# Research on Joint Resource Allocation for Multibeam Satellite Based on Metaheuristic Algorithms
## 4 Results
Except for the GA, the relationship between the algorithm's execution time and population size was approximately linear. The DE, PSO, and QPSO algorithms executed faster than the GA algorithm. In comparison to the UBA method, the NDBC method reduced the total execution time of each algorithm by 9-21%. Figure 8b depicts the mean values of each algorithm's optimized solutions for varying population sizes. The quality of the optimal solution improved proportionally to the size of the population. As depicted in the graph, the NDBC method significantly improved the optimized solution of each algorithm by 9-33%. In terms of algorithm comparison, the strength of algorithm optimization capability is ranked as QPSO, PSO, DE, and GA. However, when using UBA, this ordering is PSO, DE, GA, and QPSO. We speculate that QPSO is more sensitive to the flexibility of the search compared to other algorithms. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**GA** & **GA** & **DE** & **DE** \\ \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Tournament size & 5 & & \\ Blend Alpha & 0.2 & Ini. Mutation Prob. & 0.2 \\ Crossover prob. & 0.95 & Crossover Prob. & 0.1 \\ Mutation prob. & [0.05, 0.15] & & \\ \hline
**PSO** & **PSO** & **QPSO** & **QPSO** \\ \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Inertia weight & [0.2, 1.0] & & \\ Cognitive factor & 1.0 & & \\ Social factor & 1.0 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Algorithm parameters. Figure 7: The execution time required for the constraint handling session: (**a**) constraint handling time of a single execution (with 500 iterations and a population size of 400); (**b**) total constraint handling time generated by both methods versus the number of iterations (with a population size of 400). The results in the graph demonstrated that NDBC can reduce the execution time of the bandwidth-constrained handling session by 75–80%. Increasing the population size can result in a more optimal solution, but it also increases the execution time. In the following simulations, the population size was set to 400 in order to control the execution time of the algorithm without causing it to take too long. The maximum and minimum number of iterations were 500 and 200, respectively. After 50 iterations, the algorithm terminated if the improvement in the optimized solution was less than \(5\times 10^{-5}\). Figure 9 depicts the convergence curves with the best results from 50 executions of each algorithm under the threshold termination condition, reflecting the convergence speed and optimization capability of each algorithm. As depicted in Figure 9, GA and QPSO had faster convergence rates, reaching convergence after approximately 100 iterations. The NDBC method enhanced the quality of each algorithm's optimized solution and accelerated their convergence, and DE and QPSO showed greater potential. In combination with the algorithm execution time, QPSO can achieve the best optimized solution in the shortest time, indicating that QPSO performs better when the search flexibility is high. Figure 8: The execution time (**a**) and optimized solutions (**b**) obtained by the algorithms. The population size of each algorithm ranged from 100 to 600, and 300 iterations were performed. The dashed and solid lines in the figures represent the UBA and NDBC methods, respectively, whereas different colors represent different algorithms. Figure 9: Convergence for the best execution of the algorithms using different bandwidth constraint handling methods. The dashed and solid lines in the figure represent the UBA and NDBC methods, respectively, whereas different colors represent different algorithms. As shown in Table 5, the improvement in GA (UBA) in the mean USC metric (0.958) was 51% compared to UBUP (1.967), as opposed to 55.7% (1.9\(\rightarrow\)0.84) reported in the literature [20]. We modified the link budget model based on [20] and used a different MODCOD table such that the results slightly differed.