## 3 Description of the Integrated Improved Algorithm
### Grey Wolf Optimizer  
The grey wolf optimizer simulates the leadership hierarchy and predation mechanism within a grey wolf pack. The wolf pack is divided into four hierarchical levels, Represented by \(\alpha,\beta,\delta\) and \(\omega\), respectively. The \(\alpha\) wolf represents the leader of the pack, responsible for leading the entire pack in hunting and decision-making. The \(\beta\) wolf assists the \(\alpha\) wolf in decision-making and provides auxiliary commands. The \(\delta\) wolves obey the commands of the \(\alpha\) and \(\beta\) wolves, respectively, and perform tasks such as scouting and reconnaissance. The \(\omega\) wolves are considered as the lowest hierarchy and engage in activities around the \(\alpha,\beta\) or \(\delta\) wolves. In the mathematical simulation, the \(\alpha\) wolf represents the best solution in the algorithm, the \(\beta\) wolf represents a suboptimal solution, the \(\delta\) wolf represents the third best solution with relatively poor fitness, and the remaining solutions are considered as \(\delta\) wolves. During the hunting process of the wolf pack, three steps can be distinguished: encircling the prey, chasing the prey, and attacking the prey. In the grey wolf optimizer, the following formula can be used to update the position of the wolves, achieving the encircling of the prey:  
\[\overset{\rightarrow}{D}=\left|\overset{\rightarrow}{C}\times\overset{ \rightarrow}{X}_{P}(t)-\overset{\rightarrow}{X}(t)\right| \tag{21}\]  
\[\overset{\rightarrow}{X}(t+1)=\overset{\rightarrow}{X}_{p}(t)-\overset{ \rightarrow}{A}\times\overset{\rightarrow}{D} \tag{22}\]  
Figure 2: Effect of cubic B-spline curve. In the equation, \(\overset{\rightarrow}{D}\) represents the distance between an individual wolf and the prey. \(\overset{\rightarrow}{X}\) and \(\overset{\rightarrow}{X_{p}}\) are the position vectors of the wolf and the prey, respectively. \(t\) denotes the iteration number. \(\overset{\rightarrow}{A}\) and \(\overset{\rightarrow}{C}\) are synergy vectors determined by coefficients, and their calculation formula is as follows:  
\[\overset{\rightarrow}{A}=2\overset{\rightarrow}{a}\times\overset{ \rightarrow}{r_{1}}-\overset{\rightarrow}{a} \tag{23}\]  
\[\overset{\rightarrow}{C}=2\overset{\rightarrow}{r_{2}} \tag{24}\]  
In the equation, \(\overset{\rightarrow}{r_{1}}\) and \(\overset{\rightarrow}{r_{2}}\) are random vectors within \([0,1]\). \(\overset{\rightarrow}{a}\) is the convergence factor, which linearly decreases from 2 to 0 during the iteration process. During the chasing process, since the \(\alpha\), \(\beta\) and \(\delta\) wolves are the closest to the prey, the \(\omega\) wolf pack follows the guidance of the \(\alpha\), \(\beta\) and \(\delta\) wolves to move, thus achieving the encirclement of the prey. In the mathematical model, this can be expressed as follows:  
\[\left\{ \begin{aligned} \overset{\rightarrow}{D_{\alpha}}& =\left|\overset{\rightarrow}{C_{1}}\times\overset{\rightarrow}{X_{ \alpha}}-\overset{\rightarrow}{X}\right|\\ \overset{\rightarrow}{D_{\beta}}&=\left|\overset{ \rightarrow}{C_{2}}\times\overset{\rightarrow}{X_{\beta}}-\overset{ \rightarrow}{X}\right|\\ \overset{\rightarrow}{D_{\delta}}&=\left|\overset{ \rightarrow}{C_{3}}\times\overset{\rightarrow}{X_{\delta}}-\overset{ \rightarrow}{X}\right|\\ \end{aligned} \right. \tag{25}\]  
\[\left\{ \begin{aligned} \overset{\rightarrow}{X_{1}}& =\overset{\rightarrow}{X_{\alpha}}-A_{1}\times\overset{ \rightarrow}{D_{\alpha}}\\ \overset{\rightarrow}{X_{2}}&=\overset{ \rightarrow}{X_{\beta}}-A_{2}\times\overset{\rightarrow}{D_{\beta}}\\ \overset{\rightarrow}{X_{3}}&=\overset{ \rightarrow}{X_{\delta}}-A_{3}\times\overset{\rightarrow}{D_{\delta}}\\ \end{aligned} \right.