## 3 Proposed Methods
### Improved Grey Wolf-Harris Hawks Hybrid Algorithm
#### 3.3.3 Adaptive Weight Dynamic Switching Strategy  
To enable adaptive adjustment and ensure a smooth transition during the search process, this study introduces a switching threshold, as defined in Equation (45).  
\[\varepsilon(t)=\varepsilon_{max}\cdot e^{-\gamma\frac{t}{t}}+\varepsilon_{min} \cdot\left(1-e^{-\gamma\frac{t}{t}}\right), \tag{45}\]  
where the parameters \(\varepsilon_{max}\) and \(\varepsilon_{min}\) denote the maximum and minimum values of the switching threshold, respectively, and \(\gamma\) denotes the control parameter that regulates the rate of change. Preliminary experiments indicated that setting \(\gamma\) to 5 provides an optimal balance between exploration and exploitation. This setting enables the algorithm to effectively explore the search space while also converging to optimal or near-optimal solutions within a reasonable number of iterations.  
To evaluate the search state in real time, the optimal solution change rate is introduced, as shown in Equation (46).  
\[\Delta f=\frac{|f_{best}(t)-f_{best}(t-1)|}{f_{best}(t-1)+\eta}, \tag{46}\]  
where \(f_{best}(t)\) is the current optimal fitness, and \(\eta\) = 10\({}^{-8}\) is added to avoid zero in the denominator.  
If \(\Delta f>\varepsilon(t)\), the search is still exhibiting significant improvement, and the GWO algorithm continues to guide global exploration. Conversely, when the change rate falls below the threshold, the solution tends to stabilize, and control is gradually shifted to the HHO algorithm to enhance local exploitation.  
To prevent premature switching to the HHO algorithm caused by fluctuations in the optimal solution, the switching mechanism is further refined by introducing a weight ratio \(R(t)=\frac{\lambda_{H}}{\lambda_{C}}\) as the dominant criterion. This mechanism dynamically adjusts the search weights of both algorithms based on the relationship between \(\Delta f\) and \(\varepsilon(t)\), using a Sigmoid function as defined in Equation (47).  
\[\lambda_{H}(t)=\frac{1}{1+e^{-\theta(\varepsilon(t)-\Delta f)}},\lambda_{G}(t )=1-\lambda_{H}(t), \tag{47}\]  
where \(\theta\) is the sensitivity adjustment parameter, typically set to 10 [47].  
Based on the dynamic changes in \(\Delta f\) and \(\varepsilon(t)\), different switching conditions are defined for various search phases. In the early stage, a higher threshold allows larger solution fluctuations, promoting global exploration. In the middle stage, \(R(t)\) as the threshold gradually decreases to \(1/2\leq R(t)\leq 2.0\), and the GWO and HHO algorithms cooperate to maintain a balance between exploration and exploitation. In the late stage, the threshold reaches its minimum while the improvement rate increases, allowing the HHO algorithm to dominate and enhance local search accuracy.  
In the coordinated search phase, the GWO algorithm generates candidate solutions according to Equation (29), while the HHO algorithm selects appropriate position update strategies based on the prey escape energy \(E\) and the random factor \(r\). The final position of each individual is then determined by a weighted combination of the two solutions, as defined in Equation (48).  
\[X_{combined}(t)=\lambda_{G}(t)\cdot X_{GWO}(t)+\lambda_{H}(t)\cdot X_{HHO}(t). \tag{48}\]  
However, the weighted combination is treated as a candidate solution rather than the final output. A multi-source selection mechanism is subsequently applied to determine whether the optimal solution should be updated, as defined in Equation (49).  
\[X_{best}^{\prime}(t)=\left\{\begin{array}{ll}X_{GWO}(t)&,&if\ f_{GWO}<f_{HHO} \cup f_{combinet},\\ X_{HHO}(t)&,&if\ f_{HHO}<f_{GWO}\cup f_{combinet},\\ X_{combined}(t)&,&otherwise.\end{array}\right. \tag{49}\]