## 3 Proposed Effectiveness Evaluation Method of Safety Barrier
### System Failure Consequence Analysis  
#### 3.3.1. Determine Severity Types of Failure Outcomes and Their Weights Using IVIF-AHP  
Traditional event trees are mainly used to analyze the probabilities of different failure outcomes (e.g., fire and explosion) in a system; however, they do not take into account the severity of the failure consequences caused by outcomes. This leads to the inability to effectively quantify the effect of various protective SBs on failure consequences through the event tree alone, thus limiting the estimation of the effectiveness of SBs. Therefore, in this study, a link between the outcomes in the ET and the severity of the consequences is established, as shown in Figure 4, through which the failure outcomes can be further converted into a severity index so that the effect of SBs on the failure consequences of the system can be measured. In Figure 4, CO\(n\) refers to the failure outcomes in the ET, C\(m\) refers to the consequence severity types, n indicates that there are n types of failure consequences, and m means that consequences can be divided into m severity types. There are two main ways to evaluate the severity index of failure consequences. One is to estimate the monetary loss caused by each outcome, then multiply the money lost for each by the corresponding occurrence probability, and finally sum them up. However, the monetary measure for human casualties and environmental impacts is too complex and difficult to estimate. For this reason, the second way is adopted here, which is to categorize the severity of failure consequences into several types like production loss, safety effect, environmental effect, and equipment cost, and then assess the effect of each consequence on each severity type through expert judgment, and finally aggregate the effects. Figure 4: The corresponding relationship between failure outcomes and severity types. Considering the fact that different severity types have different levels of impact on the consequences, the weight of each severity type \(w_{i}\) needs to be assigned before determining the severity index of each failure outcome. The IVIF-AHP is used for this. The specific procedure for calculating the weights is described below. Firstly, the importance of pairwise comparison of each severity type is given by experts and converted into IVIFNs according to Table 4[48]. Secondly, the importance of each severity type judged by each expert IVIFWA\({}_{Ci(E_{k})}\) is calculated using Equation (23), and then the importance of each severity type IVIFWA\({}_{Ci}\) is obtained by aggregating experts' opinions using Equation (24). \[\text{IVIFWA}_{Ci(E_{k})}=\left( \begin{bmatrix}1-\prod_{j=1}^{n}\left(1-\mu_{ \tilde{A}_{ij}^{(k)}}^{-}\right)^{\lambda^{(k)}},1-\prod_{j=1}^{n}\left(1-\mu_ {\tilde{A}_{ij}^{(k)}}^{+}\right)^{\lambda^{(k)}}\right],\\ \prod_{j=1}^{n}\left(v_{\tilde{A}_{ij}^{(k)}}^{-}\right)^{\lambda^{(k)}}, \prod_{j=1}^{n}\left(v_{\tilde{A}_{ij}^{(k)}}^{+}\right)^{\lambda^{(k)}}\end{bmatrix} \right) \tag{23}\]  
\[\text{IVIFWA}_{Ci}=\left( \begin{bmatrix}1-\prod_{k=1}^{K}\left(1-\mu_{\tilde{A }}^{-}\right)^{\lambda_{k}},1-\prod_{k=1}^{K}\left(1-\mu_{\tilde{A}}^{+} \right)^{\lambda_{k}}\end{bmatrix} ,\\ \prod_{k=1}^{K}\left(v_{\tilde{A}}^{-}\right)^{\lambda_{k}},\prod_{k=1}^{K} \left(v_{\tilde{A}}^{+}\right)^{\lambda_{k}}\end{bmatrix} \tag{24}\]  
where \(\lambda_{k}\) is the weight of the \(k\)th expert, and \(\sum_{k=1}^{K}\lambda_{k}=1\). Thirdly, the rationality of the experts' opinions is verified according to the consistency ratio (\(CR\)) calculated with Equation (25). If \(CR\) does not exceed \(0.1\), the experts' opinions are reasonable. Otherwise, they need to be re-evaluated[49]. \[CR=\frac{RI-\frac{\sum_{i}^{n}(x)}{n}}{n-1} \tag{25}\]  
where \(RI\) is the random index, which is indicated in Table 5[50]. Finally, when passing the consistency test, the crisp weights of the severity types are obtained using Equations (26) and (27)[51].