Grey Fuzzy Comprehensive Evaluation of Bridge Risk during Periods of Operation Based on a Combination Weighting Method

Abstract

Bridge safety during operating periods is a primary concern worldwide, and the evaluation of bridge risks is a critical aspect of ensuring bridge safety. The most common methods used for bridge risk evaluations include fuzzy comprehensive evaluations, grey system theory, fault tree analysis, the Kent index method, and data envelopment analysis. However, these approaches are highly subjective and have uneven distributions when determining the weights of risk indicators. To improve the accuracy and feasibility of bridge risk evaluations for a given period of operation, we first establish bridge risk indicators and assign subjective weights to each indicator based on an analytic hierarchy process. Additionally, objective weights are assigned to each indicator according to an entropy weighting method. Then, the combined weights of each risk indicator are obtained by applying game theory principles. This enables the construction of a degree of membership matrix comprising these risk indicators, which is established according to an expert grading method and grey fuzzy theory. Finally, the evaluation results vector is calculated, allowing the risk level of a bridge to be assessed according to the principle of the maximum degree of membership. Overall, this study provides a more accurate and objective method for evaluating bridge risk during a given period of operation.

1. Introduction

With continuous social development, bridge construction is adapting to accommodate larger spans and more beautiful shapes. Owing to the wide availability of new materials and technologies, new challenges have emerged in terms of the safety of modern bridges throughout their operational lifetimes. If an accident occurs, it will trigger serious economic losses and lasting social impacts. However, with the increasing number of bridges, safety issues have become a more urgent concern in China and around the world. Therefore, it is crucial to conduct accurate and reasonable evaluations of a bridge’s risk during its period of operation.

Currently, the most common bridge risk evaluation strategies include fuzzy comprehensive evaluations, the Delphi method, fault tree analysis, and various others. In these conventional methods, the most critical step is determining the weights of risk indicators, which can be categorized as either subjective or objective. Based on experts’ subjective judgment of engineering risks and the uncertainty of gathered information, the analytic hierarchy process (AHP), Delphi method, and expert grading method have been introduced to determine the weights of risk indicators. Fang et al. [1] established a durability evaluation model for reinforced concrete arch bridges based on the degree of membership function by applying the fuzzy AHP to field test data. Tan et al. [2] proposed a grey-relational analysis decision-making method based on the weight of fuzzy AHP to optimize the selection of old bridge reinforcement schemes. This decision-making method makes the evaluation system more organized and systematic, and renders the index weights more operable and quantitative, thereby reducing the impact of subjective evaluations and enabling more objective and reliable evaluation results. Ji et al. [3] proposed a Delphi-improved fuzzy AHP method for risk evaluations of large complex bridges during construction. Chen et al. [4] established a model evaluating the risk of bridge collapse based on a combination of fault tree analysis theory and neural networks, and they applied their model to analyze a bridge project under construction. Mustesin et al. [5] proposed a framework for evaluating fire risks on bridges based on AHP. Li et al. [6] established a construction safety risk evaluation system for bridges and highways based on a combination of the cloud entropy weight method and cloud model theory. Cheng et al. [7] proposed a durability evaluation method based on fuzzy neural networks. Simulations of learning and test samples indicated that this method demonstrated good learning, prediction, and analytical capabilities. Sun et al. [8] proposed an innovative algorithm for bridge construction risk evaluations based on cluster analysis, set value statistics, and entropy methods, while considering multi-information fusion. Wu et al. [9] established a risk evaluation model for analyzing bridge construction based on five aspects (i.e., human, machine, material, methods, and environment). Li et al. [10] proposed a modular bridge risk identification method based on work breakdown structure–risk breakdown structure (WBS-RBS). However, these objective weighting methods rely on objective data and cannot reflect the importance of the expert in determining the weight, which may lead to inconsistencies relative to the actual situation. To address these challenges, fiber Bragg grating, interferometers, fully distributed sensors, and structural health monitoring technology based on deep learning have been gradually introduced into bridge monitoring [11,12,13,14].

The reports discussed above have contributed to numerous achievements related to risk evaluations of bridges during their periods of operation. Fuzzy comprehensive evaluations represent a particularly common method, which contains two major elements, i.e., the weights of each risk indicator and the membership relationship. However, the weight determination methods are generally too subjective or too dependent on objective data, and therefore they cannot adequately reflect the potential risk factors of the project itself, nor fully consider the correlations among risk factors [15]. For example, conventional weight determination only adopts a single method (either subjective or objective weighting), which results in uneven weight distribution. When evaluating the bridge risks, experts’ subjective decisions should be considered, in addition to the objective reality of the bridge. Therefore, the single weighting method is not sufficient. Moreover, distortion, jump, low resolution, strong subjectivity, and other phenomena influence the membership relationship among risk indicators when simply applying a fuzzy comprehensive evaluation. Considering the limitations of using a single evaluation method, it is necessary to combine grey system theory and fuzzy comprehensive evaluation methods to build a grey fuzzy comprehensive evaluation model of bridge risks. This enables the comprehensive consideration of the influence of various factors, which leads to a more accurate evaluation.

Therefore, this paper proposes a new weighting method comprising analytic hierarchy processes and the entropy weight method (AHP-EWM) to determine the combined weights of bridge risk indicators. Then, fuzzy comprehensive evaluation and grey theory are used to calculate the degree of membership matrix of bridge risk indicators. Finally, the combined weights are multiplied by the degree of membership matrix to obtain the risk level of a bridge during its period of operation.

2. Risk Indicators of a Bridge During its Period of Operation

The risk indicators of bridges during operational periods were established according to relevant literature and on-site investigations (

Table 1. Risk indicators of bridges in their period of operation.

Type	Specific Risk Indicators
Natural disaster	Wind-induced disaster
	Earthquake
	Geological hazard
	Flood
	Blizzard
Accident	Ship collision with bridge
	Fire
	Electric shock
	High falling objects
	Overload accident
	Man-made destruction
	Vehicle impact with bridge

).

The subjective weights, objective weights, and combined weights of the risk indicators in Table 1 are calculated below.

3. Methods of Weighting Bridge Risk Indicators in the Period of Operation

3.1. Analytic Hierarchy Process

AHP is a multi-criteria decision-making process that combines qualitative and quantitative techniques. This approach can be used to solve multi-objective decision-making problems with complex hierarchical structures, while also quantifying empirical assessments [16]. The basic principle is to divide the problems into interrelated structural levels (e.g., target level, criterion level, and indicator level). The upper levels dominate the lower levels, thus establishing the hierarchical relationship. This method is simple and practical; however, it cannot provide a new scheme for decision making, and it is not objective enough because the construction of the judgment matrix is easily affected by subjective factors. The steps for adopting an AHP are outlined herein.

(1) Establish the judgment matrix. The judgment matrix elements represent the levels of indicators and their relative degrees of importance. Judgment matrix A comprises a_ij, which represents the importance proportion scale of indicator a_i relative to indicator a_j. These indicators are compared using a scale from 1–9, as described in

Table 2. Relative importance scales.

Element	Assignment	Meaning
$a_{ij}$	1	Indicator i is as important as indicator j
	3	Indicator i is slightly more important than indicator j
	5	Indicator i is clearly more important than indicator j
	7	Indicator i is much more important than indicator j
	9	Indicator i is extremely important compared with indicator j
	2, 4, 6, 8	Intermediate values of the two adjacent judgments
	reciprocal	If the importance ratio of indicator i to indicator j is $a_{ij}$, the importance ratio of indicator j to indicator i is $a_{ij}=\frac{1}{a_{ji}}$

[17]. Then, the judgment matrix takes the form shown in Equation (1).

$$A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& \cdots & a_{nn}\end{array}\right]$$

(1)

where n is the number of indicators; a_ij > 0; a_ij = 1/a_ji; a_ii = 1.

(2) Calculate the weights of risk indicators. This work applied the arithmetic average method, with the following specific calculation steps.

Each column of matrix A is normalized according to Equation (2):

$$\overline{a_{ij}}=\frac{a_{ij}}{{\displaystyle \sum _{i=1}^n{a}_{ij}}}$$

(2)

Matrix $\overline{A}$ is added by rows, as expressed in Equation (3):

$$\overline{w_i}={\displaystyle \sum _{j=1}^n\overline{a_{ij}}},i=1,2,\cdots ,n$$

(3)

After summing, w_i is normalized, and the eigenvector W_i is obtained using Equation (4):

$$W_i=\frac{\overline{w_i}}{{\displaystyle \sum _{i=1}^n\overline{w_i}}},i=1,2,\cdots ,n$$

(4)

The calculated W_i = (w₁, w₂,…, w_n)^T is the eigenvector of the judgment matrix, which contains the weight of each risk indicator.

(3) Test the consistency of the judgment matrix. Expert grading is affected by subjective factors; thus, a consistency test can be used to investigate the degree or range of inconsistency in the judgment matrix. The consistency test of the judgment matrix is mainly based on three indicators: the consistency indicator (CI), the average random consistency indicator (RI), and the consistency ratio (CR).

The consistency indicator is calculated using Equation (5):

$$CI=\frac{{\lambda }_{\max }-n}{n-1}$$

(5)

where ${\lambda }_{\max }=\frac{1}{n}{\displaystyle \sum _{i=1}^n\frac{{\left(AW\right)}_i}{W_i}}$, such that λ_max is the largest eigenvector of the judgment matrix, and n is the order of the judgment matrix.

(4) Determine the corresponding random consistency indicators. For values of n = 1–9, the RI values are shown in

Table 3. Random consistency indicators.

n	1	2	3	4	5	6	7	8	9	10
RI	0	0	0.58	0.90	1.12	1.24	1.32	1.41	1.45	1.49

[17].

(5) Calculate the consistency ratio using Equation (6):

$$CR=\frac{CI}{RI}$$

(6)

When CR < 0.1, the consistency test result is acceptable; otherwise, the judgment matrix should be modified.

3.2. Entropy Weight Method

In traditional weighting methods, AHP and other subjective weighting strategies are often adopted, which can result in large deviations in the evaluation results due to subjective factors. The concept of entropy originates from the thermodynamic principles used to quantify the energy attenuation of a material system [18]. In information theory, entropy is a measure of the disorder of a system. The higher the degree of disorder in a system, the higher the entropy, and the higher the degree of order in the system, the lower the entropy. The steps to determine the objective weights via the entropy weight method are as follows.

(1) Determine the judgment matrix. Entropy is a measure of the disorder in the system. If the system can adopt various states, and the probability of each state is known, then the entropy of the system is defined as shown in Equation (7):

$$e=-{\displaystyle \sum _{i=1}^n{P}_i}\cdot \ln P_i$$

(7)

When P_i = 1/n (where i = 1, 2, …, n), the probabilities of various states are the same, and the maximum entropy is e_max = ln(n). For n risk factors and m evaluation indicators, the evaluation matrix is defined as B = (b_ij)_n×m (Equation (8)), where b_ij is the evaluated value of the indicator j by expert i.

$$B=\left[\begin{array}{cccc}{b}_{11}& b_{12}& \cdots & b_{1n}\\ b_{21}& b_{22}& \cdots & b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ b_{n1}& b_{n2}& \cdots & b_{nm}\end{array}\right]$$

(8)

(2) Standardize the quantified indicators.

Positive indicators are standardized according to Equation (9):

$$B_{ij}=\frac{b_{ij}-b_{\min }}{b_{\max }-b_{\min }}$$

(9)

and negative indicators are standardized according to Equation (10):

$$B_{ij}=\frac{b_{ij}-b_{\max }}{b_{\max }-b_{\min }}$$

(10)

where B_ij is the normalized data, b_ij is the original data, b_min is the minimum value of this indicator, and b_max is the maximum value of this indicator.

(3) Calculate the proportion of indicator i evaluated by expert j using Equation (11):

$$P_{ij}=b_{ij}/{\displaystyle \sum _{i=1}^n{b}_{ij}}$$

(11)

(4) Calculate the entropy $e_j$ evaluated by expert j using Equation (12). Here, we specify that when P_ij = 0, P_ijln(P_ij) = 0.

$$e_j=-\frac{1}{\ln n}{\displaystyle \sum _{i=1}^n{p}_{ij}}\cdot \ln p_{ij}$$

(12)

(5) Calculate the complementary value of the entropy, and obtain the entropy weight after normalization based on Equation (13):

$$w_j=\frac{1-e_j}{m-{\displaystyle \sum _{j=1}^m{e}_j}}$$

(13)

Here, $0\le w_j\le 1$, where $j=1,2,\cdots ,m$. Assuming that the initial weights of all indicators are equal, the entropy weight can be used to represent the weight of each indicator after adjustment.

3.3. Combination Weighting Methods

This report adopts game theory [19] to perform combination weighting. The principle of game theory is to seek Nash equilibrium [20] between subjective and objective weights and to seek agreement or compromise among different weighting methods. The goal is to minimize the deviation between the weights of the indicators and the weights obtained by different weighting methods, thereby improving the accuracy of the indicator weights. The weighting results are combined according to the following process: w₁, w₂, …, w_m represent m indicators’ weight vectors obtained based on different weighting methods. Any linear combination of them can be expressed as shown in Equation (14):

$$W={\displaystyle \sum _{i=1}^m{\alpha }_i}\cdot w_i^T$$

(14)

The core principles of combination weighting in game theory are to find the ideal weight vector and to minimize its deviation from each W_i. The resulting game model is expressed as shown in Equation (15):

$$\min {\Vert {\displaystyle \sum _{j=1}^m{\alpha }_j{w}_j^T-w_i^T}\Vert }_2$$

(15)

The matrix differential property is transformed into a linear system of equations under optimal first derivative conditions, as shown in Equation (16):

$$\left[\begin{array}{cccc}{w}_1{w}_1{}^T& w_1{w}_2^T& \cdots & w_1{w}_m^T\\ w_2{w}_1^T& w_2{w}_2^T& \cdots & w_2{w}_m^T\\ ⋮& ⋮& ⋮& ⋮\\ w_n{w}_1^T& w_n{w}_2^T& \cdots & w_n{w}_m^T\end{array}\right]\left[\begin{array}{c}{\alpha }_1\\ {\alpha }_2\\ ⋮\\ {\alpha }_m\end{array}\right]=\left[\begin{array}{c}{w}_1{w}_1^T\\ w_2{w}_2^T\\ ⋮\\ w_m{w}_m^T\end{array}\right]$$

(16)

Considering this formula and the combination weighting algorithm of game theory, Equation (17) can be constructed:

$${\alpha }_i^{\ast }=\frac{\left|{\alpha }_i\right|}{{\displaystyle \sum _{i=1}^m\left|{\alpha }_i\right|}}$$

(17)

Therefore, the combination weight of a given risk indicator is expressed in Equation (18):

$$W^{\ast }={\displaystyle \sum _{i=1}^m\left({\alpha }_i^{\ast }\times w_i^T\right)}=\left(w_1,w_2,\cdots ,w_m\right)$$

(18)

4. Grey Fuzzy Theory

Grey fuzzy theory is a risk evaluation method based on grey system theory and fuzzy mathematics theory. Grey system theory was proposed by Deng in 1982, and it is characterized by less information and incompleteness of problems [21].

A fuzzy comprehensive evaluation applies the concept of fuzzy mathematics to provide methods for evaluating practical problems. The procedure involves first determining the indicator set and evaluation grade set of the evaluated object. Then, the weight and degree of membership of each indicator are determined to obtain the fuzzy evaluation matrix. Finally, the fuzzy evaluation matrix and indicator weight vector are normalized, and the fuzzy comprehensive evaluation results are obtained [22].

4.1. Constructing the Risk Evaluation Matrix

The risk evaluation level of the bridge in its period of operation can be categorized as class I, class II, class III, class IV, or class V, and the corresponding risk levels are v = 9, 7, 5, 3, and 1, respectively. Risks falling between two levels can be represented by v = 8, 6, 4, or 2.

After the risk evaluation level is determined for a bridge in a given period of operation, experts in the field are invited to assess the risk indicators according to the evaluation levels discussed above, ranging from 1 to 10. Experts objectively and fairly grade the evaluation factors according to their own experience and professional knowledge to obtain a risk evaluation matrix. For n risk factors and m experts, the grey fuzzy evaluation matrix is defined as U = (u_ij)_n×m, where u_ij is the score given by expert i on indicator j, as shown in Equation (19).

$$U=\left[\begin{array}{cccc}{u}_{11}& u_{12}& \cdots & u_{1n}\\ u_{21}& u_{22}& \cdots & u_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ u_{m1}& u_{m2}& \cdots & u_{mn}\end{array}\right]$$

(19)

4.2. Constructing the Degree of Membership Matrix

According to the classification principle of risk evaluation grades, the grey class can be similarly divided into five grades: class I, class II, class III, class IV, and class V. Here, f₁–f₅ represent five types of membership functions, which are determined according to the aforementioned grades and used to construct the membership matrix. Additionally, x is the result of the expert’s score, such that x can be substituted into the membership function, and the calculated result can be statistically analyzed to obtain the fuzzy evaluation matrix V by following the steps below.

Class I:

$$f_1=\{\begin{array}{cc}\frac{x}{9}& x\in \left[0,9\right)\\ 1& x\ge 9\end{array}$$

(20)

Class II:

$$f_2=\{\begin{array}{cc}\frac{x}{7}& x\in \left[0,7\right)\\ -\frac{x}{7}+2& x\in \left[7,14\right]\\ 0& x>14\end{array}$$

(21)

Class III:

$$f_3=\{\begin{array}{cc}\frac{x}{5}& x\in \left[0,5\right)\\ -\frac{x}{5}+2& x\in \left[5,10\right]\\ 0& x>10\end{array}$$

(22)

Class IV:

$$f_4=\{\begin{array}{cc}\frac{x}{3}& x\in \left[0,3\right)\\ -\frac{x}{3}+2& x\in \left[3,6\right]\\ 0& x>6\end{array}$$

(23)

Class V:

$$f_5=\{\begin{array}{cc}1& x\in \left[0,1\right)\\ -x+2& x\in \left[1,2\right]\\ 0& x>2\end{array}$$

(24)

Assuming that the value of grey statistics corresponding to each risk indicator in these five defined weighted functions is n_ij, we combine these values with the risk evaluation matrix. Taking the evaluation indicator i as an example, the specific calculation process for various membership functions is as follows:

The risk indicator belongs to class I statistical values:

$$n_{i1}=f_1(u_{1i})+f_1(u_{2i})+f_1(u_{3i})+\cdots +f_1\left(u_{ki}\right)$$

(25)

This risk indicator belongs to class II statistical values:

$$n_{i2}=f_2(u_{1i})+f_2(u_{2i})+f_2(u_{3i})+\cdots +f_2\left(u_{ki}\right)$$

(26)

This risk indicator belongs to class III statistical values:

$$n_{i3}=f_3(u_{1i})+f_3(u_{2i})+f_3(u_{3i})+\cdots +f_3\left(u_{ki}\right)$$

(27)

This risk indicator belongs to class IV statistical values:

$$n_{i4}=f_4(u_{1i})+f_4(u_{2i})+f_4(u_{3i})+\cdots +f_4\left(u_{ki}\right)$$

(28)

This risk indicator belongs to class V statistical values:

$$n_{i5}=f_5(u_{1i})+f_5(u_{2i})+f_5(u_{3i})+\cdots +f_5\left(u_{ki}\right)$$

(29)

The total grey statistical value of these risk indicators is

$$n_i=n_{i1}+n_{i2}+n_{i3}+n_{i4}+n_{i5}$$

(30)

and its evaluation weight is $v_{ij}=\raisebox{1ex}{$n_{ij}$}\!\left/ \!\raisebox{-1ex}{$n_i$}\right.$, where j = 1, 2, 3, 4, or 5. Then, the grey evaluation weight vector of this evaluation factor is

$$V=(v_{i1},v_{i2},v_{i3},v_{i4},v_{i5})$$

(31)

The evaluation weight vectors of other factors can also be obtained using this process, and the membership matrix of other risk indicators is defined as shown in Equation (32):

$$V=\left[\begin{array}{cccc}{v}_{11}& v_{12}& \cdots & v_{15}\\ v_{21}& v_{22}& \cdots & v_{25}\\ ⋮& ⋮& ⋮& ⋮\\ v_{n1}& v_{n2}& \cdots & v_{n5}\end{array}\right]$$

(32)

5. Example Analysis

This study used the Dongmen bridge to conduct an example analysis; this bridge is a concrete continuous beam bridge, with a length of 175 m, a width of 23.5 m, and a span of 20 m.

5.1. Subjective Weight Calculation

Experts on the working conditions and service environment of the bridge were invited to grade the Dongmen bridge according to the scale presented in Table 2. The class I indicators of the bridge risk in its period of operation are represented by C, among which the natural disasters C₁ include wind-induced disasters (C₁₁), earthquakes (C₁₂), geological hazards (C₁₃), floods (C₁₄), and blizzards (C₁₅). The relative importance of each risk indicator is summarized in

Table 4. Relative importance of natural disasters.

C1	C11	C12	C13	C14	C15
C11	1	1/3	1/3	1/4	1/4
C12	3	1	4	3	3
C13	3	1/4	1	1/2	1/2
C14	4	1/3	2	1	3
C15	4	1/3	2	1/3	1

.

The corresponding judgment matrix C₁ is shown in Equation (33):

$$C_1=\left[\begin{array}{ccccc}1& \frac{1}{3}& \frac{1}{3}& \frac{1}{4}& \frac{1}{4}\\ 3& 1& 4& 3& 3\\ 3& \frac{1}{4}& 1& \frac{1}{2}& \frac{1}{2}\\ 4& \frac{1}{3}& 2& 1& 3\\ 4& \frac{1}{3}& 2& \frac{1}{3}& 1\end{array}\right]$$

(33)

Normalizing each column element of the matrix yields matrix P₁:

$$P_1=\left[\begin{array}{ccccc}0.0667& 0.1481& 0.0357& 0.0492& 0.0323\\ 0.2000& 0.4444& 0.4286& 0.5902& 0.3871\\ 0.2000& 0.1111& 0.1071& 0.0984& 0.0645\\ 0.2667& 0.1481& 0.2143& 0.1967& 0.3871\\ 0.2667& 0.1481& 0.2143& 0.0656& 0.1290\end{array}\right]$$

(34)

The elements in the matrix P₁ are added in rows to obtain the vector P₁′, which is then normalized to obtain the eigenvector u₁ of the judgment matrix C₁:

$$u_1=\left[\begin{array}{ccccc}0.0664& 0.4101& 0.1162& 0.2426& 0.1647\end{array}\right]$$

(35)

The maximum eigenvalue of matrix C₁ is ${\lambda }_{\max }=5.4192$, $CI=\frac{5.4192-5}{4}=0.1048$, $CR=\frac{0.1048}{1.12}=0.0936<0.1$; therefore, this matrix passes the consistency test.

The accidents affecting the bridge in its period of operation in the category C₂ include ship collisions with the bridge (C₂₁), fires (C₂₂), electrical shocks (C₂₃), falling objects (C₂₄), overload accidents (C₂₅), man-made damages (C₂₆), and vehicle collisions (C₂₇). The relative importance of each risk indicator is summarized in

Table 5. Relative importance of accidents.

C2	C21	C22	C23	C24	C25	C26	C27
C21	1	4	5	4	3	4	3
C22	1/4	1	2	3	4	3	3
C23	1/5	1/2	1	1/2	1/4	1/3	1/4
C24	1/4	1/3	2	1	1/3	1/2	1/3
C25	1/3	1/4	4	3	1	3	2
C26	1/4	1/3	3	2	1/3	1	1/3
C27	1/3	1/3	4	3	1/2	3	1

.

The corresponding judgment matrix is expressed as shown in Equation (36):

$$C_2=\left[\begin{array}{ccccccc}1& 4& 5& 4& 3& 4& 3\\ \frac{1}{4}& 1& 2& 3& 4& 3& 3\\ \frac{1}{5}& \frac{1}{2}& 1& \frac{1}{2}& \frac{1}{4}& \frac{1}{3}& \frac{1}{4}\\ \frac{1}{4}& \frac{1}{3}& 2& 1& \frac{1}{3}& \frac{1}{2}& \frac{1}{3}\\ \frac{1}{3}& \frac{1}{4}& 4& 3& 1& 3& 2\\ \frac{1}{4}& \frac{1}{3}& 3& 2& \frac{1}{3}& 1& \frac{1}{3}\\ \frac{1}{3}& \frac{1}{3}& 4& 3& \frac{1}{2}& 3& 1\end{array}\right]$$

(36)

Normalizing each column element of the matrix gives

$$P_2=\left[\begin{array}{ccccccc}0.3822& 0.5926& 0.2381& 0.2424& 0.3186& 0.2697& 0.3025\\ 0.0955& 0.1481& 0.0952& 0.1818& 0.4248& 0.2022& 0.3025\\ 0.0764& 0.0741& 0.0476& 0.0303& 0.0265& 0.0225& 0.0252\\ 0.0955& 0.0494& 0.0952& 0.0606& 0.0354& 0.0337& 0.0336\\ 0.1274& 0.0370& 0.1905& 0.1818& 0.1062& 0.2022& 0.2017\\ 0.0955& 0.0494& 0.1429& 0.1212& 0.0354& 0.0674& 0.0336\\ 0.1274& 0.0494& 0.1905& 0.1818& 0.0531& 0.2022& 0.1008\end{array}\right]$$

(37)

Adding the elements in matrix P₂ by rows gives the vector P₂′, which is then normalized to obtain the eigenvector u₂ of the judgment matrix C₂:

$$u_2=\left[\begin{array}{ccccccc}0.3351& 0.2072& 0.0432& 0.0576& 0.1495& 0.0779& 0.1293\end{array}\right]$$

(38)

The maximum eigenvalue of matrix C₂ is ${\lambda }_{\max }=7.7915$, $CI=\frac{7.7915-7}{6}=0.1319$, $CR=\frac{0.1319}{1.12}=0.0998<0.1$; therefore, this matrix passes the consistency test.

The secondary indicators include natural disasters C₁ and accidents C₂, and the relative importance of these risk factors is presented in

Table 6. Relative importance of criteria layers.

Relative Importance
C	C1	C2
C1	1	3
C2	1/3	1

.

The corresponding judgment matrix is shown in Equation (39):

$$C=\left[\begin{array}{cc}1& 3\\ \frac{1}{3}& 1\end{array}\right]$$

(39)

Normalizing each column element of the matrix yields matrix P:

$$P=\left[\begin{array}{cc}0.7500& 0.7500\\ 0.2500& 0.2500\end{array}\right]$$

(40)

The elements in matrix P are added in rows to obtain the vector P′, which is then normalized to obtain the eigenvector u of judgment matrix C₂:

$$u=\left[0.7500,0.2500\right]$$

(41)

Here, $CR=0<0.1$; therefore, the overall consistency inspection meets the requirements.

The total hierarchy permutation of the indicator layers relative to the target layer can then be calculated, as described in

Table 7. Total ranking of levels.

Second Level Risk Factors	C1	C2	Total Weight Value
	0.7500	0.2500
C11	0.0664	—	0.0498
C12	0.4101	—	0.3076
C13	0.1162	—	0.0872
C14	0.2426	—	0.1820
C15	0.1647	—	0.1235
C21	—	0.3351	0.0838
C22	—	0.2072	0.0518
C23	—	0.0432	0.0108
C24	—	0.0576	0.0144
C25	—	0.1495	0.0374
C26	—	0.0779	0.0195
C27	—	0.1293	0.0322

.

The subjective weight calculated via AHP is

$$W_1=\left[0.0498,0.3076,0.0872,0.1820,0.1235,0.0838,0.0518,0.0108,0.0144,0.0374,0.0195,0.0323\right]$$

(42)

5.2. Objective Weight Calculation

First, the grade definition evaluation set of the possible risk indicators and the loss after risk were established, as shown in

Table 8. Risk occurrence possibility levels.

Risk Occurrence Possibility Level	Risk Description
0.1	Risk will almost never occur
0.3	The risk is less likely to occur
0.5	The risk is moderate; it can happen under certain circumstances
0.7	Risk is highly likely to occur and likely to occur in most cases
0.9	Risks are highly likely to occur and, in most cases, unavoidable
0.2, 0.4, 0.6, 0.8	The median of the above possibilities

and

Table 9. Risk loss levels.

Loss Level	Loss Description
0–20	The loss caused by risk is negligible
20–40	The loss caused by risk is small and can be controlled immediately
40–60	The risk causes medium loss and damage to the bridge structure
60–80	Risk causes heavy losses
80–100	Risk causes great losses

, respectively.

Second, experts were invited to assess the probabilities of risk indicators and the losses after these risks. Then, the entropy weight method was used to calculate the objective weights of the risk indicators, as shown in

Table 10. Objective weights.

Criterion Layer	Indicators Layer	Probability	Loss	Weight
Natural disaster	Wind induced disaster	0.6	40	0.08306
	Earthquake	0.5	50	0.08374
	Geologic hazard	0.5	30	0.08350
	Flood	0.8	40	0.08306
	Blizzard	0.6	40	0.08306
Accident	Ship collision bridge	0.4	50	0.08374
	Fire	0.5	30	0.08350
	Get an electric shock	0.7	50	0.08306
	High falling objects	0.6	40	0.08306
	Overload accident	0.5	60	0.08350
	Man-made destruction	0.5	50	0.08366
	Vehicle impact Bridge	0.6	40	0.08306

.

The objective weight is:

$$\begin{array}{l}{W}_2=[0.08306,0.08374,0.08350,0.08306,0.08306,0.08374,\\ 0.08350,0.08306,0.08306,0.08350,0.08366,0.08306]\end{array}$$

(43)

5.3. Combination Weight Calculation

According to the described formulas, ${\alpha }_1=-1,{\alpha }_2=2.9892$. Using the comprehensive weighting method derived from game theory, the comprehensive weights can be calculated, as shown in

Table 11. Combination weights.

Criterion Layer	Indicator Layer	Subjective Weight	Objective Weight	Combination Weight
Natural disaster	Wind-induced disaster	0.0498	0.08306	0.0747
	Earthquake	0.3076	0.08374	0.1399
	Geologic hazard	0.0872	0.08350	0.0844
	Flood	0.1820	0.08306	0.1079
	Blizzard	0.1235	0.08306	0.0932
Accident	Ship collision with bridge	0.0838	0.08374	0.0838
	Fire	0.0518	0.08350	0.0756
	Get an electric shock	0.0108	0.08306	0.0649
	High falling objects	0.0144	0.08306	0.0658
	Overload accident	0.0374	0.08350	0.0719
	Man-made destruction	0.0195	0.08366	0.0676
	Vehicle impact on bridge	0.0322	0.08306	0.0703

.

According to Equations (13)–(17), the combination weight, $W^{*}$, can be obtained.

$$\begin{array}{l}{W}^{*}=[0.0747,0.1399,0.0844,0.1079,0.0932,0.0838,\\ 0.0756,0.0649,0.0658,0.0719,0.0676,0.0703]\end{array}$$

(44)

5.4. Comprehensive Evaluation

Five experts with significant experience in bridge engineering were invited to grade each risk indicator, leading to the matrix in Equation (45):

$$U=\left[\begin{array}{cccccccccccc}6& 9& 7& 8& 7& 7& 6& 7& 5& 8& 7& 6\\ 7& 8& 7& 8& 6& 8& 7& 7& 6& 7& 8& 7\\ 7& 9& 8& 8& 7& 8& 7& 8& 7& 8& 7& 7\\ 6& 8& 7& 7& 6& 7& 6& 7& 6& 7& 8& 8\\ 6& 9& 8& 7& 7& 6& 8& 8& 7& 8& 7& 6\end{array}\right]$$

(45)

The grades given by the experts were substituted into the risk evaluation model, and the elements 6, 7, 7, 6, and 6 in the first column were substituted into the five corresponding types of defined weighted functions in the risk evaluation model.

When u₁₁ = 6, $f_1\left(u_{11}\right)=\frac{6}{9}=0.667$, $f_2\left(u_{11}\right)=\frac{6}{7}=0.857$, $f_3\left(u_{11}\right)=-\frac{6}{5}+2=0.8$, $f_4\left(u_{11}\right)=-\frac{6}{3}+2=0$, and $f_5\left(u_{11}\right)=0$.

When u₂₁ = 7, $f_1\left(u_{21}\right)=\frac{7}{9}=0.778$, $f_2\left(u_{21}\right)=-\frac{7}{7}+2=1$, $f_3\left(u_{21}\right)=-\frac{7}{5}+2=0.6$, $f_4\left(u_{21}\right)=0$, and $f_5\left(u_{21}\right)=0$.

The statistical values of class I are as follows: $n_{11}=f_1\left(u_{11}\right)+f_1\left(u_{21}\right)+f_1\left(u_{31}\right)+f_1\left(u_{41}\right)+f_1\left(u_{51}\right)$.

The grey statistical values of the other four classes were solved by other methods, and the results are presented in

Table 12. Grey class statistics.

Indicators	f1	f2	f3	f4	f5	Total Ash Value
C11	3.556	4.571	3.600	0.000	0.000	11.727
C12	4.778	3.857	1.400	0.000	0.000	10.035
C13	4.111	4.714	2.600	0.000	0.000	11.425
C14	4.222	4.571	2.400	0.000	0.000	11.193
C15	3.667	4.714	3.400	0.000	0.000	11.780
C21	4.000	4.571	3.200	0.000	0.000	11.771
C22	3.778	4.571	3.200	0.000	0.000	11.549
C23	4.111	4.714	2.600	0.000	0.000	11.425
C24	3.444	4.429	3.800	0.333	0.000	12.000
C25	4.222	4.572	2.400	0.000	0.000	11.194
C26	4.111	4.714	2.600	0.000	0.000	11.425
C27	3.556	4.571	3.200	0.000	0.000	11.327

.

The evaluation weights belonging to the first evaluation standard are: $v_{11}=\frac{n_{11}}{n_{C_{11}}}=\frac{3.556}{11.727}=0.303$, $v_{12}=\frac{n_{12}}{n_{C_{11}}}=\frac{4.571}{11.727}=0.390$, $v_{13}=\frac{n_{13}}{n_{C_{11}}}=\frac{3.600}{11.727}=0.307$, $v_{14}=\frac{n_{14}}{n_{C_{11}}}=\frac{0.000}{11.727}=0.000$, and $v_{15}=\frac{n_{15}}{n_{c_{11}}}=\frac{0.000}{11.727}=0.000$.

Similarly, five evaluation weights corresponding to other indicators were obtained.

$$V=\left[\begin{array}{ccccc}0.303& 0.390& 0.307& 0.000& 0.000\\ 0.476& 0.384& 0.140& 0.000& 0.000\\ 0.360& 0.413& 0.227& 0.000& 0.000\\ 0.377& 0.408& 0.214& 0.000& 0.000\\ 0.311& 0.400& 0.289& 0.000& 0.000\\ 0.340& 0.388& 0.272& 0.000& 0.000\\ 0.327& 0.396& 0.277& 0.000& 0.000\\ 0.360& 0.413& 0.227& 0.000& 0.000\\ 0.287& 0.369& 0.317& 0.000& 0.000\\ 0.377& 0.408& 0.214& 0.000& 0.000\\ 0.360& 0.413& 0.227& 0.000& 0.000\\ 0.313& 0.403& 0.283& 0.000& 0.000\end{array}\right]$$

(46)

5.5. Determining the Risk Level

According to the total weight and membership matrix of each risk indicator, the project can be comprehensively evaluated by applying grey fuzzy theory. The evaluation results are as follows:

$$X=W^{*}\cdot V$$

(47)

$$X=\left[0.3583,0.3981,0.2417,0.0018,0\right]$$

(48)

The comprehensive evaluations indicate that the degree of class I is 0.3583, the degree of class II is 0.3981, the degree of class III is 0.2417, the degree of class IV is 0.0018, and the degree of class V is 0. According to the relevant principle of the maximum degree of membership, 0.3981 is the maximum value. Therefore, the risk level of the Dongmen bridge during its period of operation is predicted to be class II.

6. Conclusions

To avoid the uneven weight distribution of risk indicators during conventional bridge risk evaluations, this report combines game theory and grey fuzzy theory to propose a combination weighting method that can be applied to determine the weights of risk indicators. This newly developed comprehensive evaluation method was then applied to a real bridge risk evaluation.

(1)

Considering the potential natural disasters and accidents affecting the bridge during its period of operation, a grey fuzzy comprehensive evaluation of the bridge risk was proposed. In this approach, the subjective and objective weights of the risk indicators were combined and integrated, and the combination weights of the risk indicators were obtained. The degree of membership matrix was calculated based on expert grading and grey fuzzy theory.

(2)

The evaluation result vector was calculated by multiplying the degree of membership matrix by the combination weight. Thus, the risk level of the bridge in its period of operation could be determined according to the principle of the maximum degree of membership.

(3)

The proposed method combines qualitative and quantitative risk analysis of bridges in their periods of operation and provides an effective basis for the quantitative analysis of risk uncertainty.

(4)

A fuzzy comprehensive evaluation of the risk of an example bridge was conducted for its period of operation. The results show that the predicted risk grade of the bridge is class II, which confirmed that the developed method is practical for risk evaluations of real operational bridges.