# An Evaluation of Optimization Algorithms for the Optimal Selection of GNSS Satellite Subsets
## 5 Results and Discussion
### Multi-GNSS Optimal Satellites  
Based on the evaluation of the optimization algorithms for the GPS-only satellite constellation, the ABC algorithm proved to be the most accurate and suitable for the optimization of GNSS satellite subsets. The next step was to apply and evaluate the ABC algorithm for the optimization of the multi-GNSS satellite constellation. The same methodology as that for identifying the GPS-only satellite constellation was followed by selecting the optimal GNSS satellite subset every five minutes. GPS, GLONASS, and Galileo satellites were included for satellite selection, and four cases of satellite subset sizes (4, 8, 12, and 15 satellites) were examined. ABC was applied three times for each size of satellite subset, with different values for the parameters of (i) the number of iterations and (ii) population size (Table 2), to evaluate their impact on algorithm performance and determine which parameters provide the best results. The ABC algorithm requires the setting of three parameters: the maximum number of iterations (maxIter), population size (pSize), and abandoned limit (limit). In the case of the GPS-only satellite constellation, the maxIter, pSize, and limit values were set to 100, 100, and the product of problem dimensions and population size (\(D\times pSize\)), respectively. The value of abandoned limit (\(D\times pSize\)) is a dynamic value that automatically adapts to the problem size [64]; hence, it does not need to be adjusted. On the other hand, higher values of population size (pSize) and iteration (maxIter) can lead to better results, but they may cause slow convergence (longer processing times) [45; 66]. Increasing the value of both pSize and maxIter could be unnecessary as the algorithm results can be improved with a higher pSize and limited maxIter [66]. Based on previous studies, a pSize of up to 200 is recommended [45]. As a result, pSize and maxIter were set to the values shown in Table 2.  
Figure 15 shows the accuracy of the ABC algorithm in the GNSS satellite selection, which was determined by comparing ABC's optimal GNSS satellite selection with the actual (the TM's selection), for each satellite subset size (4, 8, 12, and 15) and for each parameter setting. In addition, Figure 15 illustrates the speed of the ABC algorithm and the TM, expressed as the required selection time for the optimal satellite selection of all 288 epochs (GNSS satellite selection per 5 min for the 24 h GNSS dataset). It can be observed that satellite selection becomes more challenging due to the size of the satellite subset, requiring higher parameter values and longer computation time. ABC showed better accuracy in selecting the optimum four GNSS satellite subset compared to the other sizes of satellite constellations (8, 12, 15) using the first two cases of parameter settings. This is due to the significant increase in the possible satellite combinations, which follows the increase in the number of satellites in the constellation (from 4 to 8, 12, and 15) (Figure 10). By increasing the value of the pSize and maxIter to 200, ABC accurately (100%) determined the optimal GNSS satellite subset for all sizes. However, the time required for selection increased by about 3.6 times with respect to the first setting case using 100 for both pSize and maxIter. Therefore, satellite selection becomes more challenging due to constellation size. This requires higher values of population size and iteration, which leads to longer computation time. This is consistent with what was reported in [45; 66], as a lower pSize and maxIter prevent achieving the best solution in complex problems, whereas increasing the value of these parameters increases the processing time. The number of possible satellite combinations of 8- and 12-satellite subsets is less than that for 15 satellites, and consequently, the selection of the optimal satellite constellation should be easier than with 15 satellites.