## 3 Multi-Objective Optimization
### Multi-Objective Hybrid Particle Swarm Optimization and Differential Evolution Algorithm
```
if\(PS^{(G+1)}>0.75\)then if\(rand>0.5\)then \(\mathbf{s}_{i}^{(G+1)}=\mathbf{s}_{i}^{(G)}+rand_{1}F\Big{(}\mathbf{x}_{\tau_{2 }}^{(G)}-\mathbf{x}_{\tau_{3}}^{(G)}\Big{)}+c_{2}rand_{2}\Big{(}\mathbf{x}_{ gbest}^{(G)}-\mathbf{x}_{i}^{(G)}\Big{)}\) else \(\mathbf{s}_{i}^{(G+1)}=\mathbf{s}_{i}^{(G)}+c_{1}rand_{1}\Big{(}\mathbf{x}_{ pbest,i}^{(G)}-\mathbf{x}_{i}^{(G)}\Big{)}+c_{2}rand_{2}\Big{(}\mathbf{x}_{ gbest}^{(G)}-\mathbf{x}_{i}^{(G)}\Big{)}\) endif else if\(PS^{(G+1)}\leq 0.75\)then if\(rand>0.5\)then \(\mathbf{s}_{i}^{(G+1)}=\mathbf{s}_{i}^{(G)}+c_{1}rand_{1}\Big{(}\mathbf{x}_{pbest,i }^{(G)}-\mathbf{x}_{i}^{(G)}\Big{)}\) \(+c_{2}F\Big{(}\mathbf{x}_{best}^{(G)}-\mathbf{x}_{i}^{(G)}\Big{)}+F\Big{(} \mathbf{x}_{\tau_{1}}^{(G)}-\mathbf{x}_{\tau_{2}}^{(G)}\Big{)}\) else \(\mathbf{s}_{i}^{(G+1)}=\mathbf{s}_{i}^{(G)}+c_{1}rand_{1}\Big{(}\mathbf{x}_{ pbest,i}^{(G)}-\mathbf{x}_{i}^{(G)}\Big{)}+c_{2}rand_{2}\Big{(}\mathbf{x}_{ gbest}^{(G)}-\mathbf{x}_{i}^{(G)}\Big{)}\) endif endif
```  
**Algorithm 2** Presudocode of the velocity update equation of the hybrid algorithm  
According to the pseudo-code given in Algorithm 2, it can be observed that based on the value of the population spacing \(PS^{(G+1)}\) the appropriate equation for the velocity update will be chosen. Therefore, for \(PS^{(G+1)}>0.75\) the diversification of the population is uneven, and the exploitation ability should be enhanced. Therefore, the improved expression in Equation (47) will be applied with the probability of 0.5, otherwise Equation (37) is applied. On the other hand, smaller value of \(PS^{(G+1)}\), e.g., \(PS^{(G+1)}\leq 0.75\), indicates that the local exploitation should be improved. Therefore, the expression (48) is applied with the probability 0.5, otherwise Equation (37) is applied. To further increase diversity of the population the crossover operator from the DE algorithm has been incorporated into the PSO algorithm. Based on the above, the pseudo-code of the proposed MHPSODE iterative procedure is presented in Algorithm 3. ```
Initialize the parameters \(N_{P}\), \(n\), \(MaxIter\)  Uniformly and randomly generate positions of the particles in the swarm  Initialize velocities of each particle in the swarm \(\mathbf{s}_{i}^{(0)}\), \(\forall i\in{1,2,\ldots,N_{P}}\)  Set the initial \(\mathbf{x}_{i,pbest}^{(0)}\) and \(\mathbf{x}_{gbest}^{(0)}\) values whileiter \(<MaxIter\)do  update the inertia weight \(w^{(G)}\) fori = 1:\(N_{P}\)do  Calculate the velocity if\(PS^{(G+1)}>0.5\)then if\(rand>0.5\)then \(\mathbf{s}_{i}^{(G+1)}=\mathbf{s}_{i}^{(G)}+rand_{1}F\Big{(}\mathbf{x}_{\mathbf{ \tau}_{2}}^{(G)}-\mathbf{x}_{\mathbf{\tau}_{3}}^{(G)}\Big{)}+c_{2}rand_{2}\Big{(} \mathbf{x}_{gbest}^{(G)}-\mathbf{x}_{i}^{(G)}\Big{)}\) else \(\mathbf{s}_{i}^{(G+1)}=\mathbf{s}_{i}^{(G)}+c_{1}rand_{1}\Big{(}\mathbf{x}_{gbest,i}^{(G)}-\mathbf{x}_{i}^{(G)}\Big{)}+c_{2}rand_{2}\Big{(}\mathbf{x}_{gbest}^{( G)}-\mathbf{x}_{i}^{(G)}\Big{)}\) endif elseif\(PS^{(G+1)}\leq 0.5\)then if\(rand>0.5\)then \(\mathbf{s}_{i}^{(G+1)}=\mathbf{s}_{i}^{(G)}+c_{1}rand_{1}\Big{(}\mathbf{x}_{gbest,i}^{(G)}-\mathbf{x}_{i}^{(G)}\Big{)}\) \(+c_{2}F\Big{(}\mathbf{x}_{best}^{(G)}-\mathbf{x}_{i}^{(G)}\Big{)}+F\Big{(} \mathbf{x}_{\mathbf{\tau}_{1}}^{(G)}-\mathbf{x}_{\mathbf{\tau}_{2}}^{(G)} \Big{)}\) else \(\mathbf{s}_{i}^{(G+1)}=\mathbf{s}_{i}^{(G)}+c_{1}rand_{1}\Big{(}\mathbf{x}_{gbest,i}^{(G)}-\mathbf{x}_{i}^{(G)}\Big{)}+c_{2}rand_{2}\Big{(}\mathbf{x}_{gbest}^{( G)}-\mathbf{x}_{i}^{(G)}\Big{)}\) endif endif  Update the position of particles \(\mathbf{x}_{i}^{(G+1)}=\mathbf{x}_{i}^{(G)}+\mathbf{s}_{i}^{(G+1)}\)  Evaluate crossover operator from the DE algorithm according to Equation (45)  Determine the personal best solution according to \(\mathbf{x}_{pbest,i}^{(G+1)}=\left\{ \begin{array}{ll}\mathbf{x}_{pbest,i}^ {(G)}&\text{if}\mathbf{x}_{pbest,i}^{(G)}\prec\mathbf{x}_{i}^{(G)}\\ \mathbf{x}_{i}^{(G)},\text{otherwise}\end{array}