## 4 Developed Multi-Objective Optimization Method Based on [PERSON]
However, it is worth noting that a well-known optimization technique like the epsilon-constraint method does come with two notable drawbacks that warrant careful consideration. When dealing with \"\(p\)\" objective functions, it becomes essential to define the range of \"\(p\)-1\" objective functions, as these objectives are treated as constraints in the problem [35]. In this paper, addressing these challenges entails the utilization of the lexicographic optimization technique to effectively ascertain the permissible range for each objective function. This approach serves to alleviate the initial limitation of this multi-objective optimization method. To confront the second issue concerning the efficiency and quality of the resulting solutions, we employ a modified epsilon-constraint technique. Consequently, this paper combines a hybrid approach involving the adapted epsilon-constraint and lexicographic optimization methods to resolve the presented multi-objective optimization dilemma characterized by conflicting objective functions. The core concept of this methodology revolves around designating one of the objective functions as the primary focus while treating all other functions as constraints within the problem, as outlined below. \[ \begin{cases}\min,f_{1}(\overline{x})\\ f_{2}(\overline{x})\leq e_{2},\ldots f_{\mathrm{P}}(\overline{x})\leq e_{ \mathrm{P}}.\end{cases}  \tag{21}\]  
where \"\(p\)\" represents the count of objective functions, and \"x\" refers to the decision variable vector. As indicated by the expression above, the goal is to minimize all objective functions. As previously mentioned, it is vital to establish the permissible ranges for the \"\(p\)-\(1\)\" objective functions. This can be accomplished by utilizing a payoff matrix. To create this payoff matrix, the problem is initially solved as a single-objective optimization problem to determine the optimal values for each objective function. Subsequently, these values are used to compute the single optimal values for the other objective functions, denoted as \(f_{i}(\overline{x}_{i}^{*})\) and \(\overline{x}_{i}^{*}\), respectively. The resulting matrix of outcomes is structured with rows that depict various combinations of objective function values. For a deeper comprehension and access to the associated equations, kindly consult the provided reference. It is worth highlighting that the remaining objective functions possess a sole optimal value. \[f_{1}(\overline{x}_{i}^{*}),f_{2}(\overline{x}_{i}^{*}),f_{i-1}^{*}(\overline{ x}_{i}^{*}),f_{i+1}(\overline{x}_{i}^{*}),\ldots f_{\mathrm{P}}(\overline{x}_{i}^{*}). \tag{22}\]  
The formation of the payoff matrix involves arranging the elements of each row (\(i\)) in a particular fashion. \[f_{1}(\overline{x}_{i}^{*}),f_{2}(\overline{x}_{i}^{*}),\ldots f_{i}( \overline{x}_{i}^{*}),f_{\mathrm{P}}(\overline{x}_{i}^{*}). \tag{23}\]  
\[Q= \begin{bmatrix}f_{1}^{*}(\overline{x}_{1}^{*})&\cdots&f_{1}(\overline{x}_{1} ^{*})&\cdots&f_{\mathrm{P}}(\overline{x}_{1}^{*})\\ \vdots&&\\ f_{1}(\overline{x}_{i}^{*})&\cdots&f_{1}^{*}(\overline{x}_{i}^{*})&\cdots&f_{ \mathrm{P}}(\overline{x}_{i}^{*})\\ \vdots&&\\ f_{1}(\overline{x}_{\mathrm{P}}^{*})&\cdots&f_{1}(\overline{x}_{\mathrm{P}}^{*} )&\cdots&f_{\mathrm{P}}^{*}(\overline{x}_{\mathrm{P}}^{*})\end{bmatrix} . \tag{24}\]  
The payoff matrix, denoted as (\(p\times q\) ), is a square matrix in which each column (\(n\)) represents the optimal value of a specific objective function (\(f_{n}\)), constrained within minimum and maximum values. Meanwhile, the utopia point (\(f^{U}\)) signifies the optimal values of all objectives within an infeasible region. \[f^{\mathrm{U}}=[f_{1}^{\mathrm{U}},\ldots,f_{1}^{\mathrm{U}},\ldots,f_{ \mathrm{P}}^{\mathrm{U}}]=[f_{1}^{*}(\overline{x}_{1}^{*}),\ldots,f_{1}^{*}( \overline{x}_{i}^{*}),\ldots,f_{\mathrm{P}}^{*}(\overline{x}_{\mathrm{P}}^{*} )]. \tag{25}\]  
On the contrary, the nadir point (denoted as \(f^{N}\) ) designates the point within the objective region where all objective functions reach their most unfavorable values.