## 2 Material and Methods
### Whale Optimization Algorithm  
[PERSON] and [PERSON] (2016) introduced the whale optimization algorithm (WOA), a metaheuristic optimization technique inspired by the social behavior of humpback whales specifically designed to tackle challenging optimization problems ([PERSON] and [PERSON], 2016). Humpback whales are known for their unique hunting strategy, wherein they can pinpoint and entirely engulf their prey. In the context of WOA, this behavior is simulated by allowing the simulated whales to adjust their positions relative to a designated optimal search agent, analogous to the target prey in their natural hunting process. This adaptation of humpback whale behavior into a computational model is encapsulated in the mathematical formulations presented in Equations (6) and (7), detailing the algorithm's mechanics of search and optimization. \[\overset{\rightarrow}{D}=\left|\overset{\rightarrow}{C.X^{*}}(t)-\overset{ \rightarrow}{X}(t)\right| \tag{6}\]  
\[\overset{\rightarrow}{X}(t+1)=\overset{\rightarrow}{X^{*}}(t)-\overset{ \rightarrow}{A.D} \tag{7}\]  
Figure 2: (**a**) One single neuron and (**b**) Single hidden layer MLP network. The location vector of the optimal solution that has been found up to this point is denoted by \(\overrightarrow{\mathbf{X}}\). \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{C}}\) coefficient vectors are calculated in Equations (8)-(10). The position vector is denoted by \(\overrightarrow{\mathbf{X}}\). \(\mathbf{t}\) represents the most recent iteration. \[\overrightarrow{A}=2.\overrightarrow{\mathbf{a}}.\overrightarrow{r}- \overrightarrow{\mathbf{a}} \tag{8}\]  
\[\overrightarrow{\mathbf{a}}=2-2\frac{t}{t_{max}} \tag{9}\]  
\[\overrightarrow{\mathbf{C}}=2.\overrightarrow{r} \tag{10}\]  
In the whale optimization algorithm (WOA), a coefficient vector \"a\" decreases linearly from 2 to 0 over the course of iterations, influencing both the exploration and exploitation phases. The vector \"r\" is a randomly generated vector within the range of [0, 1], with \"t_max\" representing the maximum number of iterations and \"t\" denoting the current iteration number. Equations (8) and (9) are designed to maintain a balance between exploration and exploitation during the optimization process. In these equations, \"r\" generates a random number, introducing stochastic elements into the population's location updates. Exploration is initiated when \(A\geq 1\), while exploitation occurs when \(A<1\), enabling the algorithm to transition between exploration and exploitation at any optimization stage. The unique bubble-net attacking strategy of humpback whales is mimicked within WOA through two main mechanisms: the spiral update position mechanism and the shrinking encircling mechanism. The coefficient vector \(A\), which influences these mechanisms, is determined by setting the value of \"a\" within the range of [\(-1\), 1] [\(-1\), 1], while the shrinking encircling mechanism reduces the value of \"a\" linearly through iterations. This approach positions the new location midway between its current location and that of the optimal search agent. To emulate the humpback whales' spiral movements toward their prey, the spiral equation defining the path between the whale and the prey's location is established. This mathematical representation captures the essence of the humpback whales' hunting strategy, applying it to the optimization process in WOA. \[\overrightarrow{X}(t+1)=\overrightarrow{\mathbf{D}}.\vec{e}^{bl}.\text{cos}(2\pi l )+\overrightarrow{\mathbf{X}}^{\ast}(t) \tag{11}\]  
\[\overrightarrow{\mathbf{D}}=\left|\overrightarrow{\mathbf{X}}^{\ast}(t)- \overrightarrow{\mathbf{X}}(t)\right| \tag{12}\]  
where \(\overrightarrow{\mathbf{D}}\) is the distance of the \(\mathbf{i}\) whale from its target (the best solution so far), \(\mathbf{l}\) is a random value within the range \([-1,\ 1]\), and \(\mathbf{b}\) is a constant used to define the logarithmic spiral's form.