## 4 Numerical Examples
### Engineering Cases  
#### 4.4.1 Case 1: Double Slope, Multilayered, Two Random Variables  
The first engineering case is a two-layered cohesive slope. Safety of the slope is controlled by the two state variables, i.e., the cohesions of its two layers. Slope dimension and layer conditions were shown in Figure 10. state variables of case 1 are shown in Table 5. Performance function evaluation was realized by slip surface stress method as described in literature [1; 36; 37]. The stress field of the slope was obtained through a FEM model with 5728 elements.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithms** & & \(\mathbf{Z_{1}}\) & \(\mathbf{Z_{2}}\) & \(\mathbf{Z_{3}}\) & \(\mathbf{Z_{4}}\) \\ \hline \multirow{3}{*}{NMMSO} & mean & 0.2795 & 0.2230 & 0.1048 & 0.1783 \\  & std & (0.0980)+ & (0.3901)+ & (0.0725)+ & (0.0707)+ \\ \multirow{3}{*}{Multi\_AMO} & mean & 0.6270 & 0.3919 & 0.7226 & 0.1808 \\  & std & (0.0549)+ & (0.1003)+ & (0.0352)+ & (0.0642)+ \\ \multirow{3}{*}{DP-MSCC-ES} & mean & 0.1255 & 0.2646 & 0.0496 & 0.1888 \\  & std & (0.1745)+ & (0.3114)+ & (0.0887)+ & (0.0546)+ \\ \multirow{3}{*}{ RSCSAESII} & mean & 0.0250 & 0.0597 & 0.0075 & 0.0083 \\  & std & (0.3040)+ & (10.5494)+ & (0.0952)+ & (0.1928)+ \\ \multirow{3}{*}{NSPSO} & mean & **0.8985** & **0.9908** & **0.9516** & **0.9333** \\  & std & (0.0337) & (0.0107) & (0.0136) & (0.0135) \\ \multirow{3}{*}{\(+/-/-\)} & \multirow{3}{*}{4/0/0} & \multirow{3}{*}{4/0/0} & \multirow{3}{*}{4/0/0} & \multirow{3}{*}{4/0/0} & \multirow{3}{*}{4/0/0} \\  & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Statistical results of \(EOR\) of different algorithms for four explicit performance functions.  
Figure 8: Optimization history for all algorithms (**a**) NMMSO. (**b**) Multi_AMO. (**c**) DP-MSCC-ES. (**d**) RSCMSAESII. (**e**) NSPSO.  
Optimization results of all algorithms were shown in Figure 11. It can be seen that the result of NSPSO have a more uniform and better coverage of the LSF, while other competitive algorithms have a clustering problem to a greater or lesser extent. The found LSF points show a clear pattern. When \(x_{2}\) (strength of layer 2) is big enough, the safety of the slope will be completely decided by \(x_{1}\) (strength of layer 1) since slip surface will not overlap with layer 2. When \(x_{2}\) is small, slip surface will go through both layers, so the LSF is related to both \(x_{1}\) and \(x_{2}\). To show the superiority of the proposed method, a comparison analysis was also conducted like Section 4.3. The resulting optimal points of all algorithms were shown in Figure 11. A huge amount of performance function calculation was carried out to obtain the true LSF of case 1 illustrated as solid lines in Figure 11. Apparently, NSPSO outperformed all competitive algorithms. Performance indicator results in Table 6 also indicates that the NSPSO produced the best coverage and diversity.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Slope Layer** & **Parameters** & **Mean** & **COV** & **Distribution Type** \\ \hline \multirow{2}{*}{Top layer} & \(c_{1}\) [kPa] & 120 & 0.3 & LogNormal \\  & \(\gamma_{1}\) [kN/m\({}^{3}\)] & 19 & 0 & Constant \\ \multirow{2}{*}{Bottom layer} & \(c_{2}\) [kPa] & 160 & 0.3 & LogNormal \\  & \(\gamma_{2}\) [kN/m\({}^{3}\)] & 19 & 0 & Constant \\ \hline \hline \end{tabular}
\end{table}
Table 5: Statistical information of soil parameters for engineer case \(1\).  
Figure 10: Geometry of the slope and statistics of soil parameters for engineer case \(1\).  
Figure 9: Results from NSPSO with different sampling density. (**a**) Lower density. (**b**) Higher density.