Symmetry-Based Urban Rail Transit Network Planning Using Two-Stage Robust Optimization

Abstract

To address the symmetry-related resilience issues of stations and lines in urban rail transit networks, we propose a two-stage robust optimization-based approach for urban rail transit network planning. In this context, resilience is conceptualized as the ability of the network to maintain its operational symmetry under normal and disruptive conditions. Firstly, we used passenger flow distributions as decision variables to construct a two-stage symmetry-based urban rail transit network planning model, aiming to simultaneously minimize the total cost and total operating time of the network while preserving its functional symmetry. Secondly, we designed a hybrid evolutionary algorithm with chromosomes having a two-layer encoding structure, where the Niched Pareto Genetic Algorithm served as the main algorithmic framework, and a Large Neighborhood Search mechanism was designed to optimize the connectivity gene layer of individuals, ensuring the symmetry of network connectivity. Finally, we conducted computational verification on randomly generated instances to confirm the effectiveness of the model and algorithm. The experimental results demonstrated that our method could find two sets of Pareto optimal solutions for cost preference and time preference, thereby preserving the operational symmetry of the network under normal and damaged conditions, as well as reducing the total operating time. This effectively improved the overall efficiency and resilience of the network. Our designed hybrid evolutionary algorithm converged to satisfactory objective values in the early iterations, exhibiting strong search and optimization performance and effectively solving the two-stage symmetry-based urban rail transit network planning model.

1. Introduction

Urban rail transit networks, as a crucial component of public transportation in some modern metropolitan areas, typically provide strong operational capabilities, including relatively high speed, punctuality, and cost-effectiveness, making them a key mode of transportation in these cities [1]. They significantly alleviate the increasingly severe problem of urban traffic congestion and play a crucial role in ensuring urban transportation safety. The planning phase of urban rail transit networks is the primary stage in constructing urban rail transit systems. The rationality of planning directly affects the overall operational effectiveness of the entire urban rail transit network. However, once rail transit lines are constructed and put into operation, they typically undergo only essential maintenance and upgrades, with significant structural changes being less common due to the high initial construction costs. Expansions or upgrades to older networks are generally carried out to conform to modern standards, but such comprehensive changes are not as frequent [2]. Planners typically assume these facilities and lines are reliable, seldom considering potential damage scenarios. With the increasing complexity of urban rail transit networks, the number of unforeseen events affecting rail transit safety is also on the rise. These events mainly include equipment failures, sabotage, natural disasters, and peak passenger flows. Due to the random timing and locations of these events, they can easily lead to network interruptions, affecting passenger travel, potentially causing passenger delays, posing threats to passenger safety, and significantly impacting the entire urban rail transit network. Consequently, there is a growing emphasis on understanding the characteristics of urban rail transit networks and their resistance to attacks [3].

To guide the planning of urban rail transit networks, scholars have conducted extensive research. These studies primarily approach the problem of network planning from a mathematical modeling perspective, abstracting it into an optimization problem. By defining objective functions and relevant constraints, scholars seek the optimal layout for urban rail transit networks. In these studies, scholars typically consider multiple factors such as travel time [4,5], construction costs [6,7,8], passenger flows [9], etc., abstracting the network planning problem into a mathematical optimization problem to achieve the optimal network layout.

Gutierrez-Jarpa et al. [6] proposed a model for urban rail transit networks generating network shapes such as star, triangle, and wheel shapes. This model was aimed at reducing construction costs while maximizing passenger flow, and it was applied to the planning of the rail transit system in Concepción, Chile. Subsequently, they refined this model by further minimizing passenger travel costs on top of reducing construction costs and maximizing passenger flow [5]. In their subsequent research [9], they also took into account the competition between urban rail transit and other public transportation modes, further enhancing the model. Canca et al. [8], on the other hand, aimed to maximize operational profit while considering constraints such as vehicle capacity and personnel scheduling. They established a mixed-integer nonlinear model for urban rail transit network planning.

Mohri et al. [10] proposed a multi-objective optimization model for urban rail transit networks, aiming to maximize population coverage and the number of paths between stations. They applied this model to urban rail transit planning in Isfahan, Iran. Krol et al. [11], based on determining the positions of stations, developed a two-stage optimization model to optimize the connections between stations, employing a genetic algorithm for solving. Additionally, Ye et al. [12] presented a two-stage model based on path search for solving the optimal layout of urban rail transit networks. In the first stage, they aimed to minimize the construction cost of urban rail transit and maximize passenger flow to find the optimal path for each OD pair. In the second stage, under the constraints of network total length and connectivity, they aimed to maximize the total passenger flow of the network.

Zhang et al. [13] addressed environmental factors by proposing a multi-objective optimization model aimed at minimizing system emissions and total travel costs, solved using a genetic algorithm. Additionally, OwaiS et al. [14] approached urban rail transit network design from the perspective of two types of network shapes: mesh and ring. Their objective was to maximize network connectivity, and they applied this model to simulate grid-based urban rail transit planning. The research findings indicate that when network connectivity reaches its maximum, the cost-effectiveness ratio of ring topology exceeds that of mesh topology by 41%.

Furthermore, scholars have extensively researched the resilience of urban rail transit networks. Angeloudis et al. [15] conducted a study on the rail transit networks of London and Paris, exploring their resilience against deliberate attacks, and found them to possess considerable resilience. Derrible et al. [16] analyzed rail transit networks in 33 global cities and discovered that resilient networks typically exhibit densely interconnected internals, a higher proportion of candidate transfer stations, and numerous alternative routes. Building upon this, Wang et al. [17] employed metrics such as reliability, network efficiency, and connectivity to comprehensively assess the rail transit networks of these 33 cities, identifying Tokyo and Rome as having particularly good resilience. Bernal et al. [18] considered random and deliberate disruption scenarios, conducting a multifaceted study on the resilience of Madrid’s rail transit network, revealing its high resilience against random disruptions. Dong et al. [19] investigated the resilience of Portland’s rail transit network, finding that increasing the number of stations can enhance network resilience, albeit with diminishing marginal returns. Shen et al. [20] took into account the heterogeneous nature of bidirectional passenger flows in rail transit networks, analyzing the resilience of Nanjing’s rail transit network.

Rodriguez-Nunez et al. [21] examined passengers’ travel route preferences and evaluated the vulnerability of the Madrid rail transit network in the event of station and line paralysis. They found that, due to the high passenger flow and relatively low number of stations, the network exhibited considerable vulnerability. Jiang et al. [22] considered factors such as passenger flow, land use, and transfers, establishing a vulnerability assessment model for rail transit networks and conducting a case study using the Shanghai rail transit system. Additionally, Lu et al. [23], from the perspective of accessibility, proposed a method for assessing the vulnerability of rail transit networks, taking into account factors such as passenger flow, travel costs, and alternative transportation, and identified significant impacts of passenger heterogeneity on the vulnerability of the Shenzhen rail transit network. On another front, Zhang et al. [24] conducted a comparative analysis of the vulnerability of rail transit networks in Shanghai and Guangzhou. The results indicated that the Shanghai rail transit network exhibited the highest vulnerability, while the Guangzhou rail transit network showed the lowest vulnerability. Finally, Sun et al. [25] introduced a weighted network cascading failure model considering passenger flow redistribution to analyze the vulnerability of the Beijing rail transit network and identify critical nodes within the network.

Robust optimization methods are widely applied in rail transportation. Recent studies have also explored robust optimization in various IoT and cloud-based environments [26,27]. Wan et al. [28] proposed a method to improve the geometry of railway turnouts through robust optimization, incorporating the variability of design parameters into the optimization problem. The study indicates that deterministic optimization fails when design variables change, whereas robust optimization enhances robust performance. Cheng et al. [29] introduced an innovative approach combining uniform design and an efficient global optimization algorithm to generate robust suspension parameters in railway vehicle models. The optimization process successfully identified the optimal and robust design of suspension parameters. Nejlaoui et al. [30] investigated the robust safety design optimization of rail vehicle systems traveling on short-radius curved tracks. By employing a multi-objective imperialist competitive algorithm and Monte Carlo methods, the sensitivity of rail vehicle safety to design parameter uncertainty was significantly reduced. Wang et al. [31] presented a variant of the distributionally robust conditional vertex p-center problem, proposing a distributionally robust optimization (DRO) method. Through two case studies of different scales, the advantages of the DRO model in the location of emergency rescue stations in high-speed rail networks were demonstrated. Xing et al. [32] developed a multi-period empty container repositioning optimization model for China’s railway express, based on the mean and variance of demand. Utilizing distributionally robust chance-constrained programming, the impact of sea–rail intermodal transport and collapsible containers on total cost was analyzed. The results showed that the use of collapsible containers and the optimization model could significantly reduce the total cost of empty container repositioning.

The above-mentioned literature provides references for urban rail transit network planning from multiple perspectives, enriching the related theoretical foundation. Existing research on urban rail transit network planning primarily employs mathematical modeling approaches, abstracting the problem into an optimization challenge by defining objective functions and related constraints to seek the optimal layout. However, these studies exhibit several limitations. While optimization models have been proposed, the considerations are often limited, insufficiently accounting for environmental factors and network resilience. Some studies focus on multi-objective optimization and path optimization but do not delve deeply into the uncertainties and disruptions encountered in practical operations. Other studies, despite exploring network resilience, inadequately address passenger flow heterogeneity and specific operational schemes. Despite these shortcomings, such research has made significant contributions, including the foundational framework for network planning, multi-objective optimization models, resilience assessment methods, and robust optimization applications. Building upon this foundation, this study aims to comprehensively consider multiple objectives, enhance network resilience, introduce uncertainty factors, and improve the feasibility and applicability of the model in practical operations. Additionally, a thorough practical evaluation of optimization strategies will be conducted to ensure the model’s effectiveness in real-world applications.

With the emergence and practice of resilience concepts, emphasis has been placed on the ability of various functional entities within cities to coordinate operations to resist and respond to risks. As a major component of urban public transportation, the resilience of urban rail transit networks has garnered significant attention [33]. Therefore, studying the characteristics and resilience of urban rail transit networks is crucial for maintaining their daily stable operation and ensuring passenger travel.

Currently, robust optimization methods are commonly used for resilient urban rail transit network planning issues, but they mainly focus on single-stage decisions, lacking independent evaluations of network performance under normal and damaged states. In practical applications, planning urban rail transit networks considering only performance under damaged conditions may result in the need for additional facilities and node connections, thereby increasing network construction costs. Resilience theory provides new research directions for urban rail transit networks to address uncertain risks, reduce the negative impacts of disturbances, and achieve long-term adaptive development.

Assessing the resilience of urban rail transit networks is of practical value and theoretical significance for reducing socioeconomic losses caused by accidents and efficiently restoring urban rail transit to its standard operating conditions [34]. This study, based on complex network theory, focuses on key stations within urban rail transit networks, including candidate transfer stations, origin stations, destination stations, and their connecting routes, while excluding ordinary intermediate stations. The network is analyzed in two distinct states, normal operating conditions and disrupted conditions, with objective functions established for minimizing total costs across these states and minimizing total operating time during normal operations [35].

The remainder of this paper is organized as follows: Section 2 introduces the problem and establishes the mathematical model. Section 3 designs the solution algorithm. In Section 4, the effectiveness of the model and solution algorithm is validated on case studies. Finally, Section 5 presents the conclusions and future research directions.

2. Problem Description and Mathematical Model

To address this issue, a mixed evolutionary algorithm is designed, incorporating various enhancement strategies to obtain urban rail transit network planning solutions with good performance under normal and damaged states. This helps in arranging the near-term and long-term station layouts and network planning of the entire urban rail transit network rationally. Therefore, in resilient urban rail transit network planning, the influence of network structure on network performance under normal conditions is equally important.

The primary research objectives of this study are as follows:

(1)

To develop a two-stage urban rail transit network planning model that aims to minimize total costs and total operating time as independent objectives, while ensuring the preservation of the network’s operational symmetry under standard and disruptive conditions.

(2)

To design a hybrid evolutionary algorithm with a dual-layer encoding structure to effectively solve the proposed model. The algorithm is built on the Niched Pareto Genetic Algorithm framework and incorporates a Large Neighborhood Search mechanism, diversity preservation strategies, and hierarchical clustering-based cross-over and mutation operations to optimize connectivity and maintain the structural symmetry of the network.

(3)

To validate the proposed model and algorithm through extensive computational experiments. These experiments include comparative analyses across different methods, datasets of varying scales, and single-objective versus multi-objective optimization results. The findings aim to demonstrate the practical applicability and effectiveness of the proposed approach in maintaining the operational symmetry and resilience of urban rail transit networks.

The framework for addressing the problem in this paper is shown in Figure 1.

The flowchart visually represents the structured approach taken to solve the problem of urban rail transit network planning. It begins with the problem establishment, where key challenges and objectives are identified. Next, a two-stage mathematical model is developed to minimize costs and operating time while ensuring network symmetry. Following model development, the focus shifts to algorithm design. Here, a hybrid evolutionary algorithm is crafted to solve the proposed model effectively. The algorithm incorporates various enhancement strategies, including Large Neighborhood Search for initial population generation, diversity preservation mechanisms, and hierarchical clustering-based cross-over and mutation operations.

2.1. Problem Description

The preliminary planning approach for urban rail transit stations, particularly subways, primarily follows the principle of “establishing stations where congestion is high and laying tracks where development is concentrated”. In large metropolitan areas such as Beijing, New York, and Tokyo, different functional zones like commercial districts (e.g., Times Square in New York and Wangfujing in Beijing), administrative centers, and tourist attractions (e.g., the Tokyo Tower area) attract a significant number of passengers, creating urgent transportation demands. Consequently, constructing such an urban rail transit network requires identifying the critical properties of candidate stations, including major passenger departure stations, transfer stations, major passenger arrival stations, and their corresponding connectivity [36].

In the initial stage of urban rail transit planning, newly built candidate stations are identified through comprehensive investigation and analysis, and these stations are typically categorized into three types:

(1)

Major passenger departure points: These stations are located in densely populated areas or regions with high transportation demand. For example, residential areas such as Brooklyn in New York or Chaoyang District in Beijing often serve as the main departure points for passengers, where people start their journeys from home or workplaces to reach various destinations across the city.

(2)

Major passenger destinations: These stations are often located in city centers, commercial hubs, or tourist attractions. For instance, stations like Grand Central Terminal in New York or Xidan in Beijing serve as primary destinations where passengers disembark to engage in commercial activities, sightseeing, or other essential activities.

(3)

Major transfer stations: These stations are positioned at the intersections of different subway lines or other transportation modes within the urban rail network. Examples include Union Station in Washington, D.C., or Shinjuku Station in Tokyo. These stations are selected based on passenger transfer demands and traffic flow, aiming to facilitate convenient transfers and optimize the interconnectivity of the urban rail transit system.

The urban rail transit network planning problem studied in this paper selects a subset of candidate departure and transfer points for construction and determines the connectivity between selected sites and other newly proposed candidate stations to minimize the construction and operating costs of the network while meeting passenger demand. Additionally, the uncertainty of node and connectivity damage is considered, and the operating costs under normal and damaged states of the network are incorporated into the optimization objectives. Therefore, the problem addressed in this paper is a two-stage robust optimization problem for resilient urban rail transit network planning, with total network cost and total network operating time as two independent optimization objectives. The total network cost is further divided into two stages: the cost in the first stage includes construction costs and operating costs under normal conditions, which involve operating and transfer costs; the cost in the second stage considers possible changes in network structure due to node and line damage, including operating and transfer costs, as well as penalty costs incurred due to unmet passenger demand. This stage occurs after the node location and connectivity decisions have been made. To handle uncertainty, robust optimization methods are applied to the costs in the second stage, setting an uncertainty set for network damage scenarios, and computing the network operating costs for this stage based on this scenario set. The objective function minimizes total operating time and total transfer time, i.e., minimization of total network operating time.

In this problem, there are two independent optimization objectives with conflicting interests. Pursuing the minimization of total network operating time tends to avoid passenger detours, prioritizing the selection of departure and transfer points along routes with the shortest total operating time. However, this operational mode may lead to a network structure that favors direct routes, increasing the likelihood of operational interruptions after damage and making it difficult to maintain the continuity of operations for other nodes, thus resulting in increased post-damage operating costs. On the other hand, pursuing the minimization of total network cost focuses more on reducing construction costs, potentially adopting single-chain network structures. However, once damaged, the penalty for operating costs will increase due to the difficulty of maintaining service continuity in single-chain configurations under damaged conditions. Therefore, this optimization problem requires finding a balance between total network cost and total operating time, balancing the objectives to find a suitable network planning solution.

The development of our model and algorithm is based on several key assumptions:

Station Selection and Passenger Flow: We assume that the identification of candidate stations for major departure points, destinations, and transfer points is accurate and representative of the real-world demand. This assumption simplifies the station selection process, focusing on areas with the highest congestion and strategic importance, which limit the applicability of the model in scenarios where demand patterns are less clear or subject to rapid changes.

Network Symmetry: The model assumes that maintaining symmetry in the urban rail network enhances resilience and efficiency. While this is effective in most urban settings, the assumption cannot fully account for highly asymmetrical urban layouts or extreme demand variations, potentially affecting the generalizability of the findings.

Uncertainty Representation: The model incorporates uncertainty through robust optimization, assuming that the set of possible damage scenarios is well defined and exhaustive. This assumption is crucial for the optimization process but cannot fully capture unanticipated disruptions or rare events, which could impact the model’s robustness in highly volatile environments.

Cost and Time Trade-offs: It is assumed that there is a clear and direct trade-off between minimizing total network operating time and total network cost. This assumption simplifies the optimization process, yet it could overlook other significant factors such as social, environmental, or political influences that could affect decision making in real-world applications.

2.2. Methodology

The urban rail transit system is characterized by a vast and intricate network structure, where the normal state is defined as the condition in which all network components, including nodes (stations) and edges (railway lines), are fully functional, and all facilities are operating smoothly to ensure the efficient movement of passengers.

However, this system is susceptible to disruptions that can cause damage to its components. Damage refers to any form of impairment to the network’s infrastructure, which can range from minor functional reductions to the complete failure of specific nodes or edges. These damages can arise from natural disasters—such as earthquakes, heavy rainfall, mudslides, floods, and strong winds—and man-made events, including large-scale gatherings, terrorist attacks, and traffic accidents that cause blockages.

A potential damage scenario represents a situation in which one or more components of the network may be compromised due to such disruptive events. Damaged conditions describe the state of the network when one or more components have suffered damage, leading to a degradation of the network’s operational capacity. For instance, a damaged condition may involve a station (node) being partially operational or a railway line (edge) being temporarily out of service.

The network failure scenario is a specific type of damaged condition where the damage is severe enough to cause a substantial breakdown in network operations. In such a scenario, key network components—like critical nodes or major edges—are damaged to the extent that they can no longer support the intended flow of passengers, leading to widespread service disruption. The components that cause failure typically include critical stations (nodes) and major connecting lines (edges); when these are damaged, the network’s ability to function is compromised.

In this paper, the uncertainty set for network failures is modeled using a budgeted uncertainty set. This approach assumes that in a network with $a$ nodes, up to $b$ nodes can be damaged to varying degrees, representing different levels of disruption. These damaged nodes are key indicators of network damage, and their failure directly correlates with the overall degradation of network performance. The set of possible damaged nodes and edges under these scenarios defines the network’s damaged state, in contrast to its normal, fully operational state.

The network’s resilience is determined by its ability to withstand and recover from these disruptions. Static resilience refers to the network’s capacity to absorb the impacts of damage, preventing catastrophic failure, while dynamic resilience relates to the system’s ability to rapidly recover from such failures and return to its normal operational state [37]. Thus, the network failure scenario $u$ can be expressed as

$$u=\left\{o_1{\chi }_1,o_2{\chi }_2,\cdots ,o_a{\chi }_a\right\},\forall o_m\in \left\{0,1\right\};{\chi }_m=\left\{0,1\right\}$$

(1)

$${\displaystyle \sum _m{o}_m\le b}$$

(2)

$$o_m\ge {\chi }_m$$

(3)

In the equation, $o_m$ represents whether node $m$ is damaged, with damage denoted by $o_m=1$ and no damage denoted by $o_m=0$; $b$ represents the upper limit of the number of damaged nodes, while ${\chi }_m$ denotes the degree of node damage within the range of $[0,1]$. When ${\chi }_m=0$, it indicates that node $m$ has not incurred any damage, with $o_m$ synchronously being 0. When ${\chi }_m=1$, it indicates that node $m$ has been completely damaged. The value of the damage level ${\chi }_m$ is only valid when certain nodes are confirmed to be damaged. For the sake of subsequent discussions, let us assume that when a node is completely damaged, the edges directly connected to that node become unusable. When a node is partially damaged, its partial functionality (as a starting point or transfer point) remains effective, and the edges connected to that node can still be used. Additionally, since network damage typically involves only a small number of nodes and rarely affects multiple nodes, especially those geographically distant, the likelihood of simultaneous damage to multiple nodes is low. Therefore, this paper sets the number of damaged nodes to 1, focusing solely on the urban rail transit network planning problem when only one node is damaged. For a given scale of network, generating several sets of $o_m$ and ${\chi }_m$ that satisfy Equations (1)–(3) at random can form an uncertainty set, $U$, of network failure scenarios.

2.3. Description of Symbols and Variables

The parameters and symbols involved in the model are explained in

Table 1. Symbol description in the model.

Symbol	Explanation	Symbol	Explanation
$c_m^{(G)}$	The construction cost of node $m$ as the origin station.	$\nu$	Urban rail transit operating speed.
$c_m^{(H)}$	The construction cost of node $m$ as the transfer station.	${\eta }_n$	Unit passenger transfer time at transfer point $n$.
$j_{mn}$	The distance between node $m,n$.	$D_G$	Set of departure stations.
$\epsilon$	The operating cost per unit distance per passenger.	$D_H$	Set of transfer stations.
$\delta$	The transfer cost per passenger.	$D_X$	Set of destination stations.
$Z$	The maximum passenger capacity per train car.	$D$	Create candidate urban rail transit stations.
$X_n$	The passenger flow volume at destination station $n$.	$P$	The set of edges forming a fully connected directed network graph from the set $D$.
$Q_m$	The passenger flow volume at departure point $m$.	$U$	Set of uncertain network disruption scenarios.
$H_n$	The maximum transfer capacity at transfer point $n$ per transfer.	$D^u$	The set of departure and transfer nodes that can still function under the $u$-th network disruption scenario.
${\varphi }_n$	The penalty cost per unit when the passenger demand at destination station $n$ is not met.	$P^u$	The set of edges for the fully connected directed network graph composed of node sets $D^u$ and $D_X$ under the $u$-th network disruption scenario.

.

In network planning problems, three aspects of decision variables are involved:

(1)

Site selection for departure and transfer points;

(2)

Determination of connectivity between nodes;

(3)

Allocation of network passenger flow.

Let ${\beta }_m^{(G)}$ and ${\beta }_m^{(H)}$ represent binary variables indicating whether node $m$ is designated as a starting point or a candidate transfer point.

$${\beta }_m^{(G)}=\left\{\begin{array}{l}1,\mathrm{Node}m\mathrm{is}\mathrm{designated}\mathrm{as}\mathrm{the}\mathrm{starting}\mathrm{station}.\\ 0,\mathrm{Conversely}\end{array}\right.$$

(4)

$${\beta }_m^{(H)}=\left\{\begin{array}{l}1,\mathrm{Node}m\mathrm{is}\mathrm{designated}\mathrm{as}\mathrm{a}\mathrm{transfer}\mathrm{station}.\\ 0,\mathrm{Conversely}\end{array}\right.$$

(5)

Let ${\alpha }_{mn}$ represent whether node $m$ is connected to node $n$. Here, $M$ is the set of departure points and transfer points, $m\in M$, while $N$ is the set of transfer points and demand points, $n\in N$.

$${\alpha }_{mn}\left\{\begin{array}{l}1,\mathrm{Node}m\mathrm{is}\mathrm{connected}\mathrm{to}\mathrm{node}n.\\ 0,\mathrm{Conversely}\end{array}\right.$$

(6)

In addition, with the network structure established, decisions regarding passenger flow allocation are necessary. Let $l_{mn}$ and $l_{mn}^u$ represent the passenger flow between arcs $(m,n)$ in the normal state of the network and the passenger flow between arcs $(m,n)$ under network failure scenario $u$. These decision variables collectively influence the structure and operation of the network. Therefore, solving the problem requires finding the optimal combination of these decision variables to minimize total costs and total operation time.

2.4. Mathematical Model

The multi-objective two-stage symmetry-based urban rail transit network model established in this paper is as follows:

$$\begin{array}{l}\underset{{\beta }_m^{(G)},{\beta }_m^{(H)},{\alpha }_{mn},l_{mn}}{\min }{\displaystyle \sum _{m\in D_{(G)}}{c}_m^{(G)}{\beta }_m^{(G)}+}{\displaystyle \sum _{m\in D_{(H)}}{c}_m^{(H)}{\beta }_m^{(H)}+}{\displaystyle \sum _{(m,n)\in P}\epsilon j_{mn}{l}_{mn}}+\\ {\displaystyle \sum _{n\in D_H}{\displaystyle \sum _{m\in D_G\cup D_H}\delta l_{mn}+\underset{u\in U}{\max }}}\underset{l_{mn}^u\in \theta ({\beta }_m^{(G)},{\beta }_m^{(H)},{\alpha }_{mn},u)}{\min }\left[{\displaystyle \sum _{(m,n)\in P^u}\epsilon j_{mn}{l}_{mn}^u}\right.+\\ {\displaystyle \sum _{n\in D^u\cap D_H}{\displaystyle \sum _{m\in D^u\cap (D_G\cup D_H)}\delta l_{mn}^u}+\left.{\displaystyle \sum _{n\in D_X}{\varphi }_n\left(X_n-{\displaystyle \sum _{m\in D^u}{l}_{mn}^u}\right)}\right]}\end{array}$$

(7)

$$\min {\displaystyle \sum _{(m,n)\in P}\frac{j_{mn}}{\nu }}\left[{\displaystyle \frac{l_{mn}}{Z}}\right]+{\displaystyle \sum _{n\in D_H}{\displaystyle \sum _{m\in D_G\cup D_H}{l}_{mn}{\eta }_m}}$$

(8)

$$s.t.\hspace{1em}{\displaystyle \sum _{m\in D_G}{Q}_m{\beta }_m^{(G)}\ge {\displaystyle \sum _{n\in D_X}{X}_n}}$$

(9)

$${\alpha }_{mn}\le {\beta }_m^{(G)},\forall m\in D_G;\forall n\in D_H\cup D_X$$

(10)

$${\alpha }_{mn}=0,\forall m\in D_H;\forall n\in D_G$$

(11)

$${\alpha }_{mn}\le {\beta }_m^{(H)},\forall m\in D_H;\forall n\in D_H\cup D_X$$

(12)

$${\alpha }_{mn}=0,\forall m\in D_X;\forall n\in D_G\cup D_H$$

(13)

$${\displaystyle \sum _{m\in D}({\alpha }_{mn}+{\alpha }_{nm})}\ge 1,\forall n\in D$$

(14)

$${\displaystyle \sum _{n\in D_H\cup D_X}{l}_{mn}\le G_m,\forall m\in D_G}$$

(15)

$${\displaystyle \sum _{n\in D_H\cup D_X}{l}_{mn}}-{\displaystyle \sum _{c\in D_G\cup D_H}{l}_{cm}=0,\forall m\in D_H}$$

(16)

$${\displaystyle \sum _{c\in D_G\cup D_H}{l}_{cm}\le X_m,\forall m\in D_X}$$

(17)

$$l_{mn}\le H_n,\forall m\in D_G\cup D_H;\forall n\in D_H$$

(18)

$${\displaystyle \sum _{n\in D_H\cup D_X}{l}_{mn}\le M{\beta }_m^{(G)}},\forall m\in D_G$$

(19)

$${\displaystyle \sum _{n\in D_H\cup D_X}{l}_{mn}\le M{\beta }_m^{(H)}},\forall m\in D_H$$

(20)

$$l_{mn}\le M{\alpha }_{mn},\forall m,n\in D$$

(21)

$$\left\{\begin{array}{l}{\beta }_m^G=\left\{0,1\right\},\forall m\in D_G\\ {\beta }_m^H=\left\{0,1\right\},\forall m\in D_H\end{array}\right.$$

(22)

$${\alpha }_{mn}=\left\{0,1\right\},\forall m\in D_G\cup D_H;\forall n\in D_H\cup D_X$$

(23)

In this case, $\theta \left({\beta }_m^G,{\beta }_m^H,{\alpha }_{mn},u\right)$ satisfies

$${\displaystyle \sum _{n\in D^u\cap (D_H\cup D_X)}{l}_{mn}^u}\le (1-{\chi }_m{o}_m)G_m{\beta }_m^{(G)},\forall m\in D_G\cap D^u$$

(24)

$${\displaystyle \sum _{n\in D^u\cap (D_H\cup D_X)}{l}_{mn}^u}\le (1-{\chi }_n{o}_n)H_n{\beta }_n^{(H)},\forall n\in D_H\cap D^u$$

(25)

$${\displaystyle \sum _{n\in D^u\cap (D_H\cup D_X)}{l}_{mn}^u}\le G_m,\forall m\in D_G\cap D^u$$

(26)

$${\displaystyle \sum _{n\in D^u\cap (D_H\cup D_X)}{l}_{mn}^u}-{\displaystyle \sum _{c\in D^u\cap (D_G\cup D_H)}{l}_{cm}^u}=0,\forall n\in D_H\cap D^u$$

(27)

$${\displaystyle \sum _{c\in D^u\cap (D_G\cup D_H)}{l}_{cm}^u}\le X_m,\forall m\in D_X\cap D^u$$

(28)

$$l_{mn}^u\le H_n,\forall m\in D_G\cup D_H;\forall n\in D_H\cap D^u$$

(29)

$${\displaystyle \sum _{n\in D_H\cup D_X}{l}_{mn}^u\le M{\beta }_m^{(G)}},\forall m\in D_G\cap D^u$$

(30)

$${\displaystyle \sum _{n\in D_H\cup D_X}{l}_{mn}^u\le M{\beta }_m^{(H)}},\forall m\in D_H\cap D^u$$

(31)

$$l_{mn}^u\le M{\alpha }_{mn},\forall m\in D^u;\forall n\in D$$

(32)

Equation (7) represents the objective function of the total network cost. In the first stage, the costs include the construction cost of departure and transfer nodes, the total operating cost, and transfer costs under normal conditions (i.e., when the network is not damaged). In the second stage, given ${\beta }_m^{(G)},{\beta }_m^{(H)},{\alpha }_{mn},u$, the costs include the operating cost incurred by network operation, transfer costs, and penalty costs for unmet passenger demand at destination stations. Here, ${\displaystyle \sum _{i\in D_G}{c}_m^{(G)}{\beta }_m^{(G)}},{\displaystyle \sum _{i\in D_H}{c}_m^{(H)}{\beta }_m^{(H)}}$ represent the construction cost of departure points and transfer nodes. Upon determining the locations of departure and transfer nodes, further decisions regarding the connectivity between nodes ${\alpha }_{mn}$ yield the corresponding network graph. By completing passenger flow allocation decisions on the network graph, total operating costs and transfer costs can be calculated. Equation (8) represents the objective function of total network operation time, assessing the operational efficiency under normal network conditions. The calculation of total network operation time includes passenger travel time and total transfer time under normal conditions. Here, $\left[{\displaystyle \frac{l_{mn}}{Z}}\right]$ denotes the minimum number of carriages required for the passenger flow passing through arc $(m,n)$. This aspect could further extend the research to train composition and departure interval issues, which are not detailed here, as the focus of this paper is on network planning. Considering the fact that network damage occurrence is a low probability event, network developers often do not prioritize minimizing operation time after network damage. Additionally, for network developers, it is often more important whether demand nodes’ needs are met rather than efficiently met, once network damage occurs. Hence, this paper focuses on optimizing the efficiency of the network under normal conditions. Equation (9) ensures that the total number of open transfer nodes in the network can meet the total demand of the network. Equation (10) imposes constraints on the connectivity of departure nodes, indicating that only open departure nodes can be connected to other transfer and demand nodes. Equation (11) restricts transfer nodes from being connected in reverse to departure nodes. Equation (12) limits the connectivity of transfer nodes, specifying that only open transfer nodes can be connected to other transfer and demand nodes. Equation (13) ensures that demand nodes are not connected in reverse to departure and transfer nodes.

Equation (14) states that there should be no isolated points in the designed network, where ${\displaystyle \sum _{m\in D}({\alpha }_{mn}+{\alpha }_{nm})}$ represents the number of edges connected to node $m$. Equations (15)–(17) represent the flow balance constraints for nodes in the network. Equation (15) ensures that the passenger flow from departure nodes does not exceed their capacity; $G_m$ denotes the capacity of departure node $m$. Equation (16) ensures that the passenger flow leaving transfer nodes equals the passenger flow entering them. Equation (17) ensures that the passenger flow received by demand nodes does not exceed their demand. Equation (18) imposes capacity constraints on transfer nodes, ensuring that the passenger flow passing through a transfer node in a single trip does not exceed its maximum transfer capacity. Equations (19) and (20) state that only open departure and transfer nodes can facilitate passenger departure and transfer (where M is an arbitrarily large positive number). Equation (21) ensures that passenger flow decisions can only be made between nodes that are planned to be connected. Equations (22) and (23) respectively impose 0–1 constraints on the decision variables for node location and node connectivity.

Equations (24)–(31) represent the constraints under network failure scenarios, constraining the decision making regarding passenger flow $l_{mn}^u$ in the presence of given ${\beta }_m^{(G)},{\beta }_m^{(H)},{\alpha }_{mn},u$. Equations (24) and (25) stipulate that damaged departure and transfer nodes can only partially function, with their retained departure and transfer capabilities affected by the degree of node damage δ. Equations (26)–(28) enforce flow balance constraints for nodes in the network under damage scenarios; Equation (26) ensures that the passenger flow from damaged departure nodes does not exceed their capacity; Equation (27) ensures that the passenger flow leaving damaged transfer nodes equals the passenger flow entering them; and Equation (28) ensures that the passenger flow received by demand nodes under damage scenarios does not exceed their capacity. Equation (29) imposes constraints on the passenger flow through damaged transfer nodes in a single transfer. Equations (30) and (31) state that only open departure and transfer nodes with partial or full capabilities, and are damaged, can facilitate passenger departure and transfer under damage scenarios. Equation (32) allows passenger flow decisions $l_{mn}^u$ to be made only between nodes that retain partial or full functionality in the damaged network.

3. Algorithm Design

The multi-objective robust optimization two-stage symmetry-based urban rail transit network planning model established in this paper involves critical decisions such as determining the location of departure and transfer points, the connectivity between nodes, and the distribution of passenger flow in normal and disrupted states of the network. The model integrates scenario-robust sets and optimizes multiple objectives, ensuring that the network maintains its operational symmetry under varying conditions. This categorizes the problem as a multi-objective two-stage robust optimization problem.

Given the complex nested structure of the problem, which includes multiple objectives, multiple stages, and scenario-robust uncertainty sets, traditional solution methods prove inadequate due to their limitations in handling such intricate optimization landscapes [38]. To address these challenges, this paper employs the Niched Pareto Genetic Algorithm (NPGA) as the primary framework for the algorithm. NPGA is particularly well-suited for this problem due to its advanced capabilities in managing multi-objective optimization tasks, maintaining a diverse set of high-quality solutions, and effectively exploring the solution space to find Pareto optimal solutions [39].

Moreover, the problem’s complexity necessitates a robust search strategy to navigate its vast and complex solution space effectively. Therefore, we have enhanced the NPGA framework with a Large Neighborhood Search (LNS) mechanism. LNS is specifically chosen for its ability to perform deep explorations of large portions of the solution space, systematically optimizing the solution by making significant changes to parts of the solution [40]. This hybrid approach ensures that the algorithm not only explores diverse regions of the solution space but also refines these solutions to achieve higher optimization performance. The combination of NPGA and LNS significantly improves the algorithm’s capability to solve the complex two-stage symmetry-based subway network planning problem, ensuring that the network’s resilience and operational efficiency are optimized under normal and disrupted conditions.

3.1. Encoding

The decision variables of the model include the location selection of departure and transfer points, the connectivity between nodes, and passenger distribution. Passenger distribution is decided after determining the locations and connectivity, and it can be solved using a nested minimum cost flow algorithm. To encode these decision variables, a dual-layer chromosome-encoding structure is designed, consisting of a location gene layer and connectivity gene layer.

The gene length of the location gene layer equals the total number of candidate departure and transfer points, using 0–1 encoding to express the selection of departure and transfer points in segments. Meanwhile, the connectivity gene layer, based on the location gene layer, expresses the connectivity between nodes in the form of an adjacency matrix. Its gene length corresponds to the adjacency matrix formed by selected departure, transfer, and destination points, with a value of 0 on the main diagonal, 1 for connectivity between different nodes, and ∞ for non-connectivity.

For example, in a city rail transit network planning problem considering three departure points, three transfer points, and three newly proposed candidate stations for encoding, the chromosome is divided into two layers. The first layer has a gene length of 6, representing the selection of departure and transfer points, while the second layer is an adjacency matrix generated based on the first layer genes, as shown in Figure 2.

This encoding structure facilitates a clear description of the site selection and connectivity between nodes, providing the algorithm with appropriate inputs. From the interpretation of the location gene layer of this chromosome, it can be observed that the first and third departure points are selected as departure points, while the second and third transfer points are chosen as active transfer points. Meanwhile, the connectivity gene layer expresses whether nodes are connected through an adjacency matrix of a directed network. Each chromosome corresponds to a city rail transit network planning scheme, and the passenger flow in the network is distributed based on valid chromosomes.

3.2. Initial Population Generation Based on Large Neighborhood Search

In generating the initial population, we combine Equation (9) to randomly generate a location gene layer that satisfies constraints. Based on the site selection situation in the location gene layer, comprising departure points and transfer points, Equations (10)–(14) are utilized to randomly determine the selected departure points, transfer points, and connectivity between demand nodes, thereby generating the connectivity gene layer. However, there are numerous legitimate connectivity schemes that meet Equations (10)–(14). If only random generation is employed to produce the initial population, it is challenging to obtain a high-quality initial population. Moreover, due to the high randomness of the connectivity gene layer, the algorithm’s optimization process may become unstable.

To address this issue, we developed a specialized Large Neighborhood Search mechanism tailored for the connectivity gene layer [41]. This mechanism aims to optimize the connectivity gene layer of each individual in the initial population, thereby improving the population’s quality. The Large Neighborhood Search mechanism involves destruction and repair operator operations.

In the destruction operator operation, we first randomly select the node number to be destroyed, which is generated from the selected departure points and transfer points. For the selected destruction node number, starting from the index row of the corresponding node in the connectivity gene layer, a portion of the current connectivity is randomly interrupted, forming the destroyed connectivity gene layer. The repair operator operation then restores the destroyed connectivity gene layer. Starting from the index row of the destruction node number, we search for nodes that are currently unconnected but can be connected to the node, randomly establishing a connection between the destroyed node and the unconnected node, thereby producing the repaired connectivity gene layer. Subsequently, the legality of the repaired connectivity gene layer is checked through constraint conditions to generate a neighborhood solution set. Taking the connectivity gene layer in Figure 2 as an example, assuming a one-gene destruction and repair operation, the specific operations are illustrated in Figure 3 and Figure 4.

In the design of the destruction and repair operators described above, the number of genes for each operation can be adjusted based on the scale of the problem. When there are more genes involved in the destruction and repair operations, a greater number of neighborhood solutions are generated, enhancing the optimization capability. However, this also leads to increased computational time. The specific settings can be dynamically adjusted according to the scale of the problem.

For the generated neighborhood solutions, we calculate the cost and runtime objective values separately using Equations (7) and (8). Subsequently, we compare these values with the initial solution and select the neighborhood solution with the smallest number of dominated solutions. If this neighborhood solution dominates the initial solution, it replaces the initial solution, and the destruction and repair operations continue on the replaced solution.

Regarding the termination condition, if the neighborhood solution does not dominate the initial solution, the iteration count is incremented by 1, and a new set of neighborhood solutions is generated for the initial solution until the algorithm meets the termination condition. Below, Algorithm 1 shows the pseudocode for the Large Neighborhood Search mechanism:

Algorithm 1 Pseudocode for Large Neighborhood Search mechanism.

Algorithm for Generating Initial Population Based on Large Neighborhood Search

Input: Gene segments for site selection, allocation status between nodes, number of departure points, number of transfer points, number of demand points.

Initialization: Number of nodes to destroy, initial cost, initial time, cost of initial destruction scene, total initial cost.

Output: Allocation status between nodes optimized through large neighborhood search.

while Iteration ≤ Max Iteration:

Randomly select a set containing a specified number of indices for nodes to destroy

Perform destruction and repair operations to obtain recover-d

for each solution in recover-d:

Calculate the first-stage cost cost1 and time time of the solution

Calculate the second-stage cost cost2 of the solution

Calculate the total cost

end for

Determine the solution that has an advantage in cost and time

Select the best solution

if the best solution has better cost and time than the initial solution then:

Update the solution to the best solution

else:

Continue iterating

end if

end while

return Optimized allocation status between nodes

3.3. Selection Operator Based on Diversity Preservation Mechanism

In designing the selection operator, this paper introduces a diversity preservation mechanism. When making selections, the diversity of individuals is considered an important factor to promote a more comprehensive and robust search process. With the introduction of the diversity preservation mechanism, the similarity between individuals can be defined as the average Euclidean distance. Specifically, given an individual $q_r$ and another individual $q_s$ in the population, the similarity $K_{rs}$ can be calculated using the following formula:

$$K_{rs}={\displaystyle \frac{1}{Y}}{\displaystyle {\sum }_{i=1}^Y‖q_r-q_{si}‖}$$

(33)

where $Y$ is the size of the population, and $q_{si}$ represents the $i$th individual in the population. This method of similarity calculation considers the average Euclidean distance of individual $q_r$ from other individuals in the population.

In the selection operator, two individuals are randomly chosen from the population for comparison. Subsequently, the average Euclidean distance of each individual from the rest of the population is computed to assess its diversity. Based on the individuals’ performance, diversity, and considering the diversity preservation mechanism, suitable individuals are selected as members of the new generation population. This operator comprehensively considers individual performance, congestion, and diversity, ensuring diversity in the new generation population through the diversity preservation mechanism. These steps are repeated until a sufficient number of individuals are selected for the new generation population. In the diversity preservation mechanism, individuals with better diversity are chosen by comparing their diversity, thereby enhancing the population’s diversity and convergence speed. Ultimately, individuals with better fitness are selected as members of the next generation population.

3.4. Cross-Over and Mutation Operators Based on Hierarchical Clustering

Due to the numerous feasible solutions in the model solved in this paper and the large solution space, suppose the problem contains $\left|D_G\right|$ departure points, $\left|D_H\right|$ transfer nodes, and $\left|D_X\right|$ destination nodes. Then, the corresponding number of gene loci is $\left|D_G\right|+\left|D_H\right|+{(\left|D_G\right|+\left|D_H\right|+\left|D_X\right|)}^2$. As the problem size grows, the solution space expands rapidly, and the algorithm’s exploration capability directly determines whether optimal solutions can be obtained. As a result, this paper devises crossover and mutation operators rooted in hierarchical clustering to augment the algorithm’s capacity for exploration [42]. The application of hierarchical clustering in cross-over and mutation offers the following advantages:

(1)

Promotion of Population Diversity: Hierarchical clustering divides the population into distinct subgroups, each with unique characteristics. By performing cross-over and mutation operations between different subgroups, the diversity of the population is promoted, avoiding the trap of local optima.

(2)

Enhancement of Global Search Ability: Partitioning the population into different subgroups enables more effective exploration of the search space, thereby improving the algorithm’s global search ability. Cross-over and mutation operations between subgroups help the algorithm converge faster to global optimal solutions.

(3)

Prevention of Premature Convergence: Hierarchical clustering prevents premature convergence by dividing the population into different subgroups, each with a specific search direction. Cross-over and mutation operations between different subgroups effectively prevent premature convergence issues.

(4)

Comprehensive Exploration of Search Space: Performing cross-over and mutation operations between different subgroups allows for a more comprehensive exploration of the search space, increasing the likelihood of finding better solutions. This comprehensive search strategy contributes to improving the quality and stability of algorithm solutions.

Specific operations include cross-over and mutation operations based on hierarchical clustering. The process of cross-over based on hierarchical clustering is as follows: first, the total cost and total running time objective values of individuals are used as input features. Then, hierarchical clustering is applied to the offspring population formed by selection operators, resulting in clustered subpopulations. Next, a higher probability is assigned to cross individuals from different subpopulations to increase population diversity. Simultaneously, a lower probability is allocated to cross individuals within the same subpopulation to maintain population stability. In the cross-over operation, a double-point cross-over method is employed, limited to the selection gene layer, as there is a strong correlation between the selection gene layer and the connection gene layer. If the resulting selection gene layer after cross-over does not satisfy specific constraints, the cross-over points are reselected until a legal selection gene layer is produced. For the resulting selection gene layer, legal connection gene layers are randomly generated based on constraint conditions, followed by the introduction of a Large Neighborhood Search mechanism to optimize the connection gene layer. Finally, the generated individuals become part of the next generation population. The schematic representation of the cross-over operator based on hierarchical clustering is illustrated in Figure 5.

The process of mutation based on clustering is as follows. Initially, individuals resulting from crossover undergo clustering according to their objective function values. Each resultant cluster is regarded as a subpopulation. Next, the centroids of each subpopulation are assessed to ascertain whether they are dominated by centroids of other subpopulations, and the domination count of each subpopulation centroid is computed. Based on the domination count of subpopulation centroids, they are categorized into high-, medium-, and low-quality subpopulations. Subsequently, different mutation strategies are applied to subpopulations of varying qualities to enhance the efficiency of the mutation process. The mutation operation exclusively pertains to the selection gene layer of the chromosome. During the mutation process, a baseline mutation rate is established. In high-quality subpopulations, to maintain the exceptional traits of population individuals, the baseline mutation rate is decreased prior to mutation. In medium-quality subpopulations, the baseline mutation rate remains unchanged for mutation. Conversely, in low-quality subpopulations, where population individuals exhibit lower quality, the baseline mutation rate is elevated to further explore the solution space. When mutating the selection gene layer of an individual, the mutation position is randomly determined, followed by a 0–1 flip operation on the gene at the mutation position. If the resulting selection gene layer after mutation does not comply with the rules, the mutation position is reselected until a legal selection gene layer is generated. Ultimately, the Large Neighborhood Search mechanism is implemented to enhance the optimization of the mutated selection gene layer’s connection gene layer. Figure 6 depicts the mutation operator grounded on clustering.

Additionally, this paper sets an operational range for the designed hierarchical clustering-based crossover and mutation operations. Within the specified operational range, the hierarchical clustering-based crossover and mutation operations are performed, while outside this range, traditional crossover and mutation operations are performed. The specific rules are as follows. Let G represent the total iterations of the algorithm. The hierarchical clustering-based crossover and mutation operations are conducted for the first a iterations, during the remaining G-a iterations, traditional crossover and mutation operations are executed. The reason for this rule is as follows: during the initial iteration stages, the distribution of population individuals is relatively dispersed, and the diversity of individuals participating in clustering is greater; thus, the exploration strategy will have a better search effect on the solution space. However, as the algorithm progresses to the middle and later stages of iteration, the population individuals evolve and gradually converge towards the Pareto front, resulting in subpopulations formed after clustering that are more inclined to exhibit non-dominant relationships with each other. In this scenario, there is no significant difference between the hierarchical clustering-based crossover operation and the conventional crossover operation. Furthermore, the hierarchical clustering-based mutation operation may fail because subpopulations cannot be assessed as high, medium, or low quality.

3.5. Algorithm Flowchart

The main algorithm framework proposed in this paper is NPGA, aimed at solving the multi-objective two-stage symmetry-based urban rail transit network planning model. The NPGA incorporates a Large Neighborhood Search mechanism to optimize the connectivity gene layer of individuals. This mechanism boosts the algorithm’s ability to explore the solution space and enhances solution diversity. Additionally, NPGA introduces a hierarchical clustering-based crossover and mutation operation, which further enhances the algorithm’s ability to explore the solution space in the early stages by clustering and crossover mutation of subpopulations, facilitating the convergence of solutions and maintaining diversity. The main algorithm flow is illustrated in Figure 7.

4. Results

4.1. Case Study Analysis

In the urban rail transit network to be planned, which focuses specifically on metro systems commonly found in large metropolitan areas, three types of key nodes are involved: departure points, transfer points, and destination points. Metro systems, characterized by their high capacity and frequent service, are crucial for densely populated urban centers where efficient and reliable public transportation is essential.

The objective of metro network planning is to strategically select sites for departure points and transfer points based on connectivity and accessibility to meet the transportation needs of the population. The operation process in a given network planning scheme includes the flow movement from departure points to destination stations, selecting the shortest or most cost-effective route for operation from departure points to destination stations, either directly or via transfer points. Transfer points in metro systems incur certain costs and time, making their optimal placement crucial for maintaining efficiency.

To illustrate the planning and analysis process, two fictional datasets were generated to simulate typical scenarios encountered in large metropolitan areas. These datasets do not represent any specific real-world location but are designed to reflect the conditions in densely populated cities with extensive metro networks. The first dataset consists of three departure points, three transfer points, and five destination points. The second dataset includes five departure points, ten transfer points, and fifteen destination points. Each unsatisfied demand node is associated with a penalty coefficient of 1500, and the passenger demand at each destination station is randomly generated within the range of [50, 110]. These fictional scenarios allow us to systematically evaluate the proposed methods and demonstrate their applicability to metro network planning.

This study comprehensively considers the multi-objective optimization objectives of network construction costs, operating costs under normal network conditions, and operating costs under network damage conditions. Evaluation indicators mainly include the following:

(1)

Network construction costs: including the site selection costs of departure points and transfer points and the construction costs of connectivity between nodes.

(2)

Operating costs under normal network conditions: involving operating and transfer costs.

(3)

Operating costs under network damage conditions: including operating and transfer costs, as well as penalty costs incurred when the demand at destination stations cannot be met.

(4)

Total network operation time: including total operation time and total transfer time.

Furthermore, in the construction of damage scenarios, the number of damaged nodes is set to 1, and the damage levels ${\chi }_m$ are 30%, 50%, and 100%. The generation of damage scenarios follows a random approach, where the node to be damaged is first randomly selected, followed by randomly determining the degree of damage. If a generated damage scenario coincides with any existing scenario in the damage scenario set, it is regenerated until the predetermined number of damage scenarios is achieved. The size of the damage scenario set in this study increases with the increase in the number of candidate departure and transfer nodes in the network. Given the dataset size $\left|D_G\right|-\left|D_H\right|-\left|D_X\right|$, let us specify that there are $3^{\left|D_G\right|+\left|D_H\right|}$ damage scenarios.

4.2. Effectiveness Analysis of the Algorithm

In this section, we utilize a randomly generated 3-3-5 dataset along with the same set of network damage scenarios for uncertainty. The 3-3-5 rail transit network is represented as an 11 × 11 matrix, as shown in

Table 2. Matrix representation of the 3-3-5 rail transit network (unit: meters).

Station ID	1	2	3	4	5	6	7	8	9	10	11
1	0	323	420	157	294	498	84	319	348	127	17
2	323	0	9	171	433	476	204	271	64	484	419
3	420	9	0	479	38	363	440	256	32	66	80
4	157	171	479	0	68	347	214	66	451	354	353
5	294	433	38	68	0	73	134	114	138	355	153
6	498	476	363	347	73	0	172	216	100	457	240
7	84	204	440	214	134	172	0	248	32	268	372
8	319	271	256	66	114	216	248	0	107	252	97
9	348	64	32	451	138	100	32	107	0	470	220
10	127	484	66	354	355	457	268	252	470	0	455
11	17	419	80	353	153	240	372	97	220	455	0

. Each element in the matrix corresponds to the distance between two stations within the network, measured in meters. This matrix provides a clear overview of the spatial relationships within the rail transit system.

The proposed hybrid evolutionary algorithm and NSGA-II (Non-dominated Sorting Genetic Algorithm II) [43] are employed separately to solve the problem and validate the effectiveness of the hybrid evolutionary algorithm. NSGA-II adopts the same encoding method as proposed in this paper. The comparison between the two algorithms is conducted with a uniform population size of 20 and 50 iterations. Ultimately, the algorithms achieve Pareto optimal solutions. NSGA-II obtains an approximate optimal solution for the problem, demonstrating that the proposed hybrid evolutionary algorithm effectively addresses the two-stage symmetry-based urban rail transit network planning model. The specific computational results are presented in

Table 3. Comparison of Pareto optimal solutions between the hybrid evolutionary algorithm and NSGA-II.

Algorithm and Optimal Solution		Total Cost Objective Value/Ten Thousand Dollars	Total Operating Time Objective Value/Hours	Optimal Location Scheme	Optimal Connectivity Relationships
Hybrid evolutionary algorithm	Pareto Solution 1	1,724,200	21.23	101,010	G1-H2, X1, X2, X3, X4, X5 G3-H2, X1, X3, X4, X5 H2-X1, X2, X3, X4, X5
	Pareto Solution 2	1,474,915	206.13		G1-H2, X1, X2, X4, X5 G3-X2, X3, X4, X5 H2-X1, X2
NSGA-II	Pareto Solution 1	3,602,970	21.74	101,010	G1-H2, X1, X3, X4, X5 G3-H2, X3, X4 H2-X1, X2, X3
	Pareto Solution 2	3,001,090	206.54		G1-H2, X1, X3, X4, X5 G3-H2, X2, X3, X4, X5 H2-X1, X3

. The crossover probability and mutation probability of our algorithm are set to 0.8 and 0.2, respectively. All case tests were conducted on a laptop computer equipped with a 6-core processor running at 3.20 GHz and 16 GB of RAM, operating on the Windows 11 operating system. The simulation environment utilized the Mat1ab2016 experimental platform.

NSGA-II obtained an approximate optimal solution for the problem, differing from the optimal result of the hybrid evolutionary algorithm. This suggests that, for this specific problem, the hybrid evolutionary algorithm is capable of finding the Pareto optimal solution, while NSGA-II provides an approximate solution, indicating that its performance may be influenced by algorithm parameters and population settings.

Based on the results obtained, solutions can be categorized into cost-preferred and time-preferred schemes. Cost-preferred schemes prioritize lower overall network costs when the total cost objective value is high, resulting in relatively lower total network operating time. Conversely, time-preferred schemes exhibit significantly prolonged total network operating time when the total cost objective value is low. In such cases, network planning leans towards time-preferred schemes, where overall network costs are higher, but total network operating time is relatively shorter.

From Table 3, it can be observed that the hybrid evolutionary algorithm designed in this paper yielded two sets of Pareto solutions, corresponding to time-preferred and cost-preferred solutions, denoted as Pareto Solution 1 and Solution 2, respectively. NSGA-II ultimately obtained three sets of Pareto solutions, where Pareto Solution 1 and Solution 3 represent time-preferred and cost-preferred solutions, while Pareto Solution 2 serves as a compromise between Solutions 1 and 3. Analyzing the optimal objective function values reveals that the cost-preferred Pareto solutions of the hybrid evolutionary algorithm dominate NSGA-II’s Pareto Solution 1 and Pareto Solution 2, whereas the time-preferred Pareto solutions of the hybrid evolutionary algorithm dominate all three sets of NSGA-II’s Pareto solutions. This indicates that, regardless of decision makers’ cost or time preferences, the hybrid evolutionary algorithm can provide superior network construction plans compared to NSGA-II. Meanwhile, Table 3 presents the optimal location schemes and the best connectivity relationships. Nodes in this Table are labeled using the format ‘G (departure points)/H(transfer stations)/X(destinations) + node number’. Connections between nodes are indicated with a dash; for example, G1-H2 represents a connection between departure point 1 and transfer station 2. If multiple connections originate from the same point, they are separated by commas. Furthermore, the optimization convergence of the two algorithms for cost-preferred solutions is depicted in Figure 8.

The hybrid evolutionary algorithm converges to the optimal value in fewer than 10 generations. Specifically, the algorithm employs a hierarchical clustering-based crossover and mutation strategy within the first 15 iterations. In other words, during the initial 15 iterations, the algorithm utilizes the hierarchical clustering-based crossover and mutation strategy, demonstrating its strong ability to explore the solution space. In comparison to the final optimization results of NSGA-II for cost-preferred solutions, although NSGA-II achieves convergence in the mid-term iterations, the obtained optimal objective function values are relatively inferior to those of the hybrid evolutionary algorithm. This indicates that the hybrid evolutionary algorithm possesses stronger optimization capabilities and can generate superior resilient subway network construction plans.

4.3. Comparative Results and Analysis of Different Scale Cases

To further validate the algorithm’s performance, this section compares the results obtained by solving two sets of data with different scales using the algorithm proposed in this paper. The Pareto optimal solutions are presented in

Table 4. Pareto optimal solutions for resilient urban rail transit network planning problems of different scales.

Optimal Solutions for Datasets of Different Scales		Total Cost Objective Value/Ten Thousand Dollars	Total Operating Time Objective Value/Hours
3-3-5	Pareto Solution 1	1,724,200	21.23
	Pareto Solution 2	1,474,915	206.13
5-10-15	Pareto Solution 1	3,778,355	37.19
	Pareto Solution 2	2,319,115	928.84

.

From the various sets of Pareto optimal solutions in Table 4, it can be observed that, under different scale datasets, the network planning problem yields two sets of planning solutions: cost-preferred type (where the total network cost is lower but the total network operation time is longer) and time-preferred type (where the total network cost is higher but the total network operation time is shorter). Depending on factors such as the urgency requirement of the operating commodities and the expected investment in network construction, network planners can opt for different network construction schemes.

4.4. Model Comparison Results and Analysis

This section introduces a single-objective two-stage symmetry-based subway network planning model and compares it with the model constructed in this paper. The single-objective two-stage symmetry-based subway network planning model only considers the minimization of the total network cost objective. Its objective function includes the network construction cost, operating cost under normal network conditions, and operating cost under damaged scenarios, calculated in the same manner as Equation (7). The solving algorithm adopts the traditional single-objective genetic algorithm. The optimal solution achieved in this paper’s model is contrasted with that of the single-objective two-stage symmetry-based subway network model, as shown in

Table 5. Comparison of optimization results with single-objective two-stage symmetry-based subway network planning model.

Optimal Solutions for Different Models		Total Cost Objective Value/Ten Thousand Dollars	Total Operating Time Objective Value/Hours
The model constructed in this paper	Pareto Solution 1	1,724,200	21.23
	Pareto Solution 2	1,474,915	206.13
Single objective two-stage model		1,443,942	206.53

. The comparison cases select the same 3-3-5 dataset and set the same uncertain set of network damage scenarios. The total network operation time of the single-objective two-stage symmetry-based subway network planning model is calculated by substituting its optimal network construction plan into Equation (8).

According to Table 5, the multi-objective two-stage symmetry-based subway network planning model constructed in this paper successfully obtained two sets of Pareto solutions. These solutions, respectively, represent time preference (Pareto Solution 1) and cost preference (Pareto Solution 2). It is noteworthy that Pareto Solution 2, which exhibits cost preference, is very close to the optimal solution of the single-objective two-stage symmetry-based subway network planning model. Additionally, the model constructed in this paper also provides time preference network construction solutions with lower network operation times. Therefore, when clients are sensitive to operational efficiency, the proposed network design model in this paper can provide multiple balanced network construction solutions.

To further compare the differences in cost preference solution optimization between the multi-objective two-stage symmetry-based subway network planning model proposed in this paper and the single-objective two-stage model, we plotted the optimization convergence of cost preference solutions for the two models, as shown in Figure 9.

The multi-objective model gradually converges to the optimal value during iterative solving. By contrast, although the single-objective model achieves convergence in the early to middle stages of iteration, the obtained optimal objective function value is less satisfactory compared to the multi-objective model. This indicates the superiority of the multi-objective model, which can provide better solutions for resilient subway network construction.

In summary, the hybrid evolutionary algorithm proposed in this paper demonstrates excellent performance in effectively solving the two-stage symmetry-based subway network planning model, offering a feasible approach for addressing real-world problems.

4.5. Performance Evaluation and Sensitivity Analysis

To validate the effectiveness of the algorithms, we applied the hybrid evolutionary algorithm developed in this study, along with the standard NSGA-II and NSGA-III, to the previously described 3-3-5 dataset. NSGA-II and NSGA-III used the same encoding method as outlined in this paper. For consistency, all algorithms were configured with a population size of 20 and run for 50 iterations, achieving optimal solutions. Additionally, we conducted a sensitivity analysis to evaluate the model’s robustness under varying total demand scenarios, focusing on demand levels of 1800, 3000, and 5000. The detailed computational results and the outcomes of the sensitivity analysis are presented in

Table 6. Comparison of optimal solutions.

Algorithms and Optimal Solutions	Total Cost Objective Value/Ten Thousand Dollars			Total Operating Time Objective Value/Hours
	1800	3000	5000	1800	3000	5000
Hybrid evolutionary algorithm	1,474,915	2,398,660	4,233,010	206.13	234.21	417.15
NSGA-II	3,001,090	3,780,940	5,015,430	206.54	241.30	423.45
NSGA-III	2,765,020	3,467,716	4,620,168	209.12	245.80	431.70

.

The optimal results obtained using NSGA-II, NSGA-III, and the hybrid evolutionary algorithm show significant differences. As indicated in Table 6, the hybrid evolutionary algorithm consistently provides superior solutions compared to NSGA-II and NSGA-III. Sensitivity analysis results reveal that as demand levels increase, total cost and total operating time for all algorithms rise. However, the hybrid evolutionary algorithm consistently delivers lower values in these metrics across varying demand levels, highlighting its significant advantage. While NSGA-III performs better than NSGA-II in terms of total cost, it still does not match the performance of the hybrid evolutionary algorithm. NSGA-II performs the worst across all scenarios, particularly in optimizing total cost.

5. Conclusions

As urbanization progresses worldwide, rapid urban population growth leads to spatial constraints and traffic congestion, which have become critical factors constraining urban development. Urban rail transit systems, with their unique underground operation mode, not only alleviate traffic congestion and improve commuter efficiency but also optimize urban spatial structure and drive regional economic development. The symmetry and reliability of rail transit networks are crucial for ensuring normal and efficient urban mobility and operation. Research on the resilience of urban rail transit systems, conceptualized as maintaining symmetry in response to operational accidents and natural disasters, is of great significance. This paper aimed to study urban rail transit network planning, supplementing the deficiencies of traditional station-line network planning, and providing technical support for the efficient and symmetric operation of urban rail transit.

We constructed a novel multi-objective two-stage symmetry-based subway network planning model, which uniquely integrates the dual goals of minimizing total network cost and total operating time as independent optimization objectives. Our approach innovatively incorporates scenario-robust optimization methods to ensure that the network achieved high resilience by maintaining symmetry, while also minimizing operational costs and times.

A significant contribution of our research is the development of a hybrid evolutionary algorithm featuring a dual-chromosome-encoding structure. This algorithm, which employs NPGA as its core framework, is enhanced by advanced Large Neighborhood Search mechanisms and hierarchical clustering-based crossover and mutation strategies. These enhancements significantly improve the algorithm’s search capabilities within the solution space, ensuring that the network’s symmetry is preserved even under varying conditions.

The results from the case studies underscore our contributions: the proposed model markedly reduces network operating costs after disruptions, thereby substantially improving resilience through symmetry maintenance. Furthermore, the hybrid evolutionary algorithm demonstrates superior optimization performance, effectively solving the two-stage symmetry-based subway network planning problem and yielding a diverse set of Pareto optimal solutions. This advancement provides decision makers with a range of design options, showcasing the practical and theoretical impact of our approach.

Although the hybrid evolutionary algorithm proposed in this paper performs well in solving the two-stage symmetry-based urban rail transit network planning model, there are still some potential challenges in its application. First, when dealing with larger-scale networks, the computational complexity of the algorithm will significantly increase, leading to longer computation times. This may impact the algorithm’s practical usability, particularly in real-time planning and emergency response scenarios. Second, although the algorithm’s effectiveness has been validated in experiments using randomly generated network damage scenarios, its robustness and adaptability in handling more complex and unpredictable real-time disruptions have not been fully tested. Third, the validation of the urban rail transit network planning problem in this paper was conducted using case studies, which may introduce a certain level of specificity, potentially limiting the generalizability of the findings.

In light of the complexities and assumptions inherent in our current model and algorithm, several avenues for future research present themselves, listed as follows:

(1)

Algorithmic Enhancements

Future work could focus on improving the efficiency of the Niched Pareto Genetic Algorithm combined with Large Neighborhood Search. Enhancements might include hybridizing with other metaheuristic approaches or integrating machine learning techniques to better predict and adapt to dynamic changes in the optimization landscape, potentially leading to faster convergence and higher-quality solutions.

(2)

Model Extensions

The current model could be extended to accommodate more diverse and complex urban layouts that exhibit asymmetrical characteristics, which are not fully captured in our symmetry-based approach. Additionally, incorporating adaptive station selection methods that dynamically respond to changes in passenger demand and urban development patterns would increase the model’s applicability in rapidly evolving urban environments.

(3)

Application to Other Transportation Networks

While the focus of this study has been on urban rail transit networks, the principles and methods developed could be applied to other types of transportation networks, such as bus systems or intercity rail. Extending the model to include multimodal transportation networks, where passengers may transfer between different types of transport, would offer a broader application of the optimization techniques discussed.

(4)

Robustness Against Rare Events

The model currently assumes a well-defined set of damage scenarios for robust optimization. Future research could explore the integration of more sophisticated uncertainty modeling techniques, such as stochastic programming or scenario generation methods that account for rare but high-impact events. This would enhance the model’s robustness and reliability in the face of unexpected disruptions.

(5)

Incorporation of Social and Environmental Factors

Expanding the optimization objectives to include social and environmental factors, such as equity in access to transportation and the environmental impact of network design choices, would make the model more comprehensive. This would better align the model with real-world decision-making processes, where such considerations are increasingly important.

By addressing these areas, future research can build upon the foundations laid in this paper, advancing the field of urban rail transit network planning and offering solutions that are more resilient and better aligned with the complex realities of urban development.