## 3 Solution Method Based on Improved BES Algorithm  
### Bald Eagle Search Optimization Algorithm  
BES is a new meta-heuristic optimization algorithm that imitates bald eagle hunting behavior. Bald eagles often feast on protein-rich foods, mainly fish. The first step in bald eagle hunting is to move to a specific area, and that area is exploring space. The bald eagle chooses a space between the land surface and deep water. After the bald eagle reaches the designated area, the search process begins. In addition, the bald eagle has excellent eyesight, which enables it to aim at fish in the water from great distances in the air. When the hunter determines his target, he begins a gradual descent to capture his victim and catch the fish. Therefore, the BES algorithm, depending on each stage of the hunt, can be specifically divided into three parts: select stage; search stage; and swooping stage. The superiority or inferiority of the bald eagle position is calculated by the fitness function.  
(1) Select stage:  
The bald eagle randomly selects the area to be searched and determines the best location to search for prey by judging the number of prey, so that it can search for prey in this space. The bald eagle position \(P_{i,new}\) updates at this stage and is represented mathematically by Equation (21).  
\[P_{i,new}=P_{best}+\beta\times r(P_{mean}-P_{i}) \tag{21}\]  
where, \(\beta\) is the parameter used to a bald eagle position change, and its value is at (1.5,2); \(r\) is a random number between (0,1). \(P_{best}\) represents the current search space selected by the vulture based on the best position; \(P_{mean}\) represents the average distribution of bald eaglesafter the previous search; \(P_{i}\) is the location of the \(i\) bald eagle. In this stage, the candidate solutions are enhanced significantly according to the average distribution position and the optimal distribution position.  
(2) Search stage:  
During the search phase, the bald eagle searches for prey within the selected search space and moves in different directions in a spiral space to speed up the search. The optimal position for a dive is mathematically expressed by Equation (22).  
\[P_{i,new}=P_{i}+y(i)\times(P_{i}-P_{i+1})+x(i)\times(P_{i}-P_{mean}) \tag{22}\]  
where, \(P_{i+1}\) is the next updated location of the next bald eagle \(i\).  
\[x(i)=\frac{xr(i)}{\max(|xr|)},y(i)=\frac{yr(i)}{\max(|yr|)} \tag{23}\]  
\[xr(i)=r(i)\times\sin(\theta(i)),yr(i)=r(i)\times\cos(\theta(i)) \tag{24}\]  
\[\theta(i)=a\times\pi\times rand \tag{25}\]  
\[r(i)=\theta(i)+R\times rand \tag{26}\]  
where, \(a\) and \(R\) are parameters controlling the spiral trajectory, and their values range is [5; 10] and [0.5, 2], respectively; \(\theta(i)\) and \(r(i)\) are the polar Angle and diameter of the spiral equation, respectively; \(rand\) is the random number in (0, 1); \(x(i)\) and \(y(i)\) correspond to the position of the bald eagle in polar coordinates, and the value is [-1, 1] to change the spiral shape. At this stage, the position moves around the center. As \"A\" and \"R\" change, BES diversifies to avoid falling into local solutions and to seek more precise solutions.  
(3) Swooping stage:  
The movements of bald eagles have different shapes. Selecting a polar to coordinate equation to describe the movements of these bald eagles during the dive, the average solution can diversify the proposed algorithm toward the optimal solution.  
\[x1(i)=\frac{xr(i)}{\max(|xr|)},y1(i)=\frac{yr(i)}{\max(|yr|)} \tag{27}\]  
\[xr(i)=r(i)\times\sinh[\theta(i))],yr(i)=r(i)\times\cosh[\theta(i))] \tag{28}\]  
\[\theta(i)=a\times\pi\times rand,r(i)=\theta(i) \tag{29}\]  
The bald eagle position update formula in the swoop is:  
\[\phi_{x}=x1(i)\times(P_{i}-c1\times P_{mean}) \tag{30}\]  
\[\phi_{y}=y1(i)\times(P_{i}-c2\times P_{best}) \tag{31}\]  
\[P_{i,new}=rand\times P_{best}+\phi_{x}+\phi_{y} \tag{32}\]  
where, \(c1,c2\in[1,2]\) increases the intensity of movement of the bald eagle towards the optimal and central points.