## 4 Particle Swarm Optimization and Experimental Verification  
Due to the limitations of the response surface, it is prone to local optimal solutions. Therefore, the GRD polynomial model obtained from the response surface analysis serves as the fitness function for PSO. PSO is used to optimize the fitness function to obtain a universal best solution. PSO is widely used in engineering, characterized by its fast convergence speed and high efficiency, and is widely used in parameter optimization [35; 36; 37; 38].  
PSO begins by initializing a swarm of particles within the realizable region, with each particle's characteristics represented by its position, velocity, and adaptation value, where the adaptation value reflects the particle's quality. During movement in the solution space, particles update their individual positions by tracking individual and group extremes [39]. The revision equations for speed and location are as follows:  
\[V_{id}^{t+1}=\omega V_{id}^{t}+c_{1}r_{1}(P_{id}^{t}-X_{id}^{t})+c_{2}r_{2}(P_{ gd}^{t}-X_{id}^{t}) \tag{20}\]  
\[X_{id}^{t+1}=X_{id}^{t}+V_{id}^{t+1} \tag{21}\]  
where \(i=1\), \(2\),...,\(n\); \(d=\)1,2,...,\(D\); \(n\) represents particle count; \(D\) represents the dimension; \(D=3\); \(t\) represents the current iteration count; \(\omega\) is the inertia weight; \(V_{id}^{t}\) is the velocity of the i-th particle; \(V_{id}^{t}\in[-V_{\max},V_{\max}]\); \(V_{\max}\) is an invariant; \(X_{id}^{t}\) is the location of the i-th particle; \(c_{1}\) and \(c_{2}\) are acceleration invariants; \(r_{1}\) and \(r_{2}\) are selected randomly from a uniform distribution between zero and one; \(P_{id}^{t}\) is the best location of the i-th particle; and \(P_{gd}^{t}\) is the best location for the population.  
The inertia weights and learning factors of the algorithm are adjusted to overcome the problem of slow convergence speed and the low optimization accuracy of traditional particle swarm algorithms. The inertia weight \(\omega\) shows the degree of influence of the previous flight speed on the current flight speed. When \(\omega\) is large, the algorithm exhibits strong global convergence capabilities. When the \(\omega\) value is small, it demonstrates strong local convergence capabilities. The \(c_{1}\) and \(c_{2}\) learning factors indicate the strength of self-learning summaries and the ability to learn from the community, respectively, with the earlier period focusing more on self-knowledge and the later period focusing more on the individual's ability to acquire social information. The expression for improving the PSO algorithm by improving the inertia weights and learning factors is as follows:  
\[\omega=\omega_{\max}-\frac{(\omega_{\max}-\omega_{\min})\cdot t}{T_{\max}} \tag{22}\]  
\[c_{1}=(a_{1}-a_{2})t/T_{\max}+a_{2} \tag{23}\]  
\[c_{2}=(b_{1}-b_{2})t/T_{\max}+b_{2} \tag{24}\]  
where \(\omega\) denotes the highest inertia weight, \(\omega_{\min}\) is the lowest inertia weight, and \(a_{1}\) and \(a_{2}\) are the initial and iterative end values of the learning factor \(c_{1}\), respectively. \(b_{1}\) and \(b_{2}\) are the initial and iterative end values of the learning factor \(c_{2}\), respectively. \(T_{\max}\) is the maximum number of iterations.