## 3 Proposed Methods
### Improved Grey Wolf-Harris Hawks Hybrid Algorithm
#### 3.3.4 Elite Retention and Perturbation Strategies  
In population-based optimization algorithms, the elite retention strategy enhances convergence by preserving the best-performing individuals in each iteration, thereby preventing the loss of high-quality solutions [48]. However, relying solely on elite retention may reduce population diversity and increase the risk of premature convergence. To address this issue, a Gaussian stochastic perturbation mechanism is introduced to perform localized searches around elite individuals, thereby enhancing search diversity and improving global exploration capability.  
After each iteration, individuals are ranked in descending order according to their fitness values, and the top \(P\%\) are selected to form the elite set. To ensure retention of the current best solution \(X_{best}(t)\), it is forcibly included in the elite set if it does not fall within the top \(P\%\). The elite set is defined in Equation (50).  
\[X_{elite}(t+1)=\{X_{1},X_{2},...,X_{m}\}\cup\{X_{best}(t)\},m=\lfloor N\times P\rfloor. \tag{50}\]  
To further enhance the exploration capability of elite individuals, Gaussian perturbation is introduced in Equation (51) to generate neighboring solutions. Building on this, Equation (52) incorporates a pair of sinusoidal functions to dynamically adjust the perturbation amplitude throughout the iteration process. In the early stage, a larger amplitude encourages broad exploration and reduces the risk of premature convergence. As the iteration progresses, the amplitude gradually decreases, enabling more focused local searches and improving solution refinement accuracy.  
\[X_{elite}^{\prime}(t+1)=X_{elite}(t+1)+N\Big{(}0,\sigma^{2}\Big{)},\sigma= \sigma_{c}\times\left(1-\frac{t}{T}\right)\times\left|sin\left(\frac{\pi t}{T} \right)\right|, \tag{51}\]  
\[\sigma=\sigma_{c}\times\left(1-\frac{t}{T}\right)\times\left|sin\left(\frac{ \pi t}{T}\right)\right|. \tag{52}\]  
After each round of perturbation, the fitness of the newly generated neighboring solution is evaluated and compared with that of the original elite individual. The better-performing solution is retained for the next iteration, as defined in Equation (53).  
\[X_{end-elite}(t+1)=\left\{\begin{array}{ll}X_{elite}^{\prime}(t+1),&if\ f\big{(}X_{elite}^{\prime}(t+1)\big{)}<f(X_{end-elite} (t)),\\ X_{end-elite}(t),&otherwise.\end{array}\right. \tag{53}\]  
During the hybrid search phase of the GWO and HHO algorithms, the fitness values of the combined candidate solutions and the elite-perturbed solutions are compared. The solution with the lowest fitness value is selected as the optimal one for the current iteration.  
To maintain a constant population size, the updated elite and non-elite individuals after perturbation are merged and ranked in ascending order of fitness. The top \(N\) individuals are then selected to form the next generation. This merging process is expressed in Equation (54).  
\[X_{new}(t+1)=SelectBest(X_{elite}(t)\cup X_{non-elite}(t),N), \tag{54}\]  
where \(X_{non-elite}(t)\) denotes the set of updated non-elite individuals, and the _SelectBest_ function represents the selection of the top \(N\) optimal solutions based on fitness values to keep the population size constant.