## 3 Proposed Methods
### Improved Grey Wolf-Harris Hawks Hybrid Algorithm  
Proposed by [PERSON] et al. [39] in 2014, GWO is a heuristic algorithm inspired by grey wolf hunting behavior and social hierarchy. It solves complex optimization problems by simulating leadership structures and collaborative strategies within the wolf pack. The population is divided into \(\alpha\), \(\beta\), \(\delta\), and \(\omega\), which correspond to the best, suboptimal, superior, and general solutions. The optimization process involves three main phases, including encircling the prey, tracking its position, and performing the attack. In the encirclement phase, each individual estimates the distance to the prey using the distance vector \(D\) and updates its position by adjusting the step size based on the control coefficients \(A\) and \(C\), thereby gradually approaching the optimal solution. \[D=\big{|}C\cdot X_{prey}(t)-X(t)\big{|}, \tag{23}\]  
\[X(t+1)=X_{prey}(t)-A\cdot D, \tag{24}\]where \(X_{prey}(t)\) represents the prey's position, i.e., the current optimal solution at iteration \(t\), while \(X(t)\) denotes the position of a grey wolf at the same iteration. \(T\) refers to the maximum number of iterations. The adjustment coefficients \(A\) and \(C\) are determined by random variables \(r_{1}\) and \(r_{2}\), uniformly distributed within the range [0, 1], with their formulations provided in Equations (25) and (26). \[A=2a\cdot r_{1}-a, \tag{25}\]  
\[C=2r_{2}. \tag{26}\]  
The convergence factor \(a\), which controls the magnitude of these coefficients, decreases linearly from 2 to 0 during the iteration process in accordance with Equation (27). \[a=2-\frac{2t}{T}. \tag{27}\]  
In the position update phase, the general grey wolf generates candidate solutions \(X_{1}\), \(X_{2}\), and \(X_{3}\) by following the guidance of the three leading wolves, as illustrated in Equation (28). \[\left\{ \begin{aligned} D_{\alpha}&=|C_{1}\cdot X_{ \alpha}-X|,X_{1}=X_{\alpha}-A\cdot D_{\alpha},\\ D_{\beta}&=|C_{2}\cdot X_{\beta}-X|,X_{2}=X_{\beta}-A \cdot D_{\beta},\\ D_{\delta}&=|C_{3}\cdot X_{\delta}-X|,X_{3}=X_{\delta}-A \cdot D_{\delta},\end{aligned} \right. \tag{28}\]  
where \(X_{\alpha}\), \(X_{\beta}\), and \(X_{\delta}\) denote the positions of \(\alpha\), \(\beta\), and \(\delta\) wolves, which correspond to the top three solutions with the highest fitness. The distances of individual wolves from \(\alpha\), \(\beta\), and \(\delta\) head wolves are represented by \(D_{\alpha}\), \(D_{\beta}\), and \(D_{\delta}\), respectively. Additionally, \(C_{1}\), \(C_{2}\), and \(C_{3}\) are the random coefficients. This process is further illustrated in Figure 3, which shows how the grey wolf updates its position based on the optimal, suboptimal, and superior solutions within the population. The final position is calculated as a weighted average of the positions of the optimal, suboptimal, and superior individuals, guiding the population toward the optimal direction and enhancing convergence reliability, as presented in Equation (29). \[X(t+1)=\frac{X_{1}+X_{2}+X_{3}}{3}. \tag{29}\]  
Due to its dynamic balance between global exploration and local exploitation, GWO is highly efficient for solving complex scheduling problems. Its simple structure, rapid conver  
Figure 3: The grey wolf location update in the grey wolf algorithm. gence, and effective global search capability make GWO a powerful tool. It demonstrates strong robustness and fast convergence in multi-objective optimization and is widely applied in path planning, intelligent scheduling, and low-carbon logistics [40; 41; 42]. However, GWO tends to fall into local optima, particularly in high-dimensional problems, which limits its solution accuracy. The HHO algorithm addresses this issue by enhancing local search capabilities and improving adaptability in escaping local optima. Combining both algorithms enhances convergence stability and search performance for multi-objective scheduling problems. Originally proposed by [PERSON] et al.