A Ship Fire Escape Speed Correction Method Considering the Influence of Crowd Interaction

Abstract

The aim of this study is to explore the effect of different personnel attributes and the relationship between different people on evacuation efficiency in the case of a passenger ship fire. As such, this study proposes a speed correction method that considers human attributes and interactions between different populations. Firstly, a hesitant fuzzy set and hesitant fuzzy average operator are adopted to quantify four kinds of personnel attributes. Secondly, considering the influence of different people, this study extracts the formula for acceleration under the interactive influence of different groups of people. At the same time, based on the first-order linear relationship between velocity and acceleration, an interactive velocity correction method is presented in the evacuation of ship personnel. Finally, this study uses the personnel evacuation simulation software Pathfinder to conduct experiments, and introduces the corrected speed and the uncorrected speed into the evacuation simulation process, respectively. The results show that the simulation results of the revised speed plan are more consistent with reality.

1. Introduction

In recent years, the density of maritime navigation has been increasing. The number of ships, as the main means of water transportation, is also increasing year by year. During navigation, ship fire is one of the main threats to navigation safety. In shipwreck incidents over the years, accidents caused by fire account for about 11% [1]. Fire on ships is particularly dangerous because rescue is more difficult and fires spread fast, and the internal structure of ships is complex. Therefore, it is difficult to evacuate and put out fires. Once a fire occurs, it will cause a lot of economic losses and seriously threaten the safety of people’s lives [2]. The occurrence of such accidents is closely related to the evacuation behavior and evacuation time [3]. Therefore, the behavior mechanism and rules of different groups in the evacuation process are studied extensively. It is beneficial to formulate scientific and efficient emergency evacuation measures. It is also of great significance to ensure the personal safety of evacuees in case of fire on ships.

Relevant studies have found that in the event of a fire, passengers will not only be affected by personal attributes such as their emergency ability, cognitive ability, psychological endurance, and value orientation, but also by different groups of people [4]. Therefore, in the ship scenario, the interactive hesitancy fuzzy integration operator is used to integrate the information of different groups. At the same time, the velocity and acceleration formulas under the influence of different crowd interaction are extracted. Then the cognitive ability and emergency response ability of the population at the fire scene are quantitatively analyzed. This study revises and supplements the effects of capacity, value orientation, psychological bearing capacity and group effect on evacuation efficiency. The models of escape acceleration and escape speed are also established, which provide theoretical basis and decision support for the evacuation of ship fire personnel.

1.1. Literature Review

Ensuring the safety of people in a fire is the fundamental goal of evacuating people. To achieve this goal, it is necessary to study the behavioral laws of evacuated people. The study of evacuation behavior is one of the nine key research directions of fire science. At present, the research on pedestrian evacuation mainly focuses on three aspects: evacuation model construction, evacuation decision-making and personnel evacuation efficiency.

Concerning the construction of the evacuation model, Treuille et al. [5] proposed a real-time crowd model with congestion in public places in multiple cities as the research object. Helbing et al. [6,7] established a social model to describe the walking behavior of the crowd in evacuation according to the calculation formula of Newton’s second theorem. Wang et al. [8] integrated human factors into emergency evacuation and analyzed the influence of various factors on evacuation behavior in different stages by building an evacuation model. Wang et al. [9] constructed an evacuation model that considers Openness, Conscientiousness, Extroversion, Agreeableness, and Neuroticism (OCEAN) to analyze the impact of passengers’ personality traits on evacuation behavior. Hu et al. [10] considered the interaction between the fire environment and evacuees from the perspective of the system and established a manual evacuation procedure.

Concerning the evacuation decision part, Feng et al. [11] proposed an evacuation decision-making model consisting of three parts: pedestrian distribution prediction model, pedestrian flow calculation model and path situation and feedback correction model. Lovreglio et al. [12] introduced an evacuation decision model predicting pre-evacuation behavior, and the model simulates the probability of evacuees’ behavioral state. Peng et al. [13] established a two-level decision-making model for emergency evacuation paths of high-rise buildings based on BIM, which realized the optimal planning of emergency evacuation paths. Sun et al. [14] used game-based theory in a small-world network context and built an evolutionary game model of evacuation decision diffusion between evacuees in the context of a complex network. Tian et al. [15] have designed a mobile-based system to collect medical and temporal data produced during an emergency response to mass casualty incidents.

Concerning the evacuation efficiency, Koo et al. [16] studied the psychological panic effect coefficient of evacuation speed by combining theoretical derivation with three-dimensional simulation technology. Chen et al. [17] adopted an improved social force model to study the influences of the total number of pedestrians, required speed, and specific location of obstacles on the evacuation efficiency of multi-exit configuration. Jeon [18] studied the impact of escape routes and emergency exits on evacuation speed under different environmental conditions. Yu et al. [19] conducted experimental and numerical simulation study on evacuation time and average evacuation speed of personnel in railway tunnels under train fire conditions.

The above research provides a theoretical basis for this research. At present, however, there are few studies on fire evacuation in the scenario of ships. The research on the behavior of personnel evacuation is not refined enough. Therefore, this study takes ships as the research scene and adopts the method of questionnaire survey to collect data. Then, the factors affecting evacuation efficiency of different groups are quantified. The relationship between the behavior and psychological characteristics of people in a fire and the evacuation speed are difficult to directly quantify into an accurate mathematical relationship, therefore, this paper uses fuzzy logic to quantify their influence, and selects the classical hesitant fuzzy weighted average operator for information integration.

1.2. Objective Contribution

The purpose of this study is to explore the influence of the interaction between different attribute groups on the evacuation efficiency in the ship fire scenario. The main research contributions are as follows.

Firstly, this study is based on fuzzy mathematics theory, using the classical hesitant fuzzy average operator. Then, the four attributes that affect crew escape on board are integrated. Through the quantification of four objective influencing factors, evacuation research is more realistic.

Secondly, a speed correction model considering the interaction of different populations is developed. The reduction in evacuation speed caused by different crowd interaction is quantified. It provides a reference for realizing evacuation research under the influence of multiple factors.

Finally, this study collects data through questionnaires and calculates the model. The simulation software is used to compare the modified speed plan with the unmodified speed plan, and more realistic simulation results are obtained.

The contents of this study are arranged as follows. In Section 2, the research ideas are described and the factors affecting the escape of personnel on board are analyzed. In Section 3, a fuzzy set containing four kinds of people is established by using fuzzy mathematics theory. Then, a speed correction method considering personnel attributes and the interaction between different groups is developed. Section 4 verifies the validity of the revised velocity model through simulation.

2. Research Foundation

2.1. Research Idea

The ship fire evacuation efficiency is affected by many factors. This study mainly considers the influence of the interaction between different groups of people on the evacuation efficiency. Then, a more realistic fire evacuation velocity model is extracted. The specific research ideas are as follows.

Based on the above research goals, this study has carried out the following work. Firstly, this study establishes the hesitant fuzzy sets of four kinds of people. The influence of the attributes of emergency response ability, cognitive ability, psychological bearing ability and value orientation is quantitatively analyzed. Secondly, this study introduces the interaction between the target population and other groups of people, combined with the classical universal gravitation formula. The acceleration formula of interaction between the influence of different attributes and the influence of different people is extracted. Finally, this study collects data through questionnaires, and uses simulation software to compare the revised speed plan with the uncorrected speed plan to verify the validity of the model (please, see Figure 1 for the research process).

2.2. Analysis of Key Factors for the Escape of Personnel on Board

In the process of fire evacuation, emergency ability, cognitive ability, psychological endurance and value orientation are the key factors that affect the survival of people [20]. The sudden stimuli of fire make the crowd react instantaneously, and the instantaneous response is closely related to the above four abilities of different people. The specific explanations of emergency ability, cognitive ability [21], psychological bearing ability and value orientation are as follows.

(i) Emergency ability: When people encounter an emergency, the brain immediately deals with it based on past experience and the ability to think for itself. Self-thinking is a subconscious response. Because children and the elderly have far less physical function than adults, once a fire breaks out, children and the elderly will become vulnerable groups. Their evacuation speed is also significantly lower than that of adults.

(ii) Cognitive ability: People with higher education levels have weaker fear, faster reactions and stronger ability to escape. On the other hand, people with lower education levels have slower reactions and weaker escape ability in the face of fire.

(iii) Psychological endurance: When a fire occurs on a ship, people will have a fear of fire due to a lack of understanding of fire. In this state, people are prone to irrational behavior. Adults have a strong psychological bearing capacity, while the elderly and children have a weak psychological bearing capacity. There is a certain gap in the psychological response of different groups of people in terms of psychological bearing capacity.

(iv) Value orientation: When a ship fire occurs, the value orientation of the elderly is conservative, which greatly affects the escape ability of the elderly. Therefore, value orientation is also one of the key factors affecting the escape speed of the crew on board.

2.3. Research Tools

When a fire occurs on a ship, panic and chaotic behavior are bound to occur in the crowd. In a ship with concentrated personnel, the personnel’s emergency ability, cognitive ability, psychological bearing ability and value orientation vary greatly. In addition, under fire conditions, the four abilities of different groups are complex and abstract, and the relationship with evacuation speed cannot be directly quantified. Therefore, this study proposes a fire escape velocity correction model based on hesitant fuzzy sets.

Due to the complexity and uncertainty of objective information and the ambiguity of human thinking, Zadeh introduced the concept of fuzzy sets [22]. Hesitant fuzzy sets are fuzzy set extensions to handle hesitant situations that were not well handled by previous tools [23]. Operator theory is an important part of fuzzy theory. Based on the arithmetic ensemble method, this study uses the classical hesitancy fuzzy weighted average operator (HFWA) and the classical hesitant fuzzy average (HFA). Then, the cognitive ability, emergency response ability, value orientation, psychological bearing ability and group effect of people in the fire scene are integrated. Furthermore, the objective information of different groups is quantified [24]. The relevant definitions of the hesitant fuzzy set weighted average operator, acceleration and velocity formulas used in this study are as follows.

Definition 1.

Let $X$ be a given finite set, then $E=\left.\left\{〈x,h_E(x)〉|x\in X\right.\right\}$ is called hesitant fuzzy set. Among them, $h_E\left(x\right)$ represents the possible membership degree of $x$ belonging to $X$, which is a subset of the interval $\left[0,1\right]$, and let $h_{1,}{h}_{2,}{h}_{3,}$ be three hesitant fuzzy elements, then their basic operations are as follows

$$\begin{array}{l}{h}_1\cap h_2={\cup }_{{\gamma }_1\subset h_1,}{}_{{\gamma }_2\subset h_2}\left\{\left.\min ({\gamma }_1,{\gamma }_2)\right\}\right.;\\ h_1\cup h_2={\cup }_{{\gamma }_1\subset h_1,}{}_{{\gamma }_2\subset h_2}\left\{\left.\max ({\gamma }_1,{\gamma }_2)\right\}\right.;\\ h_1\oplus h_2={\cup }_{{\gamma }_1\subset h_1,}{}_{{\gamma }_2\subset h_2}\left\{\left.{\gamma }_1+{\gamma }_2-{\gamma }_1{\gamma }_2\right\}\right.;\\ h_1\otimes h_2={\cup }_{{\gamma }_1\subset h_1,}{}_{{\gamma }_2\subset h_2}\left\{\left.{\gamma }_1{\gamma }_2\right\}\right..\end{array}$$

In addition, $h_j(j=1,2\cdots ,n)$ is a set of hesitant fuzzy elements, and the operation of hesitant fuzzy weighted average operator $H^n\to H$ is the mapping of $H^n\to H$, and the specific operation is as follows [25]

$$HFWA(h_1,h_2,\cdots h_n)=\underset{j=1}{\overset{n}{\otimes }}(w_j{h}_j)={\cup }_{{\gamma }_1\subset h_1,}{}_{{\gamma }_2\subset h_2}\left\{\left.1-\underset{j=1}{\overset{n}{\mathrm{\Pi }}}{(1-{\gamma }_j)}^{w_j}\right\}\right.;$$

(1)

where $w={(w_1,w_2,\cdots w_n)}^T$ is the weight vector of $h_j$, $w_j>0,{\displaystyle \sum _{j=1}^n{w}_j=1}$.

Definition 2.

To calculate the escape speed of people on board, this study introduces relevant acceleration formulas. For research convenience, the probability that the target group perceives the other group is denoted as $\theta$. The influence of the other group on the target group is denoted as $\mu$, and the influence direction of the other group on the target group is denoted as $\mathrm{sgn}\left(v_j-v_i\right)$, When $v_j$ < $v_i$, its value is −1, $v_j$ > $v_i$, its value is +1. $t_{act}$ is the instantaneous reaction time of personnel escape. The acceleration equation is obtained as

$$a=\frac{\Delta v}{\Delta t}=\frac{{\theta }_i\cdot \mu \cdot \mathrm{sgn}(v_j-v_i)}{t_{act}}.$$

(2)

Definition 3.

Considering the influence of hesitancy fuzzy average operator and based on the first-order linear relationship between velocity and acceleration, this study gives the velocity correction formula of different people under different influences when fire occurs. The influence of all attributes on a single population was denoted as $Q_i$, $v_i$ is the expected speed of group $i$. The corrected speed $v^{\prime }$ can be obtained as

$${v^{\prime }}_{}=\left(1-Q_i\right)\cdot v_i+a\cdot t_{act}.$$

(3)

3. Model Formulation

Based on the above analysis of human behavior characteristics in a passenger ship fire, this study constructs its model as follows. Firstly, fuzzy mathematical theory is applied to acquire fuzzy sets including different groups of people. Secondly, an ensemble operator with different attributes is obtained by using the classical hesitant fuzzy average operator. Thirdly, the escape speed of each crowd is obtained by combining the universal gravitation formula and the relationship between velocity and acceleration. Finally, considering various special cases, the properties and inferences are acquired. The model building process is shown in Figure 2.

3.1. Consider Different Attributes and the Speed Correction Model of the Crowd

The steps of model construction are as follows. Step 1 is to construct the hesitant fuzzy sets including an adult male, adult female, children and the elderly, and comprehensively consider the factors affecting the escape speed of people on the ship. The classical hesitancy fuzzy integration operator is used to consider and quantify various factors in Step 2. Step 3 is inspired by the classical universal gravitation formula, and extracts the formula of escape acceleration of people on board under the interaction of two factors, i.e., different attributes and different people [26]. Step 4, combined with the relationship between acceleration and velocity, further extracts the escape speed of people on board under this interactive influence [27].

Step 1. This study establishes hesitant fuzzy sets of four groups of people. For the convenience of research, in this study, $\left\{H_{i1},H_{i2},H_{i3},H_{i4}\right\}$ represents the hesitant fuzzy sets under the four attributes of the four groups of people. $N_{ijk}$ is the hesitant fuzzy elements under the four attributes of the four groups of people. ${\rho }_{ijk}$ is the percentage of $i$ crowd, $j$ attribute, and $k$ ability judgment options. Among them, $i$ represents the four groups of people, respectively; $i\in \left\{1,2,3,4\right\}$. $j$ represents the four attributes,$j\in \left\{1,2,3,4\right\}$. $k$ represents the evaluation options of the four attributes; $k\in \left\{1,2,3,4\right\}$. The hesitant fuzzy sets under the four attributes of the four groups of people are expressed as follows:

$$\begin{gathered} H_{i1}=\left\{\stackrel{N_{i11}}{\overbrace{{\rho }_{i11},{\rho }_{i11}\cdots {\rho }_{i11}}},\stackrel{N_{i12}}{\overbrace{{\rho }_{i12},{\rho }_{i12}\cdots {\rho }_{i12}}},\stackrel{N_{i13}}{\overbrace{{\rho }_{i13},{\rho }_{i13}\cdots {\rho }_{i13}}},\stackrel{N_{i14}}{\overbrace{{\rho }_{i14},{\rho }_{i14}\cdots {\rho }_{i14}}}\right\}, \\ {H}_{i2}=\left\{\stackrel{N_{i21}}{\overbrace{{\rho }_{i21},{\rho }_{i21}\cdots {\rho }_{i21}}},\stackrel{N_{i22}}{\overbrace{{\rho }_{i22},{\rho }_{i22}\cdots {\rho }_{i22}}},\stackrel{N_{i23}}{\overbrace{{\rho }_{i23},{\rho }_{i23}\cdots {\rho }_{i23}}},\stackrel{N_{i24}}{\overbrace{{\rho }_{i24},{\rho }_{i24}\cdots {\rho }_{i24}}}\right\}, \\ {H}_{i3}=\left\{\stackrel{N_{i31}}{\overbrace{{\rho }_{i31},{\rho }_{i31}\cdots {\rho }_{i31}}},\stackrel{N_{i32}}{\overbrace{{\rho }_{i32},{\rho }_{i32}\cdots {\rho }_{i32}}},\stackrel{N_{i33}}{\overbrace{{\rho }_{i33},{\rho }_{i33}\cdots {\rho }_{i33}}},\stackrel{N_{i34}}{\overbrace{{\rho }_{i34},{\rho }_{i34}\cdots {\rho }_{i34}}}\right\}, \\ {H}_{i4}=\left\{\stackrel{N_{i41}}{\overbrace{{\rho }_{i41},{\rho }_{i41}\cdots {\rho }_{i41}}},\stackrel{N_{i42}}{\overbrace{{\rho }_{i42},{\rho }_{i42}\cdots {\rho }_{i42}}},\stackrel{N_{i43}}{\overbrace{{\rho }_{i43},{\rho }_{i43}\cdots {\rho }_{i43}}},\stackrel{N_{i44}}{\overbrace{{\rho }_{i44},{\rho }_{i44}\cdots {\rho }_{i44}}}\right\}. \end{gathered}$$

Step 2. Based on the fuzzy mathematics theory, this study transforms the qualitative problem into the quantitative problem. According to the above hesitant fuzzy sets, the objective factors affecting crowd evacuation efficiency are integrated. Then, the influence of a single attribute on a single population can be denoted as $Q_{ij}$. The influence of all attributes on a single population was denoted as $Q_i$ by integrating $Q_{ij}$. Based on experience, the literature, and current research, this study assumes that the weight of the four groups is equal. According to the classical hesitant fuzzy average operator [24], $Q_i$ is obtained as

$$Q_{ij}=\left(1-{\displaystyle \prod _{i=1}^4{\left(1-{\rho }_{ijk}\right)}^{\frac{1}{4}}}\right),$$

(4)

$$Q_i=\frac{{\displaystyle \sum _{j=1}^4{Q}_{ij}}}{4},$$

(5)

respectively, where $j=1,2,3,4;i=1,2,3,4.$

Step 3. Consider that there is only a single target population, its escape speed is only affected by four attributes and its expected speed. However, in a case of multiple groups, the escape speed of the target group will be affected not only by cognitive ability, emergency response ability, value orientation and psychological endurance but also by other groups. To simplify the research work, this study divides the influences from other groups into three categories, as follows. The probability that the target group perceives the other group is denoted as $\theta$, the magnitude of the influence of the other group on the target group is denoted as $\mu$, and the direction of the influence of the other group on the target group is denoted as $\mathrm{sgn}\left(v_j-v_i\right)$. Since this study only considers the influence between the four groups of people, other influencing factors are not considered. Therefore, in this study, other unconsidered factors are denoted as ${\lambda }^{*}$ and defaulted to 0.375 [28]. $M_i$ is used to represent the number of single people, with $v_i$ representing the expectations of a single population escape velocity, with $v_i^{\prime }$ representing a single population after a reaction time of the final velocity. ${\lambda }_{ij}$ denotes mutual influence between the two groups, $t_{act}$ instantaneous response time for escape, ${\omega }_1$ is the weight of the probability that the target group feels other groups of people, and ${\omega }_2$ is the weight of the influence of other groups on the target population. Based on experience, the literature, and current research [29], this study considers that the instantaneous reaction time of adult men and women is $t_{1act}=t_{2act}=2$ s, and the elderly and children is $t_{3act}=t_{4act}=3$ s. Thus, the formula of escape acceleration ($a_i$) of a single crowd is obtained as

$$\begin{array}{cc}{a}_i\hfill & ={\displaystyle \sum _{i=1}^4\left({\lambda }_{ij}\right)}\hfill \\ & ={\displaystyle \sum _{j=1}^4\frac{\frac{4}{\pi }\cdot \arctan \left(\begin{array}{c}{\left(\frac{M_j}{M_i+M_j}\right)}^{w_1}\cdot \\ {\left(\frac{\left|v_i-v_j\right|}{\max \left\{v_i,v_j\right\}}\right)}^{w_2}\end{array}\right)\mathrm{sgn}\left(v_j-v_i\right)\cdot \left|v_j-v_i\right|}{t_{\mathrm{i}act}}}.\hfill \end{array}$$

(6)

Among them, $\mathrm{sgn}\left(v_j-v_i\right)$ represents the influence direction of other groups on the target group. In $i=\left\{1,2,3,4\right\}$, $j\in \left\{1,2,3,4\right\}$, 1, 2, 3, and 4 represent adult males, adult females, the elderly, and children, respectively. When $v_j>v_i$, the value of $\mathrm{sgn}\left(v_j-v_i\right)$ is $+1$, indicating that other groups have a positive influence on the target group. When $v_j<v_i$, the value of $\mathrm{sgn}\left(v_j-v_i\right)$ is $-1$, indicating that other groups have a negative influence on the target group. When $v_j=v_i$, the value of $\mathrm{sgn}\left(v_j-v_i\right)$ is 0, and other groups are in the same direction as the target group.

Step 4. This study integrates the influence of four attributes and other populations on the target population, substitute it into Equation (3), $v_i^{\prime }$ is obtained as

$${v^{\prime }}_i=\left(1-Q_i\right)\cdot v_i+{\displaystyle \sum _{j=1}^4\left({\lambda }_{ij}\right)}\cdot {\lambda }^{*}\cdot t_{\mathrm{i}act},$$

(7)

where, $i=1,2,3,4.$

Then, substitute Equation (7) into Equation (6), and it is obtained as

$$\begin{array}{cc}{v^{\prime }}_i\hfill & =\left(1-Q_i\right)\cdot v_i+\hfill \\ & {\displaystyle \sum _{j=1}^4\frac{\frac{4}{\pi }\cdot \arctan \left(\begin{array}{l}{\left(\frac{M_j}{M_i+M_j}\right)}^{w_1}\cdot \\ {\left(\frac{\left|v_i-v_j\right|}{\max \left\{v_i,v_j\right\}}\right)}^{w_2}\end{array}\right)\mathrm{sgn}\left(v_j-v_i\right)\cdot \left|v_j-v_i\right|}{t_{\mathrm{i}act}}}\cdot {\lambda }^{*}\cdot t_{\mathrm{i}act}\hfill \end{array}$$

(8)

where, $i=1,2,3,4.$

3.2. Supplement and Description

The corollary and properties of Equation (8) are as follows.

Corollary 1.

When $M_i$ is much larger than $M_j$, $M_i$ is regarded as the maximum value and $M_j$ as the minimum value, thus, which is substituted into Equation (8) to obtain ${v^{\prime }}_i=\left(1-Q_i\right)\cdot v_i.$

Theorem 1.

When the number of the target population is considered larger than that of other groups, the speed of the group is not affected by other groups but is only related to its expected speed and its cognitive ability, emergency response-ability, value orientation and psychological bearing capacity.

Corollary 2.

When $M_j$ is much larger than $M_i$, $M_j$ is regarded as the maximum value and $M_i$ as the minimum value, thus $\frac{M_j}{M_i+M_j}=1$, which is substituted into Equation (8) to obtain the simplified Equation (9).

$$\begin{array}{cc}{v^{\prime }}_i\hfill & =\left(1-Q_i\right)\cdot v_i+\hfill \\ & {\displaystyle \sum _{j=1}^4\frac{\frac{4}{\pi }\cdot \arctan {\left(\frac{\left|v_i-v_j\right|}{\max \left\{v_i,v_j\right\}}\right)}^{w_2}\mathrm{sgn}\left(v_j-v_i\right)\cdot \left|v_j-v_i\right|}{t_{\mathrm{i}act}}}\cdot {\lambda }^{*}\cdot t_{\mathrm{i}act}\hfill \end{array}$$

(9)

where,$i=1,2,3,4.$

Theorem 2.

When the number of other groups is much larger than the number of target groups, the impact of the number of groups can be ignored.

Corollary 3.

When $v_i$ and $v_j$ differs greatly, there will be the following two situations.

(i) 

$v_j$is regarded as the maximum value and$v_i$as the minimum value, so$\frac{\left|v_i-v_j\right|}{\max \left\{v_i,v_j\right\}}=1$are substituted into Equation (8) to obtain simplified Equation (10).

$${v^{\prime }}_i=\left(1-Q_i\right)\cdot v_i+{\displaystyle \sum _{j=1}^4\frac{\frac{4}{\pi }\cdot \arctan {\left(\frac{M_j}{M_i+M_j}\right)}^{w_1}\mathrm{sgn}\left(v_j-v_i\right)\cdot v_i}{t_{\mathrm{i}act}}}\cdot {\lambda }^{*}\cdot t_{\mathrm{i}act},$$

(10)

where, $i=1,2,3,4.$

(ii) 

Similarly,$v_i$is regarded as the maximum value and$v_j$as the minimum value, so$\frac{\left|v_i-v_j\right|}{\max \left\{v_i,v_j\right\}}=1$can be substituted into Equation (8) to obtain simplified Equation (10).

Theorem 3.

When the speed of the target crowd differs considerably from that of another crowd, the evacuation speed of the target crowd is not affected by the speed of others.

Corollary 4.

When $v_i$ = $v_j$, then $\frac{\left|v_i-v_j\right|}{\max \left\{v_i,v_j\right\}}=\left|v_i-v_j\right|=0$, substitute the sub-data into Equation (8) to obtain ${v^{\prime }}_i=\left(1-Q_i\right)\cdot v_i$.

Theorem 4.

When the speed of the target group is the same as that of other groups, the speed of the group is not affected by other groups, but is only related to its own speed and the impact of cognitive ability, emergency response-ability, psychological bearing capacity and value orientation.

4. Simulation Example

In order to verify the effectiveness of the interactive speed correction method, a comparative simulation experiment is carried out in this study. Firstly, a questionnaire survey was conducted. Based on the results of the questionnaire survey, the expected speed of four different groups of people is revised by using the interactive speed modification method. Secondly, the single deck of a ro-ro passenger ship is selected as a simulation example. The deck is then modeled by Pathfinder evacuation software. Finally, this study sets up two evacuation plans, namely, ordinary evacuation and evacuation under the speed correction of interactive influence. Through the comparison of simulation results, it is concluded that the speed correction method of interactive influence proposed in this study is in line with reality.

4.1. Personnel Evacuation Speed Correction

The age, gender, cultural background, and other factors of the people on board will not only affect their judgment of the degree of fire risk, but also affect the evacuation speed. Therefore, this study combines the previous research results to design a questionnaire on ship fire evacuation behavior [30]. The questionnaire structure of personnel evacuation behavior in a ship fire situation is shown in

Table 1. Questionnaire structure of personnel evacuation behavior in a ship fire situation.

Variable	Problem Setting	Options
basic information of personnel	gender	men; women
	age	under 20 years old; 20–60 years old: over 60 years old
cognitive ability	education level	high school diploma and below; college degree; bachelor degree; Master degree and above
emergency capability	escape response when hearing a fire alarm	look around and judge for yourself;
		ask others to determine the direction of escape;
		escape immediately;
		observe the behavior of others
value orientation	will you escape with valuables	no; possibly; not sure, must
psychological endurance	the level of panic at hearing a fire alarm	no panic; low panic; moderate panic; extreme panic

. The questionnaire topic is mainly set to investigate the cognitive ability, emergency response ability, value orientation, and psychological endurance of different groups of people. Questionnaires were randomly distributed on an online questionnaire survey platform. The respondents were then divided into different age groups. A total of 129 questionnaires were distributed, and 105 questionnaires were finally effectively recovered. Finally, the reliability of the questionnaire was tested. The Cronbach reliability coefficient α of the questionnaire was 0.67. This indicates that the data reliability of the questionnaire is good and meets the requirements of usability. Among them, the proportion of adult men, adult women, the elderly, and children surveyed is 8:8:3:2. The information summary is shown in

Table 2. Summary of questionnaire information.

Influencing Factors		Adult Male (20–60 Years Old)	Adult Female (20–60 Years Old)	Elderly (Over 60 Years Old)	Children (Under 20 Years Old)
Cognitive ability	high school diploma or below	3%	18%	14%	0%
	college degree	54%	53%	43%	54%
	bachelor degree	33%	24%	29%	46%
	master degree and above	10%	5%	14%	0%
The emergency ability	look around and judge for yourself	29%	22%	43%	38%
	ask others to determine the direction of escape	37%	36%	14%	31%
	escape immediately	20%	31%	43%	23%
	observe other people’s behavior	14%	11%	0%	8%
The value orientation	no	58%	58%	68%	70%
	probably	15%	10%	8%	10%
	not sure	19%	22%	16%	10%
	must	8%	10%	8%	10%
Mental endurance	no panic	17%	22%	29%	8%
	low panic	32%	29%	43%	31%
	moderate panic	29%	29%	14%	38%
	extreme panic	12%	20%	14%	23%

.

Based on the questionnaire data, this study brings the above data into the speed correction model and obtains the expected correction speed of four groups. The specific model calculation steps are as follows.

Step 1. Bring ${\rho }_{ijk}$ into a fuzzy set based on the questionnaire data in Table 2. It can be obtained that the impact of a single attribute on a single population is recorded as $Q_{ij}$. Then, $Q_{ij}$ is integrated to calculate the impact of all attributes on a single population, which is recorded as $Q_i$. Among them, $M_1=40$, $M_2=40$, $M_3=15$, $M_4=10$. The effects of all attributes on adult men, adult women, the elderly, and children are

$$Q_1=0.246, Q_2=0.266, Q_3=0.275, Q_4=0.277$$

Step 2. In this study, the expected speeds of adult men, adult women, children and the elderly are set to be 1.5 m/s, 1.3 m/s, 1.1 m/s, and 0.9 m/s, respectively. Combined with the questionnaire data in Table 2 and substituted into Formula (6), the escape acceleration $a_i$ of the four groups is, respectively,

$$a_1=0.205 \mathrm{m}/{\mathrm{s}}^2, a_2=0.056 \mathrm{m}/{\mathrm{s}}^2, a_3=-0.076 \mathrm{m}/{\mathrm{s}}^2, a_4=-0.236 \mathrm{m}/{\mathrm{s}}^2.$$

Step 3. Substituting $v_i$, $Q_i$, $a_i$ and $t_{iact}$ into Formula (8), the correction speed of adult males, adult females, the elderly, and children can be obtained as

$$v_1=1.28 \mathrm{m}/\mathrm{s}, v_2=1 \mathrm{m}/\mathrm{s}, v_3=0.71 \mathrm{m}/\mathrm{s}, v_4=0.38\mathrm{m}/\mathrm{s}.$$

The above results are used as examples to simulate the modified expected speed of adult men, adult women, children, and the elderly.

4.2. Simulation Model Construction

This study takes a ro-ro passenger ship as the simulation object. The ship has a length of 196.27 m, a width of 28.60 m, a seating capacity of 1588 people, a passenger quota of 1500 people, and 10 decks. The seven and eight decks of the ship belong to the passenger activity area, and both decks have independent evacuation assembly areas, vertical single channel marine evacuation system, and fully enclosed lifeboats. In this study, the eight decks of the ship are selected as the simulation model for the evacuation of ship fire personnel.

In this study, Pathfinder software is used to model the above simulation examples. The deck model is 196 m long, 28.6 m wide and 3 m high. A total of 620 people need to be evacuated on the deck, and the proportion of adult men, adult women, children, and the elderly is 8:8:3:2, who are randomly distributed on the deck. According to the internal structure of the eighth deck, the evacuation routes and exits shall be set. Considering that the vertical single channel marine evacuation system and the fully enclosed lifeboat are difficult to set up in Pathfinder software, this paper sets the personnel evacuation deck as the form of successful personnel evacuation. The 3D model of personnel evacuation on the ship’s deck is shown in Figure 3. The green column represents adult men, the blue column represents adult women, the yellow column represents children, and the black column represents the elderly. The green line indicates the evacuation exit, that is, the evacuation from the green line indicates the successful evacuation.

4.3. Comparative Analysis of Evacuation Results

In order to verify the effectiveness of the correction method, this study sets up two evacuation plans, namely ordinary evacuation and speed correction method of interactive influence. At the same time, the velocity correction methods of ordinary evacuation and interactive influence are compared and analyzed experimentally. In addition, the specific details of the plan are as follows.

Plan 1: ordinary evacuation. The range of passenger evacuation speed is set to be 0.51~1.50 m/s, and remains unchanged. The evacuation path is uniform evacuation at the exits on both sides of the deck.

Plan 2: speed correction method of interactive influence. Based on the calculation results in Section 4.3, the expected evacuation speeds of adult men, adult women, the elderly, and children are set to 1.28 m/s, 1 m/s, 0.71 m/s, and 0.38 m/s, respectively. The evacuation speed obeys the normal distribution. The expected speed of each group is compared with the corrected speed (please see Figure 4 for details). The experimental results of the two evacuation plans are compared, as shown in Figure 5.

According to Figure 5 and

Table 3. Data comparison between two plans.

Plan	Evacuation Time (s)	Evacuation Efficiency (Rate of the Curve)	Remaining Staff
1	265.28	high	0
2	349.03	low	0

, the total evacuation time of ordinary evacuation is 265.28 s, and the overall evacuation efficiency is high. The evacuation time of the interactive speed correction method is 349.03 s, and the overall evacuation efficiency is low. With the development of time, the evacuation efficiency decreases significantly. After 325 s, the evacuation efficiency is close to 0. The evacuation efficiency (curve slope) of the speed correction method of interactive influence is significantly lower than that of ordinary evacuation, and the evacuation efficiency decreases significantly with the development of time. This is due to the consideration of the negative psychology of personnel in the fire and the influence of fire smoke. As time goes on, smoke concentration and temperature will gradually increase, causing harm to the human body and resulting in reduced evacuation efficiency. After 325 s, the evacuation efficiency of the interactive speed correction method is close to 0. This is because the concentration and temperature of fire smoke are enough to threaten the lives of people at 325 s. However, considering the mutual assistance behavior of the group, people around will actively help those who are slow and unconscious, so that they can keep the same speed and continue to move.

To sum up, ordinary evacuation oversimplifies the evacuation behavior of people. It is assumed that people only evacuate evenly according to the evacuation path during the evacuation process, and the impact of fire smoke and group behavior on people is not considered, which is not in line with reality and there is a large error. The aim of the speed correction method of interactive influence is to study evacuation from the human point of view. Through the analysis of the complex psychology and group effect on evacuation behavior, the evacuation path and evacuation speed are modified and supplemented. This is more realistic.

5. Conclusions

This study adopts the HFA to research evacuation from the perspective of the people on board, analyzes the influence of cognitive ability, emergency response ability, value orientation, psychological tolerance and group effect on evacuation behavior of the people in the fire scene. Then the escape acceleration and escape speed are corrected and supplanted to make them more realistic. The innovation of this study is mainly reflected in the following three points.

First, this study considers the interaction among fire escape personnel. The main influencing factors among escaped groups are summarized as emergency ability, cognitive ability, psychological bearing ability and value orientation. The interactive effects of the four attributes of the four groups of people are introduced into the evacuation model, which makes the evacuation research more realistic.

Secondly, this study uses the hesitant fuzzy integration operator to integrate the four attributes of the four groups of people, and realizes the quantification of the interaction between the groups. Then, the acceleration formula and velocity correction modulus formula considering the interaction effect of different groups of people are extracted, and the influence from other types of people is introduced into the evacuation research.

Finally, this study collects data through questionnaires and calculates the revised speed for different populations. Then, simulation software is used to compare the revised speed plan with the uncorrected speed plan, it is concluded that the revised speed plan is more realistic and provides a reference for subsequent evacuation research.

The shortcoming of this study is that it fails to take into account the emotional contagion of panic among pedestrians, the influence of fire smoke toxicity and fire temperature as well as other factors. Further study will be carried out by combining the above influencing factors with simulation examples.