## 4 Rule-Based Large Neighborhood Search Algorithm (RLNS)
### Heuristic Rules and Various Operators
#### 4.2.5. Onboard Data First-Come-First-Served Transmission Operator
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sat** & \(m_{k}\) & \(energy_{k}\) & \(M_{k}\) & \(Energy_{k}\) \\ \hline Sat01 & 2 & 2 & 500 & 500 \\ Sat02 & 2 & 2 & 500 & 500 \\ Sat03 & 2 & 2 & 500 & 500 \\ Sat04 & 2 & 2 & 500 & 500 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Information of satellites for experiments. Figure 11: (**a**) Description of time windows for case30. (**b**) Description of time windows for case40. (**c**) Description of time windows for case50. (**d**) Description of time windows for case100. (**e**) Description of time windows for case150. (**f**) Description of time windows for case200. Figure 10: (**a**) Description of time windows for case10. (**b**) Description of time windows for case20. ### Analysis of the Results

The exact solution experiment was implemented using the CPLEX 12.9 Studio, while the heuristic algorithm solution experiment was implemented using C++ language. Both experiments were run on a personal computer with an AMD Ryzen Threadripper PRO 5975 WX 32-Cores 3.60 GHz processor, 32.0 GB (31.8 GB available) of memory, and the operating system Windows 10. The experimental results are shown in Figures 12-14. Figure 12: (**a**) Description of scheduling shcheme for case10 in RLNS. (**b**) Description of scheduling shcheme for case20 in RLNS. (**c**) Description of scheduling shcheme for case30 in RLNS. (**d**) Description of scheduling shcheme for case40 in RLNS. (**e**) Description of scheduling shcheme for case50 in RLNS. (**f**) Description of scheduling shcheme for case100 in RLNS. Figure 14: _Cont._

Figure 13: Diagram algorithm convergence for RLNS. The solution of CPLEX is not the optimal solution when the target scale is greater than 50, but a truncated solution according to the set time limit. ## 7 Discussion

Through the experimental results we found the following:

As the problem size increases, the solution time of the cplex solver increases exponentially with the problem size. And, when the problem exceeds a certain size, the solver has difficulty in finding a feasible solution within an acceptable time frame. For small-scale problems, the solution performance of the cplex solver is not much different from that of the cp solver when resources are more abundant, while for resource-constrained cases, the cplex solver is able to find the optimal solution faster without having to explore all the branches as much as possible, as in the case of the cp solver. Rule-based algorithms for large neighborhoods converge better, but have limited effectiveness in finding optimal results for large-scale problems. Neighborhood updates by large neighborhood search algorithms to explore to a better solution during neighborhood search occur more often on the task reassignment operator than on the task sequence perturbation operator, which may reflect the significant advantage of the first-come-first-scheduled rule for stochastic task sequences. Genetic algorithms have a better ability to find the optimal based on the heuristic rules of the large neighborhood algorithms, but for the case of tight resource constraints, the solution effect is not satisfactory enough; when the size of the problem increases, the solution performance of genetic algorithms is better than that of the neighborhood search algorithms. Meanwhile, during the experiments we found that the genetic algorithm is prone to premature maturity, where the algorithm has found a better solution under the heuristic rule in the first generation, while it is unable to find a better neighborhood solution in the subsequent iterations of the optimization search process. Current rule-based operators fall short in the explorability of neighborhood spaces: first of all, the rules of the repair operator designed in this paper are too simple, and there may be cases where the optimal solution cannot be searched.