## 4 Experimental Design and Results Analysis
### Hypervolume and Friedman Evaluation Index  
This study utilized the hypervolume (HV) and Friedman fraction metrics to evaluate the algorithm's performance [31] and comprehensively analyze the diversity and convergence of the Pareto front generated by the algorithm. Each solution within the Pareto front corresponds to a scheduling plan that satisfies the dynamic scheduling demands of relay satellites under different preference scenarios. The HV metric is computed by summing the areas of the rectangles formed between each point in the non-dominated solution set and a reference point, typically the maximum individual in the objective space. A higher HV value signifies superior overall algorithm performance. The HV calculation is expressed by the following Formula (27):  
\[HV=\sum_{i=1}^{n}\left(\frac{r_{1}-f_{1}(x_{i})}{f_{1}(x_{i})^{max}}\right) \times\left(\frac{f_{2}(x_{i})-r_{2}}{f_{2}(x_{i})^{max}}\right) \tag{27}\]  
In this context, \(f_{1}(x_{i})\) and \(f_{2}(x_{i})\) denote the first and second objective values of solution \(x_{i}\), respectively. Similarly, \(r_{1}\) and \(r_{2}\) are the respective coordinates of the reference point. \(f_{1}(x_{i})^{max}\) represents the maximum value of the first objective function \(f_{1}\) in all solutions \(x_{i}\). \(f_{2}(x_{i})^{max}\) represents the maximum value of the second objective function \(f_{2}\) in all solutions \(x_{i}\). Cases \(c1\) through \(c4\) lack a Pareto front, and as a result, they are not considered in the hypervolume (HV) assessment. Table 4 presents the HV values and the reference point coordinates for the 20 experimental scenarios. A normalization process has been implemented to avoid excessively high HV values. This process involves standardizing each objective value by dividing it by its maximum, which maintains uniform scaling among the objectives.  
Table 9 presents the average HV values obtained by seven algorithms over 30 runs across all experimental scenarios. In solving satellite data transmission problems, the MAPSO algorithm demonstrates a significant advantage, particularly in scenarios 15 to 24, where its HV values are consistently higher than those of other algorithms. Table 9 indicates that MAPSO effectively balances solution diversity and convergence, achieving superior solution sets. In contrast, the HV values of the MOGA algorithm are noticeably  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Experiment** & **MAPSO** & **DRSC-MOAGDE** & **DSC-MOPSO** & **MO-Ring-PSO-SCD** & **NSGA-II** & **MOEA/D** & **MOGA** \\ \hline c18 & 3.2068 & 3.3687 & 3.4322 & 4.6823 & 6.431713 & 7.801853 & 4.94989 \\ c19 & 3.9835 & 4.2311 & 4.2659 & 5.5848 & 7.315233 & 8.538053 & 5.708143 \\ c20 & 5.1150 & 5.2879 & 5.3052 & 7.0592 & 8.291603 & 9.5398 & 6.821257 \\ c21 & 6.2237 & 6.5182 & 6.4691 & 7.9553 & 9.480343 & 10.82844 & 8.05883 \\ c22 & 7.0269 & 7.2036 & 7.2855 & 8.3954 & 10.18052 & 11.6449 & 8.767617 \\ c23 & 8.2340 & 8.4617 & 8.5146 & 9.7211 & 11.46392 & 12.86323 & 10.01952 \\ c24 & 9.3054 & 9.5289 & 9.5960 & 10.8173 & 12.49769 & 13.99352 & 11.156 \\ \hline \hline \end{tabular}
\end{table}
Table 8: _Cont._lower across all scenarios, reflecting poor solution diversity and limited ability to explore the solution space effectively. The HV values of the DRSC-MOAGDE, DSC-MOPSO, MO-Ring-PSO-SCD, NSGA-II, and MOEA/D algorithms fall between those of MAPSO and MOGA, exhibiting varying performance levels.