## 3 Numerical Example
### Seismic Isolation and Optimization Parameters  
The constrained optimization processes include objective function, variables, and constraints. In the present study, the energy components and roof floor acceleration are objective functions as described in Section 2.1. As in many studies in nonlinear seismic isolation modeling, \(T_{0}\) and \(F_{Q}/W\) are used for search agents. For the constraints, isolation displacement \((D)\), effective damping \((\beta)\), and inter-story drift ratio \((ISDR)\) are chosen. The range of \(T_{0}\) was set to vary between 2 s and 4 s, which is in a typical range for many seismically isolated structures. Similarly, the range of \(F_{Q}/W\) 0.03 and 0.15 was considered for optimal parameters [17; 20; 59; 60]. The same \(T_{0}\) and \(F_{Q}/W\) ranges were used in all optimization process cases. Firstly, energy components were compared with PRA/PGA. By performing additional analyses with the R_En case, optimized isolation results and the effectiveness of the structure responses were determined. While the constraint parameters of R_En, \(D\), and \(\beta\) are applied differently, \(ISDR\) is kept constant. In a scenario with limited horizontal displacement, \(D\) is decreased while \(\beta\) is increased. Recommendations in the Turkish building earthquake code were taken into account for \(D\), \(\beta\), and \(ISDR\) restrictions [9]. In the GWO, the population size is set to 25. Each solution contains 11 nonlinear time history analyses. The termination criterion is determined as 100 iterations. For each of the cases, three independent runs were performed with each objective function. The result that best performs the objective function is presented. The summary of the seismic isolation and optimization parameters used in this study is given in Table 4.  
For the cases given in Table 4, 11 time history analyses were performed using the scaled ground motions given in Section 3.2. The peak value of the 11 ground motions was \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Case**} & \multicolumn{3}{c}{**Searching Isolation**} & \multirow{3}{*}{**Constraints**} & \multirow{3}{*}{**Ground**} & \multirow{3}{*}{**GWO**} \\  & & & & & & & & **Motion** & & **Population Size** & **Iteration** \\ \cline{2-2} \cline{6-10}  & \(T_{0}\) **(s)** & & & & & & & & **Population Size** & **Iteration** \\ \hline Hys\_En & & & & & & & & & & \\ Imp\_En & & & & \(<\)0.50 & \(<\)30 & & & & & \\ R\_En & & & & & & & & & & \\ PRA/PGA & 2–4 & 0.03–0.015 & & & \(<\)0.01 & 11 & 25 & 100 \\ R\_En\_45\_40 & & & & \(<\)0.45 & \(<\)40 & & & & & \\ R\_En\_40\_50 & & & & \(<\)0.40 & \(<\)50 & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Seismic isolation and optimization parameters. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**No** & **RSN** & **Earthquake** & **Location** & **Year** & **Mw** & **Component** & **PGA** & **PGV** & **PGD** & **Scale** \\ \hline
1 & 879 & Landers & Lucerne & 1992 & 7.28 & LCN260 & 0.73 & 133.4 & 113.93 & 1.3 \\
2 & 828 & Cape Mendocino & Petrolia & 1992 & 7.01 & PET090 & 0.662 & 88.51 & 33.22 & 1.5 \\
3 & 2114 & Denali, Alaska & TAPS Pump St. \#10 & 2002 & 7.9 & PS10-047 & 0.333 & 115.72 & 55.44 & 1.1 \\
4 & 6906 & Darfield & GDLC & 2010 & 7 & N55W & 0.765 & 116.1 & 100.39 & 1.1 \\
5 & 4451 & Montenegro, Yug. & Bar-Skuptsina Op. & 1979 & 7.1 & BSO009 & 0.368 & 52.82 & 15.98 & 1.3 \\
6 & 1165 & Kocaeli & Izmit & 1999 & 7.51 & IZT090 & 0.23 & 38.29 & 24.29 & 3.5 \\
7 & 779 & Loma Prieta & LGPC & 1989 & 6.93 & LGP000 & 0.57 & 96.1 & 41.9 & 1.1 \\
8 & 4040 & Iran & Bam & 2003 & 6.6 & BAM-L & 0.808 & 124.12 & 33.94 & 1.5 \\
9 & 1787 & Hector Mine & Hector & 1999 & 7.13 & HEC090 & 0.328 & 44.78 & 10.7 & 3.5 \\
10 & 1617 & Düzce & Lamont 375 & 1999 & 7.14 & 375-E & 0.5136 & 20.48 & 7.43 & 3.5 \\
11 & 143 & Tabas. Iran & Tabas & 1978 & 7.35 & TAB-L1 & 0.854 & 98.848 & 37.53 & 1.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Earthquake ground motions and parameters [55]. used in the objective functions.