# Adaptive Salp Swarm Algorithm for Optimization of Geotechnical Structures
## 7 Design Examples
### Spread Footing Optimization  
The first two design examples were concerned with the best design for a dry sand inner surface spread footing. Table 14 lists the other input parameters for such case studies. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Lower Bound** & **Upper Bound** \\ \hline Width of footing & \(B_{\min}=0.4\,H\) & \(B_{\max}=0.7\,H\) \\ \hline Thickness of base slab & \(X_{\min}=H/12\) & \(X_{\max}=H/10\) \\ \hline Width of toe & \(X_{\min}=0.4\,H/3\) & \(X_{\max}=0.7\,H/3\) \\ \hline Stem thickness at top & \(X_{\min}=20\) cm & - \\ \hline Steel reinforcement ratio & \(\rho_{\min}=max\left\{\frac{14}{f_{y}}.0.25\frac{\sqrt{f_{z}}}{f_{y}}\right\}\) & \(\rho_{max}=0.85\beta_{1}\frac{f_{y}^{\prime}}{f_{y}}\left(\frac{600}{600\cdot 1 7_{y}}\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Upper bound and lower bound for design variables of retaining wall. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Unit** & **Value for Example 1** & **Value for Example 2** \\ \hline Effective friction angle of base soil & degree & 35 & 30 \\ Unit weight of base soil & kN/m\({}^{3}\) & 18.5 & 18.5 \\ Young’s modulus & MPa & 50 & 35 \\ Poisson’s ratio & – & 0.3 & 0.3 \\ Vertical dead load (\(D\)) & kN & 2000 & 4200 \\ Vertical live load (\(L\)) & kN & 1000 & 2100 \\ Moment (\(M\)) & kN-m & 0.0 & 850 \\ Concrete cover & cm & 7.0 & 7.0 \\ Yield strength of reinforcing steel & MPa & 400 & 400 \\ Compressive strength of concrete & MPa & 30 & 28 \\ Factor of safety for bearing & – & 3.0 & 3.0 \\ capacity & – & 0.04 & 0.04 \\ Allowable settlement of footing & m & 0.04 & 0.04 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Input parameters for design examples 1 and 2. Tables 18 and 19 show the results of the assessments for the examples with the lowest cost. Tables 18 and 19 show that, when compared to the traditional SSA and other methods, the ASSA may be able to provide a better solution by calculating lower values of the objective functions. It can be observed that the ASSA's best price was relatively lower than that of the SSA and significantly lower than that of the PSO and FA. However, additional experiments revealed that increasing the maximum number of iterations reduced the distinctions between the algorithm results. The fact that the best solution was found in the first iteration was due to the effective modification of the algorithm proposed in this study. The modified algorithm demonstrated a much-enhanced efficacy. In order to determine the statistical significance of the comparative results between the considered algorithms in all the design examples, a nonparametric Wilcoxon's rank \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Design** & **Unit** & \begin{tabular}{c} **Optimum** \\ **Values ASSA** \\ \end{tabular} & \begin{tabular}{c} **Optimum** \\ **Values SSA** \\ \end{tabular} & \
\begin{tabular}{c} **Optimum** \\ **Values FA** \\ \end{tabular} &
\begin{tabular}{c} **Optimum** \\ **Values PSO** \\ \end{tabular} \\ \hline (\(X_{1}\)) & m & 0.7233 & 0.6947 & 0.6948 & 0.6436 \\ (\(X_{2}\)) & m & 0.2 & 0.2 & 0.25 \\ (\(X_{3}\)) & m & 0.4674 & 0.5 & 0.5 & 0.55 \\ (\(X_{4}\)) & m & 0.7778 & 0.7778 & 0.7778 \\ (\(X_{5}\)) & m & 0.2727 & 0.2727 & 0.2723 & 0.2727 \\ (\(X_{6}\)) & cm\({}^{2}\)/m & 6.67 & 6.66 & 6.66 & 6.66 \\ (\(X_{7}\)) & cm\({}^{2}\)/m & 6.75 & 6.75 & 6.75 & 6.75 \\ (\(X_{8}\)) & cm\({}^{2}\)/m & 6.75 & 6.75 & 6.75 & 6.75 \\ \hline Objective & \multirow{2}{*}{USD/m} & \multirow{2}{*}{822.73} & \multirow{2}{*}{827.02} & \multirow{2}{*}{860.42} & \multirow{2}{*}{848.17} \\ function & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 18: Optimization results for design example 3.