## 4 Numerical Example
### Optimization Results
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c} \hline \hline \multicolumn{11}{c}{\multirow{2}{*}{**Antimuable**}} & \multicolumn{1}{c}{**DEM**} & \multicolumn{1}{c}{**ES**} & \multicolumn{1}{c}{**WOA**} & \multicolumn{1}{c}{**DEM**} & \multicolumn{1}{c}{**ES**} & \multicolumn{1}{c}{**WOA**} \\ \cline{2-11} \multicolumn{1}{c}{**Ground Motion**} & \(\xi_{A}\) & \(f\) & \multicolumn{1}{c}{\(\underbrace{Time}_{\textbf{(a)}}\)} & \(\zeta_{A}\) & \(f\) & \multicolumn{1}{c}{\(\underbrace{Time}_{\textbf{(a)}}\)} & \(\zeta_{A}\) & \(f\) & \multicolumn{1}{c}{\(\underbrace{Time}_{\textbf{(a)}}\)} & \(\zeta_{A}\) & \(f\) & \multicolumn{1}{c}{\(\underbrace{Time}_{\textbf{(a)}}\)} & \(\zeta_{A}\) & \(f\) & \multicolumn{1}{c}{\(\underbrace{Time}_{\textbf{(a)}}\)} & \(\zeta_{A}\) & \(f\) & \multicolumn{1}{c}{\(\underbrace{Time}_{\textbf{(a)}}\)} \\ \hline El centre & 0.48 & 2.00 & 3821.31 & 0.50 & 2.00 & 18934.41 & 0.50 & 2.00 & 1721.31 & 0.40 & 2.00 & 3786.23 & 0.39 & 2.00 & 1867.96 & 0.39 & 2.00 & 1864.05 \\ Kobe & 0.01 & 0.95 & 2135.75 & 0.00 & 0.94 & 10582.85 & 0.01 & 0.94 & 9620.55 & 0.36 & 1.40 & 2008.20 & 0.35 & 1.38 & 1042.45 & 0.40 & 1.31 & 9208.09 \\ Lema Prieta & 0.03 & 1.09 & 2543.40 & 0.01 & 0.96 & 10689.82 & 0.00 & 0.95 & 12801.0 & 0.20 & 0.98 & 2700.10 & 0.16 & 0.89 & 1875.78 & 0.15 & 0.89 & 1235.60 \\ Northridge & 0.50 & 2.00 & 1064.68 & 0.50 & 2.00 & 5275.42 & 0.50 & 2.00 & 475.58 & 0.00 & 1.21 & 1004.97 & 0.50 & 2.00 & 5195.61 & 0.50 & 2.00 & 462.65 \\ \hline \multicolumn{11}{c}{**OLAZ**} \\ \hline El centre & 0.21 & 0.50 & 3898.77 & 0.18 & 0.91 & 19003.27 & 0.18 & 0.91 & 1677.40 & 0.33 & 0.98 & 3886.45 & 0.34 & 0.98 & 18592.44 & 0.35 & 0.97 & 1669.75 \\ Kobe & 0.07 & 1.01 & 2270.04 & 0.07 & 1.00 & 10621.03 & 0.07 & 1.00 & 1975.51 & 0.13 & 1.01 & 2133.04 & 0.37 & 1.01 & 10593.20 & 0.23 & 1.01 & 959.03 \\ Lema Prieta & 0.13 & 0.53 & 2904.0 & 0.10 & 1414.06 & 0.10 & 0.90 & 1284.84 & 0.16 & 0.86 & 2803.04 & 0.14 & 0.86 & 19053.23 & 0.16 & 0.85 & 1264.07 \\ Northridge & 0.14 & 0.53 & 1086.28 & 0.12 & 0.02 & 5844.60 & 0.11 & 0.92 & 4673.5 & 0.22 & 0.84 & 1003.32 & 0.22 & 0.83 & 5235.86 & 0.22 & 0.84 & 0.73 \\ \hline \end{tabular}
\end{table} 
Table 3: TMD design parameters optimized by objective function OA1 and OA2. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{12}{c}{\(\mu=0.02\)} & \multicolumn{12}{c}{\(\mu=0.05\)} \\ \cline{2-13} \multicolumn{1}{c}{} & \multicolumn{3}{c}{**DEM**} & \multicolumn{3}{c}{**ES**} & \multicolumn{3}{c}{**WOA**} & \multicolumn{3}{c}{**DEM**} & \multicolumn{3}{c}{**ES**} & \multicolumn{3}{c}{**WOA**} \\ \cline{2-13}
**Eurbative** & \multicolumn{3}{c}{**Edm**} & \multicolumn{3}{c}{**Time**} & \multicolumn{3}{c}{**\(\mathcal{L}_{d}\)**} & \multicolumn{3}{c}{**Time**} & \multicolumn{3}{c}{**\(\mathcal{L}_{d}\)**} & \multicolumn{3}{c}{**\(f\)**} & \multicolumn{3}{c}{**Time**} & \multicolumn{3}{c}{**\(\mathcal{L}_{d}\)**} & \multicolumn{3}{c}{**\(f\)**} & \multicolumn{3}{c}{**Time**} \\ \cline{2-13}
**Ground Motion** & \(\mathcal{L}_{d}\) & \(f\) & **(a)** & \(\mathcal{L}_{d}\) & \(f\) & **(b)** & \(\mathcal{L}_{d}\) & \(f\)