## 3 Optimization Methods
### Particle Swarm Optimization and Hybrid Particle Swarm Optimization Methods
The improved velocity formula is: \[v_{i}^{t+1}=\omega\;(t)\times v_{i}^{t}+c_{1}\times rand(\;)\times(P_{i}^{t}- x_{i}^{t})+c_{2}\times Rand(\;)\times(P_{g}^{t}-x_{i}^{t})\] (13) \[\omega\;(t)=(\omega_{1}-\omega_{2})\times\frac{(N_{t}-t)}{N_{t}}+\omega_{2}\] (14) where,\(\omega_{1},\;\omega_{2}\) are the initial and final inertia weight, respectively, and \(N_{t}\) is the total number of iterations. 2. The use of a constriction factor (\(K\)) to improve the PSO algorithm. This can effectively control the stability and trajectory of the particle search process without limiting the maximum speed of the particle movement. The updated formula for the new velocity of an individual in the HPSO method combines the merits of the methods proposed by [PERSON] et al. [96] and [PERSON] [97], and can be expressed as follows:  
Figure 2: Flow chart of the PSO algorithm. \[v_{i}^{t+1}=K\Big{\{}\omega\;(t)\times v_{i}^{t}+c_{1}\times rand(\;)\times(P_{i}^ {t}-x_{i}^{t})+c_{2}\times Rand(\;)\times(P_{g}^{t}-x_{i}^{t})\Big{\}} \tag{15}\]  
\[K=\frac{2}{\Big{|}2-\varphi-\sqrt{\varphi^{2}-4\varphi}\Big{|}}\,\varphi=c_{1}+ c_{2},\;\varphi\geq 4 \tag{16}\]
3. The crossover of the two modified operations of GA and the incorporation of the top 30% selection rule into the algorithm. The starting population is a set of design variables whose values are randomly generated within a searching space. Each individual (chromosome) of the population is a possible solution to the problem and is formed by parameters (genes) that set the design variables of the problem. All genes are grouped within a vector that represents a chromosome. The swarm of the population is evaluated using the fitness function to determine the personal and global best values of the entire swarm. Subsequently, the top 30% selection is used to sift through the chromosomes with better fitness from the swarm, while the remaining 70% of particles are put into the modified crossover process to increase the diversity of the population and to prevent the particles from being trapped in the local optimal region. The modified crossover process consists of two approaches. First, a random single point is selected to cut the chromosome into two parts. A uniform crossover method is performed on the first half, in which the parents equally contribute each of their gene values to the offspring chromosomes. This enables the mixing of the parent chromosomes at the gene level rather than the segment level. A normal crossover method is performed on the second half, in which the gene values of two parent chromosomes are interchanged at the crossover point. The details of the modified crossover operator are illustrated in Figure 3, in which \(P_{1}\), \(P_{2}\) refer to the parent chromosomes and \(C_{1}\), \(C_{2}\) indicate the generated children (offspring) chromosomes. After the recombination process during the modified crossover, the fitness values of the parent and offspring chromosomes are calculated for re-evaluations. These chromosomes with fitness values will undergo a top 50% selection again to obtain improved solutions, and the positions and velocities will be updated using a modified standard PSO with inertial weight to determine the \"_Pbest_\" and \"_Gbest_\" solutions. The selection rule for new generation particles is depicted in Figure 4. The detailed flow chart of the HPSO algorithm is shown in Figure 5.  
Figure 3: Modified crossover method.