## 5 Numerical Experiments
The PSSO has the best convergent generation number of both system reliability and running time, i.e., **190**, and **23**. For the benchmark 2, the experimental results are analyzed as below and shown in Tables 6 and 7. 1. The adopted SSO has the best performance including Max, Min, Average, and STD of system reliability, i.e., **0.9999759790**, **0.9998471140**, **0.9999484850**, and **0.0000255121**. 2. The adopted SSO has the best performance including Min, Max, and Average of running time, i.e., **0.169000**, **0.224000**, and **0.193840**. 3. The PSSO has the best performance in STD of running time, i.e., **0.013011**. 4. The PSSO has the best convergent generation number of both system reliability and running time, i.e., **468**, and **222**. For the benchmark 3, the experimental results are analyzed as below and shown in Tables 8 and 9. 1. The adopted SSO has the best performance including Max, Min, Average, and STD of system reliability, i.e., **0.999886930**, **0.999592066**, **0.999814268**, and **0.000064727**. 2. The adopted SSO has the best performance including Min, Max, and Average of running time, i.e., **0.172000**, **0.219000**, and **0.193260**. 3. The PSO has the best performance in STD of running time, i.e., **0.010277**. 4. The SSO has the best convergent generation number of system reliability, i.e., **100**. 5. The PSO has the best convergent generation number of running time, i.e., **319**. For the benchmark 4, the experimental results are analyzed as below and shown in Tables 10 and 11. 1. The adopted SSO has the best performance including Max, Min, Average, and STD of system reliability, i.e., **0.999953926**, **0.999552667**, **0.99990330**, and **0.000084258**. 2. The adopted SSO has the best performance including Min, Max, Average, and STD of running time, i.e., **0.13900**, **0.17400**, **0.15506**, and **0.00809**. 3. The PSSO has the best convergent generation number of system reliability, i.e., **6**. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Statistics** & **SSO** & **PSO** & **PSSO** & **GA** \\ \hline Max (Best) & **0.999953926** & 0.999802172 & 0.999891579 & 0.998805583 \\ Gen\({}_{\text{convergent}}\) & 141 & 821 & **6** & 999 \\ Min (Worse) & 0.999552667 & 0.998303175 & 0.998482347 & 0.857600927 \\ Average & 0.99990330 & 0.999337146 & 0.999418706 & 0.965427958 \\ STD & **0.000084258** & 0.000349942 & 0.000315562 & 0.029557328 \\ \hline \hline \end{tabular} The best solution marked in bold.  
\end{table} 
Table 10: The statistics of system reliability for benchmark 4. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Statistics** & **SSO** & **PSO** & **PSSO** & **GA** \\ \hline Min (Best) & **0.13900** & 0.14900 & 0.15600 & 0.25800 \\ Gen\({}_{\text{convergent}}\) & **629** & 768 & 644 & 999 \\ Max (Worse) & **0.17400** & 0.20400 & 0.20000 & 0.33200 \\ Average & **0.15506** & 0.17540 & 0.17604 & 0.27862 \\ STD & **0.00809** & 0.01009 & 0.00999 & 0.01268 \\ \hline \hline \end{tabular} The best solution marked in bold.  
\end{table} 
Table 11: The statistics of running time (units used: seconds) for benchmark 4. The SSO has the best convergent generation number of running time, i.e., **629**. According to the above experimental results and analysis, the adopted SSO has the best performance including Min, Max, Average, and STD of both system reliability and running time for all four benchmarks of RRAP under fuzzy constraints. The reason why PSSO fails to find the good results because it is without selecting better parameters. However, The SSO, PSO, and PSSO have equal shares on the best convergent generation number of system reliability and running time for the four benchmarks of RRAP under fuzzy constraints.