## 4 Selection of Computational Intelligence and Reinforcement Learning Solutions
### Grey-Wolf Optimization  
GWO is another nature-inspired optimization algorithm based on the leadership hierarchy and hunting mechanism of grey wolves in the wild [59]. The algorithm simulates the social hierarchy and hunting behavior of grey wolves when searching for prey. In GWO, the search agents are grey wolves, which are categorized into four groups: alpha, beta, delta, and omega, which help to guide the pack's movements.  
From a mathematical perspective, the algorithm starts by initializing a population of grey wolves, where each wolf represents a potential solution in the search space. The fitness of each solution is evaluated using a problem-specific objective function. Based on their fitness, the three best solutions are selected as the leading wolves: alpha is the best solution, while beta and delta are the second- and the third-best ones, respectively.  
All other candidate solutions are considered as omega, and they will follow the leading wolves in their hunting process. Their position update equations are as follows:  
\[\begin{split}& A^{1}=2a\times\text{rand}()-a\\ & C^{1}=2\times\text{rand}()\\ & D^{\alpha}=|C^{1}\times\alpha-X_{i}|\\ & X^{1}=\alpha-A^{1}\times D^{\alpha}\end{split} \tag{3}\]  
Similar equations are used to calculate \(X^{2}\) and \(X^{3}\) based on the positions of the beta and delta wolves, respectively. The new position of each wolf is then updated as follows:  
\[X_{i}=(X^{1}+X^{2}+X^{3})/3 \tag{4}\]  
Conceptually, the leading wolves delineate the boundaries of the search space, which typically is defined as a circle. However, the same considerations are still valid if we consider an n-dimensional search space, with a hyper-cube or hyper-sphere instead of a circle. \(A\) and \(C\) are the components that encourage the wolves to search for fitter prey, helping them avoid becoming stuck in local solutions. Different from \(A\), \(C\) does not have any component that decreases linearly, emphasizing (\(C>1\)) or de-emphasizing (\(C<1\)) the exploration of each wolf not only during the initial iterations. The algorithm repeats the hunting process for a predefined number of iterations or until convergence is achieved. The alpha wolf represents the optimal solution found by the GWO algorithm. Readers can find a description of the above steps in Algorithm 3.  
Similar to PSO, this metaheuristic has no direct relationships between the search agents and the fitness function [59]. This means that penalty mechanisms can be adopted effectively for modeling constraints, thus making GWO particularly suitable for this investigation.  
```
1:Initialize the grey wolf population \(X_{i}\) (i = 1, 2,..., n)
2:Initialize \(a\), \(A\), and \(C\),
3:Compute the fitness of each agent
4:\(X_{a}\) = best solution found by a wolf
5:\(X_{\beta}\) = second-best solution found by a wolf
6:\(X_{\delta}\) = third-best solution found by a wolf
7:while t < Max number of iterations do
8:for each wolf \(X_{i}\)do
9: Update the position of \(X_{i}\) by Equation (4)
10:endfor
11: Update \(a\), \(A\), and \(C\)
12: Compute the fitness of each agent
13: Update \(X_{\alpha}\), \(X_{\beta}\), and \(X_{\delta}\)
14:\(t=t+1\)
15:endwhile
16:Return \(X_{\alpha}\)
```  
**Algorithm 3** Grey-Wolf Optimization (GWO).