## 4 Case Studies
### An Example of Network Security Risk Assessment
#### 4.1.5 Use Weighted Average Combination Rule to Combine the Mass Functions  
On the basis of BPAs (\(m_{ij}\)) of bottom criteria, along with the comprehensive weights \(z_{ij}\), the weighed average of evidence \(\overline{m_{i}}\)\((i=1,2,3)\) can be calculated by Equation (9). Then, we can use  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Bottom Criteria** & **Uncertainty Values** & **Objective Weights** \\ \hline \(a_{11}\) & 0.1247 & 0.2847 \\ \(a_{12}\) & 0.2139 & 0.1660 \\ \(a_{13}\) & 0.2093 & 0.1697 \\ \(a_{14}\) & 0.1681 & 0.2112 \\ \(a_{15}\) & 0.2109 & 0.1684 \\ \(a_{21}\) & 0.2087 & 0.2063 \\ \(a_{22}\) & 0.1729 & 0.2491 \\ \(a_{23}\) & 0.2161 & 0.1993 \\ \(a_{24}\) & 0.1247 & 0.3453 \\ \(a_{31}\) & 0.2109 & 0.4974 \\ \(a_{32}\) & 0.2087 & 0.5026 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Objective weights of bottom criteria.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Middle Level Criteria** & **Subjective Weights** & **Bottom Criteria** & **Subjective Weights** \\ \hline \multirow{4}{*}{\(a_{1}\)} & \multirow{4}{*}{0.310} & \(a_{11}\) & 0.157 \\  & & \(a_{12}\) & 0.393 \\  & & \(a_{13}\) & 0.164 \\  & & \(a_{14}\) & 0.172 \\  & & \(a_{15}\) & 0.114 \\ \hline \multirow{4}{*}{\(a_{2}\)} & \multirow{4}{*}{0.580} & \(a_{21}\) & 0.281 \\  & & \(a_{22}\) & 0.312 \\  & & \(a_{23}\) & 0.280 \\  & & \(a_{24}\) & 0.127 \\ \hline \multirow{4}{*}{\(a_{3}\)} & \multirow{4}{*}{0.110} & \(a_{31}\) & 0.670 \\  & & \(a_{32}\) & 0.330 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Subjective weights of criteria.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Bottom Criteria** & **Subjective Weights** & **Objective Weights** & **Comprehensive Weights** \\ \hline \(a_{11}\) & 0.1570 & 0.2847 & 0.2312 \\ \(a_{12}\) & 0.3930 & 0.1660 & 0.3375 \\ \(a_{13}\) & 0.1640 & 0.1697 & 0.1440 \\ \(a_{14}\) & 0.1720 & 0.2112 & 0.1879 \\ \(a_{15}\) & 0.1140 & 0.1684 & 0.0993 \\ \(a_{21}\) & 0.2810 & 0.2063 & 0.2463 \\ \(a_{22}\) & 0.3120 & 0.2491 & 0.3302 \\ \(a_{23}\) & 0.2800 & 0.1993 & 0.2371 \\ \(a_{24}\) & 0.1270 & 0.3453 & 0.1863 \\ \(a_{31}\) & 0.6700 & 0.4974 & 0.6677 \\ \(a_{32}\) & 0.3300 & 0.5026 & 0.3323 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comprehensive weights of bottom criteria.  
Equations (7) and (8) to combine the weighted average of evidence \(\overline{m_{1}}\) 4 times to obtain \(m_{1}\). Similarly, we can derive \(m_{2}\) and \(m_{3}\). Weighted average of evidence \(\overline{m_{i}}\) and the BPAs of the bottom criteria after combination are detailed in Tables 6 and 7.  
The BPAs of bottom criteria after combination can be viewed as the mass functions of middle level criteria. Similarly, the objective and comprehensive weights of these pieces of evidence can also be obtained (see Table 8). Then, the weighted average of evidence \(\overline{m}\) is derived. After using Equations (7) and (8) to combine \(\overline{m}\) 2 times, the combination result, denoted as \(m\), which is also the mass function of network security risk assessment (see Table 9), is obtained. Meanwhile, the uncertainty of the evaluation result can be calculated by Equation (13), which is 0.0630 (quite small).