## 2 Basic Theories  
### T-S Multistate Fault Tree  
T-S multistate fault trees are inverted tree-like diagrams consisting of top events, intermediate events, and basic events that map the potential causes of system failures using a hierarchical deductive framework that can quantitatively or qualitatively estimate the probability of top events. Compared with the traditional fault tree, the T-S multistate fault tree uses a series of T-S gates with IF-THEN rules instead of the logic gates in the traditional fault tree. T-S gates allow events to have multiple fault states, making them more suited to complex components with multiple fault states or with escalating faults. T-S gate input events are multiple fault states or with escalating faults [24; 25].  
The T-S gate input event is \(x_{i}(i=1,2,\ldots,n)\), the corresponding multiple fault state can be expressed as \(x_{i}^{a_{i}}(a_{i}=1,2,\ldots,k_{i})\), [6] and its output event corresponds to the multiple fault state as \(y^{b_{j}}(b_{j}=1,2,\ldots,\eta_{j})\).  
In the case of a known IF-THEN rule \(l\), the input event \(x_{i}\) fault state in the T-S gate corresponds to: \(x_{1}=x_{1}^{a_{1}},x_{2}=x_{2}^{a_{2}},\ldots,x_{n}=x_{n}^{a_{n}}\), then \(y\) is a possible representation of state \(y^{b_{j}}\) as \(P^{l}(y^{b_{j}})\). Number \(r\) of rule \(l\) is determined by the total number \(k_{i}\) of states of \(x_{i}\). The calculation is given in Equation (1):  
\[r=k_{1}k_{2}\cdots k_{n}=\prod_{i=1}^{n}k_{i} \tag{1}\]  
In addition, the T-S gate rule can be used to assign values to the conditional probability tables of the corresponding nodes in the Bayesian network [26]. The specific process of constructing a Bayesian network based on a T-S multistate fault tree is shown in Figure 2.  
Figure 1: Theoretical framework for the approach.  
Fault Tree Structural Validity Verification  
Verifying the validity of the fault tree structure is essential for the subsequent reliability analysis method based on this fault tree [27]. By collecting expert knowledge, the content validity ratio (CVR) and content validity index (CVI) of the fault tree structure are calculated to measure the relevance and necessity of primary and intermediate events in the fault tree. The content validity ratio (CVR) is a measure of validity proposed by [PERSON] [28]. Using expert knowledge to calculate this value, experts classify each essential event according to a three-level Likert scale (1 = unnecessary, 2 = helpful but unnecessary, 3 = necessary). Then, they were calculated using Equation (2).  
\[CVR=\frac{n_{E}-\frac{N}{2}}{\frac{N}{2}} \tag{2}\]  
The \(n\) is the necessary number of experts selected (3), and \(N\) is the total number of experts.  
The calculated content validity ratio (CVR) value is compared with the standard parameters in [PERSON]'s table. If it is higher than the standard, the content validity of the fault tree structure is accepted.  
The content validity index (CVI), where the item-level CVI (I-CVI) evaluates the content validity of each item [29], and the method is used to measure the relevance of each event in the fault tree. In this method, experts are required to determine the relevance of each event. A 4-level Likert scale was used (1 = not relevant, 2 = slightly relevant, 3 = strongly relevant, 4 = very relevant). For each event, the number of experts who gave a rating of 3 or 4 divided by the total number of experts who participated was the corresponding I-CVI, and the I-CVI determined the content validity of the fault tree structure.