## 3 Proposed Methodology
### Phase 2: Generation of the First Population
\[\bar{x}^{(j)}=x^{(j)}_{\text{min}}+x^{(j)}_{\text{max}}-x^{(j)} \tag{18}\]  
\[\bar{x}^{(j)}=\text{rand}\left(\frac{x^{(j)}_{\text{min}}+x^{(j)}_{\text{max} }}{2},x^{(j)}_{\text{min}}+x^{(j)}_{\text{max}}-x^{(j)}\right) \tag{19}\]  
\[\bar{x}^{(j)}=\frac{x^{(j)}_{\text{min}}+x^{(j)}_{\text{max}}}{2}+\left(v^{(j )}\cos\left(\pi\mathcal{N}(1,0.25)\right)-v^{(j)}\sin\left(\pi\mathcal{N}(1,0. 25)\right)\right) \tag{20}\]  
\[v^{(j)}=x^{(j)}-\frac{x^{(j)}_{\text{min}}+x^{(j)}_{\text{max}}}{2}\]  
\[\varphi_{z+1}=\sin\left(\frac{0.7\pi}{\varphi_{z}}\right) \tag{13}\]0  
\[\varphi_{z+1}=\sin\left(\frac{0.7\pi}{\varphi_{z}}\right) \tag{13}\]1  
\[\varphi_{z+1}=\sin\left(\frac{0.7\pi}{\varphi_{z}}\right) \tag{13}\]2  
\[\varphi_{z+1}=\sin\left(\frac{0.7\pi}{\varphi_{z}}\right) \tag{13}\]3  
\[\alpha=\left\{ \begin{array}{lcl}s\cdot p&,&M<0.5\\ s&,&M\geq 0.5\end{array} \right.\]\[\beta=\left\{ \begin{array}{ll}s&,\quad M<0.5\\ s\cdot p&,\quad M\geq 0.5\end{array} \right.\]  
\[s=\left(\frac{1}{\sqrt{\
u}}\right)^{1+\mathcal{N}(0.05)},\text{ for Equation (\ref{eq:s1})}\]  
\[s=0.1\sqrt{\
u}+0.9\text{, for Equation (\ref{eq:s1})}\]  
\[p=\left\{ \begin{array}{ll}\frac{(s-2)M+1}{s(1-M)}&,\quad M<0.5\\ \\ \frac{2-s}{s}+\frac{s-1}{s-M}&,\quad M\geq 0.5\end{array} \right.\]  
\[M=\frac{x_{\max}^{(j)}-x_{\min}^{(j)}}{x_{\max}^{(j)}-x_{\min}^{(j)}}\text{, for Equation (\ref{eq:s1})}\]  
\[M=\frac{x_{\min}^{(j)}-x_{\min}^{(j)}}{x_{\max}^{(j)}-x_{\min}^{(j)}}\text{, for Equation (\ref{eq:s1})}\]  
\[\
u=\frac{1}{N}\sum_{i=1}^{N}\underset{\in\{\mathbf{x}_{1},\ldots,\mathbf{x}_ {N}\}-\{\mathbf{x}_{i}\}}{\text{argmin}}\sqrt{\frac{1}{D}\sum_{j=1}^{D}\left( \frac{x^{(j)}-c^{(j)}}{x_{\max}^{(j)}-x_{\min}^{(j)}}\right)^{2}}\]  
Algorithm 1 serves as a valuable tool for demonstrating the operational principles of the initialization phase within our methodology. By presenting a step-by-step procedure, it effectively showcases how the initial population of candidate solutions is generated. Through Algorithm 1, we highlight the specific techniques and strategies employed to create a diverse and representative set of individuals at the beginning of the optimization process. It is worth pointing out that if a candidate solution exceeds the boundaries of the search space after undergoing an opposition operation, it is subsequently restored to within the valid range utilizing Equation (24). \[\left\{ \begin{array}{ll}x^{(j)}\gets x_{\max}^{(j)}&,\quad x^{(j)}>x_{ \max}^{(j)}\\ x^{(j)}\gets x_{\min}^{(j)}&,\quad x^{(j)}<x_{\min}^{(j)}\end{array} \right. \tag{24}\]  
```
Input:\(D\): The dimensionality of the search space. Input:\(N\): The population size. Input:\(x_{\min}^{(1)},\ldots,x_{\min}^{(D)}\): The lower boundaries of entries \(x^{(1)},\ldots,x^{(D)}\). Input:\(x_{\max}^{(1)},\ldots,x_{\max}^{(D)}\): The upper boundaries of entries \(x^{(1)},\ldots,x^{(D)}\). Input:\(f(. )\): The multivariate function to be minimized. for\(i\gets 1\) to \(N\)do for\(j\gets 1\) to \(D\)do if(\(i=1\))then\(\sigma_{i}^{(j)}\leftarrow\) rand(0,1);  end if else\(\sigma_{i}^{(j)}\) is updated using the selected chaotic map (Equations (11)-(16) or (17);  end if  end for Compute \(\mathbf{x}_{i}\) using Equation (10);  Compute \(\bar{\mathbf{x}}_{i}\) using the selected opposition-based learning strategy (Equations (18)-(22) or (23); \(\mathbf{x}_{i}\leftarrow\underset{\text{argmin}}{\argmin}\{f(\mathbf{x}_{i}),f( \bar{\mathbf{x}}_{i})\}\);  end for
```  
**Algorithm 1**The pseudocode for the initialization phase of our methodology.