Optimizing Bus Bridging Service Considering Passenger Transfer and Reneging Behavior

Abstract

This paper addresses the design of bus bridging services in response to urban rail disruption, which plays a critical role in enhancing the resilience and sustainability of urban transportation systems. Specifically, it focuses on unplanned urban rail disruptions that result in temporary closure of line sections, including transfer stations. Under this “transfer scenario”, a heuristic-rule based method is firstly presented to generate candidate bus bridging routes. Non-parallel bridging routes are introduced to facilitate transfer passengers affected by the disruption. Meanwhile, the bridging stops visited by parallel routes are extended beyond the disrupted section, mitigating passenger congestion and bus bunching at turnover stations. Then, we propose an integrated optimization model that collaboratively addresses bus route selection and vehicle deployment issues. Capturing passenger reneging behavior, the model aims to maximize the number of served passengers with tolerable waiting times and minimize total passenger waiting times. A two-stage genetic algorithm is developed to solve the model, which incorporates a multi-agent simulation method to demonstrate dynamic passenger and bus flow within a time–space network. Finally, a case study is conducted to validate the effectiveness of the proposed methods. Sensitivity analyses are performed to explore the impacts of fleet size and route diversity on the overall bridging performance. The results offer valuable insights for transit agencies in designing bus bridging services under transfer scenarios, supporting sustainable urban mobility by promoting efficient public transit solutions that mitigate the social impacts of sudden service disruptions.

1. Introduction

Urban rail transit (URT) has rapidly developed as the backbone of public transit systems in many megacities. With the continuous expansion of URT networks, daily ridership is increasing sharply, placing severe operational pressure on URT systems. For instance, the average daily passenger volume in Shanghai has exceeded 13 million. Such a high reliance on the URT system indicates that even minor service disruptions have substantial negative impacts on a large number of rail travelers [1,2]. Especially, disruptions during peak hours can exacerbate adverse effects, such as travel delays for passengers and overcrowded rail stations, amplifying the seriousness and potential dangers. In such situations, it is imperative to evacuate stranded passengers efficiently and provide alternative transit modes to resume their travel.

Although the degraded URT network can accommodate several passengers, its capability heavily depends on the URT network’s topology and the severity of disruptions [3]. Some passengers may lack feasible detour routes and thus cannot be rerouted to other rail lines, leading them to seek alternative transit modes to reach their destinations. Buses, known for their flexibility and relatively large carrying capacity, play a vital role in emergency evacuations [4]. However, regular bus services may be unable to cope with all stranded passenger demand, especially in peak hours [5]. Thus, transit agencies generally provide a temporary replacement, called a “bus bridging service”, in response to unplanned rail disruptions, which functions as a “bridge” to restore the network connectivity. A survey conducted by Pender et al. [6] revealed that 85% of agencies relied on bus bridging services to manage unplanned disruptions. This study explores innovative bus bridging approaches for maintaining seamless and passenger-friendly transportation during unforeseen service disruptions, which is essential for achieving long-term urban sustainability goals.

Numerous scholars have focused on the bus bridging service design problem (BBSDP) with the goal of evacuating stranded passengers efficiently and effectively. The conceptual framework of the BBSDP was firstly proposed by Kepaptsoglou and Karlaftis [7]. It is a three-step procedure: first, define the bridging environment using a matrix; second, generate candidate routes by a shortest path algorithm and then select the optimal route set by a bi-level heuristic process; finally, allocate available bus resources with the objectives of minimizing bus travel times and the operational effects on the regular bus network. On this basis, a series of follow-up works have been conducted. For instance, Jin et al. [8] presented a column generation procedure to generate candidate bus routes. Then, the most effective route combination was selected by a path-based multicommodity network flow model. An optimization model was constructed to determine vehicle resources allocation and bus headways. The shortest path theory was utilized by Y. Wang et al. [2] and Deng et al. [9] to address the problem of bus bridging route generation. However, they concentrated solely on route design, and the issue of bus deployment was not considered. Assuming that the set candidate bus bridging route was known, Liang et al. [5], Luo and Xu [10] and Feng et al. [11] utilized integrated transit networks to evacuate disrupted commuters, deploying the remaining rail lines, existing bus lines and newly introduced bus bridging lines. Specifically, Feng et al. [11] proposed a method to adjust partial bus routes to better accommodate the total passenger demand based on available service capacity.

Concerning the route layout, parallel routes were designed by many researchers to restore connectivity of disrupted rail segments [12,13,14,15]. Considering various OD demands beyond the disrupted sections, nonparallel routes were introduced in many studies [2,9,16,17] to provide more convenient bus bridging services. Particularly, Gu et al. [4] directly designed a bridging service for each bus based on a predefined vehicle operation mode. Buses were not restricted to a dedicated bridging route, thereby improving the flexibility of the bridging service. Nevertheless, this flexibility may result in difficulties for drivers and make stranded passengers confused in choosing which buses to board.

With respect to the stopping mode of bus bridging routes, two typical modes were designed in the existing studies. On the one hand, the stop-by-stop route was observed as the most popular mode utilized by many scholars [2,8,9,12]. Note that the standard bus bridging route, i.e., running buses parallel with the disrupted rail segment to replicate the lost train service, is also a stop-by stop route employed as a common response to unplanned rail disruptions [8,12,16]. On the other hand, an express route with no intermediate stops was applied by Gu et al. [4] and Yin et al. [18] to make quick roundtrips for bridging buses.

Apart from route design and bus deployment, some researchers have studied the BBSDP from other aspects. Instead of designing a new bus bridging network, Jin et al. [19] proposed a modeling framework for optimizing localized integration between metro and bus systems through adjustments to the existing bus services. Pender et al. [20] investigated the issue of bus reserve location to respond better to bus bridging demands. Considering the balking and reneging behavior of impatient passengers, Y. Wang et al. [21] explored the problem of bus bridging demand modeling based on queuing theory. Under uncertain disruption duration, Zhang and Lo [22,23] studied two metro disruption management problems: determining the optimal initiation time of the bus bridging service and designing the bus bridging contract. Additionally, some recent studies utilized integrated optimization of the bridging service design and passenger assignment. For instance, Zhen et al. [15] developed a two-stage stochastic programming model to jointly optimize bus bridging routes, schedules, and passenger assignments. A tailored tabu search algorithm is designed to solve the model. Incorporating a PS–logit model to estimate the probabilities of passenger path choices, Zhu et al. [24] proposed an optimization model to simultaneously determine bus bridging route selection and vehicle deployment. Wang et al. [25] explored the joint optimization of bus bridging route design, frequency determination, and passenger assignment in an integrated metro and bus network with the objective of minimizing the overall costs to operator and passengers.

The existing studies have several limitations: (1) transfer passengers are usually included as affected passengers during URT service disruptions. However, transfer demand for these affected passengers are largely overlooked in bus bridging service design; (2) as noted by Y. Wang et al. [21], an international survey of bus bridging practices indicates that passenger reneging—where passengers abandon their wait for bridging buses—is common during urban rail transit service disruptions. However, this behavior is overlooked in existing bus bridging optimization models, potentially impacting the effectiveness of the resulting bus bridging schemes; (3) analytic methods are typically employed to determine the optimization of objectives, which fail to capture dynamic passenger and bus flows, as well as to manage time-varying constraints (e.g., berth capacity and vehicle capacity). For clarity, some related studies are compared and listed in

Table 1. Comparison of related studies.

Publications	Decision Variables	Consider Passenger Reneging Behavior	Consider Transfer Demand	Objective(s)	Solution Algorithms
Kepaptsoglou and Karlaftis [7]	Route design and vehicle deployment	No	No	Maximize passenger welfare	Genetic algorithm
Jin et al. [8]	Route selection and vehicle deployment	No	Yes	Minimize the total increase in travel time and the number of unsatisfied passengers	Column generation procedure
Luo and Xu [10]	Route selection and vehicle deployment	No	No	Minimize the expected unsatisfied demand	Sample average approximation
Dou et al. [12]	Route design and vehicle deployment	No	No	Minimize passenger inconvenience	A decomposition method
Hu et al. [13]	Route design and vehicle deployment	No	No	Minimize the evacuation time	A customized genetic algorithm
Zhen et al. [15]	Route design and vehicle deployment	No	No	Minimize the unsatisfied passenger demand	Tabu search
Chen et al. [17]	Route design and vehicle deployment	No	Yes	Minimize passenger travel cost and bus running cost	A dynamic decision framework
Zhu et al. [24]	Route selection and vehicle deployment	No	Yes	Minimize the passenger travel time and unsatisfied demand	A variable neighborhood search algorithm
This paper	Route design and vehicle deployment	Yes	Yes	Maximize the total number of served passengers and minimize waiting times	A two-stage genetic algorithm with multi-agent simulation

to highlight the existing gaps.

To overcome the existing limitations, this paper specifically focuses on designing bus bridging service in response to an unplanned urban rail disruption along with the “transfer scenario”. The contributions of this study are listed as follows. (1) A heuristic-rule based method is first presented to generate candidate bus bridging routes. Extended parallel and non-parallel bridging routes are introduced to facilitate transfer passengers and mitigate passenger congestion and bus bunching at turnover stations. (2) Considering passenger transfer and reneging behavior, a model is formulated to lead to collaborative optimization of route selection and bus deployment with the objectives of maximizing the number of passengers served and minimizing their waiting times. (3) A two-stage genetic algorithm is developed to solve the model, which incorporates a multi-agent simulation method to demonstrate dynamic passenger and bus flow within a time–space network. The simulation module realizes dynamic control of the constraints of berth capacity and vehicle capacity and accurately records the optimized results, e.g., passenger waiting times, bus bridging times. (4) A case study is conducted to validate the effectiveness of the proposed methods. Sensitivity analyses are performed to explore the impacts of fleet size and route diversity on the overall bridging performance. The results offer valuable insights for transit agencies in designing bus bridging services under transfer scenarios.

The remainder of this paper is organized as follows. Section 2 provides a description of the problem. Section 3 proposes the main methodology, including generation of candidate routes, formulation of route selection and bus deployment model, and the solution algorithm. A case study is conducted in Section 4 to verify the proposed model and algorithm. Finally, Section 5 concludes and provides suggestions for future research.

2. Problem Statement

Consider the link closure caused by an unplanned disruption, i.e., the temporary closure of a line section including several stations between two turnover stations. Our focus is on a special scenario where the closed link contains a transfer station. We refer to the scenario of link closure without a transfer station as the “no transfer scenario” and with a transfer station as the “transfer scenario” hereafter. With the continuous expansion of the URT network, the transfer scenario has become more prevalent compared to the no transfer scenario.

As illustrated in Figure 1, a link closure occurs on line 1 between two turnover stations $s_1$ and $s_2$. The remaining portion of line 1 operates in short routing mode on segments beyond $s_1$ and $s_2$. $s_3$ is a disrupted transfer station linking line 1 and line 2. Note that line 2 is not affected by the disruption and operates normally.

Under this circumstance, affected passengers whose travel routes include part or all of the closed rail link can be categorized into two groups based on whether they require a transfer at $s_3$. For non-transfer passengers, the disrupted segments on their routes are between $s_1$ and $s_2$ or between a turnover station ($s_1$ or $s_2$) and a disrupted station. For transfer passengers, the disrupted segments are between $s_3$ and a turnover station or between $s_3$ and a disrupted station. Under the no transfer scenario, a large number of passengers are stranded at turnover stations, namely $s_1$ and $s_2$. Nevertheless, besides the turnover stations, the transfer station $s_3$ also becomes a crucial point, with an abundant number of blocked commuters under the transfer scenario.

First, we establish parallel routes along the closed rail link, consisting of the standard route (Route 1 in Figure 1) and various alternative routes, like Route 2 and Route 3 in Figure 1. These routes are designed to transport passengers to their destinations or to facilitate transfers to the remaining rail network. During non-peak hours, utilizing the parallel routes alone may suffice to accommodate all passengers. However, during peak hours when the affected passenger volume is high, especially when there is a large transfer volume at $s_3$, relying solely on parallel routes could pose challenges. It may result in overcrowding at $s_1$, $s_2$ and $s_3$, especially at $s_3$, where all affected transfer passengers must board or alight bridging buses. Station capacity constraints may hinder the efficient evacuation of passengers to $s_3$, $s_1$ and $s_2$. Hence, there is a need for non-parallel routes, like Route 4 and Route 5 shown in Figure 1, connecting Line 1 and Line 2 to transport transfer passengers more efficiently. Consequently, addressing the BBSDP under the transfer scenario primarily involves generating candidate bus bridging routes, including both parallel routes and non-parallel routes. Based on the candidate routes, the subsequent work is to select the optimal route set and allocate the limited bus resources effectively. A detailed flowchart for the entire study is illustrated in Figure 2.

3. Methodology

3.1. Generation of Candidate Bus Bridging Routes

3.1.1. Determining Candidate Originating Nodes

To minimize evacuation time and minimize disruption to regular bus operations, bridging bus stops are typically positioned near metro station exits rather than at existing bus stops. Before generating candidate bus routes, it is crucial to determine the set of candidate originating bus stops. The following factors are considered when selecting originating nodes.

(1)

Distance to the depot or bus terminal: The bridging buses should be dispatched from the depots or bus terminals to the originating nodes as quickly as possible.

(2)

Requirements for turning back: The originating nodes must meet the requirements for turning back, so only URT stations with nearby road conditions suitable for turning around can be selected as originating nodes.

(3)

Distance to the disrupted section: To minimize response time, URT stations far away from the disrupted section should not be selected as originating nodes.

(4)

Stranded passenger volume: Based on the three aforementioned location-related factors, the stranded passenger volume at a station can serve as an auxiliary factor to consider. URT stations with higher number of stranded passengers should be given priority when selecting originating nodes.

The bridging buses make roundtrips between a pair of originating nodes, so the set of originating nodes also serves as the set of terminal nodes. Subsequently, the farthest nodes at both ends within the set can be identified and the nodes located between them become candidate intermediate nodes for bus bridging routes.

3.1.2. Generating Candidate Routes Based on Heuristic Rules

Given a node pair of $\left(s_i,s_j\right)$ as the originating and terminal nodes, candidate bus bridging routes can be generated by the following steps.

Step 1: connect $s_i$ and $s_j$ as a direct bus bridging route. Add this to the candidate route pool R.

Step 2: connect $s_i$ and $s_j$ as the diameter to draw a circle which contains the potential intermediate nodes (see Figure 3). The number of nodes inside the circle is denoted as $\left|N\right|$.

Step 3: generate the bridging routes by increasing the number of intermediate nodes progressively from 1 to $\left|N\right|$, and add the routes into R. The rules for adding $n(n\le \left|N\right|)$ intermediate nodes between $s_i$ and $s_j$ are explained as follows:

• The rule for searching direction

Firstly, a rectangular coordinate system is established with $s_i$ as the coordinate origin, the connecting line between $s_i$ and $s_j$ as the abscissa axis, and the vertical line passing through the origin as the ordinate axis (see Figure 4).

Then, three specific rules for adding the intermediate nodes are proposed to ensure that the searching direction of newly added nodes is towards $s_j$.

Equation (1) guarantees that the abscissa of the added node is larger than that added before:

$$x_{k+1}\ge x_k (k=1,2,\cdots ,n-1)$$

(1)

where $x_k$ is the abscissa of the $k$th added node.

Equation (2) ensures that the deviation degree between the connector of two consecutively added points and the abscissa (i.e., $\theta$ in Figure 3) should not exceed a predefined limit ${\theta }_{max}$.

$$\theta \le {\theta }_{max}$$

(2)

Equation (3) ensures that the added intermediate nodes should be farther and farther away from $s_i$, and be closer and closer to $s_j$, where $D\left(s_i,s_j\right)$ represents the distance between $s_i$ and $s_j$.

$$\left\{\begin{array}{c}D\left(s_{k+1},s_i\right)\ge D\left(s_k,s_i\right)\\ D\left(s_{k+1},s_j\right)\le D\left(s_k,s_j\right)\end{array}\right.\left(k=1,2,\cdots n-1\right)$$

(3)

2.

The rule of adjacent nodes

As illustrated in Figure 5, the adjacent nodes can neither be two points located on the same side of the disrupted section, nor two points on the intersecting line.

For each node pair in the set of candidate originating nodes, the candidate bridging routes can be generated following the above steps and rules. It should be pointed out that the candidate route pool of $\left(s_i,s_j\right)$ and $\left(s_j,s_i\right)$ is the same. The flowchart for generating the candidate bridging routes is displayed in Figure 6.

3.2. Route Selection and Bus Deployment Model

3.2.1. Assumptions

The optimization model is proposed based on the following assumptions.

Assumption 1.

Bus travel times between stations are predetermined constants, which are estimated as the average values of historical bus automatic vehicle location data [4].

Assumption 2.

The passenger demand is given, which can be forecasted using historical passenger data [26].

Assumption 3.

The available buses are assumed to be of the same type, and the capacity of buses is known and identical [4,25].

3.2.2. Notations

The notations utilized in this paper are listed in

Table 2. Notations in the paper.

Indices	Definition
Sets
R	Set of candidate bus bridging routes, $r\in R$
$R_{parallel}$	Set of parallel bus bridging routes, $r_{parallel}\in R_{parallel}$ and $R_{parallel}\subset R$
$R_{parallel}^{non}$	Set of non-parallel bus bridging routes, $r_{parallel}^{non}\in R_{parallel}^{non}$ and $R_{parallel}^{non}\subset R$
$S$	Set of bridging stops, $s\in S$
$S_r$	Set of bridging stops in route r, $s_r\in S_r$ and $S_r\in S$
$X$	Set of candidate schemes for route selection and bus resource allocation, $x\in X$
Parameters and variables
$T_{disruption}$	The duration of the disruption
$T_{\mathrm{limit}}$	The maximum waiting time passengers can afford
$r_{\mathrm{standard}}$	The standard route, $r_{\mathrm{standard}}\in R_{parallel}$
$t_{d\to o}^r$	The travel time from depot to the originating node for the bridging buses on route r
$r{t}_{s+}^r$, $r{t}_{s-}^r$	The travel time between station $s-1$ and station s of the upstream/downstream of route r
$d{t}_{x,s+}^{j,i,r}$, $d{t}_{x,s-}^{j,i,r}$	The dwell time at station s of the jth roundtrip of the ith bus on the upward/downward of route r in bridging scheme x
$t_x^{j,i,r}$	The time required for the ith bus to make the jth roundtrip on route r in scheme x
$C_{bus}$	The bus capacity
$L$	The bus load factor
$H$	The bus headway
$N_{bus}$	The total number of bridging bus resources
$C_S$	The capacity of station s
$N_x^{i,r}$	The total number of roundtrips made by the ith bus on route r in scheme x
$N_r^{\max }$	The maximum number of routes included in a bridging scheme
$t_d$	The time required for turning around
$Q_x^{j,i,r}$	The number of passengers served by the jth roundtrip of the ith bridging bus on route r in scheme x
$Q_{x,r,s}^{j,i}$	The number of passengers in the ith bridging bus for the jth roundtrip at station s on route r in scheme x
$N_{x,s}^{bus}$	The number of buses arriving simultaneously at a bridging stop s in scheme x
Decision variables
${\delta }_x^r$	A binary variable indicating whether route $r(r\in R)$ is included in scheme $x(x\in X)$, if so ${\delta }_x^r=1$, otherwise ${\delta }_x^r=0$
$N_x^r$	The number of bus resources allocated to route $r(r\in R)$ in scheme $x(x\in X)$

:

3.2.3. Model Formulations

First, we aim to maximize the bridging efficiency, which can be expressed as the total number of passengers transported by buses along each bridging route in the bus bridging scheme $x(x\in X)$ during the disruption duration $T_{disruption}$ (denoted as $g\left(x\right)$). The objective function is formulated in Equation (4).

$$Z_1=\underset{x\in X}{\max } g\left(x\right)$$

(4)

Note that passengers waiting for bridging buses may lose patience over time and convert to other transit modes if the waiting time is excessively long [21]. Considering this passenger reneging behavior, $g\left(x\right)$ depends on the number of passengers served within a specified time limit $T_{\mathrm{limit}}$, which denotes the maximum waiting time passengers can afford. As seen in Equation (5), $g\left(x\right)$ is defined as the total number of passengers served by buses on each bridging route within the bus bridging scheme $x(x\in X)$:

$$g\left(x\right)={\displaystyle \sum _{r\in R}{\displaystyle \sum _{i=1}^{N_x^r}{\displaystyle \sum _{j=1}^{N_x^{i,r}}{\delta }_x^r\ast Q_x^{j,i,r}}}}$$

(5)

where $Q_x^{j,i,r}$ denotes the number of passengers whose waiting times do not exceed $T_{\mathrm{limit}}$ and who are served by the jth roundtrip of the ith bridging bus on route r. $N_x^{i,r}$ is the total number of roundtrips made by the ith bridging bus on route r in scheme $x(x\in X)$, which can be determined using Equation (6).

$$t_{d\to o}^r+{\lambda }_x^{i,r}\times H+{\displaystyle \sum _{j=1}^{N_x^{i,r}}{t}_x^{j,i,r}}\le T_{disruption},\forall i\left|1\le i\le N_x^r\right.$$

(6)

The first and second components in Equation (6) represent the initial response time, which is the duration from the moment the bus leaves the depot to the moment the bus reaches the originating bridging stop. ${\lambda }_x^{i,r}$ denotes the departure sequence of the ith bridging bus on route r in $x(x\in X)$. The third component in Equation (6) refers to the total operational time required for buses to make roundtrips on the bridging route. $t_x^{j,i,r}$ represents the time taken by the ith bus to make the jth roundtrip on route r in scheme x, which includes travel time between intermediate nodes and bus dwell time at the nodes in both forward and backward directions and the time for turning around. In this study, $t_x^{j,i,r}$ is determined by a simulation program explained in the subsequent section.

$$t_x^{j,i,r}={\displaystyle \sum _{s_r\in S_r}\left(r{t}_{s+}^r+r{t}_{s-}^r\right)}+{\displaystyle \sum _{s_r\in S_r}\left(d{t}_{x,s+}^{j,i,r}+d{t}_{x,s-}^{j,i,r}\right)}+t_d$$

(7)

In Equation (5), $Q_x^{j,i,r}$ is the sum of passengers whose waiting times do not exceed$T_{\mathrm{limit}}$. Therefore, the determination of $Q_x^{j,i,r}$ is based on passengers’ waiting times. Let $P_s$ denote the set of passengers waiting at stop $s$($s\in S$), and $t_{p,s}^{wait}$ be the waiting time of passenger $p$($p\in P_s$) at stop s. Specifically, $t_{p,s}^{wait}$ signifies the duration from when passenger p reaches stop s to when he boards a bridging bus (see Equation (8)).

$$t_{p,s}^{wait}=A{T}_{p,s}-R{T}_{p,s}$$

(8)

Here a binary variable ${\eta }_x^{j,i,r,p}$ is introduced to indicate whether passenger p served by the jth roundtrip of the ith bridging bus on route r in scheme x has a waiting time $t_{p,s}^{wait}$ shorter than the upper limit of passengers’ tolerable waiting time $T_{\mathrm{limit}}$ (see Equation (9)). As formulated in Equation (10), if ${\eta }_x^{j,i,r,p}=1$, passenger p can be included in $Q_x^{j,i,r}$.

$${\eta }_x^{j,i,r,p}=\left\{\begin{array}{l}1,if t_{p,s}^{wait}\le T_{\mathrm{limit}};\\ 0,otherwise.\end{array}\right.$$

(9)

$$Q_x^{j,i,r}={\displaystyle \sum _{s\in S_r}{\displaystyle \sum _{p\in P_s}{\eta }_x^{j,i,r,p}}}$$

(10)

From the passengers’ perspective, they seek minimal waiting times during their trips. With this concern, another objective function is presented in this study, as outlined in Equations (11) and (12).

$$Z_2=\underset{x\in X}{\min }f\left(x\right)$$

(11)

$$f\left(x\right)={\displaystyle \sum _{r\in R}{\displaystyle \sum _{s\in S_r}{\displaystyle \sum _{p\in P_s}{\delta }_x^r\times t_{p,s}^{wait}}}}$$

(12)

Under the consideration of $T_{\mathrm{limit}}$, we assume that, if the waiting time of passenger p ($p\in P_s$) exceeds $T_{\mathrm{limit}}$, he will transfer to other transport modes or even abandon the trip. In this circumstance, the waiting time of passenger p is assigned as a larger integer (see Equation (13)), and then is reckoned within the total waiting times in Equation (12).

$$t_{p,s}^{wait}=\left\{\begin{array}{l}{t}_{p,s}^{wait}, t_{p,s}^{wait}\le T_{\mathrm{limit}};\\ \mu \times T_{\mathrm{limit}}, otherwise.\end{array}\right.$$

(13)

Combing $Z_1$ and $Z_2$, we use a linear weighting method to convert the bi-objective to a single objective and the weighting coefficients should satisfy Equation (15).

$$\max Z={\omega }_1\times {\displaystyle \frac{\left(Z_1-Z_1^{\min }\right)}{\left(Z_1^{\max }-Z_1^{\min }\right)}}+{\omega }_2\times {\displaystyle \frac{\left(Z_2^{\max }-Z_2\right)}{\left(Z_2^{\max }-Z_2^{\min }\right)}}$$

(14)

$${\omega }_1+{\omega }_2=1$$

(15)

It is assumed that the standard bridging route is included in each scheme to fulfill the demands for all origin-destination (OD) pairs. Then, we have

$${\delta }_x^{r_{\mathrm{standard}}}=1,\forall x\in X$$

(16)

Considering the transfer scenario, it is mandated that the bus bridging scheme contains non-parallel routes, as stipulated by Constraint (17).

$${\displaystyle \sum _{r_{parallel}^{non}\in R_{parallel}^{non}}{\delta }_x^{r_{parallel}^{non}}}\ge 1,\forall x\in X$$

(17)

Constraint (18) is the bus resource limitation, ensuring that the number of buses allocated to the routes equals the total available resources.

$${\displaystyle \sum _{r\in R}{N}_x^r=N_{bus}},\forall x\in X$$

(18)

Constraint (19) is the bus stop capacity limitation, designed to prevent an excessive number of buses from arriving at a stop simultaneously. Considering the dynamic nature of $N_{x,s}^{bus}$ in the bridging process, this constraint is controlled by a simulation program in this study.

$$N_{x,s}^{bus}\le C_s,\forall s\in S$$

(19)

Constraint (20) is the bus capacity limitation. Similar to $N_{x,s}^{bus}$, $Q_{x,r,s}^{j,i}$ is also dynamically changing during the bridging process, so it is also managed by the simulation program.

$$Q_{x,r,s}^{j,i}\le C_{bus}\times L,\forall x\in X,r\in R,s_r\in S_r,i\left|1\le i\le N_x^r\right.,j\left|1\le j\le N_x^{i,r}\right.$$

(20)

Constraint (21) ensures that a feasible scheme comprises no more than $N_r^{\max }$ bridging routes, for the sake of practical feasibility and operational efficiency.

$${\displaystyle \sum _{r\in R}{\delta }_x^r}\le N_r^{\max },\forall x\in X$$

(21)

Constraints (22) and (23) define the domains of variables.

$$N_{x,s}^{bus}\in N,\forall x\in X and s\in S$$

(22)

$$N_x^r\in N,\forall x\in X and r\in R$$

(23)

3.3. Solution Algorithm

3.3.1. A Two-Stage Genetic Algorithm

The genetic algorithm (GA) is known for efficiently exploring large solution spaces, navigating diverse regions, and converging towards optimal or near-optimal solutions. However, traditional GA may struggle with finding optimal solutions, especially when dealing with variables of significant variations in the model. In this study, the decision variables of route selection ${\delta }_x^r$ and bus resource allocation $N_x^r$ are interrelated and interact with each other. Using a traditional GA that find the solutions of route selection and bus deployment simultaneously, a promising solution of ${\delta }_x^r$ might be eliminated if the searched solution of $N_x^r$ is relatively weak, making it a challenging for the algorithm to explore the solution space effectively within a limited timeframe. To address this challenge, this paper proposes a two-stage GA, as illustrated in Figure 7.

In the first stage, a genetic algorithm is employed to generate optimal solutions of route selection, using an equal distribution method to determine the bus deployment. If the total bus resources cannot be evenly divided, the surplus vehicles are assigned to the standard bridging route. The top 1% solutions for route selection based on fitness function values are retained as parent populations for the next stage. In the second stage, the bus resources are reallocated combined with the retained route selection solutions to find the best solution.

• Genetic Representation

In the two-stage GA, a genotype using binary coding comprises two components. The first component represents the solution for route selection, where the decimal value of each nine-digit number in the chromosome indicates the sequence number of the selected route within the candidate route set. The second component represents the bus deployment solution, with the decimal value of each six-digit binary number indicating the number of allocated buses. An example of genotype presentation is depicted in Figure 8.

2.

Fitness function

Each individual in the population is evaluated based on a fitness function that assesses how well the solution performs in terms of the objectives of the problem. Herein, the integrated objective function, i.e., Equation (14), is utilized as the fitness measure for GA.

3.

Selection

Individuals with higher fitness values are prioritized for reproduction and have a higher chance of passing their genetic material to the next generation. In the two-stage GA, the roulette wheel method is employed to implement the selection process.

4.

Crossover

Selected individuals undergo a two-point crossover, where parts of their genetic information are exchanged to produce offspring solutions. In the crossover of the gene segments representing bus deployment solutions, two distinct integers are randomly generated within the range of $\left[1, N_r^{\max }\right]$ and used as crossover points. Subsequently, the segments between these two points are then swapped between the two parent chromosomes, creating two new offspring. Additionally, after crossover, the resulting solution may not meet Constraint (18). In such cases, the crossover process is repeated until a valid crossover is achieved.

In the crossover of the gene segments representing the route selection solutions, the same method is applied. However, to satisfy Constraint (16), which dictates that crossover operations must occur beyond gene segments representing the standard route, two different integers within the range of $\left[2, N_r^{\max }\right]$ are randomly generated at crossover points, followed by an exchange of gene segments between two chromosomes. An example of crossover operation is illustrated in Figure 9.

5.

Mutation

Mutations add diversity to the population and prevent premature convergence to suboptimal solutions. For the gene segments representing the bus deployment solutions, a new mutation strategy is proposed to satisfy the bus resource constraint. Two distinct integers, $a$ and $b$, are randomly generated within the range of $\left[1, N_r^{\max }\right]$ and used as mutation points. The decimal values of the two six-digit binary numbers (denoted as $N_x^{r,a}$ and $N_x^{r,b}$) are then compared. Assume $N_x^{r,a}$ is the smaller one of the two. Next, another integer $\zeta$ is randomly chosen within the scope of $\left[1,N_x^{r,a}\right]$. The offspring is then created by converting the decimal values of $\left(N_x^{r,a}+\zeta \right)$ and $\left(N_x^{r,b}-\zeta \right)$ into binary numbers.

For the solution of route selection, an integer within the range $\left[2, N_r^{\max }\right]$ is randomly generated as mutation point. Then a new route is randomly selected from the candidate route datasheet and is used to substitute the original one at the mutation point. An example of mutation operation is illustrated in Figure 10.

3.3.2. A Multi-Agent Simulation Method for Fitness Measurement

A multi-agent simulation method is introduced to evaluate the fitness value of a chromosome. Given a solution, the simulation mimics the bus bridging process, including train arrivals, bus departures and arrivals, and passenger boarding and alighting. The simulation follows a time-driven process with a step size of 1 min. The flowchart outlining the simulation method is depicted in Figure 11. There are three key events that drive the simulation process. First, a “train arrival event” is triggered based on the timetables of the disrupted rail line and the intersected rail line. This event functions to generate new passenger agents, representing affected rail passengers who are brought to disrupted stations by trains after the bridging service has been initiated. The number of passenger agents generated is predefined based on historical passenger data. Second, a “new bus departure event” takes place in terms of the bridging bus headway. It assumes the generation of new bus agents. Third, “agent status update event” works in each time step, updating the status parameters of both bus agents and passenger agents, such as passenger waiting time and bus running time. With well-defined inputs (e.g., $t_{d\to o}^r$, $r{t}_{s+}^r$, $r{t}_{s-}^r$, $H$) and neglecting the uncertainties in bus operations and passenger behaviors, the simulation can consistently yield a unique fitness value for each evaluated bus bridging solution.

Thereafter, we discuss the definitions and functions of the two agents: the bus agent and the passenger agent.

• Bus agent

A bus agent represents a bridging bus, which is defined as

$$B_j=\left\{r_j,directio{n}_j,ru{n}_j,run\text{\_}{S}_j,run\text{\_}{t}_j,sto{p}_j,stop\text{\_}{t}_j,stop\text{\_}{s}_j,in\text{\_}{P}_j\right\}$$

(24)

where j is the generating sequence of the bus agent, determining the total number of bus agents in the current simulation system. $r_j$ denotes the bridging route to which the jth bus agent is dispatched. $directio{n}_j$ indicates the running direction of the bus agent, as expressed in Equation (25).

$$directio{n}_j=\left\{\begin{array}{c}1,\\ -1,\end{array}\begin{array}{c}upward;\\ downward.\end{array}\right.$$

(25)

$ru{n}_j$ is the serial number of the current running section for the jth bus agent. $run\text{\_}{S}_j$ includes two serial numbers of stations on both sides of the current running section. $run\text{\_}{t}_j$ is cumulative time spent on the current section. $sto{p}_j$ is a binary variable indicating whether the jth bus agent is dwelling or not. $stop\text{\_}{t}_j$ is its cumulative dwell time at the current station. $stop\text{\_}{s}_j$ is the serial number of its current dwelling station. $in\text{\_}{P}_j$ is the set of index of passengers in the bus.

As illustrated in Figure 8, the bus agent has five functions.

Originating function: this controls the departure of a bus agent from the originating station, and initializes the abovementioned attributes for the bus agent.

Departure function: this controls the departure of a bus agent from the current station. It is triggered when either the number of passengers in a dwelling bus exceeds the bus capacity, or when all passengers complete boarding and alighting movements. Once this function works, $sto{p}_j$ and $stop\text{\_}{t}_j$ will turn to zero.

Arrival function: this realizes the bus dwelling process. When activated, it updates the attributes related to the dwelling state of a bus agent, i.e., $sto{p}_j$, $stop\text{\_}{s}_j$, $stop\text{\_}{t}_j$ and $in\text{\_}{P}_j$. Meanwhile, the arrival function triggers passenger agents to execute the alighting and boarding function.

Turnaround function: this mainly manages the turnaround process when a bus agent reaches the terminal station and the passenger alighting function is completed. Once this function works, $directio{n}_j$ will be changed.

Running function: this function is activated when $sto{p}_j$ becomes zero. It updates the attributes related to the dwelling state of a bus agent, i.e., $sto{p}_j$, $stop\text{\_}{s}_j$, $stop\text{\_}{t}_j$ and $in\text{\_}{P}_j$.

2.

Passenger agent

A passenger agent represents an affected rail passenger who requires the bus bridging service. It is defined as

$$P_i=\left\{O_i,D_i,w{t}_i,inbu{s}_i,ODmee{t}_i,leav{e}_i\right\}$$

(26)

where i is the index of the passenger agent. $O_i$ and $D_i$ represent the origin and destination stations of the ith passenger agent, respectively. $w{t}_i$ is the cumulated waiting time. $inbu{s}_i$, $ODmee{t}_i$ and $leav{e}_i$ are binary variables, as expressed in Equations (27)–(29).

$$inbu{s}_i=\left\{\begin{array}{l}1,passenger i is in a bus;\\ 0,otherwise.\end{array}\right.$$

(27)

$$ODmee{t}_i=\left\{\begin{array}{l}1,the OD demand of passenger i has been satisfied;\\ 0,otherwise.\end{array}\right.$$

(28)

$$leav{e}_i=\left\{\begin{array}{l}1,passenger i has left due to long waiting time;\\ 0,otherwise.\end{array}\right.$$

(29)

As illustrated in Figure 8, passenger agent has three functions.

Alighting function: THIS controls passenger agents getting off a bridging bus. It is triggered when $sto{p}_j$ changes from one to zero. Then, $in\text{\_}{P}_j$ is scanned to identify alighting passenger agents. If a passenger agent $P_i$ satisfies $D_i=stop\text{\_}{s}_j$, the alighting function is activated and $ODmee{t}_i$ is set to one.

Boarding function: this controls passenger agents boarding a bridging bus. For a passenger agent $P_i$, if $inbu{s}_i=0$ and $O_i=stop\text{\_}{s}_j$, it signifies that there is a bus arriving at the located station. Then, if the subsequent dwelling stations of $r_j$ include $D_i$, the boarding function is activated, and $inbu{s}_i$ is set to one.

Self-exit function: for a passenger agent $P_i$, if $w{t}_i>T_{\mathrm{limit}}$ and $inbu{s}_i=0$, $leav{e}_i$ is set to one, the self-exit function will be called. This indicates that long waiting time forces this passenger to transfer to other transport modes or cancel the trip.

4. Case Study

4.1. Case Setup

We demonstrate our model and algorithm using the Ningbo URT network, which consists of five rail lines. The case study focuses on an unplanned disruption occurring on a partial section of Line 1. As displayed in Figure 12, four stations and five sections are shut down between two turnover stations (i.e., Daqing Bridge and Zhoumeng North Road), including the transfer station of Gulou.

For the sake of presentation, the bridging nodes are numbered from 1 to 14 (see

Table 3. Serial number of bridging nodes.

Station Name	Serial Number	Station Name	Serial Number
Wangchun Bridge	1	Zhoumeng North Road	8
Zemin	2	Sakura Park	9
Daqing Bridge	3	Fuming Road	10
Ximenkou	4	Ningbo Railway Station	11
Gulou	5	Chenghuangmiao	12
Dongmenkou	6	Waitan Bridge	13
Jiangxia Bridge East	7	Zhengda Road	14

). Stations with names in bold are chosen as candidate originating nodes. Based on Section 3.1.2, 317 candidate bus bridging routes are generated, including the standard route, 110 parallel routes and 206 non-parallel routes.

Based on real-world operational data and interviews with public transit operators in Ningbo, some parameters in the proposed model are assigned values as listed in

Table 4. Assigned values of parameters in the model.

Parameters	Assigned Values	Parameters	Assigned Values
$C_{bus}$	100	$T_{\mathrm{limit}}$	1 h
$L$	90%	${\omega }_1$	0.5
$H$	1 min	${\omega }_2$	0.5
$C_s$	3	$\mu$	2
$N_{bus}$	60	${\theta }_{max}$	60°

.

4.2. Results Analysis

The use of the standard route alone is referred to as the traditional strategy, which serves as a benchmark for comparison with the optimized strategies derived from the proposed methods. As different assignments of the maximum number of routes in the bus bridging scheme (i.e., $N_r^{\max }$), the optimized results differ in route composition and bus resource allocation. The detailed results are presented in

Table 5. Results of route composition and bus resource allocation.

${\mathit{N}}_{\mathit{r}}^{\mathbf{max}}$	The Dwelling Stops of the Bus Bridging Routes	Number of Allocated Buses
2	[3;4;5;6;7;8]	31
	[3;12;8]	29
3	[3;4;5;6;7;8]	12
	[1;4;12;8]	36
	[3;5;9]	12
4	[3;4;5;6;7;8]	9
	[3;4;13;8]	8
	[2;7;8]	24
	[1;4;5]	19
5	[3;4;5;6;7;8]	4
	[3;4;5]	17
	[3;5;8]	9
	[5;6;7;10]	26
	[2;11;8]	4

.

Table 6. Comparison of the optimization results.

Objectives	Standard Route	2 Route	3 Route	4 Route	5 Route
$Z_1$	8358	11,007	11,917	11,657	12,317
$Z_2$ (h)	11,262.92	9731.55	9010.65	9038.93	8433.77
$Z$	0.2350	0.5659	0.7030	0.6851	0.7944

presents the optimization results for various bridging schemes, illustrating the performance of the proposed models with diversified route sets. The results clearly demonstrate that increasing the diversity of bridging routes significantly improves overall performance. Specifically, introducing an additional route (two-route scheme) reduces total waiting time by 13.60% and increases the number of passengers served within $T_{\mathrm{limit}}$ by 31.27%. This underscores the advantage of incorporating non-parallel routes to accommodate passengers more efficiently. Further increases in route diversity yield even better outcomes. Expanding from a two-route to a five-route scheme boosts the number of passengers served by an additional 11.90% while reducing total waiting time by 13.34%. The five-route scheme achieves the best overall performance, with a remarkable 238.04% improvement in the integrated objective $Z$ compared to the standard route. Overall, the results validate the effectiveness of the proposed model in designing diversified bridging routes to optimize passenger service during metro disruptions.

Table 7. Comparison of the number of passengers with waiting times exceeding $T_{\mathrm{limit}}$.

	Standard Route	2 Route	3 Route	4 Route	5 Route
Number of passengers with $t_{p,s}^{wait}>T_{\mathrm{limit}}$	2912	2254	2076	1847	1397
Ratio	18.55%	14.36%	13.23%	11.77%	8.90%

provides a comparison of the number of passengers whose waiting times exceed $T_{\mathrm{limit}}$, illustrating the effectiveness of the proposed models in reducing excessive waiting times. Passengers experiencing prolonged delays are more likely to abandon the bus bridging service, opting for alternative transportation modes or canceling their trips altogether. This metric is critical for evaluating the practicality of the bridging schemes. The results demonstrate a clear trend: increasing route diversity significantly reduces the number and proportion of passengers with waiting times exceeding $T_{\mathrm{limit}}$. In the standard route scenario, 2912 passengers face excessive waiting times, accounting for 18.55% of the total passenger count. In contrast, the five-route scheme reduces this figure to 1397 passengers, which is just 8.90% of the total—a 52.03% reduction compared to the baseline. Intermediate schemes also exhibit notable improvements. The consistent decline in both the absolute number and the ratio of delayed passengers underscores the effectiveness of introducing diversified routes.

Figure 13 illustrates the comparative optimization performance of three bridging strategies across different scenarios, all of which involve five bridging routes. The scenarios differ in their design approaches, progressively integrating more complex strategies to improve bus bridging performance. Scenario 1 includes only parallel routes confined within the disrupted metro section. Scenario 2 also relies exclusively on parallel routes but extends the bridging stops to areas beyond the disrupted section, offering greater spatial coverage. Scenario 3, representing our proposed strategy, builds upon Scenario 2 by introducing non-parallel routes to further diversify the bridging services.

The results show a clear progression in performance as the strategies evolve. Scenario 3 consistently outperforms Scenarios 1 and 2 across all objectives. Specifically, the number of passengers served in Scenario 3 improves by 12.54% relative to Scenario 1 and 6.38% relative to Scenario 2. This demonstrates the effectiveness of non-parallel routes in capturing passenger demand more comprehensively, particularly in areas not well-covered by parallel routes.

In terms of the total waiting time, Scenario 3 also sees a reduction of 11.47% and 0.88% compared to Scenarios 1 and 2, respectively. This indicates that, while extending stops beyond the disrupted area already brings substantial benefits in reducing passenger waiting time, the addition of non-parallel routes provides further marginal gains.

In terms of the integrated objective, it improves by 35.31% in Scenario 3 compared to Scenario 1 and 21.07% compared to Scenario 2. This result highlights the superior balance achieved between maximizing passenger coverage and minimizing waiting time under the proposed multi-strategy combination approach. In summary, the progressive enhancements observed across the scenarios emphasize the value of incorporating route diversity and broader spatial coverage in bridging service design.

4.3. GA Perfomance

The proposed two-stage GA is coded in MATLAB (R2003b) and implemented on a HP (Beijing, China) desktop with 4.2 GHz 13th Gen Intel(R) Core(TM) i5-13400 processor and 32 GB RAM. A comparison of the convergence results between the traditional GA and two-stage GA is illustrated in Figure 14 ($N_r^{\max }=5$). The final solution obtained by the traditional GA is 0.5983 after 205 iterations, while the two-stage GA obtains a final objective value 0.7944 after 513 iterations. Two-stage GA reduces the computational time for each iteration. The total CPU time of the two algorithms is similar.

As depicted in Figure 15, the two-stage GA demonstrates superior performance in enhancing solution quality. Figure 14 provides a more detailed comparison of the optimization performance between the traditional GA and two-stage GA. The results clearly indicate that both objectives show more substantial improvements when employing the two-stage GA, compared to the traditional GA. Specifically, for $Z_1$, there is a remarkable increase of 47.37% when transitioning from the standard route scheme to the 5-route combination scheme using the two-stage GA whereas, with the traditional GA, it shows a 19.38% increase.

4.4. Sensitivity Analysis

4.4.1. Number of Bridging Routes

First, we conduct a sensitivity analysis to examine the impact of route diversity on bus bridging performance. As seen in Figure 16, as the number of bridging routes increases from one to three, i.e., with the introduction of one or two additional routes alongside the standard route, $Z_1$ shows a significant increase and $Z_2$ decreases rapidly. However, when the number of bridging routes reaches four, a small rebound occurs. That is, $Z_1$ decreases slightly while $Z_2$ shows a small increase. As the number of bridging routes continues to increase from four to seven, $Z_1$ resumes an upward trend and $Z_2$ continues to decrease, but the rate of change gradually slows down. This suggests that, although adding more routes still has a positive impact on enhancing the optimization objectives, the marginal improvement is less significant. Interestingly, when the number of routes increases from seven to eight, $Z_1$ shows a slight incline and $Z_2$ also decreases slightly. This may indicate an over-supply of routes at this stage, causing some inefficiencies that result in a loss of passengers and an increase in waiting time.

In summary, as the number of bridging routes increases, the total number of served passengers ($Z_1$) also rises, while the overall waiting time ($Z_2$) decreases. However, the most significant improvements are observed with a smaller number of routes, and the benefits of additional routes diminish as route diversity further increases. This is because greater route diversity introduces added complexity in implementing the bus bridging service. Ultimately, both too many and too few routes can undermine bridging effectiveness. Under the setup of this study, a five-route combination strikes an optimal balance between performance and complexity.

4.4.2. Bus Fleet Size

To further investigate the impacts of fleet size on bridging performance, we conducted another sensitivity analysis using the five-route combination pattern, as shown in Figure 17. It can be seen that, aside from the special case at a fleet size of 40, increasing the fleet size from 20 to 70 results in a nearly linear rise in $Z_1$ and a nearly linear decline in $Z_2$. However, when the fleet size exceeds 70, the rate of change for both $Z_1$ and $Z_2$ slows down considerably, particularly for $Z_2$. This implies a possible over-supply of buses at this stage, leading to inefficiencies and resource wastage.

In summary, with the increase in available bus resources, the number of served passengers ($Z_1$) consistently rises, and the total waiting time ($Z_2$) steadily decreases. However, there is an inflection point, beyond which additional buses contribute less to performance improvements and may lead to resource wastage. Sensitivity analysis can assist transit agencies in determining the optimal fleet size to achieve the desired level of bridging service. For instance, based on our case setup, if the goal is to evacuate all affected passengers within one hour, a fleet size of at least 90 would be required, yielding an average passenger waiting time of approximately 30 min.

4.4.3. Weight Coefficients

The variations in the two objectives with different weight coefficients are depicted in Figure 18, which suggests the trade-off between the two objectives. When ${\omega }_1$ increases from 0 to 0.3, both objectives show improvement; when ${\omega }_1$ increases from 0.3 to 0.7, both objectives exhibit minor changes; when ${\omega }_1$ increases from 0.7 to 1.0, both objectives deteriorate. This suggests a positive correlation between the two objectives. Moderate weights (between 0.3 and 0.7) yield the most balanced outcome, where a relatively high number of passengers is served with a lower waiting time. Extreme prioritization of either objective leads to poor performance in both objectives. Overall, the setup of ${\omega }_1={\omega }_2=0.5$ is reasonable for this study.

5. Conclusions

This study focuses on designing a bus bridging service in response to an unplanned URT disruption that leads to a temporary closure of a line section containing a critical transfer station. By optimizing bus bridging operations, the proposed approach minimizes disruptions to passenger mobility, reducing delays and enhancing service reliability. This contributes to sustainable urban mobility by providing a reliable and efficient alternative for disrupted URT services.

First, a heuristic-rule based method is presented to generate candidate bus bridging routes. Non-parallel bridging routes are introduced to facilitate transfer passengers affected by the disruption. Additionally, the bridging stops visited by parallel routes are extended beyond the disrupted section to avoid passenger congestion and bus bunching at turnover stations. Second, we propose an integrated optimization model to addresses bus route selection and vehicle deployment issues collaboratively. The model considers berth capacity, vehicle capacity, and available bus fleet as constraints, ensuring its robustness and practicality. Passenger reneging behavior is also taken into account by incorporating passenger tolerant waiting time in defining objective functions. Third, a two-stage genetic algorithm is developed, which separates the determination of bus route selection and vehicle deployment into two stages to improve computational efficiency. A multi-agent simulation method is integrated into the two-stage GA to demonstrate the bus bridging process and passenger evacuation process. This approach enables us to identify the dynamic passenger flow and bus flow within the time-space network, providing a more accurate evaluation of variables, such as passenger waiting time and bus dwelling time, compared to deterministic methods.

Finally, a case study is conducted to validate the effectiveness of the proposed model and algorithm. The numerical results reveal the following practical insights. (1) The proposed methods significantly improve service performance by increasing the number of served passengers, reducing total waiting times, and minimizing the number of passengers experiencing excessively long waits. (2) Incorporating non-parallel bridging routes and extending the range of visited stops prove to be highly effective strategies for dispersing affected passengers and enhancing overall bridging performance. (3) Expanding route diversity plays a key role in improving bridging performance, with the most significant improvements observed when the number of bridging routes is relatively low. As the number of routes increases, the marginal gains in optimization performance diminish. In contrast, increasing the availability of bus resources consistently leads to a consistent increase in the number of passengers served and a steady decrease in the total waiting time. Sensitivity analysis is valuable for determining the optimal number of routes and fleet size. (4) A positive correlation is observed between the number of served passengers and their waiting times, suggesting that, when applying the model, setting equal weights for the two objectives can effectively balance both goals.

To ensure that each bus bridging solution yields a unique fitness value, our simulation process currently ignores the inherent uncertainties in vehicle operations and passenger behaviors. Addressing these uncertainties is an important direction for future research to enhance the robustness and realism of the approach. Additionally, our study focuses exclusively on introducing new bus routes without considering the integration with other public transportation modes, e.g., existing conventional buses, taxis, shared bikes. In emergency situations, utilizing integrated transportation modes could greatly improve the efficiency of passenger evacuation. Incorporating multi-modal transportation strategies could therefore provide a more comprehensive solution to emergency management, representing a valuable area for future exploration. Furthermore, this study only accounts for a single type of vehicle in the bus bridging tasks. In practical applications, utilizing a mix of different vehicle types could offer greater flexibility and efficiency. Exploring the use of heterogeneous vehicle fleets could enhance the effectiveness of bus bridging operations and is another promising direction for future studies.