## 3 Proposed Methodology
### Phase 2: Generation of the First Population  
Chaotic maps are employed to initialize the first population in metaheuristics. These maps provide a stochastic and highly randomized approach to generating diverse and exploratory initial solutions within the search space. By leveraging the chaotic dynamics of these maps, initial population agents are assigned initial positions in a manner that ensures wide coverage and dispersion across the solution space. This initial diversity is essential for promoting exploration and preventing premature convergence, allowing the metaheuristic algorithm to effectively explore the search space and discover promising regions that may contain optimal or near-optimal solutions. By incorporating chaotic maps in the initialization process, our methodology can enhance its ability to escape local optima and improve the overall performance and convergence characteristics. Equation (10) serves as a fundamental tool for generating the positions of individuals within the initial population. \[\mathbf{x}_{i}=\left[\varphi_{i}^{(1)}(x_{\text{max}}^{(1)}-x_{\text{min}}^{(1 )})+x_{\text{min}}^{(1)},\ldots,\varphi_{i}^{(D)}(x_{\text{max}}^{(D)}-x_{ \text{min}}^{(D)})+x_{\text{min}}^{(D)}\right]\,,\;i\in\{1,\ldots,N\} \tag{10}\]  
We compare seven distinct chaotic schemes [67], which are Tent (Equation (11)), Sinusoidal (Equation (12)), Iterative (Equation (13)), Singer (Equation (14)), Sine (Equation (15)), Chebyshev (Equation (16)), and Circle maps (Equation (17)), to determine which one exhibits the best performance. The initial term \(\varphi_{1}\) of the chaotic sequence \(\varphi_{1},\ldots,\varphi_{N}\) is a random number drawn from the interval \([0,1]\). \[\varphi_{2+1}=\left\{ \begin{array}{lcl}\frac{\varphi_{2}}{0.7}&,&\varphi_{2}< 0.7\\ \\ \frac{10}{3}(1-\varphi_{2})&,&\varphi_{2}\geq 0.7\end{array} \right. \tag{11}\]  
\[\varphi_{2+1}=2.3\varphi_{2}^{2}\sin(\pi\varphi_{2}) \tag{12}\]\[\varphi_{z+1}=\sin\left(\frac{0.7\pi}{\varphi_{z}}\right) \tag{13}\]  
\[\varphi_{z+1}=\mu\left(7.86\varphi_{z}-23.31\varphi_{z}^{2}+28.75\varphi_{z}^{3} -13.302875\varphi_{z}^{4}\right)\,\ \mu=1.07 \tag{14}\]  
\[\varphi_{z+1}=\sin(\pi\varphi_{z}) \tag{15}\]  
\[\varphi_{z+1}=\cos(z\cos^{-1}\varphi_{z}) \tag{16}\]  
\[\varphi_{z+1}=\text{mod}(\varphi_{z}+0.2-\left(\frac{0.5}{2\pi}\right)\sin(2 \pi\varphi_{z}),1) \tag{17}\]  
Although chaotic maps have proven to be useful in generating population members with higher diversity levels, they can lead to the initialization of candidate solutions that are far from the global optimum, particularly in real-world optimization problems where the global optimum is often unknown. This undesirable situation can impede the rapid convergence of solutions toward promising regions in the search space, compromising the algorithm's convergence characteristics. To address these limitations of chaotic maps, an OBL strategy is incorporated into the initialization scheme of our methodology. The purpose of this strategy is to explore the broader coverage of the search space by searching for the opposite information of the chaotic population. The inclusion of OBL allows for the simultaneous evaluation of the original chaotic population and its opposite information, thereby increasing the probability of finding fitter solutions in the search space. We compare six distinct OBL strategies, named Strategy 1 [65] (Equation (18)), Strategy 2 [68] (Equation (19)), Strategy 3 [69] (Equation (20)), Strategy 4 [70] (Equation (21)), Strategy 5 [71] (Equation (22)), and Strategy 6 [71] (Equation (23)), to determine which one exhibits the best performance. Let \(\mathbf{x}=[x^{(1)},\ldots,x^{(D)}]\) be a point in the \(n\)-dimensional space, where \(x^{(1)},\ldots,x^{(D)}\) are real numbers and \(x^{(j)}\in[x^{(j)}_{\text{min}},x^{(j)}_{\text{max}}]\,\ j=1,\ldots,D\). The opposite point of \(\mathbf{x}\) is denoted by \(\bar{\mathbf{x}}=[\bar{x}^{(1)},\ldots,\bar{x}^{(D)}]\) and can be calculated using one of Equations (18)-(22), or (23).