## 3 Theoretical Background
### Whale Optimization Algorithm
#### 3.2.1 Exploitation Phase (Encircling Prey/Bubble-Net Attacking Method)  
[PERSON] et al. designed two methods to mathematically model the bubble-net behavior of humpback whale, one of which is the shrinking encircling mechanism and the other is the spiral updating position. We then analyze the concrete implementation of these two processes from a mathematical point of view. In the shrinking encircling mechanism, [PERSON] applies the following two formulas to update the problem solution to model the movement of a whale toward a prey. \[\overrightarrow{D}=\left|\overrightarrow{C}.\overrightarrow{X^{*}}(t)- \overrightarrow{X}(t)\right| \tag{1}\]  
\[\overrightarrow{X}(t+1)=\left|\overrightarrow{X^{*}}(t)-\overrightarrow{A}.\overrightarrow{D}\right| \tag{2}\]  
where \(t\) represents the number of current iterations, \(\overrightarrow{X^{*}}\) represents the optimal solution obtained so far, \(\overrightarrow{X}\) is the current solution scheme, \(||\) is the absolute value, and. is the dot product operation between the elements. \(\overrightarrow{A}\) and \(\overrightarrow{C}\) are the coefficient vectors, which can be obtained from Equations (3)-(5):  
\[\overrightarrow{A}=2\overrightarrow{a}.\overrightarrow{r}-\overrightarrow{a} \tag{3}\]  
\[\overrightarrow{C}=2.\overrightarrow{r} \tag{4}\]  
\[a=t\frac{2}{MaxIter} \tag{5}\]  
Figure 2: The process of humpback whales hunting prey: (**a**) the whales move toward the lead whale or in random directions; (**b**) whales reach their prey by the shrinking encircling mechanism; (**c**) whales reach their prey in a spiral shape. where the value of \(A\) decreases linearly from 2 to 0 and its value is in the interval \([-a,a]\). \(r\) is a random vector between \([0,1]\). \(a\) increases linearly from 0 to 2 depending on the number of iterations. \(t\) is the number of the current iteration, and \(Maxiter\) is the maximum number of pre-set iterations. According to Equation (2), the current solution updates the position of the current solution according to the optimal solution obtained so far. Through the two vectors \(A\) and \(C\), the search range of the current solution can be controlled to be fixed within the neighborhood range of the optimal solution. In order to imitate the behavior of whales hunting prey in Figure 2b, we use the mathematical model shown in Figure 3a for modeling and analysis. It is assumed that \((X^{*},Y^{*})\) is the current global optimal solution, and the solid dots in the figure, such as \((X,Y)\), are the current solution. Figure 3a shows the possible positions from \((X,Y)\) toward \((X^{*},Y^{*})\) that can be achieved by \(0\leq A\leq 1\) in a 2D space. As mentioned above, whales also use a spiral motion to move toward prey as shown in Figure 2c. WOA uses the following formula to model this behavior. \[\overset{\rightarrow}{X}(t+1)=D^{\prime}.e^{bl}.cos(2\pi l)+\overset{ \rightarrow}{X^{*}}(t) \tag{6}\]  
where \(\overset{\rightarrow}{X^{*}}\) represents the optimal solution obtained so far, \(\overset{\rightarrow}{X}\) is the current \(i\)th solution, \(D^{\prime}=|\overset{\rightarrow}{X^{*}}(t)-\overset{\rightarrow}{X}(t)\lor\) and indicates the distance of the \(i\)th whale to the prey (best solution obtained so far), \(b\) is a constant for defining the shape of the spiral, and \(l\) is a random variable between \([-1,1]\). The approximate figure of Equation (6) and Figure 2c is shown in Figure 3b. In this 1D space, \(X_{t}\) represents the current \(i\)th solution (i.e., the whale), \(X^{*}\) represents the current optimal solution (i.e., the prey), and the distance between \(X_{t}\) and \(X^{*}\) is \(D_{i}\). The x-coordinate of the coordinate axis represents a random number \(l\), which is used to control the movement direction of the whale, and the y-coordinate represents the next position \(X_{(l+1)}\) of the current solution \(X_{t}\).