## 3 Methods of Weighting Bridge Risk Indicators in the Period of Operation  
### 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[17]. Then, the judgment matrix takes the form shown in Equation (1). \[A= \begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{m}\\ \end{bmatrix}  \tag{1}\]  
where \(n\) is the number of indicators; \(a_{ij}>0\); \(a_{ij}=1/a_{ji}\); \(a_{ii}=1\). \begin{table}
\begin{tabular}{c c} \hline
**Type** & **Specific Risk Indicators** \\ \hline \multirow{6}{*}{Natural disaster} & Wind-induced disaster \\  & Earthquake \\  & Geological hazard \\  & Flood \\  & Bizzard \\  & Ship collision with bridge \\  & Fire \\  & Electric shock \\ \multirow{2}{*}{Accident} & High falling objects \\  & Overload accident \\  & Man-made destruction \\ \multirow{2}{*}{Vehicle impact with bridge} \\ \end{tabular}
\end{table} 
Table 1: Risk indicators of bridges in their period of operation. \begin{table}
\begin{tabular}{c c c c} \hline
**Element** & **Assignment** & **Meaning** \\ \hline \multirow{6}{*}{\(a_{ij}\)} & 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 \\  & & If the importance ratio of indicator \(i\) to indicator \(j\) is \(a_{ij}\), the \\ \multirow{2}{*}{reciprocal} & Importance ratio of indicator \(j\) to indicator \(i\) is \(a_{ij}=\frac{1}{a_{ji}}\) \\  & & \\ \end{tabular}
\end{table} 
Table 2: Relative importance scales. (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):  
\[\overline{a_{ij}}=\frac{a_{ij}}{\sum\limits_{i=1}^{n}a_{ij}} \tag{2}\]  
Matrix \(\overline{A}\) is added by rows, as expressed in Equation (3):  
\[\overline{w_{i}}=\sum\limits_{j=1}^{n}\overline{a_{ij}},i=1,2,\cdots,n \tag{3}\]  
After summing, \(w_{i}\) is normalized, and the eigenvector \(W_{i}\) is obtained using Equation (4):  
\[W_{i}=\frac{\overline{w_{i}}}{\sum\limits_{i=1}^{n}\overline{w_{i}}},i=1,2, \cdots,n \tag{4}\]  
The calculated \(W_{i}=(w_{1},w_{2},\ldots,w_{n})^{T}\) 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.