## 3 Multi-Objective Optimization
### Multi-Objective Hybrid Particle Swarm Optimization and Differential Evolution Algorithm  
To successfully solve MOO problems, the EAs must effectively balance the convergence and diversity of the swarm during the optimization process. It has been previously shown that the PSO algorithm tends to be easily trapped into the local optima and to lose the diversity [36]. Therefore, to keep an effective balance between convergence and diversity the mutation operators from the DE algorithm have been introduced into the optimization procedure of the PSO algorithm. In the literature, it has been well established that the DE/rand1 and DE/rand/2 mutation operators have good exploration ability and hence, are able to successfully discover the region of global optimum [33]. On the other hand, mutation operators DE/best/1 and DE/current-to-best/1 focus their search in the region of the best solution, and therefore have better exploitation and faster convergence towards optimal solution [33]. In this regard, to improve the global exploration ability and prevent premature convergence to one of the local optima, the key elements from the DE/rand/1 mutation operator are included into the velocity update equation of the PSO algorithm. Therefore, the term \(c_{1}\Big{(}\mathbf{x}_{best,i}^{(G)}-\mathbf{x}_{i}^{(G)}\Big{)}\) in velocity equation has been substituted with the term \(F\Big{(}\mathbf{x}_{r_{2}}^{(G)}-\mathbf{x}_{r_{3}}^{(G)}\Big{)}\), and the velocity update equation has the following form  
\[\mathbf{s}_{i}^{(G+1)}=\mathbf{s}_{i}^{(G)}+rand_{1}F\Big{(}\mathbf{x}_{r_{2}}^ {(G)}-\mathbf{x}_{r_{3}}^{(G)}\Big{)}+c_{2}rand_{2}\Big{(}\mathbf{x}_{best}^{( G)}-\mathbf{x}_{i}^{(G)}\Big{)}. \tag{47}\]  
On the other hand, to improve the exploitation ability the term \(F\Big{(}\mathbf{x}_{best}^{(G)}-\mathbf{x}_{i}^{(G)}\Big{)}\) from the DE/current-to-best/1 algorithm has been introduced into Equation (37) instead of the term \(c_{2}rand_{2}\Big{(}\mathbf{x}_{gbest}^{(G)}-\mathbf{x}_{i}^{(G)}\Big{)}\). Furthermore, to maintain the required level of diversity, the term \(F\Big{(}\mathbf{x}_{\tau_{1}}^{(G)}-\mathbf{x}_{2}^{(G)}\Big{)}\) has been added to Equation (37), which becomes  
\[\mathbf{s}_{i}^{(G+1)}=\mathbf{s}_{i}^{(G)}+c_{1}rand_{1}\Big{(}\mathbf{x}_{pbest,i}^{(G)}-\mathbf{x}_{i}^{(G)}\Big{)}+c_{2}F\Big{(}\mathbf{x}_{best}^{(G)}- \mathbf{x}_{i}^{(G)}\Big{)}+F\Big{(}\mathbf{x}_{\tau_{1}}^{(G)}-\mathbf{x}_{ \tau_{2}}^{(G)}\Big{)}. \tag{48}\]  
Based on the above, to obtain a suitable diversity and convergence, which can balance the global exploration and local exploitation abilities of the particles, the normalized population spacing \(PS^{(G+1)}\) has been introduced into the algorithm. The value of the normalized population spacing of the \(i\)th particle in the population in the \((G+1)\)th iteration can be determined according to the expression  
\[PS^{(G+1)}=\sqrt{\frac{1}{N_{P}-1}\sum_{i=1}^{N_{P}}\Big{(}\bar{d}^{(G+1)}-d_{ i}^{(G+1)}\Big{)}^{2}}, \tag{49}\]  
where \(d_{i}^{(G+1)}\) denotes the minimum L1-norm distance between the ith particle and all other particles in the population, and \(\bar{d}^{(G+1)}\) represents the average minimum L1-norm distance of all particles. Therefore, employing the introduced normalized population spacing \(PS^{(G+1)}\) to combine the developed Equations (47) and (48), the expression for velocity update of the PSO algorithm can be stated with the pseudo-code given in Algorithm 2.