Based on ISM—NK Tunnel Fire Multi-Factor Coupling Evolution Game Research

Abstract

A tunnel is a complex network system with multiple risk factors interacting. At present, the cause analysis of tunnel fire accidents focuses on exploring risk sources and risk assessment, ignoring the interaction between risk factors. A single model has certain limitations. By proposing the concept of the multi-factor coupled evolutionary game of tunnel fire, integrating the natural killing model (NK) and the explanatory structure model (ISM), the evolutionary game of multi-factor coupling of tunnel fire is studied from the perspective of micro and macro analysis, qualitative and quantitative research, the coupling relationship and effect between risk factors are discussed, 100 tunnel fire accidents and 158 tunnel fire literature at home and abroad are analyzed, and 40 typical tunnel fire risk factors and 31 coupling types of fire cause factors are extracted. Using the combined ISM-NK model, a seven-level network model of tunnel fire accident risk coupling is constructed, and the degree of coupling of various types of risk factors is evaluated. The hierarchical network cascade model revealed that 4 of the 40 typical tunnel fire risk factors were the underlying risk factors, 23 shallow layers were the risk factors and direct influencing factors, and 13 were the middle-risk factors and indirect influencing factors. The NK model shows that with the increase of coupling nodes, the frequency of tunnel fire accidents also shows an upward trend, and the subjective risk factor coupled with tunnel fires have a higher frequency than the objective risk factors.

1. Introduction

With the growth of the world’s population, the rapid development of the economy, the strong demand for transportation, and the number of means of transportation continue to grow at a high speed. Therefore, in order to alleviate traffic pressure, a large number of tunnel traffic facilities need to be built around the world. While tunnel transportation facilities provide convenience for human life, safety issues also come with them. Fires, as a common accident in tunnels, continue to grow as the number of tunnels increases [1], while also causing high economic losses and adverse social impacts. A tunnel traffic system is a complex system with dynamic and fuzzy nature, involving personnel, machinery and equipment, tunnel environment, operation management and geographical influencing factors, these non-linear coupling factors lead to the complexity of tunnel fires and the difficulty related to prevention and control.

The frequent occurrence of tunnel fire accidents illustrates the urgent need to explore the correlation evolution mechanism between various risk factors in tunnel traffic systems. At present, relevant research at home and abroad focuses on the statistical analysis of tunnel fire cases [2], the setting of fire fighting facilities [3], flue gas prevention and control [4], etc., while ignoring the hierarchical, non-linear and coupled risk factors in the tunnel system. The tunnel traffic system is a three-dimensional network structure that changes with the flow of traffic, resulting in the evolution of tunnel fires being fuzzy, dynamic, and coupled [5]. When applying traditional analytical methods to analyze the tunnel fire mechanism, inevitably, the interaction of the coupling of risk factors will not be considered. At present, the research on risk coupling mainly includes navigation, transportation, mining, construction, aviation and so on, such as Zhang et al. [6] from the perspective of risk coupling of maritime pilot HOF (human-organizational factors), using human factors to analyze HFACS (classification system), constructing SD (system dynamics) coupling model of RCEs (risk factors), and analyzing the risk coupling volatility of offshore pilot HOF. Wang et al. [7] proposed a coupled driving risk assessment model, which performs coupling analysis on different traffic elements and quantitatively evaluates the driving risk of intelligent vehicles. Xue et al. [8] established a risk coupling model based on systems thinking to identify key coupling effect risk assessments of HSR (high-speed rail) projects that can identify major risk factors. Qiao et al. [9] discussed the definition, classification, coupling process and decoupling principle of multi-factor risk of underground accidents in coal mines, and proposed a multi-factor risk measurement model based on coupling theory to measure the size of the risk coupling effect. Liu et al. [10] are based on digital twins of a lifting safety risk management framework. A coupling model of digital twin lifting safety risk is established, which realizes the real-time perception and virtual and real interaction of multi-source information in the lifting process, and excavates the correlation rules and coupling relationships between risk factors. Jarrow et al. [11] and Zhou [12] analyzed the credit risk correlation of assets between multiple companies from the perspective of default probability or default correlation, and verified the existence of credit risk correlation from theoretical and empirical aspects. Shyur [13] and Liu Tangqing et al. [14] studied aviation accidents, explored their risk coupling laws and connotations, and constructed an aviation risk assessment model. In summary, the current research on risk coupling focuses on exploring risk sources and risk assessment methods. Due to the low incidence of tunnel fires, the difficulty of investigation and evidence collection, the lack of detailed records of large and small accidents, and the lack of sufficient sample size and statistical data, the research on the coupling of tunnel fire accident risks is rarely involved [15], the traditional analytical model is not considering the interdependencies between various risk factors are not comprehensive enough, and it is difficult for a single model to identify the dependence between the accident sequence and various risk factors and quantitatively evaluate the coupling relationship. For example, the ISM model is qualitatively analyzed from a macroscopic perspective, and the NK model is quantitatively analyzed from a microscopic perspective, so ISM microscopic analysis has limitations, and NK cannot build a hierarchy. Tunnel fire studies have not yet been carried out in combination with the advantages of the two models.

In view of this, this study uses the method of coupling the interpretive structural model (ISM) and the natural killing model (NK), the ISM-NK analysis model is an improvement and extension of the traditional analysis method, and the multi-factor qualitative model of tunnel fire is established from the macroscopic perspective and the coupling degree between the risk factors is quantitatively calculated from the microscopic perspective. Focus on analyzing how risk factors couple to lead to the evolution of tunnel fires. The concept of the evolutionary game [16] is introduced, and the coupling relationship and effect between 40 fire risk factors are analyzed from the perspective of the interaction of personnel, machinery and equipment, operation management, tunnel environment and regional impact, and the formation process is explained, and the coupling mechanism of tunnel risk is revealed [17]. And explain the tunnel fire formation process.

2. Materials and Methods

2.1. Tunnel Fire Risk Factor Identification and Analysis

2.1.1. Tunnel Fire Case Collection and Analysis

The PerRanent InternationaR Association of Road Congress (PIARC) believes that the statistics of tunnel fires require a statistical cycle of 5–10 years. Based on the concept of big data, this study collects tunnel fire accident cases worldwide through news report websites, relevant websites of the Ministry of Transportation and the Ministry of Emergency Management, safety management and other websites. Collect and sort out the investigation report of 100 domestic and foreign tunnel fire accidents from 1949 to 2021 (Appendix A). Considering the impact of geography on tunnel fires, statistical analysis was made on the number and proportion of accidents in various provinces and European and American countries (Figure 1a).

The comparative analysis Figure 1a,b found that 100 tunnel accidents with relevant reports and data records occurred in 22 provinces, autonomous regions and municipalities directly under the central government in Europe and the United States, other countries and countries, of which the top four occurrences were east China, foreign Europe and the United States and central China, and finally the southwest region. The analysis shows that the tunnel fire incident is mainly concentrated in the mountainous area and the economically more developed area, so the impact of the region on the tunnel fire is considered.

2.1.2. Tunnel Fire Literature Refinement and Analysis

Consult related literature publications, enter tunnel fire keywords in the web of science library, and get 158 tunnel fire literature. Combined with the systematic analysis method and questionnaire survey method, the bibliometric cause keywords (Figure 1c) and the extended keywords (Figure 1d) were extracted, and the inter-state relationship analysis was carried out (Figure 1e). Define what is involved Causes and stages of tunnel fire accidents [18], etc.

Figure 1c,d visualize the frequency of tunnel fire keywords and the research hotspots and development trends of scientific research in the field of tunnel fire in recent years. Figure 1e shows the interconnection between tunnel fire research institutions in various countries. On the basis of the previous research, 48 tunnel fire accident risk factors were summarized from the five aspects of “personnel, machinery and equipment, operation management, tunnel environment, and regional impact”. The tunnel fire risk factors were interviewed by the expert group industry, and the tunnel managers were repaired in combination with the opinions of the tunnel managers (

Table 1. Visiting members of the Expert Group.

Expert Type	Workplace	Professional Titles	Access Time	Access Mode	Interview Length
Academy specialist A	University of Science and Technology of China	professor	June 2021	Online (Email)	10 min
Academy specialist B	Wuhan University of Technology	professor	June 2021	Online (phone)	20 min
Academy specialist C	Beijing Institute Of Technology	professor	June 2021	Online (Video)	15 min
Tunnel managers D	China Railway Fourth Bureau Group Co., Ltd.	Director of Safety	August 2021	Online (Email)	10 min
Tunnel managers E	China Communications First Bureau Construction	Minister of Security	August 2021	Offline (on-site)	30 min
Tunnel managers F	China Construction Fifth Bureau	Security officer	August 2021	Offline (on-site)	20 min
Fire Engineer G	Fire and Rescue Bureau	bureau secretaries	September 2021	Online (Email)	10 min
Fire Engineer H	Fire and Rescue Bureau	director	September 2021	Offline (on-site)	10 min
Fire Engineer I	Fire and Rescue Bureau	clerk	September 2021	Offline (on-site)	40 min
Fire specialist J	Shengjing Fire Detachment	team leader	September 2021	Offline (on-site)	50 min

), which was convenient to eliminate the biases of experts with different positions, and finally obtained 40 relatively objective tunnel fire risk factors (Figure 1f).

2.2. Overview and Coupling of Explanatory Structural Models and Natural Kill Models

2.2.1. Overview of the Structural Model and the Natural Killing Model

The Explanatory Structure Model (ISM) was first proposed by American professor WarfieRd [19] in 1973, decomposing complex systems into several elements, sorting out vague and chaotic ideas, and finally building a multi-level hierarchical model structure. The Natural Killing Model (NK), proposed by KauffRan [20] on the basis of Wright [21] FitnessRandscape Theory published in 1932 to study the evolution of biological genes, is a structural simulation research method to analyze the evolution of risk factor-related effects in complex systems.

2.2.2. Interpret the Coupling of Structural Models and Natural Killing Models

Combining the ISM model and the NK model (i.e., the ISM-NK model), both models that analyze the interrelationships and interactions between various factors in a complex network [22], ISM qualitatively analyzes and explains the multi-level hierarchical structure of tunnel fire from a macroscopic perspective, and NK quantitatively analyzes the relevant effects of internal risk factors from a microscopic perspective. Establish a hierarchical network model framework for comprehensive analysis of tunnel fire risk coupling (Figure 2).

2.3. Construction of ISM Multi-Factor Step-by-Step Model for Tunnel Fire

2.3.1. Risk Factor Adjacency Matrix Construction

The relationships of the 40 fire risk factors (Figure 1f) of tunnel fire are intricate, and the multi-factor cascade model of tunnel fire ISM is constructed to analyze the structure and correlation between fire factors more clearly and intuitively. Industry expert group and tunnel manager visits to the impact relationships between 40 tunnel fire risk factors online and offline (

Table 2. Adjacency matrix F.

	S1	S2	S3	S4	S5	S6	S7	S8	—	S40
S1	0	0	0	0	0	0	0	1	—	0
S2	1	0	1	1	1	0	0	0	—	0
S3	0	0	0	1	0	0	0	0	—	0
S4	0	0	0	0	1	0	0	0	—	0
S5	0	1	0	0	0	0	0	0	—	0
S6	0	0	0	0	0	0	0	0	—	0
S7	1	0	0	0	0	0	0	0	—	0
S8	0	0	0	0	0	0	0	0	—	0
—	—	—	—	—	—	—	—	—	—	—
S40	0	0	0	0	0	0	0	0	—	0

). Relevant decision-makers have rich academic or working experience, analyze the correlation degree of 40 risk factors, combine the opinions of decision-makers and the criteria for judging the degree of association (1), and finally establish the risk factor adjacency matrix F according to the correlation between risk factors.

$${\mathrm{F}}_{\mathrm{i}\mathrm{j}}=\left\{\begin{array}{l}1\hspace{1em}When Fi has a direct effect on Fj\\ 0\hspace{1em}When Fi has no or no direct effect on Fj\end{array}\right.$$

(1)

The data in row i of the Fi adjacency matrix F, such as F1 is 0; Fj is the data in column j of the adjacency matrix F, F1 is 0.

2.3.2. Risk Factor Reachability Matrix Construction

The calculation method that describes the degree of reach between the features in the directed connection diagram through a certain length path based on the matrix form is called the risk factor reachable moment G [23]. The risk factor reachability matrix has an important evolutionary law property: if Si passes through a channel of unit length directly to Sk, and Sk passes through a channel of unit length directly to Sj, it means that Si must be able to Sj through a channel of two unit lengths. According to the law of progression, the adjacency matrix A of the risk factors is added to the unit matrix I, combined with the Boolean algebraic algorithm (2), and obtained by certain matrix calculus.

$$G={(F+I)}^{\mathrm{q}+1}={(F+I)}^{\mathrm{q}}\ne \cdots \ne {(F+I)}^2\ne {(F+I)}^1,\mathrm{q}=1,2,3\cdots$$

(2)

It describes the accessibility of pathways with a longitude length equal to or less than R-1 between the factors. For graphs with node factors, the longest path will certainly not exceed R-1. Where R is the order of matrix F, the tunnel fire impact factor is the total number of factors that determine whether the fire occurs or not. The reachability matrix G (

Table 3. Reachability matrix G.

	S1	S2	S3	S4	S5	S6	S7	S8	—	S40
S1	1	1	1	1	1	1	0	1	—	0
S2	1	1	1	1	1	1	0	1	—	0
S3	1	1	1	1	1	1	0	1	—	0
S4	1	1	1	1	1	1	0	1	—	0
S5	1	1	1	1	1	1	0	1	—	0
S6	1	1	1	1	1	1	0	1	—	0
S7	1	1	1	1	1	1	1	1	—	0
S8	1	1	1	1	1	1	0	1	—	0
—	—	—	—	—	—	—	—	—	—	—
S40	1	1	1	1	1	1	0	1	—	1

) describes the degree to which a certain length of the pathway can be reached between the node factors of the directed connection graph.

2.3.3. Determine the Hierarchy of Risk Factors

On the basis of the risk factor reachability matrix, the risk factors are divided into layers, mainly using three sets X(Fi), Y(Fi), X(Fi) ∩ Y(Fi). Reachability set X(Fi): In the row where the factor Fi is located in the reachability matrix, the reachability matrix column element is the set of 1, representing the set of all the factors that can be reached from the factor Fi. Antecedent set X(Fi): In the column where the factor Fi is located in the reachability matrix, the row elements of the reachable matrix are the set of 1 s, representing the set of all the factors that can be reached from the factor Fi. X(Fi) ∩ Y(Fi), a set where the reachable set and the anterior set intersect. According to the X(Fi) ∩ Y(Fi) = X(Fi) conditions, the hierarchical division of risk factors is determined. In the process of hierarchical division, the risk factors of the level have been determined, the relevant rows and columns in the reachability matrix are deleted, the new reachability matrix is obtained, and the risk factors of the next level are extracted from the new matrix, and so on, until all the risk factors are completed.

According to X(Fi) ∩ Y(Fi) = X(Fi), such as S1 corresponding line X(Fi) = {1,2,8,9,10,24,28}, Corresponding columns Y(Fi) = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,35,36,37,38,39,40}, X(Fi) ∩ Y(Fi) = {1,2,8,9,10,24,28} = X(Fi), thus inferring from S1 to S40.The first condition to be satisfied is I = 1, 2, 8, 9, 10, 24, 28 which means that S1, S2, S8, S9, S10, S24, S28 is the topmost layer of the system, but also the most superficial causative factor of the tunnel fire accident, and then the elements about 1, 2, 8, 9, 10, 24, 28 are removed, and I = 3,4,5,6,25,26,27 meets the requirements, which means that S3, S4, S5, S6, S25, S26, S27 are the reasons for the second layer of the system. And so on to determine the hierarchy of risk factors.

2.4. N-K Model Multivariate Coupling Evolution Game of Tunnel Fire

2.4.1. Tunnel Fire Risk Factor Coupled Evolution Game

The phenomenon in which two or more forms of motion or systems interact with each other through various interactions is called coupling [24]. Tunnel fire cause system is a multi-factor coupling complex system, whether the failure of mechanical equipment can form a tunnel fire and the size of the formation of tunnel fire risk, not only by the geographical environmental factors, but also by the personnel factors, tunnel environment and operation management factors, and there is also a mutual role and influence between the five factors, that is, the multi-factor coupling effect leads to the generation and development of tunnel fire. Coupling deduction and inference are carried out on the cause factors of tunnel fire, identification and coupling analysis of fire cause factors are carried out, the frequency of causes obtained by the ISM model is used to calculate the coupling degree of the node factors of fire causes, the influence of five factors of causes on tunnel fire is analyzed, and the deep mechanism of multi-factor coupling tunnel fire occurrence is deductively reasoned, which is convenient for grasping the key points of prevention and control in the tunnel fire cause system.

Fire risk factor analysis: In tunnel fire, the ignition source and the number of combustibles are necessary factors to cause fires, and the increase in the number of combustibles and the acceleration of oxygen diffusion rate will lead to the spread and further expansion of the fire in space. Personnel will increase the risk of fire in urban tunnels due to insufficient fire fighting ability or insufficient safety quality. Combustible and flammable materials in tunnel buildings (lighting tools, high-power ventilation equipment, electrical wiring), combustibles in tunnels, and gasoline leaking in vehicles, all act as combustion mediums, added to the intensity and range of combustion, so the fire load of combustibles loaded on vehicles, flammable and explosive materials control, tunnel refractory materials, are all-powerful factors affecting the expansion of the fire. Wind currents and high temperatures in the environment fuel the fire, and the wind speed expands the fire by affecting the spread and flow of fire fumes. Analysis of the coupling mechanism of system causative factors: In tunnel fire systems, it is often difficult for a single factor to cause fire accidents. This is because the tunnel system itself has a “threshold”, and the univariate system is extremely difficult to break the ring system and cause a fire. When the five subsystems of the personnel factor node, the mechanical equipment factor node, the tunnel environmental factor node, the operation management factor node, and the regional influencing factor node are coupled and oscillated, breaking the critical point of the equilibrium state, reaching the threshold that the system can accommodate leads to the coupling effect, causing the generation and expansion of fire accidents, the fire coupling mechanism is shown in Figure 3.

The likelihood of vehicle accidents in tunnel fire systems will vary due to population density, city size and market trade in geographical influencing factors; the emergency response ability of escape personnel will be improved by the implementation of safety management responsibility systems, safety education and publicity, and evacuation drills; and the human error of vehicle drivers will be affected by the number of tunnel lanes in the tunnel environment factors and the possibility of tunnel traffic congestion. The length and width of the tunnel in the tunnel environment will be affected by economic indicators, while the increase or decrease of traffic control will also affect the increase or decrease of the probability of vehicle accidents, and the terrain conditions will affect the size of the number of tunnel lanes.

The likelihood of vehicle accidents in tunnel fire systems will vary due to population density, city size and market trade in geographical influencing factors; the emergency response ability of escape personnel will be improved by the implementation of safety management responsibility systems, safety education and publicity, and evacuation drills; and the human error of vehicle drivers will be affected by the number of tunnel lanes in the tunnel environment factors and the possibility of tunnel traffic congestion. The length and width of the tunnel in the tunnel environment will be affected by economic indicators, while the increase or decrease of traffic control will also affect the increase or decrease of the probability of vehicle accidents, and the terrain conditions will affect the size of the number of tunnel lanes. Therefore, the five major subsystem factors in tunnel fires are coupled with each other and act together, resulting in the generation and expansion of tunnel fires [23,24].

2.4.2. Improved Multi-Factor Coupled Calculation of N-K Mode

There are 2 parameters in the N-K model: N represents the number of nodes that make up the system, and if there are N nodes in the system, and there are n branches in each node, there are a total of nN kinds of possible combinations [25,26]. K represents the number of coupling factors related to each other in the system, with the maximum value of K being N-1 and the minimum value of 0. Coupling is divided into three categories according to the actual situation of tunnel fire.

(1)

One-factor coupling. Refers to the two-by-two correlation of branch factors under a single node factor, which leads to accidents. There are five categories of univariate coupling risks, namely personnel factor node A₁, mechanical equipment factor node A₂, tunnel environmental factor node A₃, operation management factor node A₄ and geographical influencing factor node A₅, which are recorded as T₁₁(A₁), T₁₂(A₂), T₁₃(A₃), T₁₄(A₄) and T₁₅(A₅).

(2)

Dual factor coupling. Refers to the coupling association of two branch factors under any two node factors to cause an accident. There are 10 types of dual-factor couplingrisks, namely personnel-machinery, personnel-environment, personnel-management, personnel-region, machinery-environment, machinery-management, machinery-region, environment-management, environment-region, management-region. T₂₁(A₁,A₂), T₂₂(A₁,A₃), T₂₃(A₁,A₄), T₂₄(A₁,A₅), T₂₅(A₂,A₃), T₂₆(A₂,A₄), T₂₇(A₂,A₅), T₂₈(A₃,A₄), T₂₉(A₃,A₅), T₂₁₀(A₄,A₅).

(3)

Multi-factor coupling. Refers to the occurrence of an accident in which branch factors under three or more node factors are coupled with each other. As shown in Figure 4, there are 16 types of multi-factor coupling risks, namely personnel-machinery-environment, personnel-machinery-management, personnel-machinery-region, personnel-environment-management, personnel-environment-region, personnel-environment-management-region, machinery-environment-management, machinery-management-region, machinery-management-region, environment-management-region, personnel-mechanical-environment-environment-management, personnel-mechanical-environment-region, personnel-mechanical-management-region, personnel-mechanical-management-region, Personnel-Machinery-Environment-Management-Geography. The three-factor coupling risk is T₃₁(A₁,A₂,A₃), T₃₂(A₁,A₂,A₄), T₃₃(A₁,A₂,A₅), T₃₄(A₁,A₃,A₄), T₃₅(A₁,A₃,A₅), T₃₆(A₁,A₄,A₅), T₃₇(A₂,A₃,A₄), T₃₈(A₂,A₃,A₅), T₃₉(A₂,A₄,A₅), T₃₁₀(A₃,A₄,A₅). The four-factor coupling risk is T₄₁(A,A₂,A₃,A₄), T₄₂(A₁,A₂,A₃,A₅), T₄₃(A₁,A₂,A₄,A₅), T₄₄(A₁,A₃,A₄,A₅), T₄₅(A₂,A₃,A₄,A₅), The five-factor coupling risk is recorded as T₅₁(A₁,A₂,A₃,A₄,A₅).

The five node factors of tunnel fire are coupled to each other, and the increase in coupling frequency is related to the accumulation of coupling times, and the larger the coupling value, the higher the causal frequency. The node factor interaction coupling information is calculated as follows:

$$\left\{\begin{array}{cc}\hfill & T_{\left(A_1,A_2\right)}=\underset{\mathrm{v}=1}{\sum ^{\mathrm{V}}}\underset{w=1}{\sum ^W}{P}_{vw}{log}_2\left\{\frac{P_{vw}}{P_{\mathrm{v}\dots .}·P_{.w\cdots }}\right\}2\hfill \\ \hfill & T_{\left(A_1,A_2,A_3\right)}=\underset{\mathrm{v}=1}{\sum ^{\mathrm{V}}}\underset{w=1}{\sum ^W}\underset{x=1}{\sum ^X}{P}_{vwx}{log}_2\left\{\frac{P_{vwx}}{P_{\mathrm{v}\dots .}·P_{.w\cdots }·P_{..x.}}\right\}3\hfill \\ \hfill & T_{\left(A_1,A_2,A_3,A_4\right)}=\underset{\mathrm{v}=1}{\sum ^{\mathrm{V}}}\underset{w=1}{\sum ^W}\underset{x=1}{\sum ^X}\underset{y=1}{\sum ^Y}{P}_{vwxy}{log}_2\left\{\frac{P_{vwxy}}{P_{\mathrm{v}\dots .}·P_{.w\cdots }·P_{..x.}·P_{...y.}}\right\}4\hfill \\ \hfill & T_{\left(A_1,A_2,A_3,A_4,A_5\right)}=\underset{\mathrm{v}=1}{\sum ^{\mathrm{V}}}\underset{w=1}{\sum ^W}\underset{x=1}{\sum ^X}\underset{y=1}{\sum ^Y}\underset{z=1}{\sum ^Z}{P}_{vwxyz}{log}_2\left\{\frac{P_{vwxyz}}{P_{\mathrm{v}\dots .}·P_{.w\cdots }·P_{..x.}·P_{...y.}·P_{\dots {.}_z}}\right\}5\hfill \end{array}\right.$$

(3)

where v = 0, 1, …, V; w = 0, 1, …, W; x = 0, 1, …, X; y = 0, 1, …, Y; z = 0, 1, …, Z, representing the state of the five factors of personnel-equipment-environment-management-region, respectively; Pvwxyz represents the probability of coupling between the five factors of personnel-machinery-environment-management-region in the v, w, x, y, and z states, respectively; T represents the causal frequency of node coupling, and the solution of the T value also requires the frequency of five node couplings.

$$\left\{\begin{array}{l}{C}_{\mathrm{k}}^2,Single node factor coupling\hfill \\ {\displaystyle \prod _{\mathrm{j}=1}^{\mathrm{J}}{C}_{\mathrm{k}}^1,Multi-node factor coupling}\hfill \end{array}\right.$$

(4)

where j = 1, 2, …, J, representing the node layer factors contained in the coupling combination, that is, the coupling factors; k = 1, 2, …, K, representing the branch risk factor in the node layer factors contained in the coupling combination.

According to the frequency of the node factor of the tunnel fire network, the frequency of 32 coupling conditions is solved. In the deduction of tunnel fire events, each node factor has two states, occurrence (Xi = 1) and non-occurrence (Xi = 0), then the 32 cases of tunnel fire node factor coupling are divided into different combined probabilities (Figure 4), such as single-node coupling is divided into two cases of node participation in coupling and node participation in coupling (icon P02), recorded as P0 and P1; Two-node coupling is divided into two nodes are not involved in coupling, one of the nodes participate in coupling, two nodes are involved in coupling 4 cases (icon P004), denoted as P00, P01, P10, P11; and so on, there are 8 cases of three-node coupling (icon P0008), recorded as P000, P001, P010, P011, P100, P101, P110, P111; four-node coupling has 16 cases (icon P000016), denoted as P0000, P0001, P0010, P0011, P0100, P0101, P0110, P0111, P1000, P1001, P1010, P1011, P1100, P1101, P1110, P1111; There are 32 cases of five-node coupling (icon P0000032), recorded as P00000, P00001, P00010, P00011, P00100, P00101, etc. Solve for the number of occurrences of various situations to obtain the frequency of coupling combinations.

3. Results

3.1. Building the ISM Diagram

According to the above calculation and analysis, the first level node of the cause factor of the tunnel fire accident is obtained: R1 = {1, 2, 8, 9, 10, 24, 28}, the second level node: R2 = {3, 4, 5, 6, 25, 26, 27}, the third level node: R3 = {11, 12, 13, 14, 15, 16, 17, 18}, the fourth level node: R4 = {19, 20, 21, 22, 23, 29, 32, 33, 33, 36, 37}, the fifth level node: R4 = {7,35}, level 6 node: R4 = {38,39}, level 7 node: R4 = {40,31}. R1, R2, R3, R4, R5, R6, R7 are placed in each layer, and the related factors are connected according to the logical relationship of the elements in each layer, and the structural model of the cause of tunnel fire accident is constructed.

The risk factors of the R1, R2 and R3 layers are mainly from the factors of personnel and mechanical equipment, and are also the direct cause factors of the surface of tunnel fire accidents. These 23 risk factors are shallow risk factors and direct influencing factors, which can directly cause tunnel fires, of which vehicle failure, vehicle driver human error and vehicle accidents are all high-frequency causes of tunnel fire accidents. R4 and R5 are tunnel environmental and operational management factors, which are indirect causes of tunnel fires, and these 13 risk factors are affected by changes in deeper and lower causative factors, while risking and influencing shallow risk factors. R6 and R7 are geographical influencing factors, the underlying risk factors for tunnel fire accidents, and the power system factors with higher frequency.

3.2. N-K Model Calculation Result

Combined with Figure 1f and Appendix A, the frequency of 40 risk factors in 100 tunnel fire accidents is counted, such as X₁ (vehicle driver error) appears 25 times, then P_X1 = 25/100 = 0.2500, From this analogy from X₁ to X₄₀, the frequency of 40 risk factors leading to tunnel fires is obtained Figure 1f.

For example, in the case of single-node coupling of mechanical equipment, that is, the frequency of ‘01000’: P₀₁₀₀₀ = X₁ × X₂ + X₁ × X₃ + X₁ × X₄ + X₂ × X₃ + X₂ × X₄ + X₃ × X₄ = X₁(X₂ + X₃ + X₄) + X₂(X₃ + X₄) + X₃ × X₄ = 0.0877.

In single node factor coupling analysis, under different conditions, the frequency of single node coupling is different, known by the Formula (3), it is necessary to solve the value of Pv × Pw × Px × Py × Pz. Frequency P0 without the participation of personnel nodes in coupling… = P_0···· = P₀₀₀₀₀ + P₀₁₀₀₀ + P₀₀₁₀₀ + P₀₀₀₁₀ + P₀₁₁₀₀ + P₀₀₁₁₀ + P₀₁₁₁₀ + P₀₁₁₁₁ = 0.8497, the frequency of coupling with personnel nodes P_1···· = 1 − 0.8497 = 0.1503. By analogy, the probability of mechanical, environmental, administrative, and regional nodes participating in the coupling conditions can be calculated. The detailed calculation results are shown in Figure 1f.

Human-mechanical two-node coupling, i.e., frequency of ‘11000’: P₁₁₀₀₀ = X₁ × X₅ + X₁ × X₆ + X₁ × X₇ + X₁ × X₈ + … + X₄ × X₅ + X₄ × X₆ + X₄ × X₇ + X₄ × X₈ = (X₁ + X₂ + X₃ + X₄)(X₅ + X₆ + X₇ + X₈) = 0.1149.

Dual-node coupling analysis. There are 10 cases of double node coupling, there are 4 different combinations in each double node coupling, and the values of Pvw, Pvx, Pvy, Pvz, Pwx, Pwy, Pwz, Pxz, Pyz need to be calculated first. For example, the frequency of the human node and the mechanical node is completely unaffected under coupling conditions: P_00··· = P₀₀₀₀₀ + P₀₀₁₀₀ + P₀₀₀₁₀ + P₀₀₀₁₀ + P₀₀₀₀₁ + P₀₀₁₁₀ + P₀₀₀₁₁… + P₀₀₁₁₁ = 0.1889. The detailed calculation results are shown in Figure 1f (3,4,5,6).

Human-mechanical-environmental three-node coupling, i.e., the frequency of ‘11000’: P₁₁₁₀₀ = X₁ × X₅ × X₉ + X₁ × X₅ × X₁₀ + X₁ × X₅ × X₁₁ + X₁ × X₅ × X₁₂ + … + X₄ × X₈ × X₉ + X₄ × X₈ × X₁₀ + X₄ × X₈ × X₁₁ + X₄ × X₈ × X₁₂ = (X₁ + X₂ + X₃ + X₄)(X₅ + X₆ + X₇ + X₈)(X₉ + X₁₀ + X₁₁ + X₁₂) = 0.2623. The same principle can be computed to solve the frequency of other coupling cases.

Multi-node coupled analysis. In the 16 cases of multi-node coupling, it is necessary to first solve the values of the five cases of three-node coupling Pvwx, Pvwy, Pvwz, Pvxy, Pvxz, Pvyz, Pwxy, Pwxz, Pwyz, Pxyz the values of the five cases of four-node coupling Pvxyz, Pvwyz, Pvwyz, Pvwxy, Pwxyz, Pvwxyz case. The detailed calculation results are shown in Figure 1f (7,8,9).

4. Comparative Analysis and Discussion of Model Results

According to the node factor causation frequency in Figure 1f (2,3,4,5,6,7,8,9) obtained, the substitution Formula (3) calculates the coupling causal risk values of double nodes and multiple nodes: T₁₁(A₁) = 0.0585, T₁₂(A₂) = 0.0653, T₁₃(A₃) = 0.0517, T₁₄(A₄) = 0.0449, T₁₅(A₅) = 0.0381 etc., and all node coupling causation risk values are analyzed together for comparison 4-4 (10).

From the calculation results and Figure 1f (10), it can be seen that the various node coupling cause risk values are sorted: T51 > T41 > T42 > T43 > T44 > T45 > T31 > T32 > T34 > T37 > T33 > T38 > T35 > T39 > T310 > T21 > T25 > T22 > T26 > T23 > T28 > T24 > T27 > T29 > T210 > T12 > T11 > T13 > T14 > T15.

Figure 1f (10) and T value ranking show that the comprehensive coupling of five-node factors has the highest probability of tunnel fire, the risk of coupling of four-node factors is greater than the risk of coupling of three-node factors, the risk of coupling of three-node factors is greater than the risk of coupling of two-node factors, and the probability of tunnel fire occurring from single-node factor coupling is the lowest, which is in line with objective laws. Among the four-node factor risk coupling, the human-mechanical-environmental-management node coupling tunnel fire accident probability is the largest, and the mechanical-environmental-management-regional node coupling tunnel fire accident probability is the smallest. From Figure 1f (1), it can be seen that the frequency of human error and vehicle accidents caused by vehicle drivers is high, and the frequency of risk factors under personnel and mechanical equipment factors is high, while the frequency of risk factors under operation management and geographical influencing factors is generally low.

In this paper, subjective factors refer to the subjective initiative of the driver in the driving process, such as driver human error and operation management factors; objective factors refer to the material objects that exist objectively during the driving process of the vehicle, such as mechanical equipment factors and geographical influencing factors. Incomplete coupling of personnel and operations management factors is less risky than full coupling. When the coupling factors are consistent, the mechanical equipment factor is more likely to cause a fire than the tunnel environment factor, and the personnel node is more likely to cause a tunnel fire than the operation management factor. In the double-section coupling analysis, the probability of cause of fire accidents in the coupling tunnel of personnel-mechanical equipment factors was the largest, and the probability of causes of fire accidents in the coupling tunnel of operation management-geographical influencing factors was the smallest, indicating that the coupling risk of objective factors was smaller than that of subjective factors.

From the tunnel fire accident cause ISM model (Figure 5), the tunnel fire accident cause system is a 7-level step complex structure involving surface R1, subsurface R2, final surface R3, intermediate layer R4, deep R5, deep R6 and low layer R7 accident causes, reflecting the logical relationship between accident risk factors.

In the actual accident statistics, the risk coupling frequency is equal to the ratio of the number of accidents caused by each coupling situation and the number of accidents of all accidents, the sum of the frequencies is 1, in order to facilitate analysis and comparison, the risk coupling frequency is normalized, and the frequency of each coupling situation is the sum of the frequencsies of all coupling conditions. The resulting coupling frequency and ISM accident cause analysis are compared with 100 accidents at home and abroad tunnel fire accident investigation reports and 158 tunnel fire literature for comparative analysis (Figure 6).

From Figure 6, it is found that the ISM accident cause analysis and N-K multi-factor coupling risk value have a high coincidence with the statistical analysis of tunnel fire accidents at home and abroad, and with the increase of coupling nodes, the probability of tunnel fire accidents also shows an upward trend. In summary, the model analysis results are consistent with the research conclusions of Scholars such as Hu Jiawei [17], Kang Xiaolong [2], Lai Jinxing [27] that traffic accidents and spontaneous combustion of vehicles are the high-frequency causes of tunnel fires, and the hierarchical network model coupled with tunnel fire risks is highly consistent with the accident tunnel fire accident investigation report and the tunnel fire literature analysis. The ISM-NK analysis model is an improvement and extension of the traditional analysis method, which does not consider the interdependencies between various risk factors, and it is difficult to identify the dependence between the accident sequence and various causal factors and quantitatively evaluate the coupling relationship. The analysis of the hierarchical and coupling relationship of tunnel fire risk factors is conducive to the more effective allocation of resources by traffic and tunnel management departments and the strengthening of tunnel fire prevention [28]. In this study, the ISM-NK coupling model was able to analyze the interaction between tunnel fire risk factors while helping management identify risk focal areas and complex coupling relationships in high-risk systems, such as coal, chemical, and construction industries.

5. Conclusions

A tunnel traffic system is a complex network system composed of a variety of interrelated risk factors, tunnel fire causes can not be attributed solely to human error or equipment failure, but the result of the coupling of multiple risk factors. The joint coupling of multiple factors such as tunnel form, scale, traffic flow composition, traffic volume, tunnel management level and location area affects the frequency of tunnel fires. Therefore, the concept of the multi-factor coupled evolutionary game of tunnel fire is of great significance for analyzing the interaction between fire cause factors in a tunnel fire. After analyzing 100 tunnel fire accidents at home and abroad and 40 risk factors in 158 tunnel fire literature, a hierarchical network model for comprehensive analysis of tunnel fire risk coupling is established by using the integrated ISM-NK coupling concept. The coupling effect between different risk factors is studied, the dynamic process of tunnel fire will continue to be studied, and the coupling model of wind and fire will be established and analyzed at different scales.

(1)

In total, 4 of the 40 fire cause factors (such as power system and economic indicators) are the underlying risk factors of tunnel fires, 23 of the 40 fire risk factors (such as vehicle failure, vehicle driver human error and vehicle accidents) are shallow risk factors and direct influencing factors of tunnel fires, and 13 risk factors (such as the number of tunnel lanes, vehicle types, safety education and publicity) are indirect causes of tunnel fires. Strengthen the inspection and evaluation of tunnels to eliminate hidden dangers; increase safety education for drivers and passengers to ensure driving safety and vehicle safety; traffic and tunnel management departments to strengthen the management of tunnel equipment (such as emergency lighting systems, real-time broadcast systems), standardize operating procedures, and improve the safety management system.

(2)

The frequency of tunnel fires occurring in the complete coupling of the five risk factors is the highest, the probability of tunnel fire occurrence shows an upward trend with the increase of risk factors, and the coupling risk of subjective factors (personnel and operation management factors) is higher than that of objective factors coupled (equipment factors and geographical influencing factors). This conclusion shows that the traffic and tunnel management departments have increased the control of tunnel vehicles and improved the response and handling mechanism for sudden emergencies in tunnels.

(3)

The application of ISM constructs a visual multi-factor cascade model between interrelated risk factors, the NK model reveals the degree of coupling of risk factors of different coupling types, and the coupling of the ISM-NK model is more suitable for describing the complex coupling interaction between tunnel fire risk factors. Provide a theoretical basis for safety managers and decision-makers to formulate corresponding fire prevention and control measures and policies.