Multi-Objective Optimization Model for Emergency Evacuation Based on Adaptive Ant Colony Algorithm

Abstract

Evacuation in public places under emergency situations represents a significant area of management research. With the rapid development of the railway industry, the evacuation of railway stations has gradually attracted attention. This article employs the minimization of congestion degree and total evacuation time as primary objectives. In addition, the psychological behavior of individuals and the impact of congestion are sufficiently considered. Moreover, an adaptive Cauchy mutation operator is adopted for flexible population diversity. As a result, a multi-objective optimization model for the evacuation paths is established, with an improved adaptive quantum ant colony algorithm, and a comparison between the model based on adaptive quantum ant colony algorithm and the traditional ant colony model is made.

1. Introduction

In the contemporary era, emergencies commonly occur in public space. Emergency management has garnered significant interest from a diverse array of individuals and sectors. As a nexus of urban transportation, the railway station plays a pivotal role in fostering the economic growth of a vicinity. It becomes a comprehensive transportation hub, integrating transportation, interchange, and service functions.

Consequently, the evacuation of a considerable number of passengers at railway stations during an event is a challenging task, particularly given the potential for injury. With the expansion of railway networks in various countries, it is imperative to address the issue of emergency evacuation at railway stations. To this end, related research is trying to determine the most efficient and orderly scheme for evacuating passengers and staff, which includes reducing the number of casualties and minimizing property losses. Furthermore, such research will contribute to enhancing the risk response capability of railroad transportation sectors.

The evacuation process in an emergency is influenced by a multitude of factors, including the cluster effect of participants, the operational level of management, the secondary impact of emergencies, the layout of building facilities, etc. This complexity and variability introduce significant uncertainty to the decision-making related to evacuation programs [1,2]. Therefore, scholars in this field typically construct simulation and mathematical models for evacuation experiments. Moreover, in order to enhance the realism and scientific rigor of the evacuation model, scholars have introduced a variety of rules and constraints into the model, with the aim of developing a more reasonable and effective scheme.

In regard to simulation experiments, Altshuler et al. [3] employed ants to simulate herd behavior in the model with the degree of panic as a variable. Their findings corroborated the hypothesis that panic is susceptible to diffusion under insufficient information. This model provides a case of considering the individual behaviour.

For the timeliness of the model, Qian et al. [4] developed an interval parameter fuzzy evacuation management model to facilitate uncertain circumstances. Cova and Johnson [5] developed a dynamic network evacuation model that resolves conflicts of channel intersections in a static network. Li Yanfeng et al. [6] proposed a scheduling mechanism for updating routes with instant traffic information.

Regarding veracity and extension of the model, Zhou et al. [7] employed a fuzzy logic model to investigate the emergency evacuation under the terrorist attacks. Ma [8] developed a multi-objective emergency material transfer path optimization model under the user-balanced traffic assignment for the post-earthquake situation. Huang et al. [9] developed a dynamic switching model to address the uncertainty associated with initial conditions and to obtain global evacuation paths in a dynamic environment. In a study conducted by Li and Zhao [10], the advantages of crowd clustering in emergency evacuation plans was examined, particularly focus on the situation with capacity limitations. In conclusion, the simulation model can be extended as an integrated system, which typically incorporates factors such as building facility layouts, individual behaviors, and path optimization algorithms [11,12,13,14,15,16].

The application of intelligent algorithms based on soft computing offers a novel avenue for exploring solutions to evacuation-related problems. The most commonly used intelligent algorithms for solving evacuation path optimization problems include particle swarm algorithms, genetic algorithms, and ant colony algorithms, etc. Zou et al. [17] propose a multi-objective ant colony optimization model to address the problem of large-scale emergency evacuation management. Zhang Jianghua et al. [18] developed a multi-source evacuation model for addressing the priority order problem with multiple emergency source points and considering the capacity limitations of channels. They also heuristically devised an algorithm based on the network optimization method in graph theory. X. Liu et al. [19] constructed two distinct multi-objective models: one based on artifact order and the other on batch. These models were developed to address the path optimization problem. It is noticed that the researchers devised an enhanced ant colony optimization algorithm in models. Guo et al. [20] constructed a multi-objective crowd evacuation model based on an improved cuckoo search algorithm, which aims to optimize total time and path length. Zhou X. et al. [21] constructed a TDGVRP model for the time-dependent path planning problem, and an improved ant colony algorithm for calculating the traveling time of vehicles under road allocation strategies. Anshi et al. [22] developed a multi-objective vehicle scheduling model in emergency evacuation based on an enhanced genetic algorithm. Overall, intelligent algorithms were widely used in path optimization and emergency management today.

At present, there is a paucity of research examining the evacuation of rail stations, despite the abundance of studies on large public places, which have primarily focused on timeliness, with fewer of them considering the congestion’s effect. However, during the actual evacuation process, evacuees are under a high level of risk-taking as congestion increases. There are two key aspects of the emergency evacuation challenge to be faced: (1) reduce casualties and property losses, which can be achieved by minimizing the total evacuation time, i.e., timeliness; (2) the high-density crowd at the train station tends to panic in an event, which creates congestion and causes the safety risk to blow up. This research applies the adaptive ant colony algorithm into evacuation management considering the congestion degree and obtains a sufficiently effective procedure for evacuation scheme making.

2. Rail Station Personnel Evacuation Model Construction

2.1. Problem Description

Define evacuation paths as all accessible areas for evacuees. In the railway station, each stopover facility is defined as a node, and the channel connected between nodes is defined as a link. The evacuation of a railway station is characterized by dense personnel and limited channels. From the research results of M. Zhou et al. [23], the number of people to be evacuated is N_ij, and the ratio of N_ij for channel (i,j) and the capacity of channel C_ij is defined as the density of personnel p_ij(t) at time t, and p_ij(t) = N_ij(t)/C_ij, where i denotes the number of the node, j is the channel connection number of the corresponding node, L_ij denotes the channel length connected with i and j as end points.

As demonstrated by the findings of He Jianfei and Liu Xiao [24], the evacuation speed is contingent upon the density of individuals involved, and the evacuation process is no longer feasible when density reached a certain value. Furthermore, the obstruction of the people flow can result in overcrowding and trampling incidents. The aforementioned conclusion devotes its attention to the relationship between personnel density and evacuation speed. It defines the relationship between personnel density p_ij(t) and evacuation speed v_ij(t) on channel (i,j) at time t as follows:

$${\mathrm{v}}_{\mathrm{i}\mathrm{j}}=\left\{\begin{array}{ll}{\mathrm{v}}_{\mathrm{m}\mathrm{a}\mathrm{x}}{\mathrm{e}}^{-\alpha {\mathrm{p}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)}& ,{\mathrm{p}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)>0.5\\ {\mathrm{v}}_{\mathrm{m}\mathrm{a}\mathrm{x}}& ,{\mathrm{p}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)⩽0.5\end{array}\right.$$

(1)

The effect of crowding on pedestrians is primarily psychological, as evidenced by the observed adjustments in behavior in response to surrounding crowding conditions. Accordingly, when the crowd density is less than 0.5, the speed of pedestrians during evacuation can reach the maximum walking speed. However, when the crowd density is greater than 0.5, the speed of pedestrians during evacuation will be significantly affected. In the case of crowding, a is the correction factor for pedestrian speed during evacuation, where α = 0.5.

From Equation (1), if p_ij(t) < 0.5, it means that the channel (i,j) is smooth, and the channel will not be congested; however, when p_ij(t) > 0.5 is greater than 0.5, the channel (i,j) will be congested. In order to describe the congestion of each channel more comprehensively in the model, f_ij(t) is introduced as the congestion degree of channel (i,j) at time t. In this paper, we define the congestion degree of channel (i,j) at time t as the congestion degree of channel (i,j). In this paper, we define the functional relationship between p_ij(t), the density of people at time t, and f_ij(t), the congestion degree of channel (i,j), as follows:

$${\mathrm{f}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)=\left\{\begin{array}{ll}0& ,{\mathrm{p}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)<0.5\\ {\mathrm{e}}^{\beta {\mathrm{p}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)}& ,{\mathrm{p}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)⩾0.5\end{array}\right.$$

(2)

2.2. Multi-Objective Optimization Mathematical Model

The objective of personnel evacuation is to identify an optimal evacuation route within a limited time frame, which can effectively mitigate the probability of adverse events such as congestion and tripping in hazardous area of stations. In conclusion, the primary objective in the planning of the evacuation path is to minimize the evacuation time. Additionally, the degree of congestion during the evacuation process is another optimization objective.

In comparison to other public spaces, railway stations present a unique set of challenges in terms of evacuation procedures. The internal layout is often complex, with a large number of people to be evacuated from a single location. Furthermore, the connection between evacuation routes within and between stations is frequently intricate, and in some instances, there may be no available route or an inoperable route between nodes. It is thus imperative to consider the accessibility of node channels when performing optimal path optimization. This study presents a multi-objective optimization model of personnel evacuation paths, which is based on the topology of the internal nodes of the station, the attributes of the node channels, and the accessibility between the channels. This model can be effectively applied to solve the objective problem and calculate optimal feasible evacuation paths on a larger scale. The objective of the model is to identify the optimal evacuation strategy for the railway station, whereby the total evacuation time for all individuals is minimized while ensuring a balanced distribution of evacuation loads across the entire system. The pertinent parameter settings of the model are presented in

Table 1. Parameters of model.

Notation	Meaning
G	Evacuation network
t	Time
i	Index of node
M	Total number of persons in the danger zone
Pathk	Path of evacuee k
${\mathrm{s}}_0^{\mathrm{k}}$	The starting point of the path of evacuee k
lij	Length of channel (i,j)
uij	Reachability of channel (i,j), with 0 representing reachable and 1 representing not
${\mathrm{v}}_{\mathrm{i}\mathrm{j}}^{\mathrm{k}}$(t)	Velocity of evacuee k in channel (i,j)
vij(0)	Standard velocity of evacuee k in channel (i,j)
Nij(t)	Number of persons in channel (i,j) at moment t
Cij	Maximum number of people that channel (i,j) can accommodated
Tmax	Total time for evacuation network to complete full evacuation
α	Correction factor of velocity, taking 0.5
β	Correction factor of congestion, taking 0.5

.

A mathematical model is introduced as follows:

Objective function:

$${\mathrm{F}}_1=\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\sum }_{\mathrm{k}=1}^{\mathrm{M}} {\sum }_{\mathrm{i}={\mathrm{x}}_0^{\mathrm{k}}}^{\left(\mathrm{i},\mathrm{j}\right)\in \mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{k}} {\mathrm{t}}_{\mathrm{i}\mathrm{j}}^{\mathrm{k}}\right\},\hspace{1em}\hspace{1em}{\mathrm{F}}_2=\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\sum }_{\left(\mathrm{i},\mathrm{j}\right)\in \mathrm{G}} {\sum }_{\mathrm{t}=0}^{{\mathrm{T}}_{\mathrm{m}\mathrm{a}\mathrm{x}}} {\mathrm{f}}_{\mathrm{i}\mathrm{j}}\right\}$$

(3)

Constraint condition:

$$\begin{array}{ll}& {\mathrm{v}}_{\mathrm{i}\mathrm{j}}=\left\{\begin{array}{ll}{\mathrm{v}}_{\mathrm{m}\mathrm{a}\mathrm{x}}{\mathrm{e}}^{-\alpha {\mathrm{p}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)}& ,{\mathrm{p}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)>0.5\\ {\mathrm{v}}_{\mathrm{m}\mathrm{a}\mathrm{x}}& ,{\mathrm{p}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)⩽0.5\end{array}\right.\\ & {\mathrm{f}}_{\mathrm{i}\mathrm{j}}=\left\{\begin{array}{ll}0& ,{\mathrm{p}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)<0.5\\ {\mathrm{e}}^{\beta {\mathrm{p}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)},& {\mathrm{p}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)⩾0.5\end{array}\right.\hfill \\ & {\sum }_{\mathrm{t}={\mathrm{x}}_0^{\mathrm{k}}}^{\left(\mathrm{i},\mathrm{j}\right)\in \mathrm{Pathk} } {\mathrm{u}}_{\mathrm{i}\mathrm{j}}=0\\ & {\mathrm{p}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)={\displaystyle \frac{{\mathrm{N}}_{\mathrm{i}\mathrm{j}}}{{\mathrm{C}}_{\mathrm{i}\mathrm{j}}}}\\ & {\int }_0^{{\mathrm{t}}_0^{\mathrm{k}}} {\mathrm{v}}_{\mathrm{i}\mathrm{j}}^{\mathrm{k}}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}={\mathrm{l}}_{\mathrm{i}\mathrm{j}}\\ & {\displaystyle \frac{{ \mathrm{N}}_{\mathrm{i}\mathrm{j}}}{{\mathrm{C}}_{\mathrm{i}\mathrm{j}}\mathrm{t}}}\le 1\end{array}$$

(4)

In this model, we have the following: Equation (3) represents the initial optimization objective, which is to minimize the total evacuation time of all evacuees and represents the secondary optimization objective, which is to minimize the sum of the crowding degree of each channel in the evacuation process. Equation (4) illustrates the correlation between the density of individuals within the channel. Notably, we have further adopted the adaptive Cauchy operator [25] in the experiments. Moreover, it depicts the velocity of individuals traversing the channel in question. It also represents the relationship between density and crowding degree in a given moment. Meanwhile, it imposes constraints on the internal passage of the evacuation path, ensuring its accessibility. Equation (4) also defines the personnel density, that is, the ratio between the number of people to be evacuated and the number of passages. This provides a link between the passage length l_ij, the velocity v(t), and the time t. Furthermore, it constrains the number of evacuees from a passage (i,j) at the same moment to be no higher than the upper limit of the number of people that can be evacuated by the passage.

2.3. Weighted Ideal Point Method

The Weighted Ideal Point Method (WIPM) for multi-objective problems is conducted in the following manner:

$$\mathrm{g}\left(\mathrm{x}\right)=\mathrm{m}\mathrm{i}\mathrm{n}\left[{\omega }_1{\left({\displaystyle \frac{{\mathrm{f}}_1-{\mathrm{f}}_1^{\mathrm{m}\mathrm{i}\mathrm{n}}}{{\mathrm{f}}_1^{\mathrm{m}\mathrm{i}\mathrm{n}}}}\right)}^2+\cdots +{\omega }_{\mathrm{i}}{\left({\displaystyle \frac{{\mathrm{f}}_{\mathrm{i}}-{\mathrm{f}}_{\mathrm{i}}^{\mathrm{m}\mathrm{i}\mathrm{n}}}{{\mathrm{f}}_{\mathrm{i}}^{\mathrm{m}\mathrm{i}\mathrm{n}}}}\right)}^2+\cdots +{\omega }_{\mathrm{i}}{\left({\displaystyle \frac{{\mathrm{f}}_{\mathrm{i}}-{\mathrm{f}}_{\mathrm{i}}^{\mathrm{m}\mathrm{i}\mathrm{n}}}{{\mathrm{f}}_{\mathrm{i}}^{\mathrm{m}\mathrm{i}\mathrm{n}}}}\right)}^2\right]$$

(5)

where ${\mathrm{f}}_1^{\mathrm{m}\mathrm{i}\mathrm{n}}$ = min_x∈S f_i(x), i = 1,…, k₀ represents the ideal points set, ω = (ω₁,…,ω_k) is weight vector, and ${\Sigma }_{\mathrm{i}=1}^{\mathrm{k}}$ ω_i = 1, ω_i > 0, i = 1,…,k.

In this paper, the WIPM is employed to assign weights to each optimization objective and calculate the comprehensive adaptability according to the weights. This allows the objective function of the mathematical model of people evacuation in railway station to be transformed as follows:

$$\mathrm{F}\left(\mathrm{x}\right)=\mathrm{m}\mathrm{i}\mathrm{n}\left[{\omega }_1{\left({\displaystyle \frac{{\mathrm{F}}_1-{\mathrm{F}}_1^{\mathrm{m}\mathrm{i}\mathrm{n}}}{{\mathrm{F}}_1^{\mathrm{m}\mathrm{i}\mathrm{n}}}}\right)}^2+{\omega }_2{\left({\displaystyle \frac{{\mathrm{F}}_2-{\mathrm{F}}_2^{\mathrm{m}\mathrm{i}\mathrm{n}}}{{\mathrm{F}}_2^{\mathrm{m}\mathrm{i}\mathrm{n}}}}\right)}^2\right]$$

(6)

In the majority of studies examining evacuation problems, the significance of evacuation time is given greater consideration than congestion. In light of the related studies on a multi-objective evacuation path optimization model by A. Arenas et al. [26] and X. Yang et al. [27], this paper sets W1 and W2 to 0.7 and 0.3, respectively.

3. Enhanced Quantum Ant Colony Algorithm

3.1. Algorithm Description

The ant colony algorithm is a simulated evolutionary algorithm proposed by the Italian scholars Dorigo and Blum [28]. The algorithm simulates the behavior of ants discovering paths in the process of searching for food, exhibiting characteristics of distributed computation, positive feedback, and heuristic search. It is a widely utilized approach for addressing a range of optimization problems, including the traveler problem, assignment problem, workpiece sequencing problem, and vehicle path problem.

The quantum ant colony algorithm incorporates the benefits of quantum bits into the ant colony algorithm, thereby effectively addressing the limitations of traditional ant colony algorithms, which are known to exhibit slow convergence and a proclivity for local optima. However, due to the extensive scale and dynamic nature of the optimization problem associated with the evacuation of railway stations, the convergence and global requirements of the algorithm are particularly stringent. In light of these considerations, this paper primarily seeks to enhance the quantum ant colony algorithm in three key areas: (1) based on the concept of dynamic self-adaptation, the rotational angle and the amount of rotation in the quantum rotation gates are adaptively enhanced; (2) to guarantee the diversity of the population, the process of ant mutation is incorporated; and (3) ants release pheromones in accordance with the significance of the arc in which they are situated and engage in pheromone updating.

3.2. Enhanced Design of Algorithms

3.2.1. Quantum Rotation Gates

Quantum gates represent a fundamental operational component in the implementation of evolutionary operations. The selection of an appropriate method is often contingent upon the specific requirements of the problem at hand. In the context of quantum enhancement algorithms, the quantum rotation gate is typically selected as the operator for the evolution process. The following is a definition of a quantum rotation gate:

$$\mathrm{U}\left({\theta }_{\mathrm{i}}\right)=\left[\begin{array}{cc}\cos {\theta }_{\mathrm{i}}& -\sin {\theta }_{\mathrm{i}}\\ \sin {\theta }_{\mathrm{i}}& \cos {\theta }_{\mathrm{i}}\end{array}\right]$$

(7)

The process of updating a quantum bit is as follows:

$$\left[\begin{array}{c}{\alpha }_{\mathrm{i}}^{\prime }\\ {\beta }_{\mathrm{i}}^{\prime }\end{array}\right]=\mathrm{U}\left({\theta }_{\mathrm{i}}\right)\left[\begin{array}{c}{\alpha }_{\mathrm{i}}\\ {\beta }_{\mathrm{i}}\end{array}\right]=\left[\begin{array}{cc}\cos {\theta }_{\mathrm{i}}& -\sin {\theta }_{\mathrm{i}}\\ \sin {\theta }_{\mathrm{i}}& \cos {\theta }_{\mathrm{i}}\end{array}\right]\left[\begin{array}{c}{\alpha }_{\mathrm{i}}\\ {\beta }_{\mathrm{i}}\end{array}\right]$$

(8)

That is

$$\left\{\begin{array}{l}{\alpha }_i^{\prime }={\alpha }_i\cos {\theta }_i-{\beta }_i\sin {\theta }_i\\ {\beta }_i^{\prime }={\alpha }_i\sin {\theta }_i+{\beta }_i\cos {\theta }_i\end{array}\right.$$

(9)

where |${\alpha }_{\mathrm{i}}^{\prime }$|² + |${\beta }_{\mathrm{i}}^{\prime }$|² = 1, and the (α_i,β_i)^T and (${\alpha }_{\mathrm{i}}^{\prime }$,${\beta }_{\mathrm{i}}^{\prime }$)^T represent the probability of the i-th quantum bit before and after the quantum rotation gate update, respectively. Similarly, θ_i represents the rotation angle, ∆θ_i denotes the rotation angle’s size, and s(α_i,β_i) denotes the rotation angle’s direction, θ_i = ∆θ_is(α_i,β_i).

The application of an adaptive strategy to optimize and enhance the ant colony algorithm can markedly enhance the algorithm’s performance. The adaptive strategy of the quantum ant colony algorithm is to correlate the value of the rotation angle ∆θ with the iteration speed. The value of ∆θ exerts an influence not only on the convergence speed of the algorithm but also on the quality of the solution; we have ∆θ = 5exp(−t/t_max), where t indicates that the algorithm is currently in its t-th iteration. t_max means that the algorithm will be executed a maximum of tmax times and then terminated.

The s(α_i,β_i) indicates the direction of the rotation angle. This ensures that the algorithm is optimized in the optimal direction. If s(α_i,β_i) > 0, the rotation gate rotates in a counterclockwise direction; if s(α_i,β_i) < 0, the rotation gate rotates in a clockwise direction. In this paper, the variable s(α_i,β_i) is represented by s(α_i,β_i) = d_iγ_i, where γ_i = arctan(β_ib/a_ib)−arctan(β_in/a_in), and d_i = a_inβ_ib/β_inα_ibα_ib and β_1b represent the probability distributions at the i-th quantum position of the optimal individual. The arctan(β_in/α_in) denotes the current phase at the i-th quantum position of the nth individual, while arctan(β_ib/α_ib) denotes the optimal phase at the -th quantum position of the optimal individual.

3.2.2. Mutations in Individual Ants

In this paper, we build upon the concept of mutation operations in genetic algorithms as proposed by Keshanchi et al. [29] to enhance the diversity of the population in quantum ant colony algorithms. This approach aims to prevent premature convergence of the algorithm. The mutation operation is defined as follows:

$$\left(\begin{array}{ll}0& 1\\ 1& 0\end{array}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{{\alpha }_{\mathrm{i}\mathrm{j}}}{{\beta }_{\mathrm{i}\mathrm{j}}}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{{\alpha }_{\mathrm{i}\mathrm{j}}^{\prime }}{{\beta }_{\mathrm{i}\mathrm{j}}^{\prime }}}\right)$$

(10)

where (α_i,β_i)^T represents the probability of the j-th bit of the i-th individual before the mutation, whereas (α_i,β_i)^T represents the probability of the j-th bit of the i-th individual after the mutation.

3.2.3. Pheromone Updates

The fundamental principle of the ant colony algorithm is the updating of pheromones. In a quantum ant colony algorithm, the quantity of pheromone released is represented by the quantum bit. The matrix B(t) is to be constructed according to Q(t), B(t) = (${\mathrm{b}}_1^{\mathrm{t}}$, ${\mathrm{b}}_2^{\mathrm{t}}$,…, ${\mathrm{b}}_{\mathrm{m}}^{\mathrm{t}}$)^T, where ${\mathrm{b}}_{\mathrm{i}}^{\mathrm{t}}$ is an n-bit binary string of numbers.

$$\Delta {\tau }_{\mathrm{a}\mathrm{b}}^{\mathrm{i}}=\left\{\begin{array}{ll}{\displaystyle \frac{{\mathrm{Q}}^{\prime }}{{\mathrm{L}}_{\mathrm{i}}}}& , \mathrm{ant} \mathrm{i} \mathrm{passesnode} \mathrm{b} \mathrm{and} {\mathrm{p}}_{\mathrm{i}\mathrm{b}}=1\\ {\displaystyle \frac{\mathrm{Q}}{{\mathrm{L}}_{\mathrm{i}}}}& , \mathrm{ant} \mathrm{i} \mathrm{passesnode} \mathrm{b} \mathrm{and} {\mathrm{p}}_{\mathrm{i}\mathrm{b}}=0\\ 0& , \mathrm{otherwise}\end{array}\right.$$

(11)

If we have p_ib = 1, this indicates that b represents a critical point; conversely, when p_ib = 0, it indicates that b does not represent a critical point. Consequently, when p_ib = 1, the total quantity of pheromone is presumed to be Q′, which can be set to Q′ = kQ. In each iteration, the parameter of the pheromone released on the arc by each ant arriving at each node is recorded by B(t). In the event that the ant traverses the arc and is able to identify the optimal individual on B(t), the pheromone value marking the current arc will be augmented. Conversely, if the ant is unable to locate the optimal individual on B(t), the pheromone value will be diminished. In the absence of ant passage through the current arc, the pheromone value will be set to zero.

3.3. Solving Enhanced Quantum Ant Colony Algorithm

The enhanced quantum ant colony algorithm is employed to address the emergency evacuation path optimization challenge in a railway station as well as the building fire evacuation task. The implementation process encompasses eight principal stages: initialization, ant colony arrangement, binary matrix construction, solution space construction, mutation, evaluation and recording, rotation gate operation, and judgment.

3.4. Toy Example of the Enhanced Quantum Ant Colony Algorithm

In order to ascertain the efficacy of the enhanced quantum ant colony algorithm, Eil51, Eil101, and the Chinese traveler problem dataset from the TSPLIB platform have been selected as the benchmark cases for the algorithm comparison experiments presented in this paper. In parallel, a crowd evacuation network model is developed based on the railway station in W City, which is employed to assess the efficacy of the algorithm in addressing the evacuation challenge. The experiments were programmed and developed in the Matlab 2018 environment, and the code execution and value statistics were conducted on a workstation equipped with an Intel Core i9-12900K CPU and 64 GB RAM. In order to demonstrate the superiority of the enhanced quantum ACO (QACO), the fundamental ACO and chaotic ACO (CACO) are selected for illustrative purposes in this paper as control variables.

At the same time, in order not to lose the generality, each kind of algorithm experiment of each benchmark example is run independently 20 times, and the maximum number of stopping generations of the algorithms are all 200 times. The parameters of the algorithms are set in

Table 2. Algorithms parameters.

Algorithms	m	α	β	ρ	Q
ACO	31	1	6	0.9	1
CACO	30	1.5	2	0.9	-
QACO	31	1.5	2	0.9	-

, in which the signal generator of the CACO algorithm is z_i+1 = 0.4z_i(1 − z_i); the quantum multiplier k = 4 for the enhanced QACO algorithm. Notably, the parameters for QACO are further estimated by sensitivity analysis in Section 4.

The outcomes of the enhanced QACA are illustrated in Figure 1, Figure 2, Figure 3 and Figure 4. A comparison of the results yielded by the three algorithms is presented in

Table 3. Comparative analysis of three algorithms on the TSPLIB platform.

Problem	TSPLIB	Algorithm	Best Result	Worst Result	Average
Eil51	426	QCAC	441.30	467.02	458.14
		ACO	457.04	469.41	461.22
		CACO	454.16	467.59	461.87
Eil101	629	QCAC	651.17	702.01	680.59
		ACO	694.97	715.52	700.88
		CACO	665.50	706.94	685.21
CTP	15377	QCAC	15,443.19	15,972.76	15,691.25
		ACO	15,601.92	16,221.34	15,948.52
		CACO	15,592.11	16,247.70	15,813.37

and

Table 4. Comparative analysis of three algorithms for evacuation in railway station.

Problem	Convergence Iteration Number	Best Result	Worst Result	Average
ACO	154	1392.51	1457.29	1412.12
CACO	175	1375.22	1455.75	1395.78
QACO	117	1347.75	1421.12	1385.44
QACO*	106	1324.18	1406.83	1363.92

.

A comparison of the results presented in Table 3 and Table 4 reveals that QACA exhibits clear advantages in terms of convergence speed and solution quality. This suggests that QACA demonstrates superior comprehensive performance. On the one hand, CACO expands the search space by introducing chaotic variables, which can avoid the algorithm from falling into a local optimum. However, this increases the computation volume of the solution significantly, thus leading to a decrease in algorithm efficiency and solution stability. Nevertheless, the overall performance is still stronger than ACO. Conversely, the aforementioned algorithms demonstrate a negative correlation with problem size in terms of solution quality and convergence speed; that is to say, as problem size increases, solution quality and convergence speed both decline. Conversely, the aforementioned algorithms demonstrate a negative correlation with problem size in terms of solution quality and convergence speed; that is to say, as problem size increases, solution quality deteriorates, and convergence speed slows. Conversely, there is a negative correlation between the aforementioned algorithms and the problem size in terms of solution quality and convergence speed; that is to say, as the problem size increases, the quality of the solution and the speed of convergence both decline.

4. Example Validation of the Model

4.1. Parameter Optimization Strategy

The selection of key parameters in the optimization algorithm is crucial for the stability and effectiveness of the model. To ensure the rationality of the parameter choices, a comprehensive analysis was conducted. To further enhance the performance of the Quantum Ant Colony Optimization (QACO) model, a grid search methodology was employed, inspired by the approach outlined in [30]. This method systematically explored the parameter space to identify the optimal combination of parameters. For completeness, we’ve conduct experiments with the same m = 32 as stated in

Table 5. Algorithm initialized parameters with the same m = 31.

Algorithms	m	α	β	ρ	Q
ACO	31	1	6	0.9	1
CACO	31	1.5	2	0.9	-
QACO	31	1.5	2	0.9	-

, where

Table 6. Comparative analysis for evacuation in railway station.

Problem	Convergence Iteration Number	Best Result	Worst Result	Average
ACO	154	1392.51	1457.29	1412.12
CACO	181	1381.30	1456.72	1406.01
QACO	117	1347.75	1421.12	1385.44
QACO*	106	1324.18	1406.83	1363.92

summarizes the selection range for each parameter.

Each parameter combination was tested through multiple iterations to ensure the robustness and reliability of the results. The optimal model, denoted as QACO*, was selected based on its superior performance in terms of both convergence speed and solution quality. In the following experiments, the QACO and QACO* models demonstrated significant improvements in optimizing evacuation time and minimizing congestion compared to the standard QACO model. This enhancement underscores the importance of meticulous parameter tuning in achieving optimal performance for complex optimization problems, such as emergency evacuation in railway stations and fire evacuation among buildings.

Experiments on Railway Station Data

In the event of an emergency at a railway station, the evacuation network comprises 50 nodes. Each node represents a pivotal connection point within the evacuation channel of the station, with the specific configuration varying across different types of stations. The following example illustrates the calculation for a typical high-speed railway station with an evacuee population of M. If the two nodes are able to be interconnected, a straight line is drawn between them, and if both the source node and the target node contain only one element, then the following is true. The pre-evacuation path network diagram is illustrated in Figure 5. In this diagram, ⊗ represents the disaster point, the thin solid line represents inter-node access on the second level of the high-speed railway station (HSR), and the dotted line represents access on the first level of the HSR station. The objective of the evacuation task is, therefore, to identify the optimal evacuation path from node 5 to node 29, with the aim of minimizing both the total elapsed time required to complete the evacuation and the cumulative congestion of the path.

4.2. Comparative Analysis of Results

4.2.1. Enhanced QACO to CACO

In the case of this HSR station, the enhanced QACO and CACO methods are employed in order to address the issue at hand. For the purposes of this discussion, we will assume that the quantum population size is 31 and that the pheromone importance factor is α = 1. The heuristic function importance factor β = 2, the pheromone volatilization factor ρ = 0.9, the quantum multiplicity k = 4, and the number of iterations i = 200. By setting M = 100 and nodes n = 50, the final optimization results, as computed by the enhanced QACO and CACO, are illustrated in Figure 5.

As illustrated in Figure 5, the enhanced QACO algorithm is capable of solving the path A, 5 → 11 → 16 → 23 → 27 → 32 → 29, whereas the CACO algorithm is better suited to the path B, 5 → 2 → 11 → 16 → 23 → 27 → 24 → 28 → 29. A comparison of the two algorithms utilized to solve the aforementioned path is presented in

Table 7. Comparative analysis to CACO and QACO.

Problem	Lenth	Total Time	Congestion	Fitness
A	335.942	36,723.63	1.51	0.0936
B	377.227	57,909.89	2.66	0.1866

.

The results presented in Table 7 and Figure 5 reveals that when the number of iterations is held constant, the enhanced QACO solution path proposed in this study reduces the total length by 41.285 m, the total evacuation time by 21,186.26 s, and the cumulative congestion by 1.15 compared to the CACO algorithm. These findings demonstrate the high solution efficiency and optimization effect of the proposed algorithm, further substantiating its feasibility and superiority.

4.2.2. Comparative Analysis of Evacuation Efficiency with Congestion

In the context of an HSR station, which is a relatively confined space, the number of evacuation routes and the available space are both limited. Consequently, the number of evacuees has a significant impact on the overall evacuation efficiency. To further substantiate the impact of crowding on evacuation efficiency, this study conducted 20 experiments employing the enhanced quantum ant colony algorithm and the hybrid particle swarm algorithm under conditions of varying evacuee numbers (M = 30, 100, 200, and 1000, respectively).

The outcomes of the two algorithms when the number of evacuees is 1000 are illustrated in Figure 5, and the numerical comparison of the fitness function results are presented in Figure 6.

The results of the aforementioned experiments indicate that the enhanced QACO does not exhibit a discernible performance advantage in scenarios where the algorithmic iteration count is limited. Nevertheless, as the number of iterations of the algorithm increases and the complexity of the problem to be solved continues to expand, the enhanced QACO proposed in this paper gradually demonstrates its superiority in terms of convergence speed and solution quality.

4.2.3. Sensitivity Analysis on Evacuation Task

Leveraging the simulation results in Section 4.2.1 and Section 4.2.2 and the parameter ranges delineated in

Table 8. Parameter selection range.

Parameter	Search Range	Description
α	{0.3, 0.5, 0.7}	Correction factor for pedestrian speed, influencing the relation between pedestrian speed and crowd density
β	{0.3, 0.5, 0.7}	Correction factor for congestion degree, influencing the relationship between crowd density and congestion
W1	{0.50, 0.55, 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.90}	Weight assigned to total evacuation time in the multi-objective optimization
W2	{0.5, 0.45, 0.40, 0.35, 0.30, 0.25, 0.20, 0.15, 0.10}	Weight assigned to congestion degree in the multi-objective optimization, where W2 = 1 − W1
Mutation Operator	Simple in Equation (4) or Cauchy Operator	Channel factor on accessibility uij

, a systematic sensitivity analysis was conducted as follows.

Figure 7 presents a three-dimensional response surface that illustrates the coupled sensitivity of α and β with respect to average congestion and the evacuation time. The z-axis represents the normalized composite objective (0 = optimum, 1 = worst). The surface curvature confirms a significant interaction effect (α–β Sobol interaction), validating that simultaneous tuning is indispensable for achieving balanced performance between evacuation time and congestion mitigation.

4.3. Experiments on Fire Evacuation Data

To address ecological validity concerns, we implemented a high-fidelity simulation framework adapted from Liang et al.’s building fire evacuation model [30]. For comprehensive sensitivity studies, we conducted full-factorial experiments across critical parameters for our proposed QACO. Based on the article by [31], the data-generation pipeline for the fire-evacuation task can be summarized in two stages: synthetic-scene creation and data pre-processing.

The environment contains a 70 × 70 grid map (1 m² cells) with 30% obstacle density (walls/columns) and dynamic hazards on fire propagation, CO toxicity, and crowd density effects. The evacuation data is generated and processed by first tessellating a 5000 m² complex into a 70 × 70 occupancy grid, whose black cells represent walls and pillars, then extracting the convex corners of these obstacles as “vertex cells” that constitute the nodes of a reduced graph, whose edges are admitted only when a corridor of at least l m width is unobstructed, compressing the search space.

Results

For comprehensive case studies on both static and dynamic obstacles, we conducted full-factorial experiments across critical parameters for our proposed QACO as follows.

The above results in

Table 9. Experiments on building fire evacuation case with static obstacles.

Problem	Convergence Iteration Number	Best Result	Worst Result	Average
ACO	157	1495.22	1520.64	1508.31
CACO	189	1418.71	1455.75	1431.20
QACO	135	1330.25	1421.12	1398.82
QACO*	122	1329.22	1407.60	1371.30

and

Table 10. Experiments on building fire evacuation case with dynamic obstacles.

Problem	Convergence Iteration Number	Best Result	Worst Result	Average
ACO	217	1710.34	1745.62	1731.47
CACO	247	1678.26	1703.29	1686.12
QACO	186	1513.49	1617.44	1562.31
QACO*	173	1491.35	1579.31	1538.65

verify that dynamic obstacles increased convergence iterations vs. static benchmarks. W₁ = 0.75 and W₂ = 0.3 are empirically the optimal combination for build QACO*.

5. Discussion and Conclusions

5.1. Discussions on Future Works

Future research will enhance the model by integrating empirically calibrated panic models (e.g., Helbing–Farkas social-force or HiDAC) to replace the rational-agent assumption and quantify fear-driven route and time deviations, coupling the pedestrian solver to open-source CFD solvers such as the Fire Dynamics Simulator to capture transient fire, smoke, and visibility dynamics for scenario-based optimization of ventilation and signage, and extending the graph representation to 3-D networks that explicitly encode stairs, escalators, elevators, and overlapping floor plates to analyze evacuation in complex multi-level hubs like multi-storey metro stations or airport terminals, collectively yielding a more predictive framework for designers and emergency planners.

5.2. Limitations and Conclusions

This paper examines the emergency evacuation process at a railway station under an emergency situation. From the personnel density, the concepts of congestion degree and personnel evacuation speed are introduced. Thereafter, the relationship between evacuation efficiency and personnel density, as well as congestion degree, is analyzed. Finally, a multi-objective evacuation path optimization model is established. The model’s optimization objectives are twofold: to minimize the total evacuation time and to minimize the cumulative congestion degree of each channel.

Additionally, the evacuation time is considered with the congestion degree, which yields evacuation paths that are more practical and feasible. Moreover, a transformation model is devised to address the shortcomings of a traditional ant colony algorithm, such as being prone to premature convergence, and the new model is well-suited for addressing large-scale path optimization problems.

Then, a comparison between the enhanced quantum ant colony algorithm and the hybrid particle swarm algorithm is made in the context of evacuation crowd simulation. The results demonstrate that with an equivalent number of iterations (e.g., i = 200), the improved quantum ant colony algorithm exhibits a faster convergence, enhances evacuation efficiency by 36.58%, and reduces the congestion of the evacuation process by 43.23%. At the same time, as the number of evacuees increases, so too does the congestion associated with emergency evacuations. This brings a more robust convergence to the enhanced quantum ant colony algorithm. In conclusion, the multi-objective evacuation path optimization model based on the enhanced quantum ant colony algorithm in this paper demonstrates superior efficacy in solving relevant problems.

However, this paper only considers the effective length of the selected channel for the relevant analysis, without considering additional factors such as the distance between the channel and the source of danger, and the panic mentality of the pedestrians. These factors warrant further study. As previously stated, with the ongoing advancement of computer technology, simulation technology is becoming increasingly sophisticated. The question of how to apply software calculations, exemplified by intelligent optimization algorithms, to simulation technology will undoubtedly become a significant area of research.