## 2 Materials and Methods
### Outline of Computational Approach
#### 2.5.5 Gray Wolf Optimizer (GWO)  
GWO is a metaheuristic algorithm based on population, drawing inspiration from the social organization and collaborative hunting practices of gray wolves (Canis lupus). Proposed by [PERSON] et al. [48], GWO has proven to be effective in addressing various intricate optimization issues in both continuous and discrete fields. The algorithm simulates the hierarchical structure of gray wolf packs, assigning individuals to four roles--alpha (\(\alpha\)), beta (\(\beta\)), delta (\(\delta\)), and omega (\(\omega\)). The alpha wolves signify the most effective solutions discovered to date and direct the search operation, steering the remainder of the pack toward promising areas in the solution space. Beta and delta wolves enhance the optimization process by improving the search direction, whereas omega wolves concentrate on discovering different solutions. This organized framework maintains a productive equilibrium between exploration (looking for novel solutions) and exploitation (improving current solutions), boosting the algorithm's effectiveness in finding optimal solutions [48; 49]. At the heart of the GWO algorithm lies an iterative search mechanism, in which wolves adjust their locations continually by following the direction of the alpha, beta, and delta wolves. The surrounding mechanism enables wolves to modify their locations in relation to the optimal solutions recognized at every iteration, whereas the adaptive control parameter (\(\Lambda\)) manages the balance between exploration and exploitation. Through the integration of these mechanisms, GWO effectively explores the solution space, moving toward optimal solutions while reducing the likelihood of premature convergence [48]. Assuming that the total number of wolves is \(N\) and the search domain has \(K\) dimensions, the position of the \(i\)th wolf can be represented as: \(Q_{i}\ =\ (Q_{i,1}\,Q_{i,2}\,\ldots,Q_{i,K})\). To mathematically model the hierarchical structure of the wolves, the most optimal solution is identified as the alpha (\(\alpha\)) wolf. Consequently, the second and third most optimal solutions are classified as the beta (\(\beta\)) and delta (\(\delta\)) wolves, respectively. All remaining candidate solutions are considered as omega (\(\omega\)) wolves. Within the algorithm, the alpha wolf's location corresponds to the estimated prey's position [48]. The mathematical expressions of the GWO algorithm, including Equations (28)-(39), were adapted from [48]. The encircling behavior of gray wolves can be mathematically formulated as shown in Equations (28) and (29) [48]. \[D=\left|\Gamma\times Q_{p}(\tau)-Q(\tau)\right| \tag{28}\]  
\[Q(\tau+1)=Q_{p}(\tau)-\Lambda\times D \tag{29}\]  
Here, the set \(\tau\) represents the current iteration, \(Q_{p}(\tau)\) denotes the position vector of the prey, and \(Q(\tau)\) corresponds to the position vector of a gray wolf. The set \(\Gamma\) acts as a control coefficient, which is determined by the following Equation (30) [48]. \[\Gamma=2\cdot\gamma_{1} \tag{30}\]  
Here, the set \(\gamma_{1}\) is a randomly generated variable within the range \([0,1]\).The set \(\Lambda\) represents the convergence factor, which is computed using the following Equations (31) and (32) [48]:  
\[\Lambda=2\cdot\lambda\cdot\gamma_{2}-\lambda \tag{31}\]  
\[\lambda=2\bigg{(}1-\frac{\tau}{T_{\max}}\bigg{)} \tag{32}\]  
Here, the set \(\gamma_{2}\) is a randomly generated variable within the range \([0,1]\). The set \(\lambda\) serves as a control coefficient, which linearly decreases from 2 to 0 over the course of iterations, as stated in Faris, Aljarah, Al-Betar, and [PERSON] [49], where \(\lambda_{\max}=2\) and \(\lambda_{\min}=0\). When gray wolves hunt their prey, the \(\alpha\) wolf initially guides the rest of the pack in encircling the prey. Subsequently, the \(\alpha\) wolf directs the \(\beta\) and \(\delta\) wolves to execute the capture.