## 3 Multi-Strategy Ensemble While Optimization Algorithm
### Modified Spiral Updating Position Strategy  
The primary issue of the metaheuristic algorithms is that most of the searching agents prone to stagnation in local optimums owing to the defect in maintaining population diversity, and the conventional WOA algorithm is no exception. In the original spiral updating position strategy, the global optimal whale is taken as the target prey, and the other whales attempt to update their positions according to this optimal whale. However, if the present optimal whale traps into local optimal solution, the possibility of the whale swarm getting into local optimum will increase [28]. In the previous studies, the Levy flight strategy has been widely adopted by metaheuristic algorithms because of its efficient global searchability. Therefore, we introduced the modified spiral updating position strategy based on Levy flight to make up for the above drawback, and it will help the whales maintain diversity for longer and get rid of stagnation. The random step generates by the Levy flight strategy obeys the Levy distribution. The levy distribution is a type of fat-tailed distribution, and its tail is wider than Gaussian distribution. Figure 2 shows 100 consecutive steps based on Levy distribution and Gaussian distribution. It can be seen that Levy flights can generate a large step occasionally, which means a stronger disturbance effect. However, with respect to Gaussian distribution, its steps are always small. The Levy distribution is described as follows:  
\[L(s)\sim s^{-\beta},1\leq\beta\leq 3 \tag{10}\]  
where \(s\) is the step size of Levy flight, and it can also express as \(Levy(\beta)\). Generally, the algorithm proposed by [PERSON] [47] was adopted to generate random steps, which have the same behavior as [PERSON] flights. Thus, \(s\) can be calculated by the Equations (11) and (12):  
\[s=\frac{\mu}{|\
u|^{1/\beta}},\mu\sim N\big{(}0,{\sigma_{\mu}}^{2}\big{)},\
u \sim N\big{(}0,{\sigma_{\
u}}^{2}\big{)} \tag{11}\]  
\[{\sigma_{\mu}}^{2}=\left\{\frac{\Gamma(1+\beta)}{\beta\cdot\Gamma(1+\beta)/2} \cdot\frac{\sin(\pi\cdot\beta/2)}{2^{(\beta-1)/2}}\right\}^{2/\beta},\ \sigma_{\
u}^{2}=1 \tag{12}\]  
where \(N\) denotes the Gaussian distribution, and the number of \(\beta\) is 1.5. The modified spiral updating position strategy is designed as follows: when the coefficient vector \(|A|\geq 1\), as aforementioned, it is in the exploration phase, whales are supposed to explore around themselves instead of the global optimal individual. The Levy flight strategy was employed to generate a new movement when the position of the current whale individual is updated, and it performed the long-distance step occasionally so that the exploration ability was improved. The mathematical model can be expressed as follows:  
\[x^{i+1}=x^{i}+D\cdot e^{bl}\cdot\cos(2\pi l)\cdot levy(\beta),\text{if }|A|\geq 1 \tag{13}\]  
Figure 2: Trajectories based on Lévy distribution and Gaussian distribution (starting from the origin). When the coefficient vector \(|A|<\)1, the tendency of the searching process will turn to exploitation. Whales are expected to exploit within the encirclement centered on the current global optimal individual. This strategy also has a positive impact on achieving a better tradeoff between exploitation and exploration. It is defined by:  
\[x^{i+1}=x^{i}_{ghest}+D\cdot e^{bl}\cdot\cos(2\pi l)\cdot levy(\beta),\text{ if }|A|<1 \tag{14}\]  
### Modified Spiral Updating Position Strategy  
In the original WOA algorithm, when a new position is beyond the search bounds, it usually replaces it by the bound. The proposed variant will change it by a random movement towards the search bound, which generally contributes to exploration. The new positions of whales are modified in accordance with the Equation (15). \[x^{i+1}=\left\{ \begin{array}{ll}UB+rand_{5}\cdot\frac{UB-x^{i}}{x^{i}}\cdot UB,\text{ if }x^{i}>UB\\ LB+rand_{6}\cdot\left|\frac{LB-x^{i}}{x^{i}}\cdot LB\right|,\text{ if }x^{i}<LB \end{array} \right.