## 3 A Partial Multiplicative Dimensional Reduction-Based Reliability Estimation Method
### Method Presentation
\begin{gathered} G(X_{1},\ldots,X_{N},Y_{1},\ldots,Y_{M})\\ \approx G(X_{1},\ldots,X_{N},Y_{1}^{c},\ldots,Y_{M}^{c})\cdot G(X_{ 1}^{c},\ldots,X_{N}^{c},Y_{1},\ldots,Y_{M})\cdot G(X_{1}^{c},\ldots,X_{N}^{c},Y_{1}^{c},\ldots,Y_{M}^{c})^{-1}\end{gathered}  \tag{18}\]  
where \(\mathbf{X}^{c}=[X_{1}{}^{c},\ldots,X_{N}{}^{c}]^{\rm T}\) is the mean vector of \(\mathbf{X}\) and \(\mathbf{Y}^{c}=[Y_{1}{}^{c},\ldots,Y_{M}{}^{c}]^{\rm T}\) is the center point of \(\mathbf{Y}\). Therefore, the system's performance function \(G\) (\(\mathbf{X}\), \(\mathbf{Y}\)) can be finally transformed into the following separable form:  
\[G(X,Y)\approx G^{\prime}=G_{X}\cdot G_{Y}\cdot G_{c} \tag{19}\]  
where \(G_{\mathbf{X}}=G\) (\(\mathbf{X}\), \(\mathbf{Y}^{c}\)), \(G_{\mathbf{Y}}=G\) (\(\mathbf{X}^{c}\), \(\mathbf{Y}\)), and \(G_{c}=G\) (\(\mathbf{X}^{c}\), \(\mathbf{Y}^{c}\))\({}^{-1}\). Since \(G_{c}>0\) is guaranteed for most engineering cases, \(G>0\) is statistically equivalent to \(G_{\mathbf{X}}\cdot G_{\mathbf{Y}}>0\), which means that \(G_{\mathbf{X}}\) and \(G_{\mathbf{Y}}\) are identically positive or negative. On the other hand, since the chance of both \(G_{\mathbf{X}}\) and \(G_{\mathbf{Y}}\) being negative is usually negligible, the condition that \(G>0\) can be consequently replaced by the condition of both \(G_{\mathbf{X}}\) and \(G_{\mathbf{Y}}\) being positive. At this point, it is noted that the original structural system represented by \(G\) is translated to a weakest-link of two elements in which \(G_{\mathbf{X}}\) represents the element of random inputs and \(G_{\mathbf{Y}}\) represents the element of interval inputs (as shown in Figure 1). Since the safety state determination and reliability estimation of either the element of random inputs or the element of interval inputs can be properly addressed by well-established probability theory [25] or the non-probabilistic theory [26], respectively, such a task can be difficult for cases when random inputs and interval inputs are coupled, and it is evident that the advantage of the proposed method is in decorrelating the random inputs from the interval inputs, making the safety state determination or reliability estimation of probabilistic and non-probabilistic hybrid structural systems considerably easier. As for the determination of the system's safety state, due to the conservatism of non-probabilistic theory, the following strategies are proposed (shown in Figure 2): if the element of interval variables is unsafe, then the system, whose reliability is zero, is unsafe; otherwise, the system's safety state and reliability are identical to those of the element of random variables. This is because if the safety state of the element of interval values is safe, its corresponding reliability is close to 1 due to its conservative nature, and so the system's safety state and reliability can be represented by the corresponding value of the element of random variables. Further, since reliability index-based methods such as the first-order reliability method (FORM) [27] and second-order reliability method (SORM) [28] are widely used for safety state determinations, which are applicable to both random variables and interval variables, a corresponding determination method of a system's safety state is proposed in Table 1, where \(\beta\) and \(\eta\) represent the reliability indices for the element of random variables and the element of interval variables, respectively. Figure 1: Conceptional illustration of the proposed method. Figure 2: Flowchart of safety state determination. The target reliability index \(\alpha\) is taken from the specification requirements.