## 2 Methods
### Improved Grey Wolf Optimizer (IGWO)
#### 2.2.3 Adjusting the Position Update Mechanism  
6. Incorporating Step Size Proportional Weight  
This introduces a novel approach utilizing a proportional weight derived from the Euclidean distance of the step size. The formulation is as follows:  
\[\begin{cases}W_{1}=\frac{|\mathbf{x}_{1}|}{|\mathbf{x}_{1}|+|\mathbf{x}_{2}|+| \mathbf{x}_{3}|}\\ W_{2}=\frac{|\mathbf{x}_{2}|}{|\mathbf{x}_{1}|+|\mathbf{x}_{2}|+|\mathbf{x}_{3 }|}\\ W_{3}=\frac{|\mathbf{x}_{3}|}{|\mathbf{x}_{1}|+|\mathbf{x}_{2}|+|\mathbf{x}_{3 }|}\end{cases} \tag{13}\]  
where \(W_{1},W_{2}\), and \(W_{3}\) symbolize the learning rates by which the omega wolf \(\omega\) adapts from the \(\alpha\), \(\beta\), and \(\delta\) wolves, respectively. To circumvent the risk of the algorithm becoming ensnared in local optima, an improvement has been made to the traditional GWO algorithm's position update equation through the implementation of a dynamic weight strategy. This approach entails the derivation of proportional weights \(W_{1}\), \(W_{2}\), and \(W_{3}\) from the position vectors, designed to undergo modifications in each iteration. Consequently, this strategy ensures that the pack's leaders can dynamically steer the collective towards progressive movement.  
Enhancing the Significance of Position Updates by the Alpha Wolf  
In population-based optimization algorithms, such as the Genetic Algorithm (GA) [34], Particle Swarm Optimization (PSO) [35], Artificial Bee Colony (ABC) [36], and Differential Evolution (DE) [37], achieving a balance between exploration and exploitation is essential. Exploration is the process of searching for new potential solutions within the search space, whereas exploitation optimizes search efficiency by leveraging existing, high-quality solutions. The Grey Wolf Optimizer (GWO) is typically constrained by a limitation wherein non-leader wolves, other than the \(\alpha\), \(\beta\), and \(\delta\) wolves, are predominantly attracted towards the trio of leader wolves. This dynamic often leads to premature convergence, resulting in the algorithm settling for local optima without sufficiently exploring the search space. To enhance the GWO's efficacy, refining its position update mechanism is proposed to ensure  
Figure 3: Comparison chart of convergence factor.  
a more effective balance between exploration and exploitation. This refinement utilizes the insights of the three leading wolves (alpha, explorer, and tracker) to direct candidate solutions towards the global optimum. By amplifying the alpha wolf's role in position updating and the leading wolf's overall influence, the algorithm's capacity for optimization and precision in finding solutions is significantly improved.  
\[X(t+1)=\rho_{1}\times\omega_{3}\times\frac{W_{1}\times X_{1}(t)+W_{2}\times X_{ 2}(t)+W_{3}\times X_{3}(t)}{3}+\rho_{2}\times\omega_{4}\times(X_{best}-X) \tag{14}\]  
where \(\mathbf{X}_{best}\) denotes the \(\alpha\) wolf's position, and \(\omega_{3}\) and \(\omega_{4}\) refer to random numbers in the range of \([0,1]\). \(\rho_{1}\) and \(\rho_{2}\) refer to random numbers in the range of \((0,1]\), which are employed to fine-tune the balance between exploration and exploitation capabilities.