## 4 Developed Multi-Objective Optimization Method Based on [PERSON]
\[f^{\mathrm{N}}=[f_{1}^{\mathrm{N}},\ldots,f_{1}^{\mathrm{N}},\ldots,f_{ \mathrm{P}}^{N}]. \tag{26}\]  
where  
\[ \begin{cases}f_{1}^{\mathrm{N}}=(Max/\overline{x})f_{1}(\overline{x}).\\ \text{Compliance with:}\overline{x}\in\Omega.\end{cases}  \tag{27}\]The feasible region (\(\Omega\)) includes the pseudo-nadir point, which closely resembles the nadir point, and it is represented as follows:  
\[f^{\text{SN}}=[f_{1}^{\text{SN}},\dots,f_{i}^{\text{SN}},\dots,f_{\text{p}}^{ \text{SN}},\dots]. \tag{28}\]  
\[f_{\text{i}}^{\text{SN}}=\max\{f_{\text{i}}(\overline{\chi}_{1}^{*}),\dots,f_{ \text{i}}^{*}(\overline{\chi}_{1}^{*}),\dots,f_{\text{i}}(\overline{\chi}_{ \text{p}}^{*})\}. \tag{29}\]  
The range of objective functions is established by utilizing the utopia and pseudo-nadir points in accordance with the following formula:  
\[f_{\text{i}}^{\text{U}}\leq f_{\text{i}}(\overline{\chi})\leq f_{\text{i}}^{ \text{SN}}. \tag{30}\]  
The objective space depicted in Figure 8 is defined by the number of objective function dimensions. The utopia, pseudo-nadir, and nadir points aid in illustrating the Pareto frontier by segmenting the ranges of objective functions into equal intervals with the assistance of intermediate grid points denoted as \((q_{2}-1),\dots,(q_{p}-1)\). Consequently, this necessitates the solution of \(\prod_{\text{i}=2}^{p}(q_{\text{i}+1})\) subproblems,  
\[\min f_{1}(\overline{\chi}),\]  
so that  
\[f_{2}(\overline{\chi})\leq e_{2,\text{n}2},\dots,f_{\text{p}}(\overline{\chi })\leq e_{\text{p},\text{np}} \tag{31}\]  
where:  
\[e_{2,\text{n}2}=f_{2}^{\text{SN}}-\left(\frac{f_{2}^{\text{SN}}-f_{2}^{\text{ U}}}{\text{q}_{2}}\right)\times\text{n}2,\quad\text{n}2=0,1,\dots,\text{q}_{2}\]  
\[e_{2,\text{n}2}=f_{\text{p}}^{\text{SN}}-\left(\frac{f_{\text{p}}^{\text{SN}} -f_{\text{p}}^{\text{U}}}{\text{q}_{\text{p}}}\right)\times\text{np},\quad \text{np}=0,1,\dots,\text{q}_{\text{p}}. \tag{32}\]  
The approach entails addressing subproblems by incorporating the constraints from Equation (32) in addition to the primary problem's constraints. This helps in deriving the Pareto-optimal front and removing infeasible solutions. The objective function constraints are converted into equalities using the slack variable method as part of this process. Figure 8: The Pareto set, which includes the utopia point, the nadir point, and the pseudo-Nadir point. \[\left\{ \begin{array}{l}\text{Min},\left(f_{1}(\tilde{x})-\text{r}_{1}\sum_{i=2}^ {\text{P}}\left(\frac{\text{s}_{i}}{\text{r}_{i}}\right)\right).\\ \text{Subject to}\\ f_{1}(\tilde{x})+\text{s}_{i},\text{n}_{\text{i}}=\text{e}_{\text{i}},\text{n} _{\text{i}},\rightarrow\text{i}=2,\ldots,\text{PS}_{\text{i}},\text{n}_{ \text{i}}\in\text{R}^{+}.\\ \tilde{x}\in\Omega.\end{array} \right. \tag{33}\]  
The approach denoted as \(s_{2},...,s_{p}\) tackles scaling problems by introducing slack variables according to the formula in Equation (33). It also incorporates a term denoted as \(r_{1}(s_{i}/r_{i})\) into the objective function to manage discrepancies in objective scales. The objective function ranges are determined using the payoff matrix \((r_{i}=f_{i}^{S}N-f_{i}^{U})\). This method employs the augmented epsilon-constraint approach to the multi-objective optimization process. In this process, the decision maker chooses the most desirable solution based on linear membership functions defined for each objective function using the fuzzy satisfying method. \[\mu_{\text{Fi}}(x)= \begin{cases}1&\text{F}_{\text{i}}(x)\leq\text{F}_{\text{ i}}^{\text{min}}.\\ \frac{\text{F}_{\text{i}}^{\text{max}}-\text{F}_{\text{i}}(x)}{\text{F}_{ \text{i}}^{\text{max}}-\text{F}_{\text{i}}^{\text{min}}}&\text{F}_{\text{i} }^{\text{min}}\leq\text{F}_{\text{i}}(x)\leq\text{F}_{\text{i}}^{\text{max}}.\\ 0&\text{F}_{\text{i}}(x)\geq\text{F}_{\text{i}}^{\text{max}}.\end{cases}  \tag{34}\]  
The membership function assesses the degree of satisfaction of the objective function condition, where a value of 1 signifies full satisfaction, and 0 indicates no satisfaction.