## 4 Case Study and Discussion
### Risk Assessment  
1 Classification of risk factors  
Considering numerical ranges within the indirect propagation probability matrix \(C_{c}^{P}\), the potential severity matrix \(S\), and the indirect propagation time matrix \(C_{c}^{T}\), qualitative risk factor level rules are given (from light to heavy, respectively 1 to 5), as shown in Table 4 based on experience.  
2 Calculation of multi-dimensional safety risk  
Based on Table 2, the safety risk factors \(P\), \(S\), \(T\) are converted into uniform safety risk level. Equation (13) is used to calculate multi-dimensional safety risk value, as shown in Matrix (29). Total risk value that element \(i\) effect on the overall system is calculated as shown in Matrix (30).  
\[P=R_{9\times 9}=\left[\begin{array}{cccccccc}0.00&21.85&21.59&21.59&14.39&12. 36&10.78&14.30&10.78\\ 18.62&0.00&21.59&23.99&14.39&12.36&9.72&14.30&10.78\\ 15.42&19.19&0.00&22.58&19.19&14.27&15.53&19.19&17.81\\ 15.42&21.85&22.58&0.00&19.19&15.42&15.53&18.29&17.81\\ 12.36&14.39&21.59&21.59&0.00&18.62&19.19&18.29&23.06\\ 12.36&14.39&20.80&21.59&21.85&0.0&18.29&20.83&21.59\\ 9.14&9.72&17.81&17.81&19.19&14.27&0.0&19.19&23.9\\ 12.25&14.30&21.59&20.80&18.29&17.41&19.19&0.0&22.58\\ 9.14&10.78&17.81&17.81&20.83&15.42&21.85&20.29&0.0\\ \end{array}\right] \tag{29}\]  
\[R_{i}=[127.66,125.76,142.18,143.08,149.11,151.71,134.13,144.42,136.95] \tag{30}\]  
System total safety risk assessment value \(R_{N}=1255\), and based on Equation (15), safety risk ratio is shown in Matrix (31).  
\[\eta_{i}=[10.17,10.02,11.33,11.40,11.88,12.09,10.69,11.51,10.91] \tag{31}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Risk Factor Level** & \(\mathbf{C_{c}^{P}}\) & \(\mathbf{S}\) & \(\mathbf{C_{c}^{T}/s}\) \\ \hline
1 & (0, 0.3) & (0, 0.6) & (2, +\(\infty\)) \\
2 & (0.3, 0.5) & (0.6, 1.2) & (1.5, 2) \\
3 & (0.5, 0.7) & (1.2, 1.8) & (1, 1.5) \\
4 & (0.7, 0.9) & (1.8, 2.4) &  
* ALARP principle Under the ALARP principle, two risk \"boundaries\" are set based on expert experience: the intolerable boundary and the negligible boundary are 145 and 130, respectively, and the alternative values of the boundaries are 140 and 130, respectively. On the condition of alternative values, element 3, 4, 5, 6, 8 are all in the serious risk region. According to Pareto rule, when distinguishing safety-critical links, it can be considered that 80% of accidents are originated from 20% of dangerous sources, but the accumulated risk value of the elements in the serious risk region is well over 20%, which is a violation of the Pareto rule. Therefore, value of the boundaries 145, 130 are determined. The risk level of each element is presented in Figure 8.
* Safety risk entropy Based on Equation (16), entropy matrix \(H\) is presented as Matrix (32). \[H=H_{9\times 9}=\left[\begin{array}{cccccccc}0.00&0.00&0.17&0.09&0.35&0.42&0.45&0. 27&0.45\\ 1.00&0.00&0.17&0.00&0.35&0.42&0.45&0.27&0.45\\ 0.17&0.17&0.00&0.00&0.17&0.24&0.27&0.09&0.27\\ 0.09&0.00&0.00&0.00&0.17&0.17&0.27&0.24&0.27\\ 0.35&0.35&0.17&0.17&0.00&0.00&0.09&0.24&0.09\\ 0.42&0.42&0.24&0.17&0.00&0.00&0.24&0.17&0.09\\ 0.45&0.45&0.27&0.27&0.09&0.24&0.00&0.17&0.00\\ 0.27&0.27&0.09&0.24&0.24&0.17&0.17&0.00&0.00\\ 0.45&0.45&0.27&0.27&0.09&0.09&0.00&0.00&0.00\\ \end{array}\right]\] (32)  
Based on Equation (17), total safety risk entropy \(H_{i}\) of element \(i\) which effects on the overall system is calculated as shown in Matrix (33). The safety risk entropy of the single element in descending order and the accumulated the safety risk entropy are presented in Figure 9. \[H_{i}=[2.24,2.14,1.43,1.25,1.51,1.81,1.97,1.49,1.64]\] (33)  
Figure 8: Results of ALARP model chart.