# An Evaluation of Optimization Algorithms for the Optimal Selection of GNSS Satellite Subsets
## 5 Results and Discussion
### GPS Optimal Satellites
optimal subset of four satellites in two epochs. In Figures 13 and 14, the actual optimal subset of four satellites, derived by the TM, and the satellite combinations selected by ACO and PSO are displayed. The quality of these algorithms may vary from one trial to another, which could be due to the fact that these algorithms start randomly [45], which affects their results [46; 47], or because of the chosen values for the different parameters, which can also affect the algorithms' results. However, the four algorithms (ABC, ACO, GA, and PSO) can generally be considered equally accurate in terms of satellite selection. This aligns with the results presented in [29], where ABC, GA, and PSO provided approximately the same high level of accuracy in GPS satellite selection, while SA provided slightly lower accuracy in selection. In terms of satellite selection speed, the optimization algorithms were slow in selecting the optimal GPS satellites. Figure 11 presents the time required by the five optimization algorithms and TM to find the optimal GPS satellite selection for all 288 epochs, corresponding to the selection of a GNSS satellite subset of one epoch per 5 min interval, for the 24 h GNSS dataset. According to the figure, the TM, which checks all possible satellite combinations, was faster than the optimization algorithms in finding the optimal satellite subset. TM selected the optimal satellites within about 2 min in all satellite selection cases. In contrast, the five optimization algorithms took more than 2 min in all cases, with the  
Figure 12: The comparison of the performance of the optimization algorithms with respect to TM, expressed as the difference between the CNR-WGDOP of the optimal satellite constellation of each optimization algorithm and the corresponding CNR-WGDOP of TM. The results of the four cases of GPS satellite constellations (4, 5, 6, and 7 satellites) are presented. On the left axis, the CNR-WGDOP value of the optimal satellite constellation based on the TM is presented, and on the right axis is the difference between each of the optimization algorithms and the TM. required time varying depending on the number of satellites in the constellation. In general, SA was the fastest algorithm among the five, whereas GA was the slowest. The ABC, ACO, and PSO algorithms were similar in terms of required time (5 min) to complete the satellite optimization. Unlike the TM, optimization algorithms do not require a complete investigation of possible satellite combinations to determine the optimum. However, the TM was faster in finding the best GPS satellites as the optimization algorithms are computationally heavy for \"small\" problems due to them following indirect approaches that require additional processing time. For a small number of satellites that can be provided by a single system (e.g., GPS), this additional processing adds an unnecessary burden to the problem. As a result, the TM was faster in finding the optimal GPS satellites, which aligns with the outcome in [22], as the TM is suitable for selecting satellites from one system. Except for GA, the selection speed of the optimization algorithms was generally comparable. They took about 5 min to choose satellites in all selection cases. However, the GA took much longer, ranging from 14 to 24 min. This contrasts with what was  
Figure 14: Same as Figure 13, this figure presents the sky plots for epoch 81 (**left**) and epoch 215 (**right**), as well as differences between the selection of the optimal GPS satellite subset for PSO and TM. Figure 13: The sky plots of epoch 55 (**left**) and epoch 184 (**right**) presenting the selection of the optimal GPS satellite subset by the ACO and the TM, and showing the satellites commonly selected (blue) by the two methods, but also those that were differently selected by ACO (yellow) and TM (orange). stated in [46], where the GA was a fast algorithm. In addition, it contradicts the results presented by [PERSON] et al. [22], who showed that the GA outperformed ACO in terms of satellite selection speed.