## 2 Methods
### Improved While Optimization Algorithm  
The WOA algorithm is an optimization technique inspired by the migratory behavior of whales in the natural world. It possesses advantages such as a simple structure and minimal parameter tuning [25]. Analyzing the behavior characteristics of whales, the algorithm's modeling generally includes three main behaviors: encircling, bubble-net attacking, and searching [26]. The following sections provide an introduction to these three updating strategies. 1. [label=(0)]
2. Surround: Whales exhibit the capability to identify the location of their prey, where each individual whale represents a potential solution in the search space. Subsequently, other individuals in the pod surround the most optimal whale, mimicking this behavior. The mathematical model that simulates this encircling behavior is described as follows: \[X(d+1)=X^{*}(d)-A|C\cdot X^{*}(d)-X(d)|\] (5)where \(d\) represents the current number of iterations, \(X^{*}\) represents the current prey's position vector, \(A|C\cdot X^{*}(d)-X(d)|\) represents the length of the surrounding step, and \(A\) and \(C\) are the coefficient vectors, defined as follows:  
\[A=2a\cdot rand_{1}-a \tag{6}\]  
\[C=2\cdot rand_{2} \tag{7}\]  
where \(rand_{1}\) and \(rand_{2}\) represent random numbers between 0 and 1, and \(a\) is the convergence factor. As the number of iterations, \(d\), increases, the convergence factor linearly decreases from 2 to 0, that is:  
\[a=2-2d/d_{\max} \tag{8}\]  
\(d_{\max}\) represents the maximum number of iterations. 2. Bubble-Net Attacking: In this behavior, whales imitate their real-life counterparts by continuously contracting their encircling net and releasing bubbles. The mathematical model for this process is described below: (a) Contraction Encirclement Mechanism: The contraction encirclement mechanism, defined by Equations (5) and (7), results in reducing the fluctuation range of coefficient vector \(A\) as the convergence factor decreases. (b) Spiral Position Update: As a part of their hunting behavior, whales display a spiral motion while preying on their targets. The formula for updating the position is: \[X(d+1)=|C\cdot X^{*}(d)-X(d)|\cdot e^{bl}\cdot\cos(2\pi l)+X^{*}(d)\] (9) where \(|C\cdot X^{*}(d)-X(d)|\) is the distance between a whale and the current best individual on the planet. \(b\) is a constant, and the range of \(l\) is \((-1,1)\). In this simulation, whales synchronize their actions by both contracting the encircling net and updating their positions using the spiral motion. Each behavior is equally probable, with a probability set at 0.5. The mathematical model is as follows: \[X(d+1)=\left\{ \begin{array}{l}X^{*}(d)-W_{woa}\cdot A|C\cdot X^{*}(d)-X(d)| |\text{A}|<1,\text{p}<0.5\\ X_{rand}(d)-W_{woa}\cdot A|C\cdot X^{*}(d)-X(d)||\text{A}|\geq 1,P<0.5\end{array} \right.\] (10) represents a random number within the interval [0, 1]. 3. Searching for Prey: In this behavior, when the absolute value of the coefficient vector \(|A|\) is greater than 1, individual whales conduct random searches by using each other's positions. The mathematical model is as follows: \[X(d+1)=X_{rand}(d)-A\cdot|C\cdot X_{rand}(d)-X(d)|\] (11) where \(X_{rand}\) denotes the position vector of a randomly selected whale. 4. Improvement: Regarding the sensitivity of parameter selection and the issue of easily getting trapped in local optima in the original WOA, this study introduces a non-linear adaptive weighting method. The convergence factor \(a\) is a crucial parameter that affects both the search capability and convergence speed of the WOA. In traditional methods, the convergence factor \(a\) exhibits slow convergence in the early stages and is prone to becoming trapped in local optima in the later stages, affecting both the global and local search abilities and failing to fully reflect the optimization process of the algorithm.