## 4 Enhanced Multiobjective Optimization Method Based on GWO  
In essence, multiobjective optimization involves addressing multiple goals at once. In the context of this study, these goals encompass technical, economic, and reliability aspects of the system, as elaborated in the preceding section. Consequently, the optimization problem can be mathematically formulated to either minimize or maximize certain criteria [28]. \[ \begin{array}{ll}\text{Optimize:}&\text{Total}\ \text{OBJ}(\vec{X})= \{\text{obj}_{1}(\vec{x}),\text{obj}_{2}(\vec{x}),\ldots,\text{obj}_{n}(\vec {x})\}\\ \text{subject to:}&\psi_{i}(\vec{x})\geq 0,\quad i=1,2,\ldots,\
u\\ &\theta_{i}(\vec{x})=0,\quad i=1,2,\ldots,\sigma\\ &U_{i}^{\text{lb}}\leq x_{i}\leq U_{i}^{\text{ub}},\quad i=1,2,\ldots,\gamma\\ \end{array}  \tag{31}\]In the above equation, \(\
u\) represents the count of inequality constraints, \(\sigma\) denotes the number of equality constraints, \(\gamma\) stands for the number of variables, \(\psi_{i}\) signifies the ith inequality constraint, no indicates the number of objective functions, \(\vartheta_{i}\) denotes the \(i_{\text{th}}\) equality constraint, and \([U_{i}^{\text{lb}}\) and \(U_{i}^{\text{ub}}]\) represent the boundaries of the \(i_{\text{th}}\) variable. It is evident that traditional relational operators fall short in effectively comparing solutions across multiple objectives. In the current study, the predominant operator utilized is Pareto optimal dominance, which is defined as follows for minimization problems:  
\[\forall n\in\{1,2,\cdots,\gamma\}:\text{obj}_{n}(\vec{x})\leq\text{obj}_{n}( \vec{y})\quad\wedge\quad\exists n\in\{1,2,\cdots,\gamma\}: \tag{32}\]  
\[\text{obj}_{n}(\vec{x})\leq\text{obj}_{n}(\vec{y})\]  
where in Equation (32), \(\vec{x}=(x_{1},x_{2},\cdots,x_{\gamma})\) and \(\vec{y}=(y_{1},y_{2},\cdots,y_{\gamma})\). These equations demonstrate that a solution surpasses another in a multiobjective search space when it matches all objectives and outperforms in at least one objective. The principle of Pareto optimal dominance is symbolized by \(\prec\) and \(\succ\), which allow for straightforward comparison and distinction between solutions. In population-based multiobjective algorithms, solutions consist of multiple solutions. However, the exact determination of the optimal solution is challenging due to each solution being constrained by other objectives, leading to inherent conflicts between objectives. Therefore, the primary function of stochastic or population-based multiobjective algorithms is to identify the best trade-offs among objectives, resulting in what is known as a Pareto optimally set. In the gray wolf optimization (GWO) framework, the hierarchy among solutions designates the most optimal as alpha, followed by beta and delta, representing the second- and third-best solutions, respectively. The remaining solutions are collectively termed omega. Within the GWO algorithm, the strategic pursuit is orchestrated by beta and delta, while the gamma solution trails behind these key wolves. When the prey is cornered and ceases to move, the assault on the leadership of the alpha wolf commences. The orchestration of this process is facilitated through the manipulation of the reduction vector 'a'. This vector, denoted as 'A', is subject to randomization within the range [\(-\)2a, 2a]. By diminishing 'a', the coefficients of vector 'A' are likewise reduced. Should the magnitude of 'A' fall below 1, it signifies the convergence of the alpha wolf (and others) towards the prey, while a magnitude exceeding 1 indicates a movement away from the prey. The gray wolf optimization algorithm necessitates the synchronization of all wolves, compelling them to adjust their positions in accordance with the movements of the alpha, beta, and delta wolves. In the pursuit scenario, gray wolves tactically encircle their prey.