## 2 MSFGWO Algorithm Model
### GWO Algorithm  
The GWO algorithm attains its optimization objectives through emulating the predatory behavior of grey wolf populations and leveraging the cooperative mechanisms within the group [29]. Based on the hierarchical structure of the wolf pack, the \(\alpha\) wolf--as the leader--assumes a pivotal role in directing the hunting process. The \(\beta\) and \(\delta\) wolves relay the \(\alpha\)'s directives and assist in the search for prey, while the \(\omega\) wolf, positioned at the lowest rank, participates in the hunt by adhering to instructions. In the context of optimal solution identification, the \(\alpha\) wolf signifies the optimal solution, the \(\beta\) wolf represents the second-best solution, and the \(\delta\) wolf denotes the third-best solution. Meanwhile, the \(\omega\) wolf contributes to the exploration of all potential solutions. During the process of hunting prey, the distance between an individual grey wolf and their prey is calculated according to Equation (1), while their positions are updated as shown in Equation (2) [29]:  
\[\overset{\rightarrow}{D}=\left|\overset{\rightarrow}{C}\cdot\overset{ \rightarrow}{X}_{P}(t)-\overset{\rightarrow}{X}(t)\right|, \tag{1}\]  
\[\overset{\rightarrow}{X}(t+1)=\overset{\rightarrow}{X}_{P}(t)-\overset{ \rightarrow}{A}\cdot\overset{\rightarrow}{D}, \tag{2}\]  
\[\overset{\rightarrow}{A}=2\overset{\rightarrow}{a}\cdot\overset{ \rightarrow}{r}_{1}-\overset{\rightarrow}{a}, \tag{3}\]  
\[\overset{\rightarrow}{C}=2\cdot\overset{\rightarrow}{r}_{2}, \tag{4}\]  
where \(t\) is the current iteration; \(\overset{\rightarrow}{X}_{P}(t)\) and \(\overset{\rightarrow}{X}(t)\) represent the positions of the prey and wolf in the \(t\)-th iteration, respectively; \(\overset{\rightarrow}{A}\) and \(\overset{\rightarrow}{C}\) are parameters; \(\overset{\rightarrow}{a}\) represents a convergence factor; and \(\overset{\rightarrow}{r}_{1}\) and \(\overset{\rightarrow}{r}_{2}\) are random vectors. During the hunting process, the \(\omega\) wolf executes its hunting activities in coordination with \(\alpha\), \(\beta\), and \(\delta\) wolves, following their leadership. The \(\omega\) wolf dynamically adjusts its position based on the actions of the \(\alpha\), \(\beta\), and \(\delta\) wolves as detailed in Equations (5)-(7) [29]:  
\[\left\{ \begin{aligned} &\overset{\rightarrow}{D}_{a}=\left| \overset{\rightarrow}{C}_{1}\cdot\overset{\rightarrow}{X}_{a}-\overset{ \rightarrow}{X}\right|\\ &\overset{\rightarrow}{D}_{\beta}=\left|\overset{\rightarrow}{ C}_{2}\cdot\overset{\rightarrow}{X}_{\beta}-\overset{\rightarrow}{X}\right|,\\ &\overset{\rightarrow}{D}_{\delta}=\left|\overset{\rightarrow}{ C}_{3}\cdot\overset{\rightarrow}{X}_{\delta}-\overset{\rightarrow}{X}\right|\\ \end{aligned} \right. \tag{5}\]  
\[\left\{ \begin{aligned} &\overset{\rightarrow}{X}_{1}=\overset{ \rightarrow}{X}_{a}(t)-\overset{\rightarrow}{A}_{1}\cdot\overset{ \rightarrow}{D}_{a}\\ &\overset{\rightarrow}{X}_{2}=\overset{\rightarrow}{X}_{\beta} (t)-\overset{\rightarrow}{A}_{2}\cdot\overset{\rightarrow}{D}_{\beta},\\ &\overset{\rightarrow}{X}_{3}=\overset{\rightarrow}{X}_{ \delta}(t)-\overset{\rightarrow}{A}_{3}\cdot{D}_{\delta}\\ \end{aligned} \right. \tag{6}\]  
\[\overset{\rightarrow}{X}(t+1)=\frac{\overset{\rightarrow}{X}_{1}+\overset {\rightarrow}{X}_{2}+\overset{\rightarrow}{X}_{3}}{3}. \tag{7}\]  
In Equation (5), \(\overset{\rightarrow}{D}_{a}\), \(\overset{\rightarrow}{D}_{\beta}\), and \(\overset{\rightarrow}{D}_{\delta}\) represent the distances between the \(\omega\) wolf and the \(\alpha\), \(\beta\), and \(\delta\) wolves, respectively; \(\overset{\rightarrow}{C}_{1}\), \(\overset{\rightarrow}{C}_{2}\), and \(\overset{\rightarrow}{C}_{3}\) are random numbers within the range of [0, 1]; and \(\overset{\rightarrow}{X}_{a}\), \(\overset{\rightarrow}{X}_{\beta}\), \(\overset{\rightarrow}{X}_{\delta}\), and \(\overset{\rightarrow}{X}\) represent the current positions of the \(\alpha\), \(\beta\), \(\delta\), and \(\omega\) wolves, respectively.