## 3 Research Methods
### Grey Wolf Optimization Algorithm (GWO)  
The Grey Wolf Optimizer (GWO) is an optimization algorithm inspired by the collective tracking and hunting behavior of grey wolves in their pursuit of prey [26]. It was proposed by [PERSON] and others in 2014 to address intricate optimization challenges. The algorithm is based on four levels of grey wolf groups. \(\alpha\) (alpha), \(\beta\) (beta), \(\delta\) (delta), and \(\omega\) (omega) are used to simulate the social hierarchy of the wolf pack.  
Wolf pack hierarchy and social behavior were simulated as follows:  
\(\alpha\) (alpha) is the leader of the pack, and is responsible for deciding the direction of the hunt. The rest of the pack usually follows its guidance. \(\beta\) (beta), as a secondary leader, acts as an advisor and helper. \(\alpha\) wolves and \(\beta\) wolves will take over the leadership. \(\delta\) (delta), as the wolf with a certain prestige, acts under the guidance of \(\alpha\) and \(\beta\) and ensures the internal order and safety of the group. \(\omega\) (omega) is the lowest-ranking member, which mainly follows the instructions of the first three levels to complete the task, and their information is also helpful in finding prey.  
The algorithm flow is as follows:  
1. Initialization: A group of N search agents (grey wolves) is randomly generated, each representing a potential solution to the problem.
2. Lead wolf identification: This assesses the performance of every search agent, which identifies the optimal three agents as \(\alpha\), \(\beta\), and \(\delta\).
3. Finding and encircling prey: The behavior of the three-tier wolf pack members approaching prey is simulated by a mathematical model. The other members (\(\omega\)) update their positions according to the positions of the three leading wolves.
4. Attack (optimization): Constantly it updates the position of the pack in an iterative manner in the expectation of finding prey (optimal solution to the problem).
5. Termination condition: The algorithm continues its iterations until it satisfies a predetermined termination condition, such as reaching the maximum allowable number of iterations or achieving a specific level of solution quality.  
The GWO algorithm adopts the method of simulating wolf pack behavior to carry out a global optimization search. The algorithm mimics the hunting, chasing, and attacking behavior of wolves during prey pursuit. The location update formula considers the position of the current search agent, the position of the lead wolf, and the flexible adaptation of both exploration and exploitation tactics. The balanced exploration and utilization of the search space are realized by calculating the distance between wolves, which updates the position, and randomly selects the leading wolf. Compared with other optimization algorithms, GWO has fewer tuning parameters. The algorithm is simple to implement and easy to extend to high-dimensional problems. Furthermore, GWO proves to be applicable in a wide range of optimization problem scenarios, including continuous, discrete, and multi-objective optimization problems. Overall, GWO achieves a successful equilibrium between worldwide and regional optimization through the emulation of wolves' social structure and hunting tactics.