## 2 Methods
### Multi-Objective Optimization of the Satellite Selection Problem
#### 2.2.2 Archiving and Guiding Selection Mechanism  
Building on the proposed mathematical definitions of optimal Pareto non-dominated solutions, a fast satellite selection algorithm--NSWOA--is developed. This algorithm incorporates two key innovations that enhance the original WOA for multi-objective optimization problems:  
* An archiving mechanism, which maintains a dynamic set of non-dominated solutions during the optimization process;
* A guiding selection mechanism, which adaptively selects leading agents to balance convergence and diversity in the solution space.  
These enhancements improve WOA and enable effective solution comparison in the multi-objective optimization satellite selection problem.  
The archiving mechanism is designed to store the non-dominated Pareto optimal solutions generated during the optimization process. It regulates the inclusion of new satellite combinations in the archive while enabling the retrieval of existing non-dominated solutions. When a new satellite combination is evaluated, one of two possible outcomes occurs:  
* Scenario A: If the new satellite combination dominates one or more existing solutions in the archive, the dominated solutions are removed, and the new combination is added.
* Scenario B: If the new combination is non-dominated relative to the existing solutions, it is simply added to the archive.  
The selection mechanism in NSWOA is based on Pareto dominance, addressing the challenge of ranking multi-objective solutions using scalar fitness values alone. It selects guiding agents from the current archive of non-dominated solutions to steer the search process toward optimal satellite combinations. This mechanism operates in three main phases:  
Phase 1: Archive Space PartitioningBased on the maximum and minimum values in both \(f_{1}(x)\) and \(f_{2}(x)\) directions within the archive space, and using the adjustment factor \(\alpha\), the partitioning ranges are determined as follows:\(f_{1}\in(low_{1},up_{1})\cap f_{2}\in(low_{2},up_{2})\). These ranges are specifically defined as follows: \[\begin{array}{l}low_{i}=\min f_{i}-\alpha\cdot(\max f_{i}-\min f_{i}),i=1,2\\ up_{i}=\max f_{i}+\alpha\cdot(\max f_{i}-\min f_{i}),i=1,2\end{array}\] (9)
* Then, both the \(f_{1}(x)\) and \(f_{2}(x)\) directions are uniformly divided into \(n_{grid}\) cells.  
Phase 2: Grid Congestion Probability Calculation
* The congestion probability for each grid is defined as follows: \[P_{i}=N_{i}^{-\beta}\] (10) where \(N_{i}\) is the number of Pareto optimal solutions in the \(i\)-th grid, and \(\beta>1\) is the guiding selection pressure parameter, typically set to \(\beta=4\). The probabilities are then normalized.  
Phase 3: Guiding Agent Selection  
* The cumulative probability for the \(i\)-th grid is calculated as follows: \[c_{i}=\sum_{j=1}^{i}p_{j}\] (11)  
The first grid index \(i\) is identified such that \(c_{i}>r\), where \(r\) is a random probability uniformly distributed within the range [0, 1].
* From the selected grid, a non-dominated agent is randomly chosen as the guiding agent \(x_{n}^{*}\).  
This probability-biased selection method favors grids with lower solution density, thereby promoting exploration in less-crowded regions of the solution space and enhancing global search capability.