## 2 Methodology
### Optimization Algorithms
#### 2.2.1 Grey Wolf Optimizer  
Grey Wolf Optimizer (GWO) is a metaheuristic optimization algorithm imitating the hunting behavior and social hierarchy of grey wolves (_Canis lupus_) [17]. In GWO, the population is divided into four hierarchical levels:  
* Alpha (\(\alpha\)): The best solution, leading the search. * Beta (\(\beta\)): The second-best solution, assisting \(\alpha\) in leadership. * Delta (\(\delta\)): The third-best solution, subordinate to \(\alpha\) and \(\beta\). * Omega (\(\omega\)): The remaining wolves, following \(\alpha\), \(\beta\), and \(\delta\). The optimization process in GWO mimics the hunting strategy of grey wolves as shown in Figure 1, including encircling prey, searching for prey, and attacking prey. This mechanism is mathematically modeled as follows:  
Encircling prey  
The encircling behavior of grey wolves is described by  
\[\mathbf{X}(t+1)=\mathbf{X}_{p}(t)-\mathbf{A}\cdot|\mathbf{C}\cdot\mathbf{X}_{ p}(t)-\mathbf{X}(t)|, \tag{5}\]  
where \(\mathbf{X}(t)\) and \(\mathbf{X}_{p}(t)\) denote the vector position of a wolf and a prey at iteration \(t\), respectively, and the exploitation rate is controlled by coefficient \(A\), defined as follows:  
\[\mathbf{A}=2\cdot\mathbf{a}\cdot\mathbf{r}_{1}-\mathbf{a}. \tag{6}\]  
This coefficient governs the wolves to concentrate in a certain area, whereas the exploration coefficient \(C\) is defined as  
\[\mathbf{C}=2\cdot\mathbf{r}_{2}. \tag{7}\]  
This coefficient governs the random exploration vector \(\mathbf{r}_{1},\mathbf{r}_{2}\ \in[0,1]\). The standard GWO uses a linearly decreasing parameter \(\mathbf{a}\) from 2 to 0 over iterations as shown in Figure 2.  
Figure 1: Grey Wolf Optimization. Proposed Enhanced GWO with Non-Linear Decreasing Coefficient  
This paper introduces an Enhanced GWO (EGWO) by introducing a decreasing coefficient with a non-linear function, namely a cosine function. As shown in Figure 2, the value of \(\mathbf{a}\) at iteration \(t\) is given by  
\[\mathbf{a}(\mathbf{t})=2\times\frac{T-t}{T}. \tag{8}\]  
where \(T\) denotes the maximum number of iterations. The cosine is a concave function from \(0\) to \(\frac{\pi}{2}\) that provides a slower decreasing rate than a linear one. The proposed decreasing coefficient \(\mathbf{a_{c}}(\mathbf{t})\) is given by  
\[\mathbf{a_{c}}(\mathbf{t})=2\times\cos\frac{\pi t}{2T} \tag{9}\]  
For \(t\in[0,T]\), the inequality \(a(t)<a_{c}(t)\) holds. Proof. : To prove the inequality \(a(t)<a_{c}(t)\), consider the difference between \(a_{c}(t)\) and \(a(t)\), defined as  
\[\Delta(t)=a_{c}(t)-a(t). \tag{10}\]  
Substituting the expressions for \(a(t)\) and \(a_{c}(t)\), we get  
\[\Delta(t)=2\cdot\cos\biggl{(}\frac{\pi t}{2T}\biggr{)}-2\cdot\frac{T-t}{T}. \tag{11}\]  
Factoring out 2, this simplifies to  
\[\Delta(t)=2\biggl{[}\cos\biggl{(}\frac{\pi t}{2T}\biggr{)}-\frac{T-t}{T} \biggr{]}. \tag{12}\]  
Let \(f(t)=\cos\bigl{(}\frac{\pi t}{2T}\bigr{)}-\frac{T-t}{T}\). To analyze the behavior of \(f(t)\), we evaluate it at the boundaries of the interval and examine its derivative. At \(t=0\):  
\[\mathbf{A}=2\cdot\mathbf{a}\cdot\mathbf{r}_{1}-\mathbf{a}. \tag{6}\]0  
\[\mathbf{A}=2\cdot\mathbf{a}\cdot\mathbf{r}_{1}-\mathbf{a}. \tag{6}\]1 Comparison of a(i) curves in GWO. At \(t=T\):  
\[\mathbf{A}=2\cdot\mathbf{a}\cdot\mathbf{r}_{1}-\mathbf{a}. \tag{6}\]2  
To analyze the monotonicity of \(f(t)\), its derivative can be computed:  
\[\mathbf{A}=2\cdot\mathbf{a}\cdot\mathbf{r}_{1}-\mathbf{a}. \tag{6}\]3  
The derivative is  
\[\mathbf{A}=2\cdot\mathbf{a}\cdot\mathbf{r}_{1}-\mathbf{a}. \tag{6}\]4  
Factoring out \(\frac{1}{T}\), we can rewrite it:  
\[\mathbf{A}=2\cdot\mathbf{a}\cdot\mathbf{r}_{1}-\mathbf{a}. \tag{6}\]5  
Since \(\sin\bigl{(}\frac{\pi t}{2T}\bigr{)}\in[0,1]\) for \(t\in[0,T]\), the term  
\[\mathbf{A}=2\cdot\mathbf{a}\cdot\mathbf{r}_{1}-\mathbf{a}. \tag{6}\]6  
because \(\frac{\pi}{2}\approx 1.57>1\). Therefore, \(f^{\prime}(t)>0\) for all \(t\in[0,T]\), indicating that \(f(t)\) is strictly increasing on this interval.