Enhancing the Resilience of Intercity Transit System by Integrated Multimodal Emergency Dispatching and Passenger Assignment

Abstract

After the disruption of intercity railways, in order to effectively enhance system resilience and improve the sustainability of the intercity transit system, this paper studies the emergency response problem of multimodal collaboration based on the intercity multimodal transit system. Considering the constraints of the disrupted network structure, multimodal emergency resources, dynamic passenger demand, and passenger participation willingness, a bi-level optimization model is established for maximizing system resilience and minimizing the deviation of passengers’ desired arrival time. This paper integrally determines the transit capacity, timetable, and passenger quantity on each line of each mode. A hybrid genetic and ant colony algorithm is designed to solve the problem. Taking the regional disruption of the Beijing–Tianjin–Hebei intercity railway network as a case study, the research results show that 59% of demand can be met with a single attempt and 70% of the arrival time is within the planned period. Based on this resilience-enhancement strategy, the imbalance between travel demand and transit capacity can be sustainably alleviated after railway disruption.

1. Introduction

The intercity railway system mainly handles medium- and short-distance passenger flows within urban agglomerations, providing transit between cities. This system operates numerous services, with dense station distribution and long travel distances. Departure frequencies and arrival times are strictly scheduled, and during peak periods such as holidays, homecoming, and return-to-work times, it can transport thousands of intercity passengers. Regional disruptions can lead to delays for multiple trains along the line and cause passenger overcrowding at several stations. Before the completion of facility repairs, developing emergency response strategies to alleviate the increasing risk of overcrowding is essential for maintaining service sustainability. Therefore, how to formulate a multimodal alternative routing plan to address the short-term supply-demand imbalance in the railway system has become a key focus in the emergency management of intercity railways. This paper focuses on the issues of multimodal dispatching and passenger routing assignment.

Resilience reflects the system’s ability to recover to its initial state after being disturbed [1], which has been studied in ecology, environment, and so on [2,3]. The emergency management of the railway system experiences multiple stages from the occurrence of a disruption to recovery. Managers develop corresponding optimization strategies based on the scenarios of different stages. Existing studies divide this process into three stages: redundancy assessment, emergency response, and performance recovery [1,4]. The research on system resilience often develops from two directions: vulnerability and recoverability [5]. On the one hand, vulnerability research focuses on the system’s ability to maintain service levels under disturbance without emphasizing the system’s response and recovery ability. On the other hand, the recoverability depends on the repair of infrastructure, which is a long-term process [6]. The research on emergency response during the disrupting period plays an important role in bridging the two stages mentioned above. By dispatching multimodal emergency capacity to provide temporary alternative routes, it can alleviate the imbalance between supply and demand, shorten the recovery time, and achieve the goal of resilience enhancement.

This study proposes a multimodal dispatching strategy that coordinately plans the operating route, vehicle quantity, and dwell time of railway, airline, and bus transit systems to enhance the resilience of the intercity railway system by delivering in-station stranded passengers along the disrupted railway network. The dispatching strategy is modeled as a network design problem based on mixed integer nonlinear programming. Furthermore, we considered passengers’ dynamic arrival process and their routing choice along the multimodal alternative paths. Passengers are assigned based on a systematic optimal principle. By considering the congestion impact on the travel and transfer time along the route, the interaction between vehicle dispatch and passenger assignment is captured by a bi-level optimization structure. A hybrid genetic algorithm and ant colony algorithm is designed to solve the model.

The paper is structured as follows. We review the literature in Section 2. We state the problem of multimodal strategy in Section 3. The multimodal emergency corridor construction method is introduced in Section 4. The solving algorithm is shown in Section 5. The case studies and results are demonstrated in Section 6. We conclude the paper in Section 7.

2. Literature Review

Regarding the resilience characterization of the disrupted system, Mudigonda, et al. [7] measure the post-disaster resilience of the railway system using the required transportation recovery time and the change in travel time. Miller-Hooks et al. [8] evaluate the system’s recovery resilience by the proportion of expected demand that the post-disaster system can meet. Lu [9] pointed out that the number and distribution of delayed passengers can represent the ability and timeliness of rail transit resilience recovery. Tang et al. [10] evaluate and optimize the resilience of the railway system combined with bus connections in the event of multiple track disruptions. These studies all consider the long-term process of the system recovering to supply-demand balance after being disrupted, with few studies evaluating and optimizing the short-term effects of emergency response within the resilience framework. During the disruptive period, passenger demand accumulates dynamically over time, and effectively increasing capacity can meet part of the demand. Therefore, this paper uses the degree of satisfaction and timeliness of demand as quantitative indicators for the resilience framework.

During the railway-disrupting period, strategies such as stopping and short-turning are commonly used on both sides of the disruption to determine the dispatching plan for the operating trains. To address the passenger demand across disaster-affected areas, emergency trains may be deployed to alternative routes, or other forms of connections may be utilized. Regarding the deployment of emergency trains, existing studies have considered decision-making solutions for vehicle allocation, dwell time planning, and passenger reassignment. Veelenturf et al. [11] proposed adding extra services to alternative routes and replanning timetables to provide options for transferring passengers. Wagenaar et al. [12] addressed the issue by dispatching idle trains from nearby stations and planning deadhead travel time to meet excess passenger demand. Long et al. [13] determined the number of additional carriages and planned passenger reassignment, weighing passenger utility and operational costs to meet sudden surges in passenger demand. Luan and Corman [14] synchronized multiple lines by determining vehicle capacity, departure times, and arrival times, and they planned transfer passenger reassignment to improve passengers’ travel efficiency. Meanwhile, existing research emphasizes that bridging measures can increase the system’s capacity. Bus bridging, for example, can enhance connectivity between stations within the disrupted railway system and provide passengers with more flexible route options [14,15]. This paper, focusing on intercity passenger demand, aims to integrate conventional railways, high-speed railways, bus bridging, and airline routes into a closed system to construct an intercity multimodal emergency corridor. This corridor collaborates by planning departure times, dwell times, arrival times, and vehicle numbers for each route. Based on the OD distribution of stranded passengers, multimodal transferable routes are constructed, and passenger flow distribution on each chain is determined. This multimodal corridor can sustainably leverage the comparative advantages of each mode within their operational ranges while avoiding congestion in passenger flow along a single travel chain.

Existing research has pointed out that not all passengers are willing to participate in emergency transit after a disruption in rail transit. Li, et al. [16] studied the travel behavior during subway disruptions and found that passengers’ transfer choices are influenced by factors such as the uncertainty of disruption time, transfer methods, and individual attitudes. Intercity travelers are not capable of completing long-distance trips on their own and are more reliant on convenient, intuitive, and efficient intercity transportation options. Kim and Oh [17] found that the level of trust in and awareness of emergency services significantly affects passengers’ willingness to participate. García-Martínez, et al. [18] conducted a survey and found that some passengers refuse to use alternative routes due to factors such as waiting times, transfer times, and the number of transfers. Existing emergency response strategies mostly target determined passenger demand, with few studies considering the impact of in-station crowding, emergency route arrival times, and the number of transfers on passengers’ willingness to travel. Based on these considerations, this study develops a demand-oriented emergency response strategy. The emergency schedules often differ from the established timetables, causing some passengers’ arrival times to be delayed or advanced. This paper uses the minimization of arrival time differences as another objective, ensuring the smallest deviation in passengers’ arrival times.

Based on the above analysis, emergency strategies and evaluation methods for dealing with a disrupted railway system are quite abundant, but they mainly focus on railway system emergency dispatching or bus bridging. Few studies have proposed emergency response strategies aimed at improving system resilience during the disrupting period of intercity railway system disruptions, and research on integrating multimodal intercity transportation networks to construct emergency corridors is still not seen. Therefore, the three key contributions of this paper are as follows:

(1)

The paper introduces a framework for quantifying the resilience of the railway system immediately after the onset of disruptions. We use demand satisfaction and timeliness as quantitative indicators, providing a practical method for evaluating the short-term effects of emergency response within the resilience framework, focusing on the dynamic accumulation of passenger demand.

(2)

This study proposes strategies for enhancing system resilience during disruptions by integrating multimodal collaboration. We focus on optimizing multimodal vehicle dispatching and passenger assignment to alleviate the supply-demand imbalance, thereby reducing the recovery time.

(3)

A comprehensive case study is conducted based on the Beijing–Tianjin–Hebei urban agglomeration, applying the proposed multimodal emergency response strategy to a real-world intercity railway disruption scenario. The case study demonstrates the effectiveness of the proposed model in improving passenger flow sustainability and recovery from a disruption, validating the approach through practical application.

3. Problem Statement

The failure of local facilities in intercity railways will lead to most trains stopping and waiting upstream in the affected area. During peak hours and holidays, short-term facility failures will cause delays for the extensive number of trains. Passenger flow concentrates and stagnates at upstream stations, with no trains reaching downstream stations. The number of stranded passengers gradually accumulates, and the imbalance between supply and demand in the railway system becomes increasingly severe.

The resilience of the multimodal system in this paper is defined as its ability to resist and recover through collaborative assistance across the entire system. It is quantified by the demand satisfaction, where alternative routes fulfill the demand generated by the disrupted system. An example of resilience performance is illustrated in Figure 1. To improve transit capacity during the disrupting period, emergency responses such as bridging routes, detour routes, and alternative routes can meet part of the urgent travel demand. The resilience improvement effect of this strategy is represented by the red line in Figure 1. However, most travelers are still affected by factors such as service levels of the routes, information reliability, and the number of transfers and continue to wait for recovery. Therefore, an improved emergency response strategy, where multimodal collaboration provides differentiated operating routes to satisfy passengers arriving at different times in batches, can attract more passengers to participate in the emergency routes, effectively reducing the number of stranded passengers and improving system resilience, as shown by the green shaded area in Figure 1. It should be noted that during the disrupting period addressed by this strategy, achieving supply-demand balance is not possible due to the following reasons: (1) some passengers refuse to participate in the emergency routes, (2) some passengers miss the time window for the emergency routes, and (3) the backup carriage capacity is insufficient to meet the high passenger demand.

The intercity transportation system can be abstracted as an interdependent network composed of facilities, transportation hubs, transport routes, and passenger flow paths. The multimodal transit network is layered by mode of transportation, and the dependencies are reflected by passenger flow transfers at transportation hubs. As shown in the simple intercity multimodal network in Figure 2, when the intercity railway is disrupted (red lines in Figure 2b), alternative direct routes and travel chains are reconstructed for passenger flow by integrating conventional rail, high-speed rail, and airline networks (directed lines in Figure 2c).

The multimodal transport system is topologically represented as a multilayer dependent network, with connections divided into operational connections a_n ∈ A₁ and transfer connections b_n ∈ A₂. Each station (node) has multidimensional attributes $s_n$ = (C_i, m, m_n), where C_i is the city identifier C_i ∈ C, m is the mode m ∈ M, and m_n is the n-th route of mode m, m_n ∈ J. Transportation hubs are denoted by C_i^t ∈ C^t. Each multimodal transport route is a set composed of operational and transfer connections, represented as l_n = (a₁, a₂,…, b₁, a_n).

To evaluate the resilience improvement in the proposed model, various coupling strategies among modes are tested within the same scenario for comparison. The modes involved include conventional railway, high-speed railway, bus, and airline. The tested strategies are as follows: conventional railway only (Strategy R), high-speed railway and bus (Strategy H + B), conventional railway, high-speed railway, and bus (Strategy R + H + B), and conventional railway, high-speed railway, bus, and airline (Strategy R + H + B + A). In practice, Strategy R involves reallocating intercity railway passengers to slower but higher-capacity trains. Strategy H + B combines short-turning and bus bridging strategies. Strategies R + H + B and R + H + B + A are comprehensive strategies that are implemented when the necessary facilities and resources are available.

To ensure the implementation of this strategy, the following preparations are assumed to be known in the model:

(1)

The total travel demand can be obtained from the railway ticketing system, and real-time arrival numbers and OD (origin-destination) data can be obtained from the station entry and ticketing system.

(2)

It is assumed that the transmission of facility failures has ended, and the intact post-disaster facility network is known, along with the number of available conventional rolling stock, high-speed rolling stock, buses, and airplanes, as well as the number of station tracks and airport runways.

(3)

The travel times of rail sections and airlines are fixed, and the arrival times are reliable.

(4)

Buses are used to transfer stranded passengers from the station to other stations or airports within the city. This strategy does not consider integrating highways for intercity transport.

(5)

This strategy only considers passenger transfers within transportation hubs by walking, meaning the multimodal emergency corridor is a closed system.

The proposed method first establishes a multimodal emergency corridor model from the perspectives of demand change, multimodal operation rules, and passenger flow utility. It then solves the optimal route selection, carriage and airplane dispatching, and passenger flow distribution through bi-level programming. The upper level aims to optimize resilience and minimize arrival time variation, while the lower level focuses on the system-optimal passenger flow distribution problem. The following assumptions are made for model development:

(1)

An S-curve is used to describe the passenger arrival process at each station, determining the real-time number of passengers at the station [19].

(2)

Passenger participation is influenced by the accumulated number of passengers at the station, arrival time differences, and transfer times, following an exponential distribution.

(3)

The utility of the passenger flow path is related to vehicle occupancy rates, the number of transfers, waiting times, and walking times.

4. Multimodal Emergency Corridor Construction Method

4.1. Demand Change Model

The number of passengers at a station at any given time is the sum of the passenger demand from all trains passing through the station. The S-curve is used to describe the cumulative process of passenger arrival. The actual number of arriving passengers is the sum of passenger demand for each train. The demand from train $\kappa$ from station $s_n$ to $s_{n+i}$ (denoted as $\overline{d_{\kappa }^{s_n{s}_{n+i}}}$) is known, and the number of passengers from station $s_n$ to $s_{n+i}$ at time t, $d_{s_{n+i}}^{s_n}\left(t\right)$, is given by Equation (1). The accumulated number of passengers at station $s_n$ at time t, $d^{s_n}\left(t\right)$, is given by Equation (2).

$$d_{s_{n+i}}^{s_n}\left(t\right)={\sum }_{\kappa }\left(\overline{d_{\kappa }^{s_n{s}_{n+i}}}\ast {\displaystyle \frac{1}{1-e^{\alpha \left(t-H\right)}}}\right),\forall s_n,s_{n+i}\in S$$

(1)

$$d^{s_n}(t)=\sum _{s_{n+i}}{d}_{s_{n+i}}^{s_n}\left(t\right),\forall s_n\in S$$

(2)

The actual participation of passengers is influenced by the accumulated number of passengers at the station $d^{s_n}\left(t\right)$, the arrival time difference ${\delta }_t^{OD}$, the waiting time t, and the number of transfers ${\delta }_a\in \mathit{N}$. The arrival time difference ${\delta }_t^{OD}$ is defined in Equation (3) as the standard deviation between the arrival times of the alternative route $t_{OD}^{m_n}$ and the original route $t_{OD}^{m_n^{\ast }}$. The expected travel demand from station $s_n$ to $s_{n+i^i}$ denoted as $q_{s_{n+i}}^{s_n}$, is given by Equation (4).

$${\delta }_t^{OD}=\sqrt{{\sum }_{m_n}(t_{OD}^{m_n}-t_{OD}^{m_n^{\ast }}{)}^2}$$

(3)

$$q_{s_{n+i}}^{s_n}(t)=d_{s_{n+i}}^{s_n}(t)\ast e^{-{\displaystyle \frac{{\delta }_t^{OD}+{\delta }_a}{d^{s_n}\left(t\right)+t}}},\forall s_n,s_{n+i}\in S$$

(4)

The total expected demand at station $s_n$ at time t and $q^{s_n}\left(t\right)$ is the sum of the expected demand for all downstream destinations, as shown in Equation (5).

$$q^{s_n}\left(t\right)={\sum }_{s_{n+i}}{q}_{s_{n+i}}^{s_n}\left(t\right),\forall s_n\in S$$

(5)

4.2. Multimodal Operation Rules

4.2.1. Railway System

The number of empty seats $K_{s_n}^{m_n}$ when mn arrives at ${\mathrm{s}}_{\mathrm{n}}$ is related to the number of vehicles $N_{m_n}$, the capacity of the vehicles k_m, and the number of passengers on board $f_{s_n{s}_{n+i}}^{m_n}$. The number of alighting passengers $q_{al}^{m_n{s}_n}$ at ${\mathrm{s}}_{\mathrm{n}}$ is given by Equation (6), and the number of boarding passengers $q_{ab}^{m_n{s}_n}$ at $s_n$ is given by Equation (7). Therefore, the number of empty seats $K_{s_n}^{m_n}$ at station $s_n$ is the difference between the vehicle capacity and the number of alighting passengers, as shown in Equation (8). The capacity of $m_n$ is given by $K_{s_1}^{m_n}=N_{m_n}\ast k_m$.

$$q_{al}^{m_n{s}_n}={\sum }_{s_n}{f}_{s_n{s}_{n+i}}^{m_n},\forall s_n\in S, m_n\in M$$

(6)

$$q_{ab}^{m_n{s}_n}={\sum }_{s_{n+i}}{f}_{s_n{s}_{n+i}}^{m_n},\forall s_n\in S, m_n\in M$$

(7)

$$K_{s_n}^{m_n}=K_{s_{n-1}}^{m_n}-q_{ab}^{m_n{s}_{n-1}}+q_{al}^{m_n{s}_{n-1}},\forall s_n\in S, m_n\in M$$

(8)

The train dwell time $t_{dwell}^{m_n{s}_n}$ includes the time spent for station arrival and passenger boarding/alighting, as shown in Equation (9), where $v_{ped}$ represents the average boarding/alighting speed of passengers. The progression time is understood as the waiting time when the number of trains arriving at $s_n$ exceeds the available tracks ($\overline{N_R}$) and their arrival times are less than the safety time $t_{dwell}^A$, requiring the later-arriving trains to wait.

$$t_{dwell}^{m_n{s}_n}=\left\{\begin{array}{cc}{t}_{dwell}^R+{\displaystyle \frac{q_{ab}^{m_n{s}_n}+q_{al}^{m_n{s}_n}}{v_{ped}}}& \bigcap m_n{|}_{s_n\in m_n}\le \overline{N_R},t_{Arr}^{m_n{s}_n}-t_{Arr}^{m_n^{\prime }{s}_n}<t_{dwell}^A\\ {\displaystyle \frac{q_{ab}^{m_n{s}_n}+q_{al}^{m_n{s}_n}}{v_{ped}}}& else\end{array}\right.$$

(9)

The arrival time of line $m_n$ at station ${\mathrm{s}}_{\mathrm{n}}$, denoted by $t_{Arr}^{m_n{s}_n}$, is given by Equation (10), and the departure time $t_{Dep}^{m_n{s}_n}$ is given by Equation (11). This timetable represents the departure and arrival times for the entire journey across different stations $s_1, s_2,\dots s_n$ for the mode $m_n$.

$$t_{Arr}^{m_n{s}_n}=t_{Dep}^{m_n{s}_1}+{\sum }_{i=1}^{n-1}{t}_{s_i{s}_{i+1}}^{m_n}+{\sum }_{i=1}^{n-1}{t}_{dwell}^{m_n{s}_i},\forall s_n\in S, m_n\in M$$

(10)

$$t_{Dep}^{m_n{s}_n}=t_{Arr}^{m_n{s}_n}+t_{dwell}^{m_n{s}_n},\forall s_n\in S, m_n\in M$$

(11)

4.2.2. Airline

The arrival time for air routes is the sum of the travel time and the waiting time. When the number of aircraft exceeds the allocated number of runways ($\overline{N_A}$), the aircraft must wait for a certain period before taking off, denoted as $t_{dwell}^A$, as given by Equation (12). This waiting time is necessary to ensure safe dispatching and coordination when multiple aircraft need to use the same runway.

$$t_{Arr}^{m_n{s}_n}=t_{Dep}^{m_n{s}_{n-1}}+(N_{m_n}-\overline{N_A})\ast t_{dwell}^A+t_{I_{n-1}{I}_n}^{m_n},\forall s_n\in S, m_n\in M$$

(12)

4.2.3. Bus

The bus used for shuttle services between city stations has its travel time $t_{bus}^{s_n{s}_{n+i}}$ influenced by the load factor, as shown in Equation (13). In this equation, ${\overline{ul}}_{m{m}^{\prime }}$ represents the average distance between the two stations, and $v_{bus}$ is the speed limit for the bus within the city. The load factor affects the travel time because higher occupancy may lead to slower travel due to stops, traffic, or congestion, whereas lower occupancy allows for faster movement at the set speed limit.

$$t_{bus}^{s_n{s}_{n+i}}={\displaystyle \frac{{\overline{ul}}_{m{m}^{\prime }}}{v_{bus}(1-q_{ab}^{m_{n^{\prime }}{s}_n}/q_{ab}^{s_n})}},\forall {\mathrm{s}}_{\mathrm{n}}:s_n\cap s_{n+i}=C_i$$

(13)

4.3. Passenger Utility

The perceived travel time $t_{trip}^{s_n{s}_{n+i}}$ for passengers is given by Equation (14), where the first term represents the actual travel time, and the second term accounts for the extended perception time due to the load factor. This extended perception time reflects the discomfort or delay passengers feel when the vehicle is near full capacity, as crowding can make the journey feel longer due to factors like limited space, slower boarding or alighting, and possible delays.

$$t_{trip}^{s_n{s}_{n+i}}=(t_{Dep}^{m_n{s}_{n+i}}-t_{Arr}^{m_n{s}_n})\ast (1+{\epsilon }_1({\displaystyle \frac{f_{s_n{s}_{n+i}}^{m_n}}{N_{m_n}\ast k_m}}{)}^{{\epsilon }_2}),\forall s_n,s_{n+i}\in S$$

(14)

The passenger’s perceived transfer time $t_{transfer}^{s_n{m}_n{m}_{n^{\prime }}}$ consists of walking time and waiting time, both extended by parameters ε₃ and ε₄, as shown in Equation (15). The walking time $t_{walk}^{s_n{m}_n{m}_{n^{\prime }}}$ is calculated using reference [20], as shown in Equation (16), where $\overline{wl}$ is the average walking distance and $f_{s_n}^{m_n{m}_{n^{\prime }}}$ represents the number of passengers transferring at station $s_n$.

The waiting time $t_{wait}^{s_n{m}_n{m}_{n^{\prime }}}$ is given by Equation (17), which corresponds to the time difference between the arrival times of two trains at the transfer station. This waiting time reflects the passengers’ perception of the wait for the next available vehicle, which may be influenced by the length of time between the arrivals of the connecting services.

$$t_{transfer}^{s_n{m}_n{m}_{n^{\prime }}}={\epsilon }_3{t}_{walk}^{s_n{m}_n{m}_{n^{\prime }}}+{\epsilon }_4{t}_{wait}^{s_n{m}_n{m}_{n^{\prime }}},\forall s_n\in S, m_n,m_{n^{\prime }}\in M$$

(15)

$$t_{walk}^{s_n{m}_n{m}_{n^{\prime }}}={\displaystyle \frac{\overline{wl}}{v_{ped}}}\mathit{exp}\left(0.2{\left(f_{s_n}^{m_n{m}_{n^{\prime }}}/\overline{wl}\right)}^2\right),\forall s_n\in S, m_n,m_{n^{\prime }}\in M$$

(16)

$$t_{wait}^{s_n{m}_n{m}_{n^{\prime }}}=t_{Arr}^{m_{n^{\prime }}{s}_n}-t_{Arr}^{m_n{s}_n},\forall s_n\in S, m_n,m_{n^{\prime }}\in M$$

(17)

Therefore, the overall utility $u_{l_n^{OD}}$ of a passenger’s journey on path $l_n^{OD}$ from origin to destination (OD) is the sum of the perceived travel time, the perceived transfer time, and the number of transfers, as shown in Equation (18). This utility reflects the combined impact of the total journey time, the additional time incurred during transfers, and the inconvenience of having to switch between different modes of transportation. The formula provides a comprehensive measure of how a passenger perceives the entire travel experience, considering both travel and transfer-related factors.

$$u_{l_n^{OD}}={\sum }_{s_n{s}_{n+i}}{t}_{trip}^{s_n{s}_{n+i}}+{\sum }_{s_n}{t}_{transfer}^{s_n{m}_n{m}_{n^{\prime }}}+{\delta }_a$$

(18)

4.4. Bi-Level Optimization Model

4.4.1. Objectives of the Upper-Level Mode

In this study, an emergency response period is defined as $T_{duration}=[Min\left(t_{Dep}^{m_n{s}_1}\right),Max(t_{Arr}^{m_n{s}_n}\left)\right]$, which spans from the departure time of the first line to the arrival time at the final destination station of the last line. The demand satisfaction rate r(t) is defined as the ratio of the number of passengers actually participating in the transportation to the total travel demand, as shown in Equation (19). Here, ${\delta }_{s_n}^{m_n}\left(t\right)$ is the determination of whether the line m_n has reached station $s_n$, as indicated in Equation (20).

The area of the “resilience triangle” is computed as shown in Equation (21). This approach measures the resilience of the transportation system by considering the proportion of demand satisfied over the response period, which is visualized through a triangular area that captures the effectiveness of the emergency response strategy in addressing demand.

$$r(t)={\displaystyle \frac{{\sum }_{m_n}{q}_{ab}^{m_n{s}_n}{\delta }_{s_n}^{m_n}\left(t\right)}{{\sum }_{s_n}{d}^{s_n}\left(t\right)}}$$

(19)

$${\delta }_{s_n}^{m_n}(t)=\left\{\begin{array}{c}\begin{array}{cc}0,& t\le t_{Arr}^{m_n{s}_n}\end{array}\\ \begin{array}{cc}1,& t>t_{Arr}^{m_n{s}_n}\end{array}\end{array}\right.$$

(20)

$$R={\int }_{Min\left(t_{Dep}^{m_n{s}_1}\right)}^{Max\left(t_{Arr}^{m_n{s}_n}\right)}[1-r(t\left)\right]dt$$

(21)

The difference in arrival time is defined as the variance between the actual arrival time of the passengers and the scheduled arrival time, as shown in Equation (22). This measure captures how much the actual arrival times deviate from the planned schedule, reflecting the variability and potential delays in the transportation system during the emergency response phase.

$$D={\sum }_{l_n^{OD}}{({\sum }_{s_n\in l_n^{OD}}\left(t_{Arr}^{m_n{s}_n}+t_{walk}^{s_n{m}_n{m}_{n^{\prime }}}+t_{wait}^{s_n{m}_n{m}_{n^{\prime }}}\right)-t_{OD}^{m_n^{\ast }})}^2$$

(22)

4.4.2. Objectives of the Lower-Level Mode

The total utility $U$ of the system is defined as the product of the path utility $u_{l_n^{OD}}$ and the flow $f_{l_n^{OD}}$, as shown in Equation (23). The conversion relationship between path flow and segment flow is given by: $f_{s_n{s}_{n+i}}^{m_n}={\sum }_n{\sum }_{(O,D)}{f}_{l_n^{OD}}\ast {\delta }_{l_n^{OD}}^{m_n{s}_{\eta }{s}_{\eta +i}}$, where ${\delta }_{l_n^{OD}}^{m_n{s}_n{s}_{n+i}}$ is a binary conversion parameter. It equals 1 if the segment ($s_n$, $s_{n+1}$) is part of the path $l_n^{OD}$, and 0 otherwise. This formula describes how the flow on a specific segment contributes to the overall system utility by linking the path flow with the corresponding network segment flow.

$$U={\sum }_{l_n^{OD}}{u}_{l_n^{OD}}{f}_{l_n^{OD}}$$

(23)

4.4.3. Bi-Level Optimization

The bi-level optimization model, as shown in Equations (24)–(29), has the following structure:

The upper-level objective function combines two objectives using a linear weighted method, as shown in Equation (24). The decision variables are the number of cars and aircraft scheduled for each line $N_{m_n}$.

The lower level focuses on optimizing the system’s passenger flow distribution. The objective function is shown in Equation (25). The decision variables are $q_{ab}^{s_n}$, the actual passenger flow departing from station $s_n$; $f_{s_n{s}_{n+i}}^{m_n}$, the passenger flow on each path; and $f_{s_n}^{m_n{m}_{n^{\prime }}}$, the transfer passenger flow between modes.

Equation (26) ensures that the actual departure flow at each station $s_n$ is properly allocated across the available paths. Equation (27) guarantees that the departing flow at a station does not exceed the expected total demand for that station. Equation (28) ensures that the in-transit passenger flow does not exceed the vehicle capacity of the train or other vehicles. Equation (29) ensures that the total number of cars and aircraft allocated to each mode does not exceed the available backup resources for that mode.

$$\mathrm{upper}: \begin{array}{cc}\mathit{min}& F_1={\lambda }_{1}R+{\lambda }_{2}D\end{array}$$

(24)

$$\mathrm{lower}: \begin{array}{cc}\mathit{min}& F_2=U\end{array}$$

(25)

$$\begin{array}{cc}s.t.& q_{ab}^{s_n}={\sum }_{m_n}{\sum }_{i=1}{f}_{s_n{s}_{n+i}}^{m_n}\end{array},\forall s_n\in S$$

(26)

$$q_{ab}^{s_n}\le q^{s_n},\forall s_n\in S$$

(27)

$$\begin{array}{cc}{q}_{ab}^{m_n{s}_n}\le K_{s_n}^{m_n},& \forall s_n\in S,m_n\in M\end{array}$$

(28)

$${\sum }_n{N}_{m_n}\le N_m^{max}, \forall m_n\in M$$

(29)

5. Algorithm

In the railway disruption scenario, the number of passengers at the station accumulates gradually over time. Therefore, in terms of model solving, the algorithm should aim to improve its efficiency while ensuring that the solution approximates the optimal result. This is necessary to meet the emergency response time requirements for plan generation [21]. Additionally, bi-level programming has been proven to be an NP-hard problem [22], so heuristic algorithms are needed to seek numerical solutions. As noted by Akopov et al. [23,24] and Kaleybar et al. [25], hybrid heuristic algorithms are both efficient and effective in solving mixed integer programming problems. Recently, Liu et al. [19] proposed a bi-objective genetic-ant colony hybrid algorithm for multimodal vehicle dispatching and route planning. This paper improves upon that algorithm to solve the problem of constructing multimodal emergency corridors for resilience enhancement.

The flowchart of the heuristic hybrid algorithm is shown in Figure 3. The upper level uses a single-objective genetic algorithm to allocate the number of vehicles $N_{m_n}$ for each line m_n, thereby determining the capacity of the network connections. Each gene represents the number of vehicles on a line, and a chromosome represents a vehicle allocation scheme. Since each mode of transportation has a different number of backup vehicles, the chromosome crossover may exceed the available number. Therefore, before crossover, the allocation ratio $p_{Gene}^{m_n}={\displaystyle \frac{N_{m_n}}{N_m^{\mathit{max}}}}$ is calculated in advance. After crossover, the number of vehicles $N_{m_n}^{\prime }$ is converted as shown in Equation (30).

$$N_{m_n}^{\prime }=\left\{\begin{array}{c}\begin{array}{cc}⌈p_{Gene}^{m_n}\ast N_m^{\mathit{max}}⌉& {\sum }_n{p}_{Gene}^{m_n}\le 1\end{array}\\ \begin{array}{cc}⌈{\displaystyle \frac{p_{Gene}^{m_n}\ast N_m^{\mathit{max}}}{{\sum }_n{p}_{Gene}^{m_n}}}⌉& {\sum }_n{p}_{Gene}^{m_n}>1\end{array}\end{array}\right.$$

(30)

The lower level uses the ant colony algorithm to determine the actual evacuated passenger flow, $q_{ab}^{s_{\eta }}$, as well as the passenger flow distribution for running and transfer connections, $f_{s_n{s}_{n+i}}^{m_n}$,$f_{s_n}^{m_n{m}_{n^{\prime }}}$. Each ant represents a passenger, and the ants select paths based on migration probabilities, as shown in Equation (31), where μ represents the impedance information, τ represents the pheromone, and α and β are the importance parameters for each. The impedance information for the path is given by Equation (32), and the pheromone update rules are shown in Equations (33) and (34), where ρ is the pheromone evaporation parameter.

$$p_{l_n^{OD}}={\displaystyle \frac{{\tau }_{l_n^{OD}}^{\alpha }\times {\mu }_{l_n^{OD}}^{\beta }}{{\sum }_n{\tau }_{l_n^{OD}}^{\alpha }\times {\mu }_{l_n^{OD}}^{\beta }}}$$

(31)

$${\mu }_{l_n^{OD}}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{u_{l_n^{OD}}}},& follow Equation \left(27\right), \left(28\right)\\ 0,& Otherwise\end{array}\right.$$

(32)

$${\tau }_{l_n^{OD}}(t+1)=(1-\rho )\times {\tau }_{l_n^{OD}}\left(t\right)+\rho \times \Delta {\tau }_{l_n^{OD}}$$

(33)

$$\Delta {\tau }_{l_n^{OD}}={\displaystyle \frac{\underset{n}{max}\left(u_{l_n^{OD}}{f}_{l_n^{OD}}\right)-u_{l_n^{OD}}{f}_{l_n^{OD}}}{\underset{n}{max}\left(u_{l_n^{OD}}{f}_{l_n^{OD}}\right)-\underset{n}{min}\left(u_{l_n^{OD}}{f}_{l_n^{OD}}\right)}}$$

(34)

6. Computational Experiment

6.1. Description of Experiment

Based on the 2030 Vision Plan and the “Intercity Railway Network Planning Diagram for the Beijing–Tianjin–Hebei Region,” a multimodal corridor is constructed, as shown in Figure 4. This study uses a real case from 1 May 2021, where, due to strong winds in Baoding city, an overhead contact line obstruction between Dingzhou East and Baoding East along the Jing-Guang High-Speed Railway led to delays for some trains on the up and down lines. In this context, this study considers a partial failure of intercity railway facilities in the southwest direction from Beijing and applies this method to restore connectivity between core cities, specifically facilitating cross-disaster area connectivity from north to south.

In this study, the failure period is assumed to be during the peak hours of 10:00–12:00, affecting 10 intercity railways and 7 cities in the Beijing–Tianjin–Hebei region. The timetable is shown in

Table 1. Affected timetable during disruption period.

	Zhangjiakou	Chengde	Beijing	Baoding	Shijiazhuang	Xingtai	Handan
Route 1	10:00	-	11:00	11:40	12:20	12:50	13:10
Route 2	10:05	-	11:05	-	12:20	-	13:00
Route 3	-	10:10	11:10	11:50	12:30	13:00	13:20
Route 4	-	10:15	11:15	-	12:30	13:05	13:25
Route 5	10:30	-	11:30	-	-	-	13:55
Route 6	10:40	-	11:30	-	-	-	13:40
Route 7	-	11:00	12:00	-	-	-	14:15
Route 8	-	11:15	12:20	-	-	-	14:35
Route 9	-	11:30	12:35	13:15	13:55	-	14:35
Route 10	12:00	-	13:05	13:45	14:25	-	15:05

. Based on the actual case from 1 May 2021, where high wind conditions caused delays on the Jingguang High-Speed Railway due to contact network issues between Dingzhou East and Baoding East, this study considers the failure of local facilities in the southwest direction of Beijing’s intercity railways and adopts a multimodal emergency plan to restore connectivity across disaster zones between core cities. It is assumed that passenger demand for different train services follows a normal distribution during peak hours. The transportation modes considered include trains, high-speed trains, and airplanes, with capacities of 100 passengers for trains, 70 passengers for high-speed trains, and 100 passengers for airplanes. Walking speed is set at 1.5 m/s, the average transfer walking distance is 100 m, buses travel at 40 km/h, and the average station distance is 20 km. The platform safety dwell time is set to 30 min. To ensure the system’s smooth operation, the total backup capacity for each mode is assumed to be 10 train carriages, 20 high-speed train carriages, and 5 airplanes. Passenger utility-related parameters and upper-layer objective function parameters are set to ${\epsilon }_1$ = 0.15, ${\epsilon }_2$ = 4, ${\epsilon }_3$ = 2, ${\epsilon }_4$ = 1.5, and ${\lambda }_1$ = 0.5, ${\lambda }_2$ = 0.5, respectively. The upper-layer objective function combines two goals using linear weighting to optimize the allocation of vehicles for each line, while the lower layer uses an ant colony algorithm to optimize passenger flow distribution and path selection. This approach aims to maximize the efficiency of the system, ensuring that intercity travel demands are met during peak hours while avoiding overloads and restoring connectivity between core cities.

6.2. Result Analysis

The computation for the multimodal coordination strategy is performed using an Intel^® Core^TM i5-9300H CPU with a 2.40 GHz processor. The parameters for the ant colony algorithm are set as α = 3, β = 2, and ρ = 0.8, with 50 iterations. The convergence graph is shown in Figure 5a, which demonstrates convergence after 20 iterations. To validate the parameter values, different combinations of coefficients are presented in various colors. As shown in Figure 5a, as ρ increases, instability also increases, and convergence takes longer. However, as α and β decrease, the improvement becomes less noticeable. The genetic algorithm iterates for 30 generations, with the population size of 20. The convergence graph is shown in Figure 5b, where convergence is achieved after 25 iterations. To validate the parameter values, different combinations of coefficients are presented in various colors. As shown in Figure 5b, as the population size increases, convergence takes longer. However, with our parameter settings, convergence to an acceptable solution is achieved in a reasonable time.

To demonstrate the improvement in our algorithm, several traditional hybrid algorithms are tested, as shown in

Table 2. Comparison of algorithms.

Algorithms	Minimum Solution	Computational Time
Genetic + Ant colony	459,888	5564 s
Genetic + Particle swarm	533,094	7988 s
Particle swarm + Ant colony	485,678	6924 s

. The particle swarm algorithm is used as a benchmark, as it is widely employed in solving mixed integer programming problems. Our hybrid algorithm outperforms others in both solution quality and computational time. When applying the particle swarm algorithm at the lower level, the solution quality increases by 16%, and the computational time increases by 43%. Conversely, when applying the particle swarm algorithm at the upper level, the solution quality increases by 5%, and the computational time increases by 24%.

The multimodal passenger assignment plan generated by the algorithm is shown in Figure 6. A total of 14 high-speed train carriages, 5 conventional train carriages, and 5 aircraft are utilized. The flow for each segment is indicated next to the corresponding connections, and the passenger multimodal coordination transport plan is detailed in

Table 3. Multimodal coordination evacuation plan.

Origin	Destination	Alternative Route	Demand	Satisfied	Rate	Departure	Arrival
Zhangjiakou	Baoding	Intercity-Conventional	121	76	63%	10:00	13:05
Zhangjiakou	Shijiazhuang	Airline-Intercity	199	127	64%	10:15	13:53
Zhangjiakou	Xingtai	Airline-Intercity	83	51	62%	10:17	14:34
Zhangjiakou	Handan	Airline-Intercity	150	92	61%	10:05	14:38
Chengde	Baoding	Intercity-Conventional	117	84	72%	9:57	13:02
Chengde	Shijiazhuang	Intercity	185	118	64%	10:15	13:38
Chengde	Xingtai	Intercity	74	50	67%	10:08	14:03
Chengde	Handan	Airline	150	86	57%	10:03	12:13
Beijing	Baoding	Conventional	191	115	60%	11:18	13:13
Beijing	Shijiazhuang	Conventional	307	77	25%	11:20	16:10
		Intercity		138	45%	11:05	13:21
Beijing	Xingtai	Conventional	114	33	29%	11:19	17:23
		Intercity		57	50%	11:10	13:58
Beijing	Handan	Airline	223	56	25%	11:25	11:35
		Conventional		47	21%	11:19	17:55
Baoding	Shijiazhuang	Conventional	212	112	53%	13:17	17:55
Baoding	Xingtai	Conventional	70	8	11%	13:15	17:24
		Intercity		49	70%	13:11	14:23
Baoding	Handan	Conventional	108	19	18%	13:18	17:59
		Intercity		80	74%	13:03	14:31
Shijiazhuang	Xingtai	Intercity	135	120	89%	14:01	14:26
Shijiazhuang	Handan	Intercity	224	177	79%	13:28	14:16
Xingtai	Handan	Intercity	56	51	91%	13:56	14:12

. High-speed railways serve as the primary mode for post-disaster response, buses are used for city station-to-station transfers, while conventional railways and airplanes are employed for transport across disaster areas.

The resilience performance of the multimodal coordination strategy proposed in this study, compared to various modal combinations, is shown in Figure 7. The passenger departure and arrival processes are presented separately in the subfigures. According to Figure 7a, in the early phase (0–100 min), the proposed strategy effectively mitigates the imbalance between supply and demand. In the medium phase (100–200 min), the proposed strategy achieves the highest demand satisfaction rate of around 50%. In the later phase (200–500 min), the proposed strategy satisfies 59% of the travel demand in 200 min. It is important to note that the rate then decreases as additional passengers arrive at the station, leading to an increase in total demand. Therefore, in the early phase, the demand satisfaction rate continues to decrease. However, high-efficiency modes (airplanes and high-speed trains) can quickly reach the stranded stations, and the multimodal coordination strategy significantly slows the development of supply-demand imbalance. In the middle phase, as the rate of passenger arrivals slows, the lowest point of supply-demand imbalance occurs. Due to the fast transportation efficiency, large capacity, and multiple routes offered by the multimodal coordination strategy, the system provides higher post-disaster survival rates. In the later phase, due to the addition of efficient transportation modes, passengers at the terminal stations can board the emergency routes more quickly, so the proposed strategy reduces the time required to meet demand. Additionally, the multimodal coordination strategy has smaller deviations in arrival times, allowing more passengers to participate in emergency routes. However, some passengers remain at the station due to missed service time windows or significant time deviation. As shown in Figure 7b, the passenger arrival rate increases rapidly in the early stages for our strategy (blue lines), with up to 75% of passengers satisfied within the first 300 min.

The solutions proposed by different strategies and their comparison with the initial arrival time are shown in Figure 8. The multimodal emergency corridor constructed by the proposed strategy includes multiple routes, with 70% of the arrival times within the initial arrival time range. The bus shuttle strategy also has arrival times within this period, but close to the latest arrival time. The arrival times for the railway alternative result in noticeable travel delays.

7. Conclusions

This study presents a sustainable approach to enhancing the resilience of transportation systems during post-disaster recovery by introducing a multimodal emergency corridor strategy. The model developed integrates demand, supply, and transportation factors to optimize route selection, vehicle and aircraft dispatching, and passenger assignment. A bi-level optimization approach was used to construct the multimodal emergency corridor model, addressing challenges in multimodal route selection, vehicle and airplane dispatching, and passenger assignment. The proposed multimodal coordination strategy successfully meets 59% of one-way travel demand, with 70% of passengers arriving within their scheduled time frames. To assess the effectiveness of emergency response measures on system resilience, a dynamic indicator based on demand satisfaction rates during the failure phase was introduced. This study provides valuable insights into the development of flexible, scalable emergency transportation strategies that can address both localized and large-scale disruptions.

The hybrid genetic algorithm and ant colony optimization algorithm is particularly effective for the transportation system considered, as it combines genetic algorithm’s global search capabilities with ant colony optimization’s local optimization strength, achieving high solution quality and computational efficiency. This hybrid approach is well-suited for post-disaster recovery, where quick yet accurate decisions are needed.

However, the model assumes resource availability, which may not always be realistic in large-scale disasters. It also focuses primarily on regional disaster scenarios with partial facility failures. Future research will aim to extend the model to more complex disaster scenarios, integrate urban transit with intercity systems, and explore real-time adaptive decision-making to further improve resilience in large-scale disruptions.