Enhancing Concurrent Emergency Response: Joint Scheduling of Emergency Vehicles on Freeways with Tailored Heuristic

Abstract

Scheduling decisions for concurrent emergency response (CER) across multiple disaster sites presents numerous difficulties. The main challenge is to minimize human casualties while taking into account the rationality of resource allocation across different disaster sites. This paper establishes a joint scheduling model for emergency vehicles on freeways in the context of CER. The model aims to minimize the transportation time, dispatch cost, and casualty risk, by using the resource site scheduling scheme as the decision variable, addressing multiple disaster and resource sites. Specifically, a casualty risk function based on the rescue waiting time was designed to balance the competing needs among various disaster sites, enhance equitable resource allocation, and reduce the probability of casualties. To achieve global convergence in a high-dimensional solution space, a tailored heuristic algorithm called adaptive dual evolutionary particle swarm optimization (ADEPSO) is proposed. The numerical results show that the scheduling scheme proposed by the ADEPSO algorithm satisfies all constraints and demonstrates significant advantages in large-sized instances. Compared to the two basic algorithms, ADEPSO provides a more cost-effective scheme and reduces the average rescue waiting time. Moreover, integrating the casualty risk function significantly decreases the average rescue waiting time at both high- and low-priority disaster sites, thereby directly lowering the casualty risk.

1. Introduction

Freeways are essential transportation corridors in modern economies and societies, and any disruption or damage to the road network can have serious repercussions. Moreover, freeways serve as critical routes for rescue and evacuation following large-scale disaster events [1]. Major natural disasters or human-made disasters, though infrequent, have extensive and severe impacts [2]. Wen these regional emergencies occur, the emergency management sector encounters additional challenges, including the integration of information, the coordination of resources, and the dispatch of materials [3]. For instance, the 2023 Sichuan Jinyang ‘8–21’ flash flood and the Gansu ‘12–18’ earthquake in China were large-scale disasters that necessitated simultaneous rescue efforts at multiple locations, resulting in concurrent emergency response (CER). To prevent the spread of a catastrophe, multiple emergency management agencies must respond simultaneously by supplying resources and dispatching personnel to various affected locations. This concurrent response presents greater challenges compared to a single emergency response (SER) [4]. Therefore, effectively managing CER, coordinating rescue forces on freeways and their surroundings, and minimizing disaster losses are of great practical significance.

Generally, after an emergency occurs on a freeway, the corresponding road management department is directly responsible for dispatching nearby rescue forces. The most crucial aspect of this process is ensuring rapid arrival at the scene. For this reason, topics such as travel time prediction in emergencies [5] and emergency vehicle path planning [6] have been widely discussed by scholars. Further, when facing a CER situation, to avoid blind rescue efforts, the relevant road management department will swiftly establish an emergency response team to make resource scheduling decisions for the region. The specific response procedure is illustrated in Figure 1. At this stage, the focus shifts to how to efficiently schedule resources to maximize rescue effectiveness, while also considering factors such as social impact.

Two challenges arise in decision-making for CER: (i) The challenge of equitable resource allocation. Resource competition can occur among multiple disaster sites, with varying degrees of urgency. Addressing the competing needs between different priority disaster sites to ensure a prompt and reasonable response is a significant challenge; (ii) The difficulty of rapidly configuring multiple resources and disaster sites simultaneously. Due to the heterogeneity of events, more types of resources are needed at each disaster site. This requires coordinating multiple resource sites to respond collectively, necessitating the handling of more complex dispatch information in the decision-making process.

In this paper, the optimization of the emergency vehicle (EV) scheduling problem for the concurrent emergency response is addressed through model construction and algorithm development, aligned with the actual emergency rescue work requirements of freeways. The main contributions of this paper can be summarized as follows:

(1)

Joint scheduling model for CER situation: from the perspective of the freeway emergency response process, a joint scheduling model for EVs is developed to address concurrent emergencies involving multiple disaster and resource sites.

(2)

Casualty risk function for rescue equity: considering the rescue equity problem in CER situations, a casualty risk function based on the rescue waiting time is designed to effectively coordinate the competition among disaster sites.

(3)

Tailored optimization algorithm for rapid decision-making: to achieve the rapid configuration of multiple resources and disaster sites, a heuristic algorithm termed adaptive dual evolutionary particle swarm optimization (ADEPSO) is tailored for the proposed scheduling model and its superiority in large-sized instances is verified.

(4)

The numerical results demonstrate the effectiveness of the model and algorithm presented in this paper, promoting the equitable distribution of emergency rescue resources and reducing the risk of casualty.

The rest of this paper is organized as follows. Firstly, a literature review of related work is presented in Section 2. Then, we detail concurrent emergency response scenarios and propose a scheduling model in Section 3. An improved solution algorithm is introduced in Section 4. The numerical experiment and results are shown in Section 5, and finally, Section 6 presents the conclusions along with future research directions.

2. Literature Review

Based on the loading and transportation functions of vehicles, the emergency resource scheduling problem is often specified as an EV scheduling problem. Unlike ordinary logistics scheduling, which mainly focuses on transportation costs, emergency vehicle scheduling occurs after incidents, requiring an immediate response by relevant departments to quickly develop a scientific and reasonable scheduling plan [7]. This ensures that disaster sites receive the necessary materials promptly. Therefore, most studies often use transportation time as the optimization objective [8].

2.1. EV Scheduling Problem

According to the number of emergencies (or disaster sites) to be responded to, this can be categorized into single emergency response scheduling (SER) [9] and concurrent emergency response scheduling (CER) [10,11]. Although both have the same optimization objective, scheduling decisions for CER are more difficult due to its large accident coverage and high losses [12]. In the early days, a deterministic strategy was often used in rescue response, where the nearest EVs were dispatched according to the first-in-first-out rule [13]. This strategy has two significant advantages: (a) it is easy to implement, and (b) it ensures the fastest response to rescue calls. However, this static scheduling strategy does not necessarily remain applicable, as related research gradually shifts to concurrent responses from multiple resource sites to multiple disaster sites. Carter et al. [14] proved in 1972 that this scheduling strategy does not result in an optimal mean response time. Schmid [15] also used an approximate dynamic programming model to show that a more flexible scheduling and repositioning strategy reduces the mean response time by 12.89%. Therefore, the application of mathematical models to optimize the emergency response process has become mainstream nowadays [16]. Liu et al. [17] proposed an optimization model based on a mixed-integer linear programming (MILP) formulation and a risk-based traffic optimization formulation to determine the optimal emergency dispatch routes using a multi-stage optimization algorithm. Their approach aims to enhance the efficiency of emergency traffic and coordinate the traffic flow of the road network during rescue and evacuation processes. Bodaghi et al. [18] proposed an emergency response model based on the integration of GIS and the MILP model, providing an emergency resource scheduling solution to minimize the weighted emergency response completion time.

In contrast to the SER situation, the CER situation faces multiple disaster sites competing for resources, making it particularly important to manage the relationships between the disaster sites fairly and reasonably [19]. On the one hand, some scholars focus on how to distribute supplies equitably, as the discrepancy between the supply and demand of emergency resources in an emergency presents a significant challenge to emergency rescue work. Wang et al. [20] studied the scheduling and allocation of emergency resources with the COVID-19 pandemic as the research background. Based on the degree of urgency, they constructed a multi-objective optimization model to maximize the satisfaction of disaster victims, minimize the total cost, and ensure the equitable distribution of supplies. On the other hand, achieving differentiated scheduling by prioritizing each disaster site is also a common approach. Tian et al. [21] studied the issue of EV scheduling for forest fires. They addressed the varying demands of different fire sites and the competition for resources among them by establishing rescue priorities. Their approach aimed to minimize the overall firefighting time and the number of fire engines used. Zhu et al. [22] devised a relative deprivation cost metric to determine the rescue priority of each disaster site and optimized the rescue routes of EVs to achieve humanitarian rescue.

2.2. EVs Scheduling Problem Solving Methods

The EV scheduling problem is similar to the traveling salesperson problem, which is also an NP-hard problem [23]. The computational effort of generating accurate algorithms grows exponentially with the size of the problem, especially since CER requires the processing of more complex scheduling information. Therefore, heuristic algorithms are commonly used for solutions; these include genetic algorithms [24], differential evolutionary algorithms [25], and simulated annealing algorithms [26]. Researchers often enhance these basic heuristic algorithms to adapt them to problems with specific characteristics and their established mathematical models. Liu et al. [27] proposed an improved particle swarm algorithm to solve the task coordination model by abstracting the CER task into a multipoint-to-multipoint transportation delivery problem within a multimodal traffic network. Yoon et al. [28] constructed a Markov Decision Process (MDP) model to dynamically determine the type of rescue vehicles to dispatch based on resource availability. They designed a greedy threshold heuristic algorithm to solve the model and found through experiments that it closely matches the optimal strategy.

Table 1. Overview of research on emergency vehicle scheduling.

	Type	Model	Optimization Objectives	Solution Algorithm	Competition Consideration
[6]	SER	MDP	Transportation time, Road network influence	Reinforcement learning	_
[9]	SER	Multi-objective non-linear integral planning model	Rescue time, Rescue cost	Genetic algorithm	_
[4]	CER	MILP	Transportation time, Resource cost	Differential evolution	_
[10]	CER	Optimization model	Response delay	Genetic algorithm	_
[11]	CER	MILP	Resource shortage	Particle Swarm Optimization	_
[20]	CER	MILP	Satisfaction of the victims, Resource cost, Fairness of distribution	Bee colony optimization	√
[22]	CER	MILP	Relative deprivation cost, Transportation cost	Ant colony optimization	√

lists several studies related to EV scheduling, summarizing the types of response, the models, the optimization objectives, the solution algorithms, and whether or not the competition between multi-disaster sites was considered in each study.

In summary, this paper explores the challenges of concurrent emergency response based on existing research. Scholars have noted the competing needs between disaster sites and utilized equitable supply distribution to maintain balance. However, the arrival time of rescue forces, a critical factor affecting injury mortality rates and causing panic, has rarely been considered [29]. Simultaneously, by the ‘no free lunch’ theorem [30], we argue that it is imperative to tailor the algorithms for the models presented in this paper to enhance the quality of the solutions. To fill these research gaps, we developed a joint scheduling model for the CER situation and designed a tailored solution algorithm. These efforts will enhance the efficiency of emergency response and aid in decision-making for management units.

3. Problem Statement and Model Construction

This section provides a detailed introduction to the concurrent emergency response problem and its mathematical model. In particular, the assumptions that help abstract the joint scheduling problem under the CER situation into a mathematical model are presented in Section 3.1. A list of the notations used throughout the study is given in Section 3.2, and Section 3.3 offers a detailed introduction to the construction of the joint scheduling model.

3.1. Problem Statement and Hypothesis

Figure 2 illustrates an example of the CER situation on a freeway, including multiple resource and disaster sites. For the convenience of numerical analysis, this paper makes the following hypotheses:

(1)

The road network area contains M resource sites $S=\left\{S_1\dots S_M\right\}$, N disaster sites $A=\left\{A_1\dots A_N\right\}$, and K types of EVs $G=\left\{G_1\dots G_K\right\}$.

(2)

No more than K types of EVs are required at each disaster site.

(3)

Resource sites shall not be allocated resources over their reserves, and disaster sites shall not receive fewer resources than needed.

(4)

The distance between each resource site and the disaster site, the disaster site’s response level, and the vehicle dispatch cost are known.

(5)

The average arrival time of EVs denotes the rescue waiting time.

3.2. Notations

The notations for the relevant variables and parameters in the model constructed in this paper are shown in

Table 2. Related variables and parameters.

Type	Symbols	Definition
Set	$S$	Resource sites, $S=\left\{S_1\dots S_M\right\}$
	$A$	Disaster sites, $A=\left\{A_1\dots A_N\right\}$
	$G$	Types of EVs, $G=\left\{G_1\dots G_K\right\}$
	$P_i$	Scheduling priority for disaster site $i$, $P_i=\left\{0,1\right\}$
Variable	$w_i$	Scheduling scheme for a single resource site $w_i={\left[\begin{array}{ccc}{w}_{11}& \cdots & w_{1k}\\ ⋮& \ddots & ⋮\\ w_{n1}& \cdots & w_{nk}\end{array}\right]}_{N\times K}$
	$w_{ijk}$	Total scheduling scheme, the number of category $k$ EVs dispatched from resource site $i$ to disaster site $j$, ${\left\{w_{ijk}\right\}}_{M\times N\times K}=\left\{w_1\dots w_M\right\}$
	$Z_1~Z_3$	Optimization objective
	$\overline{t_i}$	Average time for disaster site $i$ to receive all EVs
	$x_{ij}$	Number of EVs dispatched from resource site $i$ to disaster site $j$
	$\theta \left(x\right)$	Penalty term
Parameter	$C_{dispatch}$	Vehicle dispatch cost
	$T_{transit}$	Transportation cost, i.e., transportation time
	$f_{risk}$	Casualty risk
	$L_0~L_4$	Equilibrium coefficients
	$t_H$	Psychological threshold time
	$\tau$	Surge risk
Condition matrix	$C_{ik}$	EVs dispatch cost matrix $C_{ik}={\left[\begin{array}{ccc}{c}_{11}& \cdots & c_{1K}\\ ⋮& \ddots & ⋮\\ c_{M1}& \cdots & c_{MK}\end{array}\right]}_{M\times K}$
	$T_{ij}$	EVs transportation time cost matrix $T_{ij}={\left[\begin{array}{ccc}{t}_{11}& \cdots & t_{1N}\\ ⋮& \ddots & ⋮\\ t_{M1}& \cdots & t_{MN}\end{array}\right]}_{M\times N}$
	$D_{jk}$	Demand matrix for disaster sites $D_{jk}={\left[\begin{array}{ccc}{d}_{11}& \cdots & d_{1K}\\ ⋮& \ddots & ⋮\\ d_{N1}& \cdots & d_{NK}\end{array}\right]}_{N\times K}$
	$S_{ik}$	Reserve Matrix for resource sites $S_{ik}={\left[\begin{array}{ccc}{s}_{11}& \cdots & s_{1K}\\ ⋮& \ddots & ⋮\\ s_{M1}& \cdots & s_{MK}\end{array}\right]}_{M\times K}$

.

3.3. Model Construction

The primary objective of EV scheduling in CER situations is to minimize the transportation time, considering socio-economic benefits and humanitarian concerns. In this paper, an optimization model is developed with the objectives of minimizing the transportation costs, dispatch costs, and risk of casualties. Here, transportation costs represent the transportation time, while dispatch costs represent the economic benefits. Together, they constitute the comprehensive cost of dispatch. The casualty risk function is designed to harmonize the competition between different priority disaster sites. The joint EVs scheduling model is as follows:

$$\begin{array}{c}min{Z}_1=T_{transit}={\displaystyle {\displaystyle \sum }_{i=0}^M}{\displaystyle {\displaystyle \sum }_{j=0}^N}{\displaystyle {\displaystyle \sum }_{k=0}^K}{w}_{ijk}·T_{ij}\hfill \\ min{Z}_2=C_{dispatch}={\displaystyle {\displaystyle \sum }_{i=0}^M}{\displaystyle {\displaystyle \sum }_{j=0}^N}{\displaystyle {\displaystyle \sum }_{k=0}^K}{w}_{ijk}·C_{ik}\hfill \\ min{Z}_3=f_{risk}\hfill \end{array}$$

(1)

$s.t$.

$${\displaystyle \sum }_{j\in N}{w}_{ijk}\le S_{ik}, \forall i\in M,\forall k\in K$$

(2)

$${\displaystyle \sum }_{i\in M}{w}_{ijk}\ge D_{jk}, \forall j\in M,\forall k\in K$$

(3)

$$w_{ijk}\ge 0, \forall i\in M, \forall j\in N, \forall k\in K$$

(4)

$$T_{ij}\ge 0, \forall i\in M, \forall j\in N$$

(5)

$$S_{ik}\ge 0, \forall i\in M, \forall k\in K$$

(6)

The above model is a MILP model. Here, $w_{ijk}$ is an integer decision variable representing the total scheduling scheme, which consists of the sub-scheduling scheme $w_i$ for a single resource site, as shown in Figure 3. Equation (1) is the model’s objective function, comprising transportation time, vehicle dispatch cost, and casualty risk function. The casualty risk function $f_{risk}$ is introduced in Section 3.3.1. The constraint term (2) ensures that the allocation scheme of the resource site does not exceed its reserve and constraint term, and (3) ensures that the receipts at the disaster site meet or exceed its demand. Equations (4)–(6) indicate that the scheduling scheme, transportation time, and resource site storage all satisfy the non-negativity constraints.

3.3.1. Design of Casualty Risk Functions

In the study of casualty risks from sudden accidents, it has been found that the longer the rescue forces take to arrive at the disaster site, the higher the mortality rate of the injured [31,32]. Additionally, in the face of disasters, the public is prone to panic, and the spread of this panic can create pressure and even affect the progress of rescue efforts. A study by Cao et al. [33] found that disasters can cause significant psychological stress and panic among personnel, negatively impacting rescue and evacuation operations. Therefore, in CER situations, coordinating the rescue waiting time (RWT) at multiple disaster sites is important to reduce casualties and alleviate victims’ psychological panic.

Inspired by the aforementioned studies and field research with freeway management departments, this paper hypothesizes that higher-priority disaster sites correspond to more severe injuries among victims. Based on the impact of the rescue waiting time on mortality rates for different injury severities, a graph (Figure 4) illustrating the relationships between RWT and the risk of casualties was created. This graph is used to design the casualty risk function. Victims of high-priority events ($P_i=1$) are more vulnerable, and the risk of casualties rises sharply when the RWT exceeds a certain threshold. Therefore, a monotonically increasing function is used as the basis, while an exponential function segmentation is used at the threshold position to reflect the sharp rise in risk. Meanwhile, victims of low-priority ($P_i=0$) events can theoretically tolerate a longer rescue time, so there is no need for a sharp increase at the threshold position.

Formula for the risk of casualties at a single disaster site:

$$f\left(\overline{t_i},P_i\right)=\{\begin{array}{cc}{k}_H\overline{t_i}+{\left(a\left(e^{b\overline{t_i}}-1\right)+\tau \right)}_{\left\{\overline{t_i}\ge t_H\right\}}, & P_i=1\\ k_L\overline{t_i}, \hfill & P_i=0\end{array}$$

(7)

where $k_H$ and $k_L$ are the slopes of the function for high and low priorities, satisfying $0<k_L<k_H$; $a$ and $b$ are constants; $t_H$ represents the psychological threshold time; and $\tau$ denotes the surge risk. $\overline{t_i}$ denotes the average time for all EVs to arrive at disaster site $i$, calculated by Equation (8):

$$\overline{t_i}={\displaystyle \frac{{\sum }_{j\in M}{t}_{ji}·x_{ji}}{{\sum }_{j\in M}{x}_{ji}}}$$

(8)

The casualty risk function for the scheduling scheme is as shown in Equation (9):

$$f_{risk}={\displaystyle \sum }_{i\in N}f\left(\overline{t_i},P_i\right)$$

(9)

It is noteworthy that this function already encompasses the casualty risk. Researchers can fine-tune the parameters based on the actual scenario. Moreover, since it represents the baseline parameters, our model remains adaptable to diverse research requirements by addressing this concern.

3.3.2. Constraint Handling

When addressing combinatorial optimization problems with constraints, a common approach is to convert constraints into penalty functions, which are then incorporated into the objective function [34]. According to the penalty function method principle, when a solution fails to meet the constraints, the presence of the penalty term causes the objective function value to become exceedingly large or small, effectively eliminating the solution. The penalty function below is designed to address the two primary inequality constraints in this paper: (1) resource sites shall not be allocated resources over their reserves, and (2) disaster sites shall not receive fewer resources than needed:

$$g_1={\displaystyle \sum }_{i=1}^M{\displaystyle \sum }_{k=1}^K{\left[\min \left(0, S_{ik}-{\displaystyle \sum }_{j=1}^N{w}_{ijk}\right)\right]}^2$$

(10)

$$g_2={\displaystyle \sum }_{j=1}^N{\displaystyle \sum }_{k=1}^K{\left[\min \left(0,{\displaystyle \sum }_{j=1}^N{w}_{ijk}-D_{jk}\right)\right]}^2$$

(11)

Finally, the penalty term is set to $\theta \left(x\right)=g_1+g_2$, and $L_0~L_4$ are used as equilibrium coefficients to standardize the following quantities: $L_0·T_{transit}+L_1·C_{dispatch}+L_2·f_{risk}+L_3·g_1+L_4·g_3$.

4. Tailored Heuristic

Particle Swarm Optimization (PSO) was first introduced by the American social psychologist Kennedy [35] in 1995 and is an optimization algorithm based on Swarm Intelligence (SI). Given its strong universality and ease of implementation, the PSO algorithm has been widely applied in various engineering optimization problems. However, the PSO algorithm’s search strategy may cause premature convergence in high-dimensional search spaces, reducing the search accuracy and convergence speed. The handling of emergency information is crucial for effective emergency response [36]. The CER problem necessitates managing complex information in scheduling decisions, demanding the greater accuracy and convergence performance of the solution algorithm. Hence, this paper introduces an Adaptive Dual Evolutionary Particle Swarm Algorithm (ADEPSO) tailored to the coding features of the scheduling scheme. It combines the crossover operator and the balance factor concept from the Beluga Whale Optimization (BWO) algorithm to enhance the algorithm performance.

4.1. PSO Algorithm

Supposing that the particle swarm consists of $m$ particles in an $n$-dimensional search space, the position of the $i$th particle is $x_i=\left[x_1^i,x_2^i,\dots ,x_n^i\right]$, and its velocity is $v_i=\left[v_1^i,v_2^i,\dots ,v_n^i\right]$. The current iteration number is $T$. The particle’s own history best solution is called $p_{best}$, and the global best solution is $g_{best}$. The equations for updating the particle’s velocity and position are as follows:

$$v_i^{T+1}=w{v}_i^T+c_1{r}_1\left(p_{best}-x_i^T\right)+c_2{r}_2\left(g_{best}-x_i^T\right)$$

(12)

$$x_i^{T+1}=x_i+v_i^{T+1}$$

(13)

where $c_1$ and $c_2$ are learning factors, which are non-negative constants and generally set to 2. $r_1$ and $r_2$ are random numbers in the range $\left[0,1\right]$, and $w$ is the inertia weight.

4.2. Tailored PSO Algorithm

4.2.1. Adaptive Decomposition Based on Coding Features

As shown in Figure 3, the total scheduling scheme consists of a combination of scheduling schemes for each resource site. The corresponding encoding of the decision variables is similarly composed of the encoding of each sub-scheme. The dimensionality of the decision variables depends on the number of resource sites, disaster sites, and EV type. Increasing the coded dimensionality of these variables is likely to decrease the search accuracy and speed. For this reason, the total scheduling scheme is decomposed into various sub-scheduling schemes based on the number of resource sites. Each sub-scheduling scheme is updated independently, resulting in a reduction in dimensionality during the updating process. This approach is expected to enhance the search accuracy of the algorithm. The sub-scheduling scheme, also called a sub-particle, is illustrated in Figure 5. The equations for the velocity and position update are as follows:

$$v_{ij}^{T+1}=w{v}_{ij}^T+c_1{r}_1\left(p_{best}^j-{\epsilon }_{ij}^T\right)+c_2{r}_2\left(g_{best}^j-{\epsilon }_{ij}^T\right)$$

(14)

$${\epsilon }_{ij}^{T+1}={\epsilon }_{ij}+v_{ij}^{T+1}$$

(15)

where $p_{best}^j$ denotes the value of sub-particle $j$ at the position corresponding to the original $p_{best}$. Similarly, $g_{best}^j$.

4.2.2. Dual Evolution

After a single evolution, sub-particles merge into a new particle. This newly formed particle must also undergo further exploration and exploitation to guide the evolution of the population. These two distinct levels of evolution are referred to as dual evolution. At the beginning of the iteration, extensive exploration should be conducted to ensure particle diversity. As the number of iterations increases, more exploitation at the particle level is necessary to ensure the algorithm’s global convergence capability. This paper introduces the balance factor from the BWO algorithm to achieve secondary balancing between the exploration and exploitation stages at the particle level. The formula is as follows:

$$B_f=B_0\left(1-{\displaystyle \frac{T}{2T_{max}}}\right)$$

(16)

where $B_0$ is a random number within $\left[0,1\right]$, $T$ is the current iteration number, and $T_{max}$ is the maximum number of iterations. The particle level exploration strategy follows the basic PSO algorithm using Equations (12) and (13). Simultaneously, to enhance the exploitation potential through sub-particle level evolution, the exploitation strategy uses the Roulette Wheel Selection (RWS) algorithm. This algorithm selects parents from the reverse order of particles, individual bests, and global bests for two-point crossover.

4.2.3. ADEPSO Algorithm

The ADEPSO algorithm first performs adaptive decomposition and dimensionality reduction based on the encoding characteristics of decision variables. Subsequently, evolution at the sub-particle level is crucial for maintaining population diversity, while evolution at the particle level guides the population towards optimal positions. The improved algorithm’s flowchart is depicted in Figure 6 and the specific process is described as follows:

5. Numerical Results

The model construction and algorithm design described above are meticulously tailored to address the CER problem. The model parameters were established in consultation with traffic management authorities. This section presents numerical experiments conducted to assess the effectiveness of these approaches.

5.1. Numerical Case

To validate the performance of the model and algorithm, numerical simulation experiments were used to test different problem instances. This was assuming that there are multiple emergency demands within a certain area of the freeway ($A_1$, $A_2$, $A_3$, $A_4$, $A_5$), with different response levels. The types ($G_1$, $G_2$, $G_3$, $G_4$) and numbers of EVs required for each disaster site, as well as their response levels, are as shown in

Table 3. Demand for EVs and response level.

	EVs Type	Fire Truck $({\mathit{G}}_1$)	$\mathbf{Ambulance} ({\mathit{G}}_2$)	$\mathbf{Police} \mathbf{Car} ({\mathit{G}}_3$)	$\mathbf{Engineering} \mathbf{Rescue} \mathbf{Vehicle} ({\mathit{G}}_4$)	$\mathbf{Response} \mathbf{Level} {\mathit{P}}_{\mathit{i}}$
Disaster Sites
$A_1$		2	2	3	1	1
$A_2$		1	2	2	1	0
$A_3$		3	3	2	4	1
$A_4$		1	2	2	3	0
$A_5$		1	2	1	2	0

. Within the road network, there are five resource sites ($S_1$, $S_2$, $S_3$, $S_4$, $S_5$) that can respond to emergency demands, and the reserves for the resource sites are shown in

Table 4. Resource reserves at site.

	EVs Type	Fire Truck $({\mathit{G}}_1$)	$\mathbf{Ambulance} ({\mathit{G}}_2$)	$\mathbf{Police} \mathbf{Car} ({\mathit{G}}_3$)	$\mathbf{Engineering} \mathbf{Rescue} \mathbf{Vehicle} ({\mathit{G}}_4$)
Resource Sites
$S_1$		2	2	4	2
$S_2$		2	4	2	4
$S_3$		3	4	5	2
$S_4$		4	2	4	4
$S_5$		3	5	2	3

. The dispatch costs of EVs are shown in

Table 5. Dispatch cost.

	EVs Type	Fire Truck $({\mathit{G}}_1$)	$\mathbf{Ambulance} ({\mathit{G}}_2$)	$\mathbf{Police} \mathbf{Car} ({\mathit{G}}_3$)	$\mathbf{Engineering} \mathbf{Rescue} \mathbf{Vehicle} ({\mathit{G}}_4$)
Resource Sites
$S_1$		10	15	10	20
$S_2$		15	15	10	25
$S_3$		20	10	5	20
$S_4$		15	20	5	15
$S_5$		20	10	10	25

. Assuming that EVs travel along the shortest path on the freeway, the transportation time is shown in

Table 6. Transportation time (min).

	Resource Sites	${\mathit{S}}_1$	${\mathit{S}}_2$	${\mathit{S}}_3$	${\mathit{S}}_4$	${\mathit{S}}_5$
Disaster Sites
$A_1$		41	48	44	78	24
$A_2$		52	38	33	19	89
$A_3$		43	62	16	54	22
$A_4$		29	73	43	56	70
$A_5$		35	32	22	60	51

. The instances of two categories for numerical simulation are detailed in

Table 7. Categories of test instances.

Test Instances	Resource Sites	Disaster Sites	EVs Type	The Dimensions of the Solution Space
Small-sized	$S_1~S_3$	$A_1~A_3$	$G_1~G_4$	$3\times 3\times 4=36$
Large-sized	$S_1~S_5$	$A_1~A_5$	$G_1~G_4$	$5\times 5\times 4=100$

. Relevant parameter settings: $L_0$ = 0.8, $L_1$ = 0.2, $L_2=L_3=L_4=10$, $t_H=30 \min$, $a,b=2, 0.1$ and $\tau =10$.

To further verify the superiority of the ADEPSO algorithm tailored to the CER problem, we also employ PSO and the Genetic Algorithm (GA) to solve the model. The parameters for each algorithm are shown in

Table 8. Parameters of three algorithms.

Algorithm	Parameter Setting
ADEPSO	population size = 50, $T_{max}=300$, $c_1=c_2=2$, $w=0.9~0.2$
PSO	population size = 50, $T_{max}=300$, $c_1=c_2=2$, $w=0.9~0.2$
GA	population size = 50, $T_{max}=300$, mutation and crossover probability = 0.1, 0.9

. All algorithms are implemented based on Python 3.7 and run on a computing platform with an AMD Ryzen 7 4800H CPU at 2.90 GHz and 16 GB of memory.

5.2. Results and Analysis

The findings in the numerical results are analyzed from three perspectives: EV scheduling scheme, algorithm performance, and casualty risk.

(1)

EVs scheduling scheme analysis

Due to space limitations, only the scheduling scheme for the large-sized instance is presented.

Table 9. Comparison of rescue indicators.

	Comprehensive Cost	ARWT (min)	Percentage of Reduction
ADEPSO	3300.6	37.98	26.3%, 10%
GA	3908.7	40.39	12.7%, 4%
PSO	4476.4	42.18	/

compares the comprehensive cost and average rescue waiting time (ARWT) of the schemes found by the three algorithms. The ADEPSO algorithm proposed in this paper solves the CER problem with a significantly better solution quality than the basic PSO and GA algorithms, and finds a better scheme in terms of the comprehensive cost and ARWT. In particular, the GA solves the problem with a 12.7% lower comprehensive cost and a 4% lower ARWT than the PSO. Furthermore, the ADEPSO achieves a comprehensive cost 26.3% lower and an AWRT 10% lower than the PSO.

Table 10. Comparison of three optimal schemes.

			Resource Site	${\mathit{S}}_1$	${\mathit{S}}_2$	${\mathit{S}}_3$	${\mathit{S}}_4$	${\mathit{S}}_5$
		Disaster Site		${\mathit{G}}_1$$,{\mathit{G}}_2$$,{\mathit{G}}_3$$,{\mathit{G}}_4$
Algorithm	ADEPSO	$A_1$		1 1 0 0	0 0 0 1	0 1 3 0	0 0 0 0	1 0 0 0
		$A_2$		0 1 1 0	0 0 0 1	0 1 0 0	1 1 1 1	0 0 0 0
		$A_3$		0 0 0 0	0 0 0 0	2 2 1 2	0 0 0 0	1 1 1 2
		$A_4$		0 0 2 0	0 0 0 0	0 0 0 0	1 0 0 3	1 2 0 0
		$A_5$		1 0 0 1	0 1 0 1	0 0 0 0	0 0 1 0	0 0 1 0
	PSO	$A_1$		1 1 0 0	0 0 0 1	0 1 3 0	1 0 0 0	1 0 0 0
		$A_2$		0 1 1 0	0 0 1 0	0 1 0 0	1 1 0 1	0 0 0 1
		$A_3$		0 0 0 0	0 0 0 0	2 1 1 2	0 0 0 0	1 2 1 2
		$A_4$		0 0 2 0	0 0 0 1	1 0 0 0	0 0 0 2	0 2 0 0
		$A_5$		0 0 0 1	0 1 0 1	0 0 0 0	1 0 1 0	0 1 0 0
	GA	$A_1$		1 1 0 0	0 0 0 1	0 1 3 0	1 0 0 0	1 0 0 0
		$A_2$		0 1 1 0	0 0 1 0	0 1 0 0	1 1 0 1	0 0 0 1
		$A_3$		0 0 0 0	0 0 0 0	2 1 1 2	0 0 0 0	1 2 1 2
		$A_4$		0 0 2 0	0 0 0 1	1 0 0 0	0 0 0 2	0 2 0 0
		$A_5$		0 0 0 1	0 1 0 1	0 0 0 0	1 0 1 0	0 1 0 0

displays the optimal schemes derived by the three algorithms. The ADEPSO algorithm’s scheduling scheme is more efficient because each resource site dispatches EVs only to the three to four closest disaster sites. In contrast, the schemes obtained by the other two algorithms are more decentralized, as some resource sites dispatch EVs to all disaster sites.

(2)

Algorithm performance analysis

Table 11. Comparison of algorithms’ computational performance.

	Computational Complexity	Large-Sized	Small-Sized
		Processing Time (s)
ADEPSO	$O(T_{\max }\cdot n\cdot (m+C_{fit}))$	23.4	2.12
PSO	$O(T_{\max }\cdot n\cdot C_{fit})$	21.6	1.62
GA	$O(T_{\max }\cdot n\cdot C_{fit})$	19.8	1.86

presents the computational complexity of ADEPSO in comparison to two basic algorithms, as well as their energy consumption across two types of instances, measured in processing time. $n$ denotes the population size and $C_{fit}$ denotes the computational complexity for evaluating the fitness of each individual. $m$ represents the number of sub-particles each individual particle is decomposed into, which correlates with the number of resource sites utilized: more resource sites leads to a higher computational complexity. The dual evolution process of decomposition and reconstruction in ADEPSO results in a higher computational complexity compared to the other two algorithms. However, the observed time difference of approximately 0.5 s in a small-sized instance and about 3.6 s in a large-sized instance has a negligible impact on the overall performance. Our algorithm provides superior solutions and does not cause significant delays in real-time decision-making.

Figure 7a displays the convergence performance of the algorithm on small-sized instances. It is evident that although PSO exhibits the fastest convergence rate, it falls into a local optimum. Moreover, the convergence performance of both GA and ADEPSO is similar, indicating that traditional heuristic algorithms possess effectiveness in small-sized instances.

In the comparison of large-sized instances (Figure 7b), the PSO converges after approximately 150 iterations, while both the ADEPSO and GA converge after around 100 iterations. During the search process, the PSO encounters two local optimums at 20 and 110 iterations, respectively, and the GA encounters one local optimum at around 50 iterations. In contrast, the ADEPSO algorithm does not encounter a local optimum and its fitness value is significantly lower. This observation indicates the success of the tailored ADEPSO algorithm proposed in this paper for the CER situation. It demonstrates stronger global convergence capabilities and the ability to discover superior solutions within the high-dimensional solution space. Specifically, the ADEPSO performs adaptive decomposition and dimensionality reduction according to the coding features. The separate evolution of the sub-particles in the early stages effectively avoids the issue of ‘premature convergence’. Additionally, the introduction of a balance factor for the second evolution ensures the particle’s diversity and enhances the global convergence in later iterations.

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Casualty risk analysis

Figure 8a compares the ARWT in two scenarios where the ADEPSO algorithm solves with or without the casualty risk function. In the test without the casualty risk function, the ARWT at each disaster site initially exhibits random variation during the iteration. Ultimately, this leads to a relatively minor reduction in the ARWT, attributed to the competing needs between disaster sites. In contrast, utilizing the casualty risk function designed in this paper enables the algorithm to consistently decrease the ARWT from 45 min to approximately 38 min throughout the solution process. This difference is attributed to the casualty risk function, which helps to overcome the negative effects of competing needs, demonstrating its ability to coordinate concurrent emergency responses.

To further understand the impact of the casualty risk function on the degree of coordination at different priority disaster sites, Figure 8b illustrates the trend of the ARWT at both high and low-priority disaster sites. It is obvious that the ARWT of the high-priority site decreased rapidly after a brief rise, from about 43 min to 29 min, a reduction of 32.6%. Meanwhile, after an initial fluctuation, the ARWT of the low-priority site stabilizes at a lower level of 40 min, decreasing by 11.1%. These phenomena indicate that the casualty risk function designed in this paper enables high-priority disaster sites to receive rescue as soon as possible while also considering the low-priority site, effectively reducing the risk of casualties at each disaster site.

6. Conclusions

This paper focuses on studying EV scheduling in the context of CER on freeways. We first establish a joint scheduling model for EVs with multiple disaster and resource sites. The optimization objectives include the transportation time, dispatch cost, and casualty risk. This model can assist relevant departments in formulating EV dispatching plans for CER situations on freeways. Secondly, to achieve the equitable allocation of resources, considering the varying degrees of urgency at disaster sites, a casualty risk function based on the rescue waiting time was designed to balance the competing needs between the sites. This function improves the equitable allocation of resources, thereby reducing the probability of human casualties. Finally, to achieve the rapid configuration of multiple resources and disaster sites, an ADEPSO algorithm is tailored.

In this paper, the connotation and significance of concurrent emergency response are discussed, and the emergency management processes of freeways in real-world scenarios are analyzed to promote the integration of theoretical research with practical applications. Numerical experiments were conducted to validate our study. The results highlighted that incorporating a casualty risk function into the scheduling model greatly enhanced the fairness of rescue operations. Specifically, the average rescue waiting time at high-priority disaster sites decreased by 32.6%, while at low-priority sites it decreased by 11.1%. Additionally, the proposed tailored algorithm significantly outperformed generalized algorithms. Compared to the PSO and GA algorithms, our tailored algorithm reduced the overall cost by 26% and 16%, respectively. More importantly, the analysis of specific schemes revealed that the tailored algorithm substantially improved the scenarios’ rationality.

The research works in this paper have significant application value for freeway emergency management during concurrent emergency responses. It can assist managers in coordinating rescue efforts and developing efficient and equitable resource scheduling schemes. Future directions could consider the following: (1) designing superior equilibrium functions for competing needs among various disaster sites, and (2) incorporating traffic control to better support emergency responses.