## 4 Optimization Techniques and Procedures
### Grey Wolf Optimizer  
The grey wolf optimizer (GWO) was inspired by the hunting behavior of grey wolves in nature. Grey wolves usually live in a pack of 5-12 members with their dominant social hierarchy [20]. The alpha (\(\alpha\)) wolf, the pack leader, makes decisions about hunting. The beta (\(\beta\)) wolf helps the alpha wolf in making decisions. The omega (\(\omega\)) wolves, the lowest in ranking, are responsible for submitting information to all the wolves. Others, called delta (\(\delta\)) \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Response** & \(a\) & \(b\) & \(c\) & \(d\) & \(e\) & \(f\) & \(g\) & \(h\) & \(i\) & \(j\) \\ \hline MRR & \(-2.381\) & \(0.0123\) & \(0.112\) & \(0.621\) & \(0.0003\) & \(-0.003\) & \(0.0143\) & \(0.00003\) & \(-0.007\) & \(-0.065\) \\ NP & \(389\) & \(-0.551\) & \(-41.8\) & \(-49.5\) & \(0.0011\) & \(0.131\) & \(1.23\) & \(0.00154\) & \(1.561\) & \(4.64\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Regression coefficients for MRR and NP models. wolves, should respect alpha and beta wolves and dominate the omega wolves. The three steps in the hunting process of grey wolves are (1) tracking the prey, (2) encircling the prey, and (3) attacking the prey. GWO is simulated with the mathematical representation of the hunting process of the grey wolves to solve complex engineering problems. The optimal solution to the problem is considered the prey. To mathematically model the hunting process of these grey wolves when designing GWO, the alpha (\(\alpha\)) wolf is considered the fittest solution. Consequently, the second- and third-best solutions are represented by beta (\(\beta\)) and delta (\(\delta\)) wolves, respectively. The remaining candidate solutions are considered to be omega (\(\omega\)) wolves. The first step in the hunting process is encircling the prey. The mathematical formulation to mimic the encircling process for the prey and wolf are represented in the Equations (4) and (5):  
\[\overset{\rightarrow}{D}=\left|\overset{\rightarrow}{C}*\overset{\rightarrow }{X}_{p}\left(t\right)-\overset{\rightarrow}{X}\left(t\right)\right| \tag{4}\]  
\[\overset{\rightarrow}{X}\left(t+1\right)=\overset{\rightarrow}{X}_{p}\left(t \right)-\overset{\rightarrow}{A}*\overset{\rightarrow}{D} \tag{5}\]  
where \(\overset{\rightarrow}{D}\) is a distance vector between the position of the prey and grey wolf. \(\overset{\rightarrow}{C}\) and \(\overset{\rightarrow}{A}\) are coefficient vectors, \(t\) is the current iteration, and \(\overset{\rightarrow}{X}_{p}\) and \(\overset{\rightarrow}{X}\) are the positions of prey and a randomly chosen grey wolf, respectively [20]. The coefficients \(\overset{\rightarrow}{C}\) and \(\overset{\rightarrow}{A}\) are determined using the following formulas:  
\[\overset{\rightarrow}{A}=2\overset{\rightarrow}{a}*\overset{\rightarrow}{ r_{1}}-\overset{\rightarrow}{a} \tag{6}\]  
\[\overset{\rightarrow}{C}=2*\overset{\rightarrow}{r_{2}} \tag{7}\]  
where \(\overset{\rightarrow}{a}\) is a decrease from 2 to 0 linearly over the course of iterations and \(\overset{\rightarrow}{r_{1}}\) and \(\overset{\rightarrow}{r_{2}}\) are random vectors (0, 1). A grey wolf is in the position of \(\overset{\rightarrow}{X}\), updating its position \(\overset{\rightarrow}{X}\left(t+1\right)\) according to the position of the prey \(\overset{\rightarrow}{X}_{p}\). The position can be updated with respect to the current position by adjusting the values of the \(\overset{\rightarrow}{A}\) and \(\overset{\rightarrow}{C}\) vectors. This process enables the GWO to search the n-dimensional solution space of the given problem efficiently.