## 3 Multi-Objective Optimization Algorithms
### Optimized multi-Objective Particle Swarm Optimization (OMOPSO)
\begin{array}{l}Y_{1}=0.5[(1-\beta)X_{1}+(1+\beta)X_{2},\\ Y_{2}=0.5[(1+\beta)X_{1}+(1-\beta)X_{2},\end{array} \right. \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Abbreviation** & **Means** \\ \hline \multicolumn{3}{c}{**OMOPSO**[16]} \\ \hline \(g\) & Iteration number \\ \(\mathrm{gmax}\) & Maximum number of iterations \\ \(\in\) & Is the value of the bounding size of the \(\in-\)_archive_ \\ \hline \multicolumn{3}{c}{**NSGA-II**[45]} \\ \hline M & Is the size of random population \\ \hline \multicolumn{3}{c}{**SPEA2**[46]} \\ \hline P & Population \\ A & External archive \\ \hline \multicolumn{3}{c}{**MOSFP**[19]} \\ \hline \(\in\) & Is the value of the bounding size of the \(\in-\)_archive_ \\ \(i\) & Iteration number \\ \(i_{\text{Max}}\) & Maximum number of iterations \\ \(\mathrm{SoW}\) & Set of Winners \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Abbreviations of previous mentioned algorithms based on their resources. where \(\beta\) is a random variable in the range of 0 and 1. \(X\) is the value of chromosome while \(Y\) is the value of chromosome after the crossover. The probability distribution of variable \(\beta\) can be calculated by:  
\[\left\{ \begin{array}{c}P(\beta)=0.5(\eta_{c}+1)\beta^{\eta_{c}},0\leq\beta \leq 1,\\ p(\beta)=0.5(\eta_{c}+1)\frac{1}{\beta^{\eta_{c}+2}},\beta>1,\end{array} \right. \tag{2}\]  
where \(\eta_{c}\) is the distribution index. Mutation operator is any changes on the variable or gene of a chromosome that can produce better value. NSGA-II and SPEA2 use a polynomial mutation, while MOSFP and OMOPSO use uniform and non-uniform mutation. 1. Polynomial mutation: this mutation is proposed by [PERSON] et al. [PERSON] et al. (2014). This mutation can be summarized via the following equation: \[p^{\prime}=\left\{ \begin{array}{c}p+\delta_{L}(p-x_{i,j}(L)),for:u\leq 0.5,\\ p+\delta_{R}(x_{i,j}(U)-p),for:u>0.5,\end{array} \right.\] (3) where \(p\) is the parent solution \(p\in[x^{(L)},x^{(L)}]\), where \(x^{(L)}\) is the lower bound value, while \(x^{(L)}\) is the upper bound value of a variable. The (\(U\)) symbol is a random number in the range of 0 and 1. The two parameters \(\delta_{L}\) and \(\delta_{R}\) are calculated as follows [PERSON] et al. (2014): \[\left\{ \begin{array}{c}\delta_{L}=(2u)^{1/(1+n_{m})}-1,for:u\leq 0.5,\\ \delta_{R}=1-(2(1-u))1/(1+n_{m}),for:u>0.5,\end{array} \right.\] (4) The parent point \(p=3.0\) in a bounded range of 1 and 8 with \(n_{\rm m}=20\). 2. Uniform mutation of value \(x\) used in MOSFP and OMOPSO can be summarized in the following equation [PERSON] et al. (2014): \[x_{i,j}=x_{i,j}^{(L)}+u(x_{i,j}^{(U)}-x_{i,j}^{(L)}),\] (5) where, \(x_{i,j}\) is the position of sperm or particle, \(x^{(L)}\) is the lower bound value, \(x^{(U)}\) is the upper bound value of sperm or particle and (\(U\)) is a random number in the range of 0 and 1. 3. Non-uniform mutation of value \(x_{i,j}\) use in MOSFP and OMOPSO can be summarized in the following equation [PERSON] et al. (2014): \[x^{\prime}_{i,j}=\left\{ \begin{array}{c}x_{i,j}+\Delta(t,x^{(U)}-x_{i,j}), if:u=0,\\ x_{i,j}+\Delta(t,x_{i,j}-x^{(L)}),if:u=1,\end{array} \right.\] (6) where, \(x^{(L)}\) is the lower bound value, \(x^{(U)}\) is the upper bound value of sperm or particle, (\(u\)) is a