## 4 Experimental Design and Results Analysis
### Stability Analysis  
BIG-O notation, typically employed to characterize the time complexity of algorithms, is not well suited for quantifying the computational complexity of evolutionary algorithms (EAs). This inadequacy stems from EAs not navigating within differentiable spaces but instead seeking global optima in complex, multimodal problem landscapes. Such landscapes often feature many local optima, which can entrap the algorithm and result in early convergence. In these contexts, the algorithm's internal complexity is of lesser importance. As the search process defies straightforward optimization, the computational complexity, or search time, is potentially infinite. Consequently, assessing the computational complexity of the MAPSO algorithm necessitates stability analysis. The stability analysis method quantifies evolutionary algorithms' time taken and success rate in discovering feasible solutions within a controlled simulation environment [30]. For its innovative approach and effectiveness over other metrics, this research utilizes success rate (SR), mean fitness evaluations (MFE), and mean search time (MST) as the criteria for stability analysis. The collective performance of seven algorithms is evaluated by determining the values of SR, MFE, and MST. The fundamental goal of the MAPSO algorithm is to identify feasible solutions for the SDTSP. Its success rate (SR) is computed using Equation (24), where \(aa\) signifies the count  
Figure 8: (**a**) One station, 40 satellites; (**b**) Two stations, 50 satellites; (**c**) Three stations, 60 satellites, 90 satellites Pareto; (**d**) Four stations, 90 satellites. of independent trials that successfully yield a feasible solution, and \(tt\) denotes the overall number of trials conducted:  
\[SR=\frac{aa}{tt}\times 100 \tag{24}\]  
A further aspect of stability assessment involves monitoring the frequency of fitness evaluations executed by the algorithm. In detail, the fitness evaluation counter (FE) is incremented with every invocation of the objective function by the algorithm. The search concludes once the algorithm effectively locates a viable solution, at which point the FE counter's tally is noted. Following this methodology, the mean number of fitness evaluations (MFE) across successful trials is computed using Equation (25):  
\[MFE=\frac{1}{aa}{\sum_{i=1}^{aa}fes_{[i]}} \tag{25}\]  
Another critical aspect of stability analysis is examining the time required by the algorithm to find a feasible solution. Once a feasible solution is located, the search halts, and the search time (ST) is recorded. The average search time (MST) for successful runs is calculated using Equation (26):  
\[MST=\frac{1}{aa}{\sum_{i=1}^{aa}st_{[i]}} \tag{26}\]  
The methods for calculating SR, MFE, and MST are based on the stability analysis described in [27], with detailed steps outlined in Algorithm 5. The statistical results for the SR, MFE, and MST of seven algorithms across 24 scenarios are presented in Tables 6-8. ```
Input: feasible solution defined for the problem Output:\(SR\), \(MFE\), MST
1. for\(i=1\) to run // total count of independent runs
2. While\((fes<\ max\ FEs)\) do
3. Start the timer, then initiate the evolutionary search steps
4. If\((fes>\ 0.3\ *\ max\ FEs\) and best_solution\(\ \leq\ FS)\)
5. Stop the timer and record the values of \(ST[i]\) and \(FE[i]\)
6. Increment the variable \(aa\) by 1
7. Exit the while loop
8. else
9. Increment the fes count in \(FE[i]\)
10. endif
11. endwhile
12. endfor
13. Compute \(SR\), \(MFE\), and \(MST\) using Formulas (24)-(26)
```  
**Algorithm 5** Stability Analysis  
Table 6 shows that the MAPSO algorithm achieves a high success rate across all scenarios, indicating its ability to find feasible solutions in every case. Table 6 demonstrates that MAPSO possesses strong robustness and reliability, effectively addressing satellite data transmission problems of varying scales and complexities. The DRSC-MOAGDE and DSC-MOPSO algorithms also achieve high success rates in smaller-scale scenarios, surpassing MAPSO.