## 4 Results
### Testing Results  
To evaluate the generalization ability of the four algorithms, we applied the trained TPHO-MRC, TPHO-Linear, TPHO-Profit, and TPHO-EFT algorithms to 60 test scenarios. We also compared the proposed algorithms with two existing metaheuristic algorithms: SA [26] and A-ALNS [30]. The average results for total profit (TP), standard deviation (SD), and coefficient of variation (CV) for five test scenarios for each satellite task size are listed in Table 6, and the results are intuitively presented in Figure 7. Table 6 and Figure 7 show that TPHO-MRC obtains the highest total profit in all test scenarios, and the CV values of the TPHO-based algorithms are lower than 3%, demonstrating the stability of the algorithms. TPHO-MRC demonstrates a performance improvement exceeding 16% compared with the A-ALNS algorithm in scenarios involving 1200 observation tasks. A two-tailed Wilcoxon signed-rank test [61] was performed to pair six algorithms to ascertain the significance of the results. Notably, the results of the 3-300 and 4-300 scenarios are not included because most algorithms obtain the optimal solutions. Table 7 details the significant test results. The results indicate that TPHO-MRC significantly outperforms other algorithms at the 1% level of significance, and algorithms based on TPHO significantly outperform the two metaheuristic algorithms, demonstrating the effectiveness of the proposed TPHO.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Number of**} & \multirow{2}{*}{**Value**} & \multirow{2}{*}{**TPHO-MRC**} & \multirow{2}{*}{**TPHO-Linear**} & \multirow{2}{*}{**TPHO-Profit**} & \multirow{2}{*}{**TPHO-EFT**} & \multirow{2}{*}{**A-ALNS**} & \multirow{2}{*}{**SA**} \\ \cline{3-4} \cline{6-9}  
The average results for the runtimes of the six algorithms are listed in Table 8. Algorithms based on TPHO show better time efficiency than metaheuristic algorithms. The runtimes of the four TPHO-based algorithms are similar. TPHO-MRC could complete scheduling within 71 s for scenarios with up to 4 AEOSs and 1200 tasks, constituting less than 3% of the computational duration required by the A-ALNS algorithm. Figure 8 illustrates the runtime trends. As shown in Figure 8, the runtime of A-ALNS increases sharply as the number of AEOSs and tasks increases. The average runtime of A-ALNS for the 4-300 scenarios is 0.193, because all feasible tasks have been scheduled in the initial solution of the algorithm so that the algorithm terminates after generating the initial solution. By contrast, the runtimes of the TPHO-based algorithms increase slightly as the number of AEOSs and tasks increases. This indicates that the TPHO-based algorithms can efficiently complete scheduling for larger-scale MAEOSPs.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Number of**} & \multicolumn{5}{c}{**Runtime(s)**} \\ \cline{2-7}
**AEOSs and Tasks** & **TPHO-MRC** & **TPHO-Linear** & **TPHO-Profit** & **TPHO-EFT** & **A-ALNS** & **SA** \\ \hline
2-300 & 3.099 & 2.798 & 2.930 & 4.136 & 92.514 & 9.369 \\
2-600 & 10.907 & 10.022 & 11.217 & 10.116 & 174.249 & 35.997 \\
2-900 & 21.876 & 20.664 & 20.122 & 19.019 & 252.129 & 78.477 \\
2-1200 & 35.800 & 34.665 & 29.784 & 30.826 & 1514.861 & 142.777 \\
3-300 & 5.741 & 4.456 & 5.805 & 5.726 & 24.709 & 12.303 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Testing runtimes.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithm** & **TPHO-Linear** & **TPHO-EFT** & **TPHO-Profit** & **A-ALNS** & **SA** \\ \hline TPHO-MRC & 0.001 & 0.002 & 0.001 & 0.001 & 0.001 \\ TPHO-Linear & & 0.161 & 0.032 & 0.001 & 0.001 \\ TPHO-FFT & & 0.097 & 0.001 & 0.001 \\ A-ALNS & & & & 0.001 & 0.001 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Results of significance test.  
Figure 7: The average results with error bars for total profit.