# A Spring Search Algorithm Applied to Engineering Optimization Problems
## 7 Experimental Results and Discussion
### Algorithms Used for Comparison
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{_GA_} & \multicolumn{2}{c}{_PSO_} & \multicolumn{2}{c}{_GSA_} & \multicolumn{2}{c}{_TLEO_} & \multicolumn{2}{c}{_GOA_} & \multicolumn{2}{c}{_GWO_} & \multicolumn{2}{c}{_SHO_} & \multicolumn{2}{c}{_EPO_} & SSA \\ \hline \multirow{2}{*}{F\({}_{1}\)} & \multirow{2}{*}{Ave} & \(1.95\times 10^{-12}\) & \(4.98\times 10^{-49}\) & \(1.16\times 10^{-16}\) & \(3.55\times 10^{-42}\) & \(2.81\times 10^{-41}\) & \(7.86\times 10^{-10}\) & \(4.61\times 10^{-23}\) & \(5.71\times 10^{-28}\) & \(6.24\times 10^{-35}\) \\  & std & \(2.01\times 10^{-11}\) & \(1.40\times 10^{-48}\) & \(6.10\times 10^{-17}\) & \(1.06\times 10^{-41}\) & \(1.11\times 10^{-41}\) & \(8.11\times 10^{-49}\) & \(7.37\times 10^{-23}\) & \(8.31\times 10^{-29}\) & \(9.17\times 10^{-38}\) \\ \hline \multirow{2}{*}{F\({}_{2}\)} & \multirow{2}{*}{Ave} & \(6.53\times 10^{-18}\) & \(7.29\times 10^{-48}\) & \(1.70\times 10^{-41}\) & \(3.23\times 10^{-45}\) & \(3.96\times 10^{-41}\) & \(5.99\times 10^{-30}\) & \(1.20\times 10^{-34}\) & \(6.20\times 10^{-40}\) & \(7.8\times 10^{-45}\) \\  & std & \(5.10\times 10^{-17}\) & \(1.84\times 10^{-40}\) & \(9.29\times 10^{-48}\) & \(8.57\times 10^{-48}\) & \(1.41\times 10^{-41}\) & \(1.11\times 10^{-47}\) & \(1.30\times 10^{-44}\) & \(3.23\times 10^{-40}\) & \(3.48\times 10^{-45}\) \\ \hline \multirow{2}{*}{F\({}_{3}\)} & \multirow{2}{*}{Ave} & \(7.27\times 10^{-10}\) & \(1.40\times 10^{11}\) & \(4.16\times 10^{-42}\) & \(4.91\times 10^{-49}\) & \(4.31\times 10^{-41}\) & \(9.19\times 10^{-45}\) & \(1.00\times 10^{-44}\) & \(2.05\times 10^{-19}\) & \(2.63\times 10^{-25}\) \\  & std & \(7.36\times 10^{-40}\) & \(7.13\times 10^{-40}\) & \(1.56\times 10^{-42}\) & \(3.89\times 10^{-49}\) & \(8.97\times 10^{-40}\) & \(6.16\times 10^{-44}\) & \(4.10\times 10^{-44}\) & \(9.17\times 10^{-30}\) & \(9.83\times 10^{-27}\) \\ \hline \multirow{2}{*}{F\({}_{4}\)} & \multirow{2}{*}{Ave} & \(9.17\times 10^{11}\) & \(6.00\times 10^{-41}\) & \(1.12\times 10^{10}\) & \(1.87\times 10^{11}\) & \(8.80\times 10^{-41}\) & \(8.73\times 10^{-41}\) & \(2.02\times 10^{-14}\) & \(4.32\times 10^{-18}\) & \(4.65\times 10^{-28}\) \\  & std & \(5.67\times 10^{11}\) & \(1.72\times 10^{-40}\) & \(9.98\times 10^{-41}\) & \(8.21\times 10^{-40}\) & \(2.50\times 10^{-41}\) & \(1.19\times 10^{-41}\) & \(2.43\times 10^{-14}\) & \(3.98\times 10^{-19}\) & \(4.68\times 10^{-29}\) \\ \hline \multirow{2}{*}{F\({}_{5}\)} & \multirow{2}{*}{Ave} & \(5.55\times 10^{-42}\) & \(4.93\times 10^{-41}\) & \(3.85\times 10^{-41}\) & \(7.37\times 10^{-42}\) & \(1.18\times 10^{-42}\) & \(8.91\times 10^{-42}\) & \(2.97\times 10^{-41}\) & \(5.07\times 10^{-40}\) & \(5.41\times 10^{-41}\) \\  & std & \(4.16\times 10^{11}\) & \(3.89\times 10^{-41}\) & \(3.47\times 10^{-41}\) & \(1.98\times 10^{-40}\) & \(1.43\times 10^{-42}\) & \(2.97\times 10^{-42}\) & \(1.84\times 10^{-40}\) & \(4.80\times 10^{-41}\) & \(5.05\times 10^{-40}\) \\ \hline \multirow{2}{*}{F\({}_{6}\)} & \multirow{2}{*}{Ave} & \(3.15\times 10^{-40}\) & \(9.23\times 10^{-40}\) & \(1.08\times 10^{-16}\) & \(4.88\times 10^{-40}\) & \(3.15\times 10^{-41}\) & \(8.18\times 10^{-47}\) & \(6.58\times 10^{-41}\) & \(7.01\times 10^{-49}\) & \(8.0\times 10^{-42}\) \\  & std & \(9.98\times 10^{-40}\) & \(1.78\times 10^{-40}\) & \(4.00\times 10^{-17}\) & \(9.75\times 10^{-40}\) & \(9.98\times 10^{-40}\) & \(1.70\times 10^{-41}\) & \(3.38\times 10^{-40}\) & \(4.99\times 10^{-40}\) & \(5.22\times 10^{-30}\) \\ \hline \multirow{2}{*}{F\({}_{7}\)} & \multirow{2}{*}{Ave} & \(6.79\times 10^{-40}\) & \(6.92\times 10^{-40}\) & \(7.68\times 10^{-40}\) & \(2.02\times 10^{-40}\) & \(5.37\times 10^{-41}\) & \(7.80\times 10^{-41}\) & \(7.21\times 10^{-40}\) & \(3.33\times 10^{-40}\) \\  & std & \(3.29\times 10^{-4

\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & \multicolumn{2}{c}{_GA_} & PSO & _GSA_ & _TLBO_ & _GOA_ & _GWO_ & _SHO_ & _EPO_ & _SSA_ \\ \hline \multirow{2}{*}{Fe\({}_{\rm 3}\)} & \multicolumn{1}{c}{Ave} & \(-5.11\times 10^{22}\) & \(-5.01\times 10^{22}\) & \(-2.75\times 10^{22}\) & \(-3.81\times 10^{22}\) & \(-6.92\times 10^{22}\) & \(-4.69\times 10^{22}\) & \(-6.14\times 10^{22}\) & \(-8.76\times 10^{22}\) & \(-12.1\times 10^{22}\) \\  & std & \(4.37\times 10^{21}\) & \(4.28\times 10^{21}\) & \(5.72\times 10^{21}\) & \(2.83\times 10^{21}\) & \(9.19\times 10^{21}\) & \(3.94\times 10^{21}\) & \(9.32\times 10^{21}\) & \(5.92\times 10^{21}\) & \(9.14\times 10^{22}\) \\ \hline \multirow{2}{*}{Fe\({}_{\rm 5}\)} & \multicolumn{1}{c}{Ave} & \(1.23\times 10^{21}\) & \(1.20\times 10^{21}\) & \(3.35\times 10^{21}\) & \(2.23\times 10^{22}\) & \(1.01\times 10^{22}\) & \(4.85\times 10^{22}\) & \(4.34\times 10^{21}\) & \(6.90\times 10^{21}\) & \(8.76\times 10^{22}\) \\  & std & \(4.41\times 10^{21}\) & \(4.01\times 10^{21}\) & \(1.19\times 10^{21}\) & \(3.25\times 10^{21}\) & \(1.89\times 10^{21}\) & \(3.91\times 10^{21}\) & \(1.66\times 10^{22}\) & \(3.04\times 10^{22}\) & \(4.81\times 10^{22}\) \\ \hline \multirow{2}{*}{Fe\({}_{\rm 10}\)} & \multicolumn{1}{c}{Ave} & \(5.31\times 10^{21}\) & \(5.20\times 10^{21}\) & \(8.25\times 10^{20}\) & \(1.55\times 10^{201}\) & \(1.15\times 10^{20}\) & \(2.83\times 10^{22}\) & \(1.63\times 10^{22}\) & \(1.63\times 10^{22}\) & \(8.03\times 10^{22}\) \\  & std & \(1.11\times 10^{21}\) & \(1.08\times 10^{21}\) & \(1.90\times 10^{20}\) & \(8.11\times 10^{20}\) & \(8.77\times 10^{21}\) & \(4.34\times 10^{22}\) & \(3.14\times 10^{22}\) & \(2.74\times 10^{22}\) & \(3.34\times 10^{22}\) \\ \hline \multirow{2}{*}{Fe\({}_{\rm 11}\)} & \multicolumn{1}{c}{Ave} & \(3.31\times 10^{21}\) & \(1.75\times 10^{21}\) & \(1.76\times 10^{21}\) & \(1.09\times 10^{21}\) & \(1.33\times 10^{21}\) & \(3.86\times 10^{22}\) & \(9.20\times 10^{22}\) & \(9.13\times 10^{22}\) & \(9.81\times 10^{22}\) \\  & std & \(1.86\times 10^{21}\) & \(2.79\times 10^{22}\) & \(2.30\times 10^{22}\) & \(2.81\times 10^{22}\) & \(3.45\times 10^{22}\) & \(1.26\times 10^{22}\) & \(1.78\times 10^{22}\) & \(1.85\times 10^{22}\) & \(1.66\times 10^{22}\) \\ \hline \multirow{2}{*}{Fe\({}_{\rm 23}\)} & \multicolumn{1}{c}{Ave} & \(2.91\times 10^{21}\) & \(3.91\times 10^{21}\) & \(2.30\times 10^{21}\) & \(3.82\times 10^{21}\) & \(7.35\times 10^{21}\) & \(1.86\times 10^{22}\) & \(2.26\times 10^{22}\) & \(1.29\times 10^{22}\) & \(1.05\times 10^{22}\) \\  & std & \(1.67\times 10^{21}\) & \(2.70\times 10^{21}\) & \(2.41\times 10^{22}\) & \(2.44\times 10^{22}\) & \(3.82\times 10^{22}\) & \(1.93\times 10^{22}\) & \(2.45\times 10^{22}\) & \(1.15\times 10^{22}\) & \(1.85\times 10^{22}\) \\ \hline \multirow{2}{*}{Fe\({}_{\rm 12}\)} & \multicolumn{1}{c}{Ave} & \(7.08\times 10^{21}\) & \(7.08\times 10^{21}\) & \(7.06\times 10^{22}\) & \(7.02\times 10^{22}\) & \(7.02\times 10^{22}\) & \(7.02\times 10^{22}\) & \(7.02\times 10^{22}\) & \(7.02\times 10^{22}\) & \(7.02\times 10^{22}\) & \(7.02\times 10^{22}\) \\  & std & \(2.97\times 10^{21}\) & \(1.32\times 10^{21}\) & \(9.07\times 10^{21}\) & \(9.40\times 10^{21}\) & \(1.10\times 10^{21}\) & \(7.75\times 10^{21}\) & \(7.07\times 10^{21}\) & \(6.76\times 10^{22}\) & \(5.50\times 10^{21}\) \\ \hline \multirow{2}{*}{Fe\({}_{\rm 24}\)} & \multicolumn{1}{c}{Ave} & \(5.79\times 10^{21}\) & \(6.07\times 10^{21}\) & \(6.73\times 10^{21}\) & \(2.58\times 10^{21}\) & \(9.93\times 10^{21}\) & \(3.43\times 10^{21}\) & \(3.49\times 10^{21}\) & \(1.86\times 10^{21}\) & \(1.47\times 10^{21}\) \\  & std & \(2.76\times 10^{21}\) & \(4.81\times 10^{21}\) & \(3.36\times 10^{21}\) & \(1.61\times 10^{21}\) & \(8.74\times 10^{21}\) & \(2.27\times 10^{22}\) & \(2.04\times 10^{22}\) & \(1.04\times 10^{22}\) & \(1.98\times 10^{21}\) & \(1.34\times 10^{21}\) \\ \hline \multirow{2}{*}{Fe\({}_{\rm 13}\)} & \multicolumn{1}{c}{Ave} & \(1.26\times 10^{21}\) & \(1.75\times 10^{21}\) & \(1.76\times 10^{21}\) & \(1.09\times 10^{21}\) & \(1.33\times 10^{21}\) & \(8.65\times 10^{21}\) & \(9.20\times 10^{22}\) & \(9.13\times 10^{21}\) & \(9.81\times 10^{22}\) \\  & std & \(1.86\times 10^{21}\) & \(2.79\times 10^{22}\) & \(2.30\times 10^{22}\) & \(2.81\times 10^{22}\) & \(3.45\times 10^{22}\) & \(1.26\times 10^{22}\) & \(1.78\times 10^{22}\) & \(1.85\times



### Welded Beam Design

The mathematical model of a welded beam design was adapted from [31].