## 5 Examples
For this example, it is hypothesized that the membership functions of the fuzzy variables \(L\), \(E\), \(P_{1}\), \(P_{2}\) and \(P_{3}\) are symmetric triangular distributions, while the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & \(\tau_{Z_{1}}\) & \(\tau_{Z_{2}}\) & \(\tau_{Z_{3}}\) & \(N^{\circ}\) \\ \hline MCS & \(1.162\times 10^{-3}\) & \(3.228\times 10^{-3}\) & \(1.336\times 10^{-3}\) & \(4\times 10^{8}\) \\ SDP & \(1.189\times 10^{-3}\) & \(3.124\times 10^{-3}\) & \(1.256\times 10^{-3}\) & 1500 \\ Error & \(2.32\%\) & \(3.23\%\) & \(5.98\%\) & — \\ \hline \hline \end{tabular}  
* Number of samples.  
\end{table} 
Table 1: Analysis results of the FFP-IM index, for example, 1. Figure 4: Example 1: Comparison of FFP-IM results obtained using the MCS and SDP methods. membership functions for \(A_{i}(i=1,2,\cdots,10)\) are Gaussian distributions. The membership functions for these variables are defined in the following:  
\[\mu_{L}(L)=\left\{ \begin{array}{l}(L-0.9)/\leq 0.1\ \ 0.9\leq L\leq\ 1\\ (L-1.1)/-0.1\leq L\leq 1.1\end{array} \right. ;\]  
\[\mu_{E}(E)=\left\{ \begin{array}{l}(E-90)/10\ \ \ 90\leq E\leq 100\\ (E-110)/-10\ 100\leq E\leq 110\end{array} \right. ;\]  
\[\mu_{P_{1}}(P_{1})=\left\{ \begin{array}{l}\left(P_{1}-7.6\times 10^{4} \right)/\left(4\times 10^{3}\right)\ 7.6\times 10^{4}\leq P_{1}\leq\ 8\times 10^{4}\\ \left(P_{1}-8.4\times 10^{4}\right)/\left(-4\times 10^{3}\right)\ 8\times 10^{4}\leq P_{1}\leq 8.4 \times 10^{4}\end{array} \right. ;\]  
\[\mu_{P_{2}}(P_{2})=\left\{ \begin{array}{l}\left(P_{2}-0.95\times 10^{4} \right)/\left(5\times 10^{2}\right)\ 0.95\times 10^{4}\leq P_{2}\leq\ 10^{4}\\ \left(P_{2}-1.05\times 10^{4}\right)/\left(-5\times 10^{2}\right)\ 10^{4}\leq P _{2}\leq 1.05\times 10^{4}\end{array} \right. ;\]  
\[\mu_{P_{3}}(P_{3})=\left\{ \begin{array}{l}\left(P_{3}-0.95\times 10^{4} \right)/\left(5\times 10^{2}\right)\ 0.95\times 10^{4}\leq P_{3}\leq\ 10^{4}\\ \left(P_{3}-1.05\times 10^{4}\right)/\left(-5\times 10^{2}\right)\ 10^{4}\leq P _{3}\leq 1.05\times 10^{4}\end{array} \right. ;\]  
\[\mu_{A_{i}}(A_{i})=\exp\left\{-\frac{\left(A_{i}-10^{-3}\right)^{2}}{2\times \left(10^{-4}\right)^{2}}\right\}\,(i=1,2,\ldots,10\right).\]  
The restriction condition dictates that the vertical shift in node 3 must not surpass 0.0043 m. In compliance with this condition, the LSF \(Y=g=0.0043-|\Delta_{2}|\) can be assembled, where \(\Delta_{2}\) is an implicit function of the input variables. This function is ascertained through FEA, as demonstrated in Figure 6.  
Figure 5: Schematic diagram of ten-bar structure. Figure 6: Finite element model of ten-rod bar structure. The FFP-IM index was evaluated using both the MCS and SDP methods, with the results detailed in Table 2 and Figure 7. The MCS method required \(5\ \times\ 10^{8}\) calls to the finite element model, resulting in a computational time of \(1.424\ \times\ 10^{9}\) s. In contrast, the SDP method achieved comparable accuracy with only \(3200\) calls, significantly reducing the computational time to \(9.110\ \times\ 10^{3}\) s. As illustrated in Figure 7, the SDP method demonstrated equivalent precision to the MCS approach but with markedly lower computational overhead. This enhanced efficiency is particularly advantageous in practical engineering applications, where computational resources are often a limiting factor, thereby significantly improving the feasibility and scalability of the proposed methodology. Observations from Table 2 and Figure 7 indicate that the fuzzy variable \(E,\ L,\ A_{1},\ A_{3},\ A_{7},\ A_{8},\) and \(P_{1}\) have major influences on the FFP and should be paid more attention to them. The influences of the fuzzy variables \(P_{2},\ A_{9},\ A_{5},\ P_{3},\ A_{2},\ A_{4},\ A_{6},\ A_{10}\) on the FFP are very small, so the uncertainty of these variables can be ignored, which can be regarded as deterministic variables.