# A Spring Search Algorithm Applied to Engineering Optimization Problems
## 7 Experimental Results and Discussion
### Algorithms Used for Comparison
The results to this optimization problem are presented in Tables 9 and 10. The SSA provides an optimal solution at (0.205411, 3.472341, 9.035215, 0.201153) with a corresponding fitness value equal: 1.723589. ### Tension/Compression Spring Design Problem

The mathematical model of this problem was adapted from [31]. The results to this optimization problem are displayed in Tables 11 and 12. The SSA provides the optimal solution at (0.051087, 0.342908, 12.0898), with a corresponding fitness value of 0.012656987. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Algorithms**} & \multicolumn{6}{c}{**Optimum Variables**} & \multirow{2}{*}{**Optimum Cost**} \\ \cline{2-2} \cline{5-8}  & \(b\) & \(m\) & \(p\) & \(l_{1}\) & \(l_{2}\) & \(d_{1}\) & \(d_{2}\) \\ \hline _SSA_ & 3.50123 & 0.7 & 17 & 7.3 & 7.8 & 3.33421 & 5.26536 & 2994.2472 \\ _EPO_ & 3.50159 & 0.7 & 17 & 7.3 & 7.8 & 3.35127 & 5.28874 & 2998.5507 \\ _SHO_ & 3.506690 & 0.7 & 17 & 7.380933 & 7.815726 & 3.357847 & 5.286768 & 3001.288 \\ _GOA_ & 3.500019 & 0.7 & 17 & 8.3 & 7.8 & 3.352412 & 5.286715 & 3005.763 \\ _GWO_ & 3.508502 & 0.7 & 17 & 7.392843 & 7.816034 & 3.358073 & 5.286777 & 3002.928 \\ _TLBO_ & 3.508755 & 0.7 & 17 & 7.3 & 7.8 & 3.461020 & 5.289213 & 3030.563 \\ _GSA_ & 3.600000 & 0.7 & 17 & 8.3 & 7.8 & 3.369658 & 5.289224 & 3051.120 \\ _PSO_ & 3.510253 & 0.7 & 17 & 8.35 & 7.8 & 3.362201 & 5.287723 & 3067.561 \\ _GA_ & 3.520124 & 0.7 & 17 & 8.37 & 7.8 & 3.366970 & 5.288719 & 3029.002 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison results for speed reducer design problem. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithms** & **Best** & **Mean** & **Worst** & **Std. Dev.** & **Median** \\ \hline _SSA_ & 2994.2472 & 2997.482 & 2999.092 & 1.78091 & 2996.318 \\ _EPO_ & 2998.5507 & 2999.640 & 3003.889 & 1.93193 & 2999.187 \\ _SHO_ & 3001.288 & 3005.845 & 3008.752 & 5.83794 & 3004.519 \\ _GOA_ & 3005.763 & 3105.252 & 3211.174 & 79.6381 & 3105.252 \\ _GWO_ & 3002.928 & 3028.841 & 3060.958 & 13.0186 & 3027.031 \\ _TLBO_ & 3030.563 & 3065.917 & 3104.779 & 18.0742 & 3065.609 \\ _GSA_ & 3051.120 & 3170.334 & 3363.873 & 92.5726 & 3156.752 \\ _PSO_ & 3067.561 & 3186.523 & 3313.199 & 17.1186 & 3198.187 \\ _GA_ & 3029.002 & 3295.329 & 3619.465 & 57.0235 & 3288.657 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Statistical results for speed reducer design problem. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Algorithms**} & \multicolumn{4}{c}{**Optimum Variables**} & \multirow{2}{*}{**Optimum Cost**} \\ \cline{2-2} \cline{5-6}  & \(h\) & \(I\) & \(t\) & \(b\) & \\ \hline _SSA_ & 0.205411 & 3.472341 & 9.035215 & 0.201153 & 1.723589 \\ _EPO_ & 0.205563 & 3.474846 & 9.035799 & 0.205811 & 1.725661 \\ _SHO_ & 0.205678 & 3.475403 & 9.036964 & 0.206229 & 1.726995 \\ _GOA_ & 0.197411 & 3.315061 & 10.00000 & 0.201395 & 1.820395 \\ _GWO_ & 0.205611 & 3.472103 & 9.040931 & 0.205709 & 1.725472 \\ _TLBO_ & 0.204695 & 3.536291 & 9.004290 & 0.210025 & 1.759173 \\ _GSA_ & 0.147098 & 5.490744 & 10.00000 & 0.217725 & 2.172858 \\ _PSO_ & 0.164171 & 4.032541 & 10.00000 & 0.223647 & 1.873971 \\ _GA_ & 0.206487 & 3.635872 & 10.00000 & 0.203249 & 1.836250 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Comparison results for welded beam design problem. ### Rolling Element Bearing Design Problem

The mathematical model of this problem is adapted from [92]. The results of this optimization problem are included in Tables 13 and 14 and prove that the SSA provides an optimal solution at (125, 21.41890, 10.94113, 0.515, 0.515, 0.4, 0.7, 0.3, 0.02, 0.6) with a corresponding fitness value equal to 85,067.983.