*2.2. Improved White Shark Optimizer (IWSO)*

Due to the traditional WSO algorithm's drawbacks, the following aspects will be improved in this paper:


#### 2.2.1. Circle Chaotic Mapping

Circle chaotic mapping has gained considerable attention owing to its simple structure and strong uniformity. It exhibits complex, unpredictable and random behaviors, and is often employed to enhance the diversity of the population. Circle chaotic mapping outperforms other kinds of chaotic mappings like logistic chaotic mapping and tent chaotic mapping in terms of ergodic uniformity, randomness and diversity.

In the traditional WSO algorithm, the initialization population of white sharks is generated randomly, which may lead to the disadvantages of uneven population distribution, poor diversity and slipping into the regional optimum easily in the later iteration. For this, the circle chaotic mapping method is employed to generate the initial circle population of white sharks, which is then combined with a random population. The resulting group is evaluated, and the best sharks are selected to form the optimal white shark population of the next generation. The optimized white shark individuals are more similar to the initial optimal solution than the sharks in the random population and initial circle population, which evens out white shark population distribution, broadens the algorithm's search range and improves its efficacy.

Let *wi* denote the individuals within the white shark population, and then the initialization formula for the white shark population using the circle chaotic mapping method can be expressed as follows [25]:

$$w\_{i+1} = mod(w\_i + 0.2 - \frac{0.5}{2\pi} \cdot \sin(2\pi \cdot w\_i), 1) \tag{7}$$

where *mod* indicates remainder.

#### 2.2.2. Adaptive Weight Factor

When white sharks hunt for prey amid the ocean's depths randomly, they may not be close enough to the optimal prey, which may lead to an imbalance between exploration and exploitation. So, the adaptive weight factor is used to update the best white shark's position in this paper, which makes the algorithm have outstanding exploration ability in the earlier iteration and excellent exploitation ability in the later iteration. By introducing the adaptive weight factor in the process of white shark hunting prey, it is beneficial to balance the capacity for both exploration and exploitation. The adaptive weight factor proposed in this paper is expressed as follows:

$$\alpha = 0.2 + \frac{1}{0.6 + \varepsilon^{(-f(w\_i)/\mu')^k}} \tag{8}$$

where *f*(*wi*) represents the *i*-th white shark's fitness value. *μ* represents the white shark's best fitness value in the first iteration. *α* is the dynamic nonlinear factor, which is used to update the best white shark's position. The refined formula is presented below:

$$
\omega\_{k+1}^i = \begin{cases}
\mathfrak{a} \cdot \omega\_k^i \cdot \neg \oplus \omega\_0 + \mathfrak{a} \cdot \mathfrak{a} + \mathfrak{l} \cdot \mathfrak{b}; & rand < m\upsilon \\
(1 - \mathfrak{a}) \cdot \omega\_k^i + v\_k^i/f; & rand \ge m\upsilon
\end{cases} \tag{9}
$$

As can be seen from the formula, the best white shark's position is adjusted by the adaptive weight factor adaptively, so that the capacity for both exploration and exploitation can be balanced.
