## 3 Proposed Methods
### Improved Grey Wolf-Harris Hawks Hybrid Algorithm
#### 3.3.2 Convergence Factor and Flight Step Size Optimization  
During the trapping phase of the GWO algorithm, the original implementation uses a linearly decreasing convergence factor \(a\), as defined in Equation (27). However, this linear strategy may cause the search range to shrink prematurely in later iterations, reducing global exploration capability. To enhance search flexibility and avoid early convergence, this study introduces a stochastic disturbance based on the gamma distribution to adaptively adjust the convergence factor. The revised formulation is given in Equation (42).  
\[a=\frac{2}{1+e^{0.025(\frac{-1}{2})}}+\vartheta\cdot gamrnd, \tag{42}\]  
where \(\vartheta\) is the perturbation adjustment factor.  
In addition, the adjustment coefficient \(C\), which maintains stochastic fluctuations throughout the iteration process, does not fully capture the dynamic nature of the search phase. This limitation may lead to excessive dispersion during the early stage of the search. To address this issue, an adaptive convergence factor is introduced to refine the calculation of coefficient \(C\) in the GWO algorithm, as defined in Equation (43).  
Figure 4: Two methods for initializing the population distribution. (**a**) Random initialization; (**b**) Tent method initialization.  
\[C=4r_{1}\mu,\mu=4r_{2}r_{3}\left(\frac{t}{T}\right), \tag{43}\]  
where \(r_{1},\;\;r_{2},\;\;r_{3}\in[0,1]\) is a random factor used to improve search diversity.  
Figure 5 illustrates that the dynamically adjusted value of coefficient \(C\) fluctuates significantly in the early stages, enhancing global exploration. As the iteration progresses, the value gradually stabilizes, supporting a balanced transition between exploration and exploitation.  
The Levy flight mechanism, proposed by the mathematician [PERSON], is a stochastic process characterized by a power-law distribution. In the local exploitation phase of the HHO algorithm, a fixed step size is typically used. In contrast, the original [PERSON] flight strategy introduces search jumps through randomly varying step sizes. Although its heavy-tailed distribution helps balance local intensification and global exploration, the step scaling factor is usually constant. This lack of adaptability may cause excessive variations in the search range during later iterations, which can undermine convergence stability and degrade solution quality. To address this issue, a dynamic step size adjustment mechanism is introduced. This allows the step length to gradually decrease with the iteration process, thereby adaptively controlling the search scope. The adjustment formulation is provided in Equation (44).  
\[S=\frac{S_{0}}{1+\beta\left(\frac{t}{T}\right)}, \tag{44}\]  
where \(S_{0}\) is the initial step size and \(\beta\) controls the step contraction rate.  
To further enhance search efficiency, each round of Levy flights incorporates a dynamic step size adjustment. If the fitness of the newly generated solution does not surpass that of the current best solution, the search position is updated using a modified step size. When no improvement is observed over three consecutive iterations, a more refined local search is triggered to intensify exploitation.  
Figure 5: Variation in adjustment coefficient \(C\) during the iteration process.