## 4 Solving Method  
### Bi-Objective Optimization  
Consider a bi-objective maximization problem formulated as follows:  
\[\left\{ \begin{aligned} &\underset{g\in G}{\max}f_{1}(g)\\ &\underset{g\in G}{\max}f_{2}(g),\end{aligned} \right. \tag{2}\]  
where the solution \(g\) belongs to a non-empty, compact set \(G\). The two objective functions, \(f_{1}(g)\) and \(f_{2}(g)\), are upper semi-continuous functions to be maximized over this set. These objectives can be represented by a single objective vector \(f(g)=(f_{1}(g),f_{2}(g))^{\mathrm{T}}\). Any such vector generated by a solution \(g\in G\) is termed a feasible objective vector. The universe of all such achievable outcomes is the set \(\mathbb{Y}=\left\{\left(f_{1}(g),f_{2}(g)\right)^{\mathrm{T}}\middle|g\in G\right\}\). Within this framework, the goal is to identify Pareto-optimal solutions. A feasible solution \(g\) is designated as Pareto optimal if it generates a non-dominated objective vector. A vector \(\vec{f}(g)\in\mathbb{Y}\) is defined as non-dominated if no other feasible vector in \(\mathbb{Y}\) offers a superior outcome. Specifically, there exists no other vector that is at least as good in all objectives (i.e., \(f_{i}(g)>\overline{f_{i}}(g)\) for \(i=1,\ 2\)) and strictly better in at least one objective (i.e., \(f_{i}(g)>\overline{f_{i}}(g)\) for some \(i\)). This definition aligns with the principle that a Pareto-optimal solution cannot be improved with respect to one objective without a simultaneous degradation in at least one of the others [52]. The locus of all such Pareto-optimal solutions, formally described by the set \(\Psi=\{g\in G|(f_{1}(g),f_{2}(g))\in\mathbb{Y}\}\), is designated as the Pareto frontier. In previous research, the weighted sum method has been one of the most extensively applied approaches for addressing the multi-objective optimization problem of marine shipping route planning and ship allocation (e.g., [53; 54; 55]). This method aggregates multiple objectives, such as navigation risk, fuel consumption, travel time, and operational costs, into a single objective function by assigning specific weights to each objective. These weights reflect the relative importance of each objective and are typically determined through expert judgment, as exemplified by standardized allocations of 0.4 for risk, 0.3 for fuel consumption, and 0.3 for travel time in one study [53]. Alternatively, weights may be derived using systematic techniques such as the Analytic Hierarchy Process (AHP) combined with entropy-based methods to ensure robustness in complex scenarios like Arctic rescue operations [54]. In some cases, weights are dynamically adjusted during algorithm execution, accompanied by sensitivity analysis to assess their impact on the solution, as seen in hub-and-spoke network optimization for short sea shipping [55]. The weighted sum method is advantageous due to its simplicity, as it allows for an easy assessment of the trade-offs between different objectives based on assigned weights. It also facilitates the interaction between decision makers' subjective preferences and the system's objective information through the adjustment of the weight combinations for various objective functions. Nevertheless, this approach has certain limitations when applied to multi-objective optimization problems. As an a priori technique, it necessitates predetermined weights for each objective function, meaning that the optimal result obtained is heavily dependent on these assigned weights. Therefore, when the relative importance of each objective is uncertain or cannot be established beforehand, which is common in hazardous waste management system planning due to the complexity of balancing environmental risks, economic costs, and social impacts, the weighted sum method becomesless effective. In the context of a posteriori decision making, [PERSON] and [PERSON] [56] noted that it cannot produce a well-distributed set of Pareto solutions or a comprehensive array of points along the Pareto frontier.