## 2 Basic Theories
### Fuzzy Theory  
The T-S gate rules in the T-S fault trees are usually formulated by expert knowledge [30], and the T-S gate rules correspond to conditional probability tables in subsequent Bayesian networks. Since experts cannot precisely calculate the probability parameters \(P^{l}(y^{b_{j}})\) in the T-S gate rule, natural language, such as \"possible\" and \"impossible\", is usually used to describe the probability parameters. However, these languages are fuzzy, so the fuzzy theory deals with this uncertain information. However, these languages carry fuzziness, so fuzzy theory is used to deal with this uncertain information. In this paper, fuzzy theory and expert linguistic judgments are used to estimate the conditional probability parameter \(P^{l}(y^{b_{j}})\) in the T-S gate rule. It is first necessary to determine the linguistic rating levels and to replace the expert linguistic rating values with triangular or trapezoidal fuzzy numbers [31]. Table 1 gives the corresponding fuzzy number forms and \(\lambda\)-cut sets for the evaluated languages. Figure 2: T-S multistate fault tree construction Bayesian network flow chart. After obtaining the evaluation information from fuzzy numbers from multiple experts, the arithmetic averaging method and the combination of expert weights are often used to synthesize the evaluation information. In order to avoid the intense subjectivity of experts, the objective weighting method of Depended Uncertain Ordered Weighted Averaging [32] (DUOWA) is used; this method determines the weights according to the degree of difference between the evaluation information of each expert and the average evaluation information, avoiding the uncertainty caused by subjective weighting. If \(n\) experts are evaluating the probability parameter \(P^{l}(y^{b_{j}})\) of the T-S gate rule \(l\), the trapezoidal fuzzy number of the evaluation information using the DUOWA operator to synthesize the expert evaluation information is:  
\[F_{k,l,y^{b_{j}}}=(F_{k,l,y^{b_{j}}}^{1},F_{k,l,y^{b_{j}}}^{2},F_{k,l,y^{b_{j} }}^{3},F_{k,l,y^{b_{j}}}^{4})\]  
Each expert's evaluation weight values are calculated and obtained by the following Equation (3):  
\[w(F_{k,l,y^{b_{j}}},F_{a})=\frac{s(F_{k,l,y^{b_{j}}},F_{a})}{\sum\limits_{k=1} ^{n}s(F_{k,l,y^{b_{j}}},F_{a})} \tag{3}\]  
where \(F_{a}\) is the arithmetic mean of \(n\) trapezoidal fuzzy numbers and \(s(F_{k,l,y^{b_{j}}},F_{a})\) is the similarity between \(F_{k,l,y^{b_{j}}}\) and \(F_{a}\). Comprehensive expert evaluation based on the evaluation weights of the experts:  
\[W_{l,y^{b_{j}}}=\sum\limits_{k=1}^{n}w_{k,l,y^{b_{j}}}F_{k,l,y^{b_{j}}} \tag{4}\]  
where \(w_{k,l,y^{b_{j}}}\) is the evaluation weight of each expert calculated by Equation (3). The information obtained by combining the expert evaluation information is still a fuzzy number. It is necessary to fuzzify the fuzzy number solution [33] into a clear T-S gate rule l with probability parameter p. The area-mean method is shown in Equation (5):  
\[P^{l}(y^{b_{j}}_{W})=\frac{a_{W}+b_{W}+c_{W}+d_{W}}{4} \tag{5}\]  
The \(P^{l}(y^{b_{j}}_{W})\) obtained above will be applied to the subsequent Bayesian network conditional probability table as the probability parameter \(P^{l}(y^{b_{j}})\) in the T-S gate rule \(l\). The \(P^{l}(y^{b_{j}})\) sum must be 1. Therefore, it is necessary to normalize \(P^{l}(y^{b_{j}}_{W})\), which can be obtained from Equation (6):  
\[P^{l}(y^{b_{j}}_{W})=\frac{a_{W}+b_{W}+c_{W}+d_{W}}{4} \tag{6}\]