# An Efficient Reliability Method with Multiple Shape Parameters Based on Radial Basis Function
## 4 Case Study
### A Series System with Four Branches  
For example 1, a series system with four branches was used to validate the proposed model. In the system, the input valuables \(x_{1},x_{2}\) obeyed a standard normal distribution, and the performance function of example 1 is as follows:  
\[g(x_{1},x_{2})=\min\left\{\begin{array}{l}3+0.1(x_{1}-x_{2})2-(x_{1}+x_{2})/ \sqrt{2}\\ 3+0.1(x_{1}-x_{2})2+(x_{1}+x_{2})/\sqrt{2}\\ (x_{1}-x_{2})+6/\sqrt{2}\\ (x_{2}-x_{1})+6/\sqrt{2}\end{array}\right. \tag{23}\]  
First, set the \(\alpha=1,\varepsilon=10^{-5},n_{mc}=10^{6}\) and the initial DoE to eight fixed sample points. Then, launch MSRBF-MCS with different types of kernel functions for example 1. Figure 6 shows the results of the above simulation, in which (a), (c), and (e) show the LSFs and the newly added sample points (triangle dots) of MSRBF-MCS + G, MSRBF-MCS + I and MSRBF-MCS + IM, respectively, and (b), (d), and (f) show their convergence curves. As we can see in Figure 6, the shape of LSF by MSRBF-MCS + M is most similar to that of the real RBF. In contrast, the shape of LSF by MSRBF-MCS + G and MSRBF-MCS + IM is distorted. Especially, the area enclosed by the LSF of MSRBF-MCS + IM is significantly larger than that of the true LSF, and this is why the failure probability by MSRBF-MCS + IM is obviously smaller than the failure probability by direct MCS. In addition, compared with the convergence curves of MSRBF-MCS + G and MSRBF-MCS + IM, the convergence curve of MSRBF-MCS + M has a smaller fluctuation and similarly high prediction accuracy. Therefore, it can be concluded that MSRBF-MCS + M has the best performance. Then, the effect of different \(\alpha\) on the performance of MSRBF-MCS + M is examined by setting \(\alpha=0.1,0.3,0.5,0.7\) and \(0.9\), respectively.  
In fact, many scholars have used this example to validate their proposed reliability model, among them, the recent metamodels ARBFM-MCS [25], CVRBF-MCS [25], RBF-GA [29], and LIF [35] have achieved good results. A comparison between the proposed metamodel and the above metamodels is listed in Table 2. From Table 2, one can imply, in this example, a positive correlation between the prediction accuracy of the MSRBF-MCS + M and the value of \(\alpha\) is significant. The prediction error of the algorithm is consistently greater than 4% when the alpha is 0.1, 0.3, and 0.5, while it immediately drops to within 2% when the alpha is 0.7, 0.9, and 1. This indicates that enhancing the effect of the penalty function \(d_{penalty}(X)\) to some degree may help to improve the performance of the method. MSRBF-MCS + M performs optimally when \(\alpha=1\), with a mean of \(n_{call}=36.3\) (smaller than the results of ARBFM-MCS and CVRBF-MCS, but larger than the results of RBF-GA) and prediction error \(e_{pr}=1.43\%\) (only smaller than the results of CVRBF-MCS). Although the method in this paper does not provide the best results for all of the perspectives, it is prone to achieving a good balance in terms of efficiency and prediction accuracy.  
Figure 6: LSFs and convergence curves of example 1. (**a**) LSF of MSRBF-MCS + G. (**b**) Convergence curve of MSRBF-MCS + G. (**c**) LSF of MSRBF-MCS + M. (**d**) Convergence curve of MSRBF-MCS + M. (**e**) LSF of MSRBF-MCS + IM. (**f**) Convergence curve of MSRBF-MCS + IM.