# Operation Risk Assessment of Urban Dense Cable Passageway Based on Fuzzy-Analytic Hierarchy Process
## 3 Operational Risk Assessment Methodology for Urban Dense Cable Corridors  
The fuzzy-hierarchical analysis method combines the advantages of fuzzy logic and hierarchical analysis, and it is able to deal with the uncertainty and ambiguity that exists in the process of operational risk assessment of urban dense cable channels. Based on the actual situation and an analysis of the objectives, this research on the operational risk assessment method of urban dense cable channel is divided into the target layer, criterion layer, and indicator layer, as shown in Figure 2.  
The indicator system of the urban dense cable channel operation risk assessment method contains 3 elements in the guideline layer and 18 elements in the indicator layer. Figure 2: Indicator system for operational risk assessment methodology of urban dense cable corridors. Taking the constructed risk assessment system hierarchy as the basis, the judgment matrix is constructed as shown in Equation (1):  
\[A= \begin{pmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{mn}\end{pmatrix}  \tag{1}\]  
where \(A\) is the judgment matrix of the guideline layer or indicator layer for the operational risk assessment of urban dense cable channels. In the guideline layer, it is the judgment matrix composed of the risk parameters of the cable body, the risk parameters of the cable channel, and the risk parameters of the external environment of the cable channel. In the indicator layer, it is the judgment matrix composed of the six indicators included in the guideline layer, respectively. \(a_{ij}\) is the relative weight of element \(i\) relative to element \(j\) in the guideline layer or the indicator layer. It has the following properties. \[\left\{ \begin{aligned} a_{ij}>0\\ a_{ij}=1/a_{ji}\\ a_{ii}=1\end{aligned} \right. \tag{2}\]  
According to the principle of scale and judgment of two-by-two comparison, the following criteria for comparison of risk assessment elements can be derived by using fuzzy mathematical theoretical methods, as shown in Table 1.  
The judgment matrix \(A\) is solved via the square root method. The product of its risk assessment elements per row is calculated as shown in Equation (3). \[M_{i}=\prod_{j=1}^{n}a_{ij} \tag{3}\]  
where \(M_{i}\) is the product of the risk assessment elements in each row, where \(i=1\), \(2\), \(\ldots\), \(n\). Then, we find the \(n\)-th root of \(M_{i}\) as shown in Equation (4). \[\overline{W_{i}}=\sqrt[n]{M_{i}} \tag{4}\]  
where \(\overline{W_{i}}\) is the \(n\)-th root vector of \(M_{i}\), which is normalized to it, as shown in Equation (5). \[W_{i}=\overline{W_{i}}/\sum_{i=1}^{n}\overline{W_{i}} \tag{5}\]  
\[W=(w_{1},w_{2},\ldots,w_{n})^{\text{T}} \tag{6}\]  
where \(W\) is the eigenvector corresponding to the largest eigenvalue of the judgment matrix \(A\), i.e., the weight vector, and \(W_{i}\) is the \(i\)-th element of \(W\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Scale Value** & **Sense** \\ \hline
1 & Risk assessment element \(i\) is as important as \(j\) \\
3 & Risk assessment element \(i\) is slightly more important than \(j\) \\
5 & Risk assessment element \(i\) is more strongly important than \(j\) \\
7 & Risk assessment element \(i\) is strongly more important than \(j\) \\
9 & Risk assessment element \(i\) is extremely important compared to \(j\) \\
2, 4, 6, 8 & Intermediate values of the above adjacent judgments \\ The inverse of 1-9 & If the ratio of the importance of risk assessment element \(i\) to element \(j\) \\  & is \(a_{ij}\), then the ratio of the importance of element \(j\) to element \(i\) is \(1/a_{ij}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Risk assessment element scale values and their meanings. \[AW= \begin{pmatrix}a_{11}&\cdots&a_{1n}\\ \vdots&\ddots&\vdots\\ a_{n1}&\cdots&a_{nm}\end{pmatrix} \times \begin{pmatrix}w_{1}\\ \vdots\\ w_{n}\end{pmatrix}