## 2 Methodology
### Multi-Objective Optimization Strategy  
In this study, to address the conflicts among different optimization objectives, we propose using an optimized Non-dominated Sorting Genetic Algorithm II (NSGA-II) to solve multi-objective constrained optimal path problems. The NSGA-II is a powerful decision space exploration engine based on Genetic Algorithm (GA), primarily used to solve multi-objective optimization problems (MOOPs) [28]. Among the branches of multi-objective optimization problems, combinatorial optimization problems (COPs) are considered some of the most challenging and complex. Since most COPs are NP-hard, their computational complexity increases significantly as the problem size grows. Consequently, approximate methods such as metaheuristics are preferred over classical methods for solving such problems [29; 30]. Multi-objective optimization involves finding the best possible solutions for multiple objectives within a given domain. A MOOP consists of a set of \(n\) decision variables, \(k\) objective functions, and a set of constraints comprising m inequality constraints and p equality constraints. The optimization objective is to find solutions that satisfy all constraints while optimizing the objective functions. \[Min/Max\ y=f(x)=(f_{1}(x),f_{2}(x),\ldots,f_{k}(x)),k\geq 2 \tag{1}\]  
\[Subject\ to\ g_{i}(x)\leq 0,i=1,2,\ldots,m \tag{2}\]  
\[h_{j}(x)\leq 0,j=1,2,\ldots,m \tag{3}\]  
Formally, \(x=(x_{1},x_{2},\ldots,x_{n})\in D\) is the n-dimensional decision vector in \(X\subseteq R^{n}\), and \(y\) is the \(k\)-dimensional objective vector in \(R^{k}\). \(f\) is defined as the mapping function, \(g_{i}\) is the \(i\)-th inequality constraint, and \(h_{j}\) is the \(j\)-th equality constraint. Thus, Equations (2) and (3) determine the set of all feasible solutions X, so it can also be written as a set of different feasible solutions \((x_{1},x_{2},\ldots,x_{n})\in X\). Suppose \(x_{1},x_{2}\in X\) are two feasible solutions to a multi-objective problem. A solution \(x_{1}\) can be regarded as superior to \(x_{2}\) if the following condition is satisfied: \(x_{1}\) is superior to \(x_{2}\) in at least one of the objectives and no worse than the others. In this case, \(x_{1}\) is said to dominate \(x_{2}\). \[\left\{ \begin{array}{l}x_{1}\prec x_{2}\Leftrightarrow f_{i}(x_{1})\leq f _{i}(x_{2}),\forall i\in\{1,2,\ldots,m\}\\ \exists j:f_{j}(x_{1})\leq f_{j}(x_{2})\end{array} \right. \tag{4}\]  
The \(j\)-th value of the objective function for the decision vector x is denoted by \(f_{j}(x)\). The solution space is denoted by \(X\), \(\prec\) represents the dominance relation. Let \(x\in\psi\subseteq X\), if all other solutions in \(\psi\) do not dominate \(x\) then \(x\) is nondominated with respect to the subset \(\psi\). This means that \(x\) is nondominated with respect to \(\psi\). Methods for identifying Pareto-optimal solutions include NSGA-II, the Pareto Adaptive Algorithm (APA), and others, which are employed in this study. Optimization problems frequently exhibit not a single solution but a set of solutions, where improving one objective function necessarily entails a reduction in another. Such a solution is termed a nondominated or Pareto-optimal solution, and all Pareto-optimal solutions constitute a Pareto-optimal set (PS). In this context, a Pareto-optimal solution \(x\in X\) is nondominated with respect to the entire solution space \(\psi\). The set of all Pareto-optimal solutions constitutes the Pareto-optimal set (PS). The objective vector corresponding to the Pareto-optimal set is defined as the Pareto frontier, as illustrated in Figure 3.  
Figure 3: Pareto domination. The NSGA-II is an enhanced iteration of the Non-dominated Sorting Genetic Algorithm (NSGA), widely utilized in multi-objective optimization problems due to its elitist properties, the absence of the necessity to share parameters, and its rapid computation speed [31].