## 3 Optimization Methods  
### Particle Swarm Optimization and Hybrid Particle Swarm Optimization Methods  
The particle swarm optimization (PSO) method was proposed by [PERSON] and [PERSON] in 1995 [26], and it employs the social behavior of birds or fish when searching for food as the searching solution concept of the optimization method. Particles correspond to an individual in a school of birds or fish: when a flock of bird searches for food, the moving velocity and direction of each individual are affected by three factors, namely inertial velocity, individual experience, and group influence. Inertial velocity refers to the flying velocity of birds during search, and it affects the search direction and velocity of an individual. Individual experience refers to the accurate direction and velocity of an individual based on past experience. Group influence refers to the effect of the group behavior on individuals, and individuals move toward the individual closest to the food at the moment. In the D-dimensional search space, each particle exhibits a position vector \(\mathbf{x}_{i}=(x_{i1},\ x_{i2},\ \ldots,\ x_{i\ D})\) with a velocity vector \(\mathbf{v}_{i}=(v_{i1},\ v_{i2},\ \ldots,\ v_{i\ D})\). Particles are originally initialized in a uniform random manner through the search space, and the velocity is also randomly initialized. The new position (\(x_{i}^{i+1}\)) and new velocity (\(v_{i}^{i+1}\)) of each particle movement are as follows. \[v_{i}^{t+1}=v_{i}^{t}+c_{1}\times rand(\ )\times(P_{i}^{t}-x_{i}^{t})+c_{2} \times Rand(\ )\times(P_{g}^{t}-x_{i}^{t}) \tag{11}\]  
\[ \begin{array}{l}x_{i}^{t+1}=x_{i}^{t}+v_{i}^{t+1}\\ -V_{\max}\leq v_{i}^{t}\leq V_{\max}\end{array}  \tag{12}\]  
where \(i\) corresponds to each individual, \(t\) is the number of iterations (generations), \(x_{i}^{t}\) is the position of an individual, \(v_{i}^{t}\) is the velocity of an individual, \(P_{i}^{t}\) is the best position (_Pbest_) of an individual \(i\) at \(t\)-th iteration, \(P_{g}^{t}\) is the global best position (_Gbest_) of the swarm at \(t\)-th iteration, \(c_{1}\) and \(c_{2}\) are the acceleration coefficients or learning factors, and they represent the cognitive and social component, respectively, and are used to decide whether particles prefer moving toward a _Pbest_ position or _Gbest_ position. Most studies utilize learning factors of 2, which indicates that the same importance is given to cognitive searching and social searching; \(rand(\ )\) and \(Rand(\ )\) are two uniformly distributed random numbers between 0 and 1. The drawbacks of the original PSO method is the requirement for specifying the velocity \(v_{i}^{t}\) within \([-V\text{max},\,V\text{max}]\). The flow chart of the PSO algorithm is shown in Figure 2.  
The HPSO algorithm was proposed by [PERSON] et al. in 2010 [33], and it incorporates the concepts of inertia weight (\(\omega\)) proposed by [PERSON] et al. [96] and constriction factor (\(K\)) proposed by [PERSON] [97]. [PERSON] et al. [33] proposed the restriction of the comparison of the fitness value to the top 30% of the best particles and keep them, and the remaining particles are crossed-over and selected using a similar approach as the GA to prevent the particles from being trapped in the local optimal region and to improve the convergence and diversity of the original PSO. The three improvements of the HPSO to the original PSO method are:  
1. The use of the linear inertia weight (\(\omega\)) proposed by [PERSON] et al., that is, multiply the original velocity by an inertia weight to linearly change the original velocity to increase the search ability of particles. This increases the versatility of PSO in the search and is closer to the optimal solution.