# Adaptive Salp Swarm Algorithm for Optimization of Geotechnical Structures
## 4 Model Verification  
A set of numerical reference test functions was used in this section to compare and confirm the achievement and effectiveness of the proposed adaptive salp swarm algorithm (ASSA). In the empirical evidence literature, these functions have commonly been used to determine the performance of optimizers [51; 52]. The mathematical models and characteristics of these test functions are shown in Tables 2 and 3. This standard set was divided into two categories: (1) unimodal functions with a single global best for testing the algorithm convergence pace and enslavement ability and (2) multimodal functions with multiple local minimums and a global ideal for testing an algorithm's local optima avoidance and exploratory capacity. MATLAB R2020b was used to create the suggested algorithms. All these functions should be minimized. Furthermore, all the functions had dimensions of 30. Three-dimensional drawings of these benchmark functions are illustrated in Figures 2 and 3. Figure 1: Flowchart of ASSA. The proposed ASSA was compared to the original SSA, as well as to some well-known optimization methods, such as the genetic algorithm developed by [53], the particle swarm optimization (PSO) proposed by [54], the firefly algorithm (FA) introduced by [55], the multiverse optimizer (MVO) developed by [56], and the tunicate swarm algorithm (TSA) introduced by [52]. For all methodologies, the sizes of the solutions (\(N\)) and the maximum number of iterations (\(t_{max}\)) were set to 30 and 1000, respectively, in order to make fair comparisons between them. Because the results of a single run of a metaheuristic method are stochastic, they may be incorrect. As a result, statistical analysis should be performed in order to provide a fair comparison and evaluate an algorithm's efficacy. To address this issue, 30 runs for the mentioned methods were performed, with the results presented in Tables 4 and 5. \begin{table}
\begin{tabular}{c c c c} \hline
**Function** & **Range** & \(f_{min}\) & \(n\) **(Dim)** \\ \hline \(F_{1}(X)=\sum_{i=1}^{n}x_{i}^{2}\) & \([-100,\;100]^{n}\) & 0 & 30 \\ \hline \(F_{2}(X)=\sum_{i=1}^{n}|x_{i}|+\prod_{i=1}^{n}|x_{i}|\) & \([-10,\;10]^{n}\) & 0 & 30 \\ \hline \(F_{3}(X)=\sum_{i=1}^{n}\left(\sum_{j=1}^{i}x_{j}\right)^{2}\) & \([-100,\;100]^{n}\) & 0 & 30 \\ \hline \(F_{4}(X)=\max\limits_{i}\left\{|x_{i}|,\;1\leq i\leq n\;\right\}\) & \([-100,\;100]^{n}\) & 0 & 30 \\ \hline \(F_{5}(X)=\sum_{i=1}^{n-1}\left[100\big{(}x_{i+1}-x_{i}^{2}\big{)}^{2}+(x_{i}- 1)^{2}\right]\) & \([-30,\;30]^{n}\) & 0 & 30 \\ \hline \(F_{6}(X)=\sum_{i=1}^{n}\left([x_{i}+0.5]\right)^{2}\) & \([-100,\;100]^{n}\) & 0 & 30 \\ \hline \(F_{7}(X)=\sum_{i=1}^{n}ix_{i}^{4}+random[0,1)\) & \([-1.28,\;1.28]^{n}\) & 0 & 30 \\ \hline \end{tabular}
\end{table} 
Table 2: Description of unimodal benchmark functions. \begin{table}
\begin{tabular}{c c c c} \hline
**Function** & **Range** & \(f_{min}\) & \(n\) **(Dim)** \\ \hline \(F_{8}(X)=\sum_{i=1}^{n}-x_{i}\sin\left(\sqrt{|x_{i}|}\right)\) & \([-500,\;500]^{n}\) & 428.9829 \(\times\)\(n\) & 30 \\ \hline \(F_{9}(X)=\sum_{i=1}^{n}[x_{i}^{2}-10\cos(2\pi x_{i})+10]\)Figure 2: 3-D versions of unimodal benchmark functions. Figure 3: 3-D versions of multimodal benchmark functions. Tables 4 and 5 show that, for all the functions, the ASSA could provide better solutions in terms of mean value of the objective functions than the conventional SSA, as well as the other optimization techniques. The results also showed that the mean and standard deviation of the ASSA were significantly lower than those of the other strategies, indicating that the algorithm was stable. The ASSA outperformed both the standard method and alternative optimization approaches, according to the findings. In order to obtain significant effectiveness between two or more algorithms, a nonparametric [PERSON]'s rank sum test is often used [57]. In this study, a pairwise comparison was performed using the best results from 30 runs of each algorithm.