## 2 Methodology
### Hybrid Metaheuristic Algorithms  
Due to the successful performance of metaheuristic algorithms in optimizing regular predictive models, in this study, four recently developed algorithms are applied to the ANN. The considered optimizers are the whale optimization algorithm (WOA), league champion optimization (LCA), moth-flame optimization (MFO), and ant colony optimization (ACO), which are used as potential search methods for finding the optimal solution to a given problem. In the case of this study, a general MLP is given as the problem, and, concerning a cost function, the algorithms aim to find the best weights and biases for the network. The WOA was designed by [PERSON] and [PERSON] [36], inspired by the bubble-net hunting of humpback whales. Three major stages of this algorithm are (a) shrinking encircling hunt, (b) exploitation (i.e., bubble-net attacking), and (c) exploration (i.e., searching for the prey). More information about the WOA can be found in previous studies [37; 38; 39; 40]. The LCA was proposed by [PERSON] [41], based on sporting competitions in sports leagues. Considering the league schedule programming and some relationships for determining the winner/loser team in an artificial league, the most appropriate solution is found. The LCA was detailed in previous studies [42; 43; 44]. As a novel nature-inspired optimization technique, the MFO was suggested by [PERSON] [45]. The pivotal idea of this algorithm is the navigation method of moths, which is known as transverse orientation. The candidate solutions in the MFO are moths, and their positions in space express the problem's variables. The optimization process of this optimizer was well described in previous studies [46; 47; 48]. Lastly, the name ACO implies a population-based optimization technique which mimics the foraging behavior of ant herds. It was presented by [PERSON] and [PERSON] [49]. In this algorithm, artificial ants guide each other to  
Figure 1: The structure of a multi-layer perceptron (MLP) neural network. achieve a proper (i.e., short) path leading to a promising food source. For more details, please refer to References [50; 51; 52]. The pseudo-code of the WOA, LCA, MFO, and ACO algorithms are shown below. ```
Initialize the whale population \(X_{i}\) (\(i\) = 1, 2,..., \(n\)) Calculate the fitness of each search agent \(X^{*}\) the best search agent while (\(t<\) maximum number of iterations) for each search agent  Update \(a\), \(A\), \(



```
While iteration < max iteration Update flame number Obj = fitness function (Motls); if Iteration = 1 Sort the moths based on their objective functions; update the flames Iteration = 0; else Sort the moths based on their objective functions and flames from last iteration; update the flames end linearly decrease the convergence constant for j = 1: Number of mothls for k = 1: Number of variables, update r and t Calculate the distance of moth from each flame; update the values of the variables of moth from the corresponding flame end Iteration = iteration + 1; end
```

**Algorithm 4**. The pseudo-code of the ant colony optimization (ACO) algorithm [56]

```
Initialization: Algorithm parameters; Ant population size K; Maximum number of iteration NMax; Generation: Generating the pheromone matrix for the ant k; Update the pheromone values and set x* = k; i = 1; Repeat for k = 1 to K Compute the cost function for the ant k; Compute probability move of ant individual; if(k) < f(x*) Then Update the pheromone values; Set x* = k; End if End for Until I > NMax;
```

**Algorithm 5** Algorithm parameters;

## 3 Data Collection

By implementing a two-dimensional (2D) axisymmetric finite element method, a shallow footing settled on a two-layered soil was analyzed in different conditions. The settlement (\(U_{y}\)) was derived in each stage.