pj_max = M.

Moreover, for constraints (13), we can take the following property.

We consider the statement f = max_x∈χ f(x), where $f:{\mathbb{R}}^n⟶\mathbb{R}$ linear, and we assume that x ∈ χ, where χ is a given bounded set, f = max_x∈χ f(x) is equivalent to 23 $\begin{array}{}f\ge f\left(x\right).\end{array}$

Therefore, constraints (13) are equivalent to the inequality: 24 $\begin{array}{}\eta \ge p{w}_i\left(k\right),\forall k\in K.\end{array}$

Based on the above analysis, the optimization model is equivalently reformulated as a MILP model, which can be solved by the GUROBI solver. This model can be formulated as the following optimization model: 25 $\begin{array}{}\left\{\begin{array}{c}\text{Objective\hspace{0.17em}function}\left(14\right),\\ \text{s}.\text{t}.\\ \text{Constraints}\left(1\right)-\left(9\right),\left(11\right)-\left(12\right),\left(18\right)-\left(22\right),\left(24\right).\end{array}\right.\end{array}$

4.2. Variable Neighborhood Search Algorithm

Mladenović and Hansen [37] proposed an algorithm for solving combinatorial and global optimization problems called the VNS. VNS is a metaheuristic algorithm obtained by performing a systematic change of neighborhood in a local search algorithm. Contrary to other local search methods, VNS does not follow a trajectory but rather explores increasingly distant neighborhoods of the current extant solution and jumps to a new solution when there is an improvement. In this way, favorable features of the extant solution are already at their optimal values and are improved in the new solution. Optimal values have been reached, and these features will be preserved and used to obtain promising neighboring solutions. After that, a local search procedure will be applied iteratively to obtain locally optimal solutions from these neighboring solutions. There are three important parts of the VNS algorithm: (1) initial solution, (2) shaking, and (3) local search. The steps of VNS are shown in Algorithm 1.

4.2.1. Initial Solution

Before the algorithm is optimized, it is necessary to randomly generate the initial solutions for train carriage flexible release and passenger flow control. The generation of initial solutions is based on the following two principles:

• (1)

  As mentioned in constraints (8) and (9), the number of released carriages at the departure station should be more than the minimum value, and the train carriages cannot be closed again after release.

• (2)

  As mentioned in constraints (10) and (11), the intensity of passenger flow control at each station should be controlled within a certain range, especially the control intensity at important stations.

Following the above two principles, the initial solution is generated.

4.2.2. Shaking

The special characteristic of VNS is the creation of a set of neighborhood structures for shaking, which is used to avoid the drawbacks of local optimality of heuristic algorithms. The shaking process is triggered when the local search fails to find a better solution. It is used to jump out of the current neighborhood and get a completely new solution by shaking. In this paper, we use CROSS to construct the shaking process, i.e., when the local search fails to find a more optimal solution, we randomly exchange passenger flow control strategies of two trains and train carriage flexible release strategies of two trains, so as to obtain a new solution.

Algorithm 1: Variable neighborhood search.

•  

  Initialization

• 1.

   A set of neighborhood structures N_l(l = 1, …, l_max) for shaking,

• 2.

   Find an initial solution c, co;

• 3.

   Setting an objective function of f(c, co)

•  

  Repeat:

•  

   Forl = 1Tol_maxDo

• (1)

    Shaking

•  

     Generate a random solution c^′ from the neighborhood of c,

•  

     Generate a random solution co^′ from the neighborhood of co;

• (2)

    Local search

•  

     Apply Algorithm 3 with c^′, co^′ as initial solution, denote the obtained local optimum by c^″, co^″;

• (3)

    Move or not

•  

     Iff(c^″, co^″) < f(c, co), Thenc⟵c^″, co⟵co^″, l⟵1;

•  

     Otherwisel⟵l + 1

•  

  Until Stopping criteria

4.2.3. Local Search

Local search is a VND process that alternates the search using neighborhood structures consisting of different actions. Therefore, in this paper, we set up three neighborhood structures.

4.2.3.1. Swap-1

There are two decision variables in this paper, so they need to be treated sequentially. First, we generate a new solution for passenger flow control. We take out the solution of Train a and swap it with the solution of Train b to get a new neighborhood solution. The values of Train a and Train b are arbitrary, and we can change the values of a and b to get more neighborhood solutions, as shown in Figure 5. Second, we use the same method to generate neighborhood solutions for train carriage flexible release.

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4.2.3.2. Rewind

We use the same rewind strategy for the solution of passenger flow control and the solution of train carriage flexible release, i.e., randomly selecting Train a and Train b in the solution, and rearranging the train solutions of Train a and Train b and the train solutions between them in reverse order. That is to say, the solution of Train a is exchanged with the solution of Train b, the solution of Train a + 1 is exchanged with that of Train b − 1, and this rule is executed for the rest trains in turn. Then, we change the values of a and b to get more neighborhood solutions, as shown in Figure 6.

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4.2.3.3. Swap-2

This neighborhood uses different methods for passenger flow control and train carriage flexible release. For train carriage flexible release, we randomly choose Train a and Train b and put them in front of the first train. Train a is changed to solution of Train 1, and Train b is changed to the solution of Train 2. At this time, the solution of Train 2 is changed to Train 3, the solution of Train 2 is changed to solution of Train 4, and the solutions of other trains are sequentially arranged backwards. Then, we change the values of a and b to get more neighborhood solutions, as shown in Figure 7. The neighborhood solution of passenger flow control adopts the strategy of station-to-station swap, i.e., we take out the solution of Station a and swap it with the solution of Station b to get a new neighborhood solution, especially note that the important stations can only be swapped with the important stations. Then, we change the values of a and b to get more neighborhood solutions, as shown in Figure 8.

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In addition, due to the vibration and local search process, it is necessary to exchange the solutions of two trains or two stations. The number of passenger flow control may be greater than the number of waiting passengers, resulting in a negative number of passengers waiting at the platform. In order to avoid this situation, when the model constraint is calculated, the number of stranded passengers is judged. If pj_i(k) < 0, the number of stranded passengers pw_i(k) is set to a very large value. Because of the poor performance in this case, this part of the solution is filtered out in the solution process to avoid infeasibility.

The process of VND is when the existing solution is improved; the process goes back to the first neighborhood structure and researches it, which is described in detail in Algorithm 2.

Algorithm 2: Local search.

•  

  Initialization

• 1.

   A set of neighborhood structures N_κ(κ = 1, …, κ_max) for local search,

• 2.

   Input an initial solution c, co

• 3.

   Setting an objective function of f(c, co)

•  

  Repeat:

•  

   Forκ = 1Toκ_maxDo

•  

    Find best solution c^′ from the κ^th neighbourhood N_κ(c) of c;

•  

    Find best solution co^′ from the κ^th neighbourhood N_κ(c) of co

•  

    Iff(c^′, co^′) < f(c, co)Then⟵c^′, co⟵co^′, κ⟵1

•  

    Otherwiseκ⟵κ + 1

•  

  Until stopping criteria

5. Numerical Experiments

Two experiments will be conducted in this section to test the performance of the proposed method. One is a small-scale case study, and the other is a real-world experiment from Chengdu Metro Line 2 to verify the collaborative optimization strategy and the applicability of the algorithm designed in this paper to large-scale cases.

5.1. A Small-Scale Case

5.1.1. Experiment 1: Collaborative Optimization Strategy Applied to Small-Scale Cases

For the small-scale case, a single-directional metro line with 6 trains and 10 stations is considered, while the train is equipped with 6 carriages, and the train capacity of each carriage is set as cc = 200. The maximum intensity of passenger flow control is set as c_max = 60, and the maximum intensity of control at important stations is set as $c_{\max }^{imp}=50$. The minimum number of released carriages is set as co_min = 3. The weights ω₁, ω₂ are specified as 1,1. To simplify the case, the passenger alighting rate is only related to the station. The number of arriving passengers and passenger alighting rate are shown in Figure 9.

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The distribution of stranded passengers without strategy is shown in Figure​ 10(a). It can be clearly seen that when the train travels from Station 1 to Station 10, the number of stranded passengers at Station 8 is the largest, while there are almost no stranded passengers at other stations. This phenomenon reflects the inequity mentioned in Section 3.1. To explain this phenomenon in detail, Figure 11 illustrates the train loading rates without strategy. It is evident that the train loading rate from Station 1 to Station 8 generally exhibits an increasing trend. At Station 7, the loading rate reaches 91%, leaving only 9% of the train’s capacity available. When the train arrives at Station 8, the remaining capacity is insufficient to accommodate all arriving passengers, resulting in a significant number of stranded passengers. To address this issue, the collaborative optimization model proposed in this paper is employed to optimize the scenario.

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After implementing the collaborative optimization strategy, the distribution of stranded passengers is shown in Figure 10(b). The number of stranded passengers at Station 8 decreases significantly from 8951 to 4578, while the number of stranded passengers at upstream stations (e.g., Station 5 and Station 7) increases. The variance of the number of stranded passengers across all stations decreases significantly from 0.072 to 0.030, enhancing equity by 58.33%. The total number of stranded passengers increases from 8951 to 9264, reducing efficiency by 3.49%. In this case, the number of controlled passengers is 717, among which the passenger flow control intensity from Station 2 to Station 7 is greater than that at Station 8, limiting the number of upstream passengers entering the stations and providing passengers at Station 8 with more opportunities to board the train. The number of released carriages is 268, and the number of reserved carriages is 92 from Station 1 to Station 7. Upon arrival at Station 8, all reserved carriages are released to accommodate waiting passengers. The situation of carriage release and passenger flow control is shown in Figure 12. It can be seen that the implementation of collaborative optimization strategy can obviously improve equity.

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In order to explain the reason for the improvement of equity with this strategy further, the train loading rate after the implementation of the collaborative optimization strategy is analyzed. The detailed data are shown in Figure 13. The loading rate in the yellow section decreases, with notable changes at Station 7; specifically, the loading rates for the first three trains decrease from 91% to 67%, and the fourth train’s rate drops to 83%. To maintain operational efficiency, the loading rate of the last two trains at Station 7 reached 100%. This observation indicates that the collaborative optimization strategy effectively reduces the loading rates at upstream stations, allowing trains to reserve capacity for passengers at Station 8, thereby significantly decreasing the total number of stranded passengers at this station. This analysis further demonstrates the effectiveness of the collaborative optimization strategy in enhancing equity.

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5.1.2. Experiment 2: Implement Passenger Flow Control and Train Carriage Flexible Release Separately

To demonstrate the advantages of the collaborative optimization model proposed in this paper, we conduct two comparative experiments. The first experiment only implements passenger flow control strategy, and the second one only implements the train carriage flexible release strategy.

After implementing passenger flow control strategy, the distribution of stranded passengers is shown in Figure 14(a). The number of stranded passengers at Station 8 decreases from 8951 to 8253. Besides, the variance of the number of stranded passengers decreases from 0.072 to 0.060, improving equity by 16.67%. The total number of stranded passengers increases from 8951 to 9374, resulting in a 4.73% decrease in efficiency.

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After implementing train carriage flexible release strategy, the distribution of stranded passengers is shown in Figure 14(b). The number of stranded passengers at Station 8 decreases from 8951 to 3990. Besides, the variance in the number of stranded passengers decreases from 0.072 to 0.021, improving equity by 70.83%. The total number of stranded passengers increases from 8951 to 11510, resulting in a 28.59% decrease in efficiency.

Table 4 presents detailed data in terms of equity and efficiency. Figure 15 compares the number of stranded passengers under different strategies. The experimental results indicate that all three strategies contribute to a more balanced distribution of stranded passengers between different stations. From the equity perspective, the train carriage flexible release strategy results in the lowest number of stranded passengers at congested stations and the smallest variance. From the efficiency perspective, the collaborative optimization strategy achieves the lowest total number of stranded passengers. Combining these two aspects, while the train carriage flexible release strategy slightly outperforms the collaborative optimization strategy in terms of equity, the latter offers significantly higher efficiency. Therefore, the collaborative optimization strategy proves to be the most effective when both equity and efficiency are considered.

Table 4 Data comparison in terms of equity and efficiency.

	Maximum number of stranded passengers at the station	Total number of stranded passengers	Variance	Equity	Efficiency
Without strategy	8951	8951	0.072	—	—
Passenger flow control	8253	9374	0.060	+16.67%	−4.73%
Train carriage flexible release	3990	11510	0.021	+70.83%	−28.59%
Collaborative optimization	4578	9264	0.030	+58.33%	−3.49%

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5.1.3. Experiment 3: Sensitivity Analysis

In the objective function of the collaborative optimization model, two key weights significantly influence the optimization results: the efficiency weight and the equity weight. To further illustrate the influence of these two weight coefficients in the collaborative optimization model, nine numerical experiments are carried out with different weight values for small-scale cases; the comparison of sensitivity analysis data is shown in Table 5. To ensure the consistency of the test, all scenarios are solved by the GUROBI solver, and the following conclusions can be drawn:

• (1)

  As the equity weight increases, the maximum number of stranded passengers at the stations decreases, the total number of stranded passengers at all stations increases, and the variance becomes smaller.

• (2)

  As the efficiency weight increases, the total number of stranded passengers at all stations decreases, the maximum number of stranded passengers at the station increases, and the variance becomes larger.

• (3)

  When the ratio of efficiency to equity reaches a certain level (e.g., ω₁/ω₂ ≤ (1/10)), the maximum number of stranded passengers at the station and the total number of stranded passengers at all stations no longer change.

Table 5 Comparison of sensitivity analysis data.

Scenarios	ω1	ω2	Z	T1	T2	Variance	The number of controlled passengers	The number of released carriages
Scenario 1	1	1	13842	9264	4578	0.030	717	268
Scenario 2	1	0.1	9642	9194	4480	0.031	270	266
Scenario 3	1	0.5	11434	9194	4480	0.031	219	267
Scenario 4	1	10	48548	11588	3696	0.020	486	241
Scenario 5	1	100	381188	11588	3696	0.020	447	244
Scenario 6	0.1	1	4854.8	11588	3696	0.020	273	244
Scenario 7	0.5	1	9023	9282	4382	0030	446	265
Scenario 8	10	1	96420	9194	4480	0.031	213	265
Scenario 9	100	1	919545	9109	8645	0.066	195	275

Furthermore, we compare the distribution of stranded passengers in nine scenarios, as shown in Figure 16. When the optimization model prioritizes efficiency, the line chart of stranded passengers at the station is more prone to peak (e.g., Scenario 9). Conversely, when the optimization model prioritizes equity, the line chart of stranded passengers at the station becomes smoother (e.g., Scenario 5 and Scenario 6). The implementation of the passenger flow control strategy and train carriage flexible release strategy is shown in Figures 17 and 18. We can observe that when the optimization prioritizes efficiency, the number of controlled passengers decreases, while the number of released carriages increases. Conversely, when the optimization pays more attention to equity, the opposite changes will occur.

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5.1.4. Experiment 4: Algorithm Calculation Time Analysis

To verify the proposed VNS solution approach, we further employ the VNS algorithm to find good solutions of small-scale cases. After 1.6 s, the approximate optimal solution is obtained, controlling 561 passengers, releasing 268 carriages, reserving 92 carriages, and achieving an objective function value of 13,846. The total number of stranded passengers is 9,464, the maximum stranded passengers at the station are 4,382, and the variance is 0.029. This approach improves equity by 59.72% while reducing efficiency by 5.73% compared with that without strategy. Table 6 shows the data comparison of the GUROBI solver and VNS on the small-scale case. It can be seen that the optimization effects of the GUROBI solver and VNS are both similar. To test the sensitivity of the calculation time of the two algorithms to the case size, five different scenarios are set up by varying the number of trains. The calculation time comparison of the two algorithms in different scenarios is shown in Table 7. The calculation time of the GUROBI solver is more sensitive to the case size. When the number of trains is fewer than eight, both the solver and VNS can obtain good solutions in a short time. However, as the number of trains increases to 10, the solver’s calculation time significantly increases, becoming 12 times that of the VNS algorithm. As the number of trains increases to 12, the solver’s calculation time reaches 3800 s. Obviously, with the increase in the scale of the problem, the calculation time of the solver becomes too long. In contrast, the VNS algorithm can solve cases of the same size in a shorter time and achieve an approximately optimal solution.

Table 6 Data comparison of the GUROBI solver and VNS on the small-scale case.

	Without strategy	GUROBI solver	VNS
Objective function	—	13842	13846
Maximum number of stranded passengers at the station	8951	4578	4382
Total number of stranded passengers at all stations	8951	9264	9464
Passenger flow control intensity	—	717	561
Total number of released carriages	—	268	268
Variance	0.072	0.030	0.029
Equity	—	+58.33%	+59.72%
Efficiency	—	−3.49%	−5.73%

Table 7 Calculation time comparison of the GUROBI solver and VNS.

Train number	GUROBI	VNS
Objective function	Solution time	GAP	Objective function	Solution time	GAP
6 trains	13842	1.43	0	13846	1.60	0
7 trains	18218	3.51	0	19435	3.75	6.7%
8 trains	23362	2.23	0	25573	6.50	9.5%
10 trains	35738	1468.3	0	36407	22.4	1.9%
12 trains	50536	3840.7	0	51234	126.7	1.4%

5.2. Numerical Experiments on Chengdu Metro Line 2

To further demonstrate the effectiveness of the proposed VNS algorithm for large-scale cases, a real-world case study on the Chengdu Metro Line 2 is conducted in this subsection. Chengdu Metro Line 2 is a metro line with a total length of 42.3 km with 32 stations, as shown in Figure 19. For convenience, the stations are renamed from Station 1 to Station 32 in the form of serial numbers. The research time considered for this case is from 7:00 a.m. to 9:00 a.m., which is the morning rush hour of the Chengdu Metro system. The passenger demand in the experiment is the historical data of a weekday in 2018 collected from the automatic fare collection (AFC) system. The distribution of arriving passengers during the morning rush hour of Chengdu Metro Line 2 is shown in Figure 20 with a total of 82,815 passengers. The passenger alighting rate is used to describe the alighting passengers, as shown in Figure 21. Clearly, there is an imbalance in the spatial and temporal distribution of passenger flow on Chengdu Metro Line 2. In such cases, it can easily lead to inequity due to the mismatch between supply and demand during peak periods. Figure 22(a) reveals the distribution of stranded passengers without strategy. It can be observed that passenger congestion is the most severe at Station 10 (SHLD) and Station 11 (BGL). The number of stranded passengers is 11631 at Station 10 and 29699 at Station 11. The variance of the number of stranded passengers at all stations is as high as 0.30, indicating that the stranded passengers at different stations are significantly inequitable.

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To further explain the phenomenon of inequity, the change of the train loading rate at each station is analyzed, as shown in Figure 23. It can be seen that the train loading rate initially increases and then decreases from Station 1 (XP) to Station 32 (LQY). In the process of traveling from Station 1 to Station 8 (YXLJ), the train loading rate is below 100%, and after Station 9 (YPTX), five trains (Train 18-19 and Train 22–24) are fully loaded. When the trains travel to Station 10, the train capacity is insufficient to carry all waiting passengers, resulting in 17 consecutive trains (Train 15–31), reaching the maximum capacity and a large number of passengers being stranded. After the trains travel to Station 11, the train capacity is seriously insufficient; 23 trains (Train 13, Train 15–36) are fully loaded, leaving a large number of stranded passengers. The above findings indicate a serious inequity in Chengdu Metro Line 2.

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Therefore, we apply the collaborative optimization model to improve the equity of Chengdu Metro Line 2. The experiment is set up with 40 trains, and each train contains 6 carriages. The capacity of each carriage is set as cc = 240. The maximum intensity of passenger flow control is set as c_max = 100, and the maximum intensity of control at important stations is set as $c_{\max }^{\text{imp}}=50$, where Station 15 (TFGC) and Station 16 (CXL) are important commercial centers. The minimum number of released carriages is set as co_min = 4. The weights ω₁, ω₂ are specified as 1,1. The strategy under the above indicators is defined as collaborative optimization strategy 1. Due to the fact that the case in this section is extremely large, the GURUBI solver cannot solve it effectively in a short time. Therefore, the proposed VNS algorithm is employed to solve the model.

Figure 24 demonstrates the iteration curve of the objective function, from which we can observe that the model tends to converge after 29,000 iterations; the approximate optimal solution is obtained. The approximate optimal target value is 64,931. The implementation of train carriage flexible release strategy is shown in Figure 25. The number of released carriages is 7,628, and the number of reserved carriages is 52. As the trains travel from upstream to downstream stations, the reserved carriages are gradually released, and the number of released carriages reaches the maximum at Station 8. The implementation of passenger flow control strategy is shown in Figure 26. The total number of controlled passengers is 8814. It can be seen that the passenger flow control intensity from Station 1 to Station 9 is gradually increasing, especially at Station 9, reaching 1474 passengers, and at Station 10, reaching 1403 passengers, while the control is not carried out at Stations 11 and beyond. The distribution of stranded passengers with collaborative optimization strategy is shown in Figure 22(b). The number of stranded passengers at Station 11 greatly reduces from 29,699 to 21,552, while Station 10 does not change significantly, only increasing from 11,631 to 13,205.

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The comparison results of stranded passengers at the stations with and without the strategy are shown in Figure 27. The variance of the number of stranded passengers at the station decreases significantly from 0.30 to 0.19, and the equity increases by 38.33%. Figure 28 compares the number of stranded passengers on the trains with and without the strategy. The number of stranded passengers on each train has increased. The total number of stranded passengers increases from 41,488 to 43,419, reducing the efficiency by 4.65%.

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To explain the reason for the collaborative optimization strategy 1 improving equity in this case, the train loading rate with collaborative optimization strategy is further analyzed, as shown in Figure 29. After implementing the optimization strategy, when the train arrives at station 11, the loading rate of most of the trains decreases. Especially, Train 17 decreases from 100% to 73%. During the period from Train 15 to Train 31, the maximum remaining capacity of the train increases from 0% to 27%. At this time, the number of passengers that the train can carry is higher than that without strategy. Therefore, the number of stranded passengers at Station 11 is reduced, which can balance the number of stranded passengers at upstream and downstream stations.

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In the collaborative optimization model, the maximum intensity of passenger flow control c_max and the minimum number of released carriages co_min may have a great influence on the improvement of equity. Theoretically, when c_max increases or co_min decreases, the equity will be improved more obviously, while the efficiency will be reduced to a greater extent. On the contrary, the equity will be slightly improved, and the efficiency will be effectively guaranteed. Because the decreasing passenger flow control intensity and the increasing number of released carriages have the same effect, so choose one of them as collaborative optimization strategy 2 to test this theory. The passenger flow control parameters of collaborative optimization strategy 2 are set as follows: c_max = 60, $c_{\max }^{\text{imp}}=40$. Other parameters and variables are unchanged.

Compared to the current practical application, the maximum number of stranded passengers decreases from 29,699 to 25,766, and the variance increases from 0.30 to 0.23 after implementing the collaborative optimization strategy 2. The equity increases by 23.33% compared with that without strategy. The total number of stranded passengers at all stations decreases from 41488 to 43,090, and the efficiency decreases by 23.33% compared with that without strategy. Table 8 compares the optimization effects of collaborative optimization strategy 1 and strategy 2. It can be seen that with the decrease of passenger flow control intensity, the improvement of equity also decreases, but the reduction of efficiency decreases.

Table 8 Data comparison of collaborative optimization Strategy 1 and Strategy 2.

	Without strategy	Collaborative optimization Strategy 1	Collaborative optimization Strategy 2
Objective function	—	64973	68856
Maximum number of stranded passengers at the station	29699	21554	25766
Total number of stranded passengers at all stations	41488	43419	43090
Passenger flow control intensity	—	8814	6992
Total number of released carriages	—	7628	7628
Variance	0.30	0.19	0.23
Equity	—	+38.33%	+23.33%
Efficiency	—	−4.65%	−3.86%

From above, different passenger flow control intensities can affect equity and efficiency. Similarly, the minimum number of released carriages is the same. Therefore, in the study of the metro system, the trade-off between equity and efficiency can be achieved not only by adjusting the weights representing equity and efficiency in the objective function but also by changing the intensity of passenger flow control and the minimum number of released carriages.

6. Conclusions

To alleviate the transportation inequity problem caused by the mismatch between supply and demand in the oversaturated metro system, this paper proposes a collaborative optimization approach that dynamically allocates train capacity and implements passenger flow control from supply and demand perspectives. The primary objective of the proposed model is to enhance transportation equity in the metro system while maintaining efficiency. To achieve this, we develop a MINLP model to minimize the total number of stranded passengers and the maximum number of stranded passengers at the station, which is further transformed into a MILP model and solved by the GUROBI solver. Due to the limitations of commercial solvers on case scale and calculation time, we also design an efficient VNS algorithm to calculate a large-scale case and get good solutions. Finally, numerical experiments are conducted to demonstrate the effectiveness and efficiency of the proposed algorithms based on a small-scale case and a real case from the Chengdu metro system. The results show that this optimization strategy has a significant improvement in transportation equity while ensuring travel efficiency.

Our future work will focus on the following two areas. (1) This study did not consider train headway, so one future research direction is to integrate our work with train timetable optimization. (2) Since this collaboration problem only considers one metro line, our future research would like to focus on solving this collaboration problem in metro networks, which is relevant from a global optimization perspective.

Data Availability Statement

We fully understand the requirements of the journal for data sharing, but due to the policy or confidentiality agreement of our laboratory, we cannot provide the original data. We have fully described the experimental design, analysis, and results, as well as the process of data analysis and processing. If editors and reviewers have questions about specific data, we will try our best to provide more detailed explanations.

Conflicts of Interest

The authors declare no conflicts of interest.

Author Contributions

Jinyang Zhong and Hao Huang: writing–original draft, writing–review and editing, validation, methodology, formal analysis, and conceptualization. Jinyi Pan: writing–original draft, writing–review and editing, methodology, and formal analysis. Lan Liu: writing–original draft, writing–review and editing, methodology, formal analysis, supervision, and investigation. Yibo Shi: writing–original draft, writing–review and editing, validation, methodology, investigation, formal analysis, and conceptualization. All authors are accountable for the contents and conclusions of the article.

Funding

This work was supported by the Science and Technology Project of Sichuan Province (2020YFSY0020).

Acknowledgments

This project received research funding support from the Science and Technology Project of Sichuan Province (2020YFSY0020). The authors thank Chengdu Metro for providing the AFC data.

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