## 3 Multi-Objective Optimization  
### Multi-Objective Particle Swarm Optimization Algorithm  
The PSO belongs to a group of nature-inspired evolutionary algorithms (EAs), whose mechanisms are inspired by bird-swarming behavior in the process of searching for food [14]. The optimization process of the PSO algorithm starts with the creation of the swarm of \(N_{P}\) particles in the swarm, written as the set \(P^{(G)}=\{\mathbf{x}_{i}\mid\mathbf{x}_{i}\in\mathbb{F}\}\), \(i\in\{1,2,\ldots,N_{P}\}\), where \(\mathbb{F}\) denotes the feasible solution space, and \(G\) is the current generation index in the initial generation \(G=0\). The current position \(\mathbf{x}_{i}^{(G)}\) of each particle is influenced by the appropriate speed vector \(\mathbf{s}_{i}^{(G)}\). The vector \(\mathbf{s}_{i}^{(G)}\) is thus influenced by the best position that each particle has previously discovered, denoted as \(\mathbf{x}_{gbest,i}^{(G)}\), and the best position discovered by the entire swarm of particles \(\mathbf{x}_{gbest}^{(G)}\). These parameters represent particles' individual intelligence and the group swarm's intelligence. In each generation, the position of particles is changed according to  
\[\mathbf{s}_{i}^{(G+1)} =\mathbf{s}_{i}^{(G)}+c_{1}rand_{1}\Big{(}\mathbf{x}_{gbest,i}^{( G)}-\mathbf{x}_{i}^{(G)}\Big{)}+c_{2}rand_{2}\Big{(}\mathbf{x}_{gbest}^{(G)}- \mathbf{x}_{i}^{(G)}\Big{)}, \tag{29}\] \[\mathbf{x}_{i}^{(G+1)} =\mathbf{x}_{i}^{(G)}+\mathbf{s}_{i}^{(G+1)}, \tag{30}\]  
where \(c_{1}\) represents the cognitive and \(c_{2}\) social coefficients, and \(rand_{1}\) and \(rand_{2}\) are random numbers in the range \(rand_{i}\in(0,1)\), \(i=1,2\), such that \(rand_{1}\
eq rand_{2}\). To update the personal best position discovered by each particle in each generation in application to multi-objective optimization, we modify the original PSO in a way that the modification of  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Variable** & **Lower Bound** & **Upper Bound** & **Type** \\ \hline \(z_{a}\) & 17 & 26 & Integer \\ \(z_{b}\) & 22 & 60 & Integer \\ \(z_{g}\) & 40 & 120 & Integer \\ \(m\) & 2 & 40 & Discrete \\ \(x_{a}\) & \(-0.5\) & 1 & Continuous \\ \(x_{b}\) & \(-0.5\) & 1 & Continuous \\ \(x_{g}\) & 0.0 & 0.5 & Continuous \\ \hline \hline \end{tabular}
\end{table}
Table 2: Details of the design variables.  
the personal best position of each particle occurs when the present personal best solution is dominated by the current position, as expressed by the subsequent equation  
\[\mathbf{x}_{pbest,i}^{(G+1)}=\begin{cases}\mathbf{x}_{pbest,i}^{(G)},&\text{if } \mathbf{x}_{pbest,i}^{(G)}\prec\mathbf{x}_{i}^{(G)}\\ \mathbf{x}_{i}^{(G)},&\text{otherwise}\end{cases}. \tag{31}\]  
To update the global best position \(\mathbf{x}_{gbest}^{(G)}\) due to the conflicting nature of the objectives in a multi-objective optimization problem, and when there is no single solution to be used, an external archive is employed. In this archive, a set of non-dominated solutions obtained in the optimization process is memorized. Then, in each generation, the crowding distance is employed to evaluate the diversity of solutions in the archive, and the solution \(\mathbf{x}_{gbest}^{(G)}\) is selected using the binary tournament [31; 32].