## 3 Construction of the Risk Assessment Model
### Triangular Fuzzy Analytic Hierarchy Process
#### 3.2.4 Calculation of Single-Level Indicator Weights  
Based on the obtained triangular fuzzy matrix, the overall fuzziness of each factor is computed. Let \(\text{M}_{\text{ij}}\) denote the value of factor i relative to factor j in the triangular fuzzy matrix. The comprehensive fuzziness of the i-th factor is then given by:  
\[\text{P}_{\text{i}}=\sum_{\text{j}=1}^{\text{n}}\text{M}_{\text{ij}}\otimes \left(\sum_{\text{i}=1}^{\text{n}}\sum_{\text{j}=1}^{\text{n}}\text{M}_{\text {ij}}\right)^{-1} \tag{4}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Comparison of the Importance** & **Equal** & **Equal** & **Moderately** & **Strongly** & **Highly** & **In Between** \\
**between \(\text{x}_{\text{i}}\) and \(\text{x}_{\text{j}}\)** & **Importance** & **Importance** & **Important** & **Important** & **Important** & **In Between** \\ \hline \(\widetilde{\text{r}}_{\text{ij}}\) Simplified triangular fuzzy numbers & 1 & 3 & 5 & 7 & 9 & 2,4,6,8 \\ \hline \(\text{r}_{\text{ij}}\) Triangular fuzzy numbers & (1,1,2) & (2,3,4) & (4,5,6) & (6,7,8) & (8,9,9) & \(\left(\text{r}_{\text{ij}}-1,\text{r}_{\text{ij}},\text{r}_{\text{ij}}+1\right)\) \\ \hline \hline \end{tabular} Note: \(\widetilde{\text{r}}_{\text{j}}\) represents the fuzzy ratio of the importance degree of factor \(\text{x}_{\text{j}}\) to factor \(\text{x}_{\text{i}}\), \(\widetilde{\text{r}}_{\text{ij}}=\frac{1}{\widetilde{\text{r}}_{\text{ij}}}\) indicates an equal importance, and similarly, \(\text{r}_{\text{ij}}=\frac{1}{\widetilde{\text{r}}_{\text{ij}}}=\left(\frac{ 1}{\widetilde{\text{r}}_{\text{ij}}+1},\frac{1}{\widetilde{\text{r}}_{\text{ ij}}},\frac{1}{\widetilde{\text{r}}_{\text{ij}}-1}\right)\) holds.  
\end{table}
Table 3: Triangular fuzzy numbers and their meanings.  
In the equation, \(\otimes\) represents the multiplication of two triangular fuzzy numbers. \(\sum\limits_{j=1}^{n}{\rm{M}}_{ij}\) and \(\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{n}{\rm{M}}_{ij}}\) represent the summation operation of triangular fuzzy numbers, which can be obtained using Equations (5) and (6), respectively.  
\[\sum\limits_{j=1}^{n}{\rm{M}}_{\bar{\rm{q}}}=\left(\sum\limits_{j=1}^{n}{\rm{I} }_{ij}\sum\limits_{j=1}^{n}{\rm{m}}_{\bar{\rm{q}}j},\sum\limits_{j=1}^{n}{\rm{ u}}_{ij}\right) \tag{5}\]  
\[\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{n}{\rm{M}}_{ij}}=\left(\sum\limits_{ i=1}^{n}{\rm{I}}_{ij},\sum\limits_{i=1}^{n}{\rm{I}}_{ij},\sum\limits_{i=1}^{n}{\rm{m}}_{ij} \sum\limits_{i=1}^{n}{\rm{u}}_{ij}\right) \tag{6}\]  
Correspondingly, \(\left(\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{n}{\rm{M}}_{ij}}\right)^{-1}= \left(\frac{1}{\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{n}{\rm{u}}_{ij}}}, \frac{1}{\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{n}{\rm{u}}_{ij}}},\frac{1}{ \sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{n}{\rm{u}}_{ij}}}\right)\) can be obtained.  
By calculating the above equation, the comprehensive fuzziness of each factor can be obtained. Assuming that \({\rm{P}}_{1}=({\rm{I}}_{1},{\rm{m}}_{1},{\rm{u}}_{1})\) and \({\rm{P}}_{2}=({\rm{I}}_{2},{\rm{m}}_{2},{\rm{u}}_{2})\) are the comprehensive fuzziness of two factors. Then, the possibility degree of \({\rm{P}}_{1}\geq{\rm{P}}_{2}\) can be defined as follows:  
\[\mu({\rm{P}}_{1}\geq{\rm{P}}_{2})=\ \sup_{y\geq x}\bigl{[}\min\bigl{(}\mu_{{\rm{P}} _{1}}({\rm{x}}),\mu_{{\rm{P}}_{2}}({\rm{x}})\bigr{)}\bigr{]} \tag{7}\]  
The geometric interpretation of the possibility degree of  
After normalizing the weight vector \(\mathrm{w_{o}}\) using Equation (11), we obtain \(\mathrm{w=}\Big{(}\mu(\mathrm{d_{1}})^{\prime},\mu(\mathrm{d_{1}})^{\prime}, \mathrm{L},\mu(\mathrm{d_{n}})^{\prime}\Big{)}\), where \(\sum\limits_{\mathrm{i=1}}^{n}\mu(\mathrm{d_{i}})^{\prime}=1\).  
\[\mu(\mathrm{d_{i}})^{\prime}=\frac{\mu(\mathrm{d_{i}})}{\sum\limits_{\mathrm{ i=1}}^{n}\mu(\mathrm{d_{i}})} \tag{11}\]  
The magnitude of the single-layer indicator weights can be calculated through the above steps.