## 2 Methods
### Principles of GWO  
The Grey Wolf Optimizer (GWO) is a state-of-the-art swarm intelligence optimization algorithm introduced by [PERSON] et al. [23] in 2014, drawing inspiration from the collaborative and adaptive prey-hunting behavior observed in grey wolf packs. This algorithm effectively mirrors the social hierarchy and dynamic hunting strategy of grey wolves, streamlining the optimization process. Unlike other swarm intelligence algorithms such as Particle Swarm Optimization and the Artificial Bee Colony Algorithm, the GWO stands out for its operational simplicity, minimal parameter requirements, and robustness. Its superior convergence speed and precision in finding solutions have made it a favored approach for solving nonlinear continuous optimization problems, with successful applications across various domains. Grey wolves, apex predators of the food chain, are among the largest members of the canine family. They are inherently social animals, typically forming packs that consist of a familial unit with 5 to 12 individuals. These packs have a stringent social hierarchy, categorized into \(\alpha\) leaders, \(\beta\) sub-leaders, \(\delta\) members, and \(\omega\) fringe members, ensuring organized and efficient group dynamics. Figure 1 clearly delineates the grey wolf pack's adherence to a strict social hierarchy. Alpha wolves, at the pinnacle, exhibit the strongest management and leadership abilities, orchestrating decisions within the pack regarding hunting plans, timings and locations for activities, and distribution of food. Beta wolves, as the alphas' main aides, assist in decision-making and organizing group endeavors. They may also temporarily assume leadership in the alpha's absence. Betas yield authority over all members except the alpha, facilitating communication by relaying orders and ensuring their execution. Delta wolves occupy the third tier, obeying alpha and beta commands while possessing command over lower-ranking wolves, chiefly responsible for reconnaissance, guarding, hunting, and caregiving. Age or diminished adaptability can lead to demotion to this rank. Omega wolves, at the lowest echelon, focus on fostering intrapack relations and tending to pups. The pack's hunting dynamics showcase a pronounced hierarchical structure, pivotal for maximizing hunting efficacy. This study explores grey wolves' predatory patterns and collaborative dynamics, emphasizing the alpha's crucial role in leadership and pack coordination. Utilizing a collaborative strategy to detect, track, and enclose prey, the wolves form a surrounding circle. As this circle tightens, the alpha commands the beta and delta wolves to initiate the attack, with the rest of the pack adjusting the enclosure dynamically in response to the prey's movements, exerting pressure from different angles until capture is achieved. Within the Grey Wolf Optimizer (GWO) framework, three wolves designated as \(\alpha\), \(\beta\), and \(\delta\) display superior search capabilities, guiding the group towards areas within the search space that potentially contain optimal solutions. The remaining wolves, identified as \(\omega\), dynamically adjust their positions in response to the movements of \(\alpha\), \(\beta\), or \(\delta\). The following section outlines three crucial definitions pertinent to the algorithm. Definition 1: The distance between individual grey wolves and the prey. Within the Grey Wolf Optimizer (GWO) framework, it is imperative for grey wolves to initially encircle their prey. This necessitates the calculation of the distance from each wolf to the prey, mirroring the spatial separation between the wolves and their target throughout the hunting sequence. \[\mathbf{D}= \left|\mathbf{C}\cdot\mathbf{X_{\mathrm{p}}}(t)-\mathbf{X}(t)\right| \tag{1}\] \[\mathbf{C}=2\cdot\mathbf{r_{1}} \tag{2}\]  
where \(\mathbf{D}\) signifies the distance between a grey wolf and the prey. The variable \(t\) indicates the current iteration count.