## 3 Methodology
### Archive Update  
The archive update step in Algorithm 2 determines which solutions will be added to archive \(A\). The performance of multi-objective algorithms is closely tied to selecting the external archive set. Therefore, choosing solutions that exhibit good convergence and diversity is crucial. The clone process is illustrated in Figure 3. Below are the detailed descriptions. ```
Input: External repository \(A\), population \(P\), population size \(N\), data Output: Updated repository \(A\)
1. \(C\gets A\cup P\)
2. If\(A\) is not empty then
3. \(C\leftarrow\text{Normalize}(C)\)
4. \(C\leftarrow\text{EvaluateFitness}(C)\)
5. \(Cnd\leftarrow\text{SelectNonDominated}(C)\) // Keep only non - dominated solutions \(Cnd\) in \(C\)
6. else
7. if\(\text{size}(Cnd)\ >N\)then
8. \(A\leftarrow\text{SelectMaxMinAngle}(Cnd,\ A)\)
9. else
10. \(A\gets C\)
11. endif
12. Return \(A\)
```  
**Algorithm 2** Archive Update ProcessThe archive update operation updates the external archive A through the following steps:  
1. Merge archives: Combine archive \(A\) with population \(P\), creating a joint population \(C\) that includes both \(A\) and \(P\). 2. Normalization: Normalize each solution in the joint population \(C\), standardizing the objective values to a range between 0 and 1. 3. Calculation fitness: Calculate the fitness value for each solution in \(C\) using Equation (15): \[\mathit{fit}\big{(}c_{j}\big{)}=\sqrt{\sum_{i=1}^{m}f_{i}(c_{j})^{2}}\] (15)
4. Non-dominated sorting: Pareto dominance filters out non-dominated solutions from \(C\), forming the non-dominated solution set \(Cnd\). * If the number of solutions in \(Cnd\) exceeds the population size \(N\), use the minimum angle selection strategy to select solutions added to \(A\); * If the number of solutions in \(Cnd\) is less than or equal to \(N\), directly add all solutions in \(Cnd\) to \(A\). Return updated archive: return the updated archive \(A\). The \(\mathit{fit}\big{(}c_{j}\big{)}\) is the fitness value of the \(c_{j}\) solution in the joint population \(C\). Fitness value serves as a standard for evaluating the convergence of the solutions. Subsequently, the non-dominated solution set \(Cnd\) is selected from \(C\)-based Pareto dominance criteria. Because in MAPSO, most solutions are non-dominated, the minimum angle selection strategy is adopted when the Pareto dominance criterion is no longer adequate. The steps of the minimum angle selection strategy, as detailed in Algorithm 3, can be divided into three main parts:  
1. Eliminate dominance-resistant solutions (DRS): * First, remove DRS from the non-dominated solutions, \(Cnd\). These solutions are highly inferior in at least one objective and difficult to dominate; * Calculate the minimum vector angle for each \(\delta_{ij}\) solution in \(Cnd\), representing the smallest angle between the solution and other solutions; * Add \(m\) extreme solutions to archive \(A\). Extreme solutions are those in \(Cnd\) that have the smallest angle with the reference vectors (1, 0, \(\ldots\), 0), (0, 1, \(\ldots\), 0), \(\ldots\), (0, 0, \(\ldots\), 1). 2. Remove minimum vector angle of solutions: * Remove solutions with the smallest vector angles from \(Cnd\) until the number of solutions in \(Cnd\) reaches N-m;  
Figure 3: Illustration of archive update. During the removal process, select solutions with lower fitness values, meaning those with fitness values lower than other solutions;
* Removed solutions are added to the removed solution set \(R\). The angle calculation method directly affects particle distribution and search space exploration.