## 2 Methodology
### Hybrid Metaheuristic Algorithms
A total of 901 analyses were carried out by 15-node triangular elements, where the effective variables were unit weight (\(\frac{kN}{m^{3}}\)), friction angle, elastic modulus (\(\frac{kN}{m^{2}}\)), dilation angle, Poisson's ratio (\(v\)), applied stress (\(\frac{kN}{m}\)), and setback distance (\(m\)). Figure 2 illustrates the distribution pattern of these factors. The descriptive statistics of the dataset are also presented in Table 1. As can be seen, the minimum and maximum values obtained for the settlement were 0 and 0.10 \(m\), respectively. Similar to a previous study [23], it was deemed that, if the \(U_{y}\) is less than 0.05 \(m\), the system fails; otherwise, it remains stable. The failure and stability of the system are represented by values 1 and 0, respectively. The gathered dataset (without normalization) was then randomly divided into two groups, namely, training (for development of intelligent models) and testing (for evaluating the prediction capability of the models). In this regard, 721 samples (i.e., 80% of whole data) were specified to the first group, and the remaining 180 samples (i.e., 20% of whole data) served as the testing data (addressing unseen soil conditions). Figure 2: Distribution of bearing capacity influential factors: (**a**) unit weight, (**b**) friction angle, (**c**) elastic modulus, (**d**) dilation angle, (**e**) [PERSON]’s ratio, (**f**) applied stress, (**g**) setback distance, and (**h**) settlement. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Features**} & \multirow{2}{*}{**Symbol**} & \multicolumn{8}{c}{**Descriptive Index**} \\ \cline{3-11}  & & \multicolumn{2}{c}{**Mean**} & \multicolumn{2}{c}{**Standard Error**} & \multicolumn{2}{c}{**Median**} & \multicolumn{2}{c}{**Mode**} & \multicolumn{2}{c}{**Standard Deviation**} & \multicolumn{2}{c}{**Sample Variance**} & \multicolumn{2}{c}{**Skewness**} & \multicolumn{2}{c}{**Minimum**} & \multicolumn{2}{c}{**Maximum**} \\ \hline Friction angle & X1 & 36.75 & 0.13 & 36.00 & 36.00 & 3.91 & 15.28 & -0.14 & 30.00 & 42.00 \\ Dilution angle & X2 & 8.28 & 0.09 & 8.00 & 8.00 & 2.61 & 6.83 & -0.39 & 3.40 & 11.50 \\ Unt weight (8N(m\({}^{3}\))) & X3 & 20.44 & 0.02 & 20.50 & 20.50 & 0.65 & 0.43 & -0.95 & 19.00 & 21.10 \\ Elastic modulus (8N(m\({}^{3}\))) & X4 & 41,007.68 & 546.65 & 35,000.00 & 35,000.00 & 16,048.72 & 20,026.192.50 & 0.22 & 17,050.00 & 65,000.00 \\ Poisson ratio (\({}^{3}\)) & X5 & 0.29 & 0.00 & 0.29 & 0.29 & 0.03 & 0.00 & 0.14 & 0.25 & 0.33 \\ Softback distance & X6 & 4.19 & 0.07 & 5.00 & 5.00 & 2.08 & 4.31 & -0.13 & 1.00 & 7.00 \\ Applied stress (8N(m\({}^{3}\))) & X7 & 289.74 & 7.89 & 246.65 & 0.00 & 26.97 & 5615.92 & 1.26 & 0.00 & 113,26.65 \\ Settlement (m) & Y & 0.04 & 0.00 & 0.03 & 0.00 & 0.03 & 0.00 & 0.46 & 0.00 & 0.10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Descriptive statistics of the used dataset. ## 4 Results and Discussion

To meet the objective of the study (i.e., investigating the optimization capability of the abovementioned metaheuristic algorithms), the algorithms should be coupled with the ANN. The aim of this work was to let this algorithm find the most appropriate matrix of the weights and biases for the ANN. To this end, firstly, an ANN with one hidden layer containing six neurons (determined by a trial-and-error process) was proposed as the base model. Thus, regarding the number of input/output parameters, the considered MLP took the form \(7\times 6\times 1\). Note that, in the present study, the activation functions of the hidden and output neurons were set to be \"tangent-sigmoid (i.e., Tansig)\" and \"puelin\", respectively. Next, it was mathematically synthesized with the WOA, LCA, MFO, and ACO algorithms to create WOA-ANN, LCA-ANN, MFO-ANN, and ACO-ANN neural ensembles. ### Hybridizing the ANN Using Metaheuristic Algorithms

After creating the ensembles, a population-based trial-and-error process was carried out to achieve the best-fitted complexity of the metaheuristic algorithms.