# Enhanced Path Planning by Repositioning the Starting Point +
## 4 Experimental Evaluation
### Evaluation Results
#### 4.2.3 Finding the Best Centering Approach  
To determine the optimal approach among those outlined in Section 3.2 for centering the starting point, the mTSP was initially executed to obtain distances without adjusting the starting point. Subsequently, the mTSP was performed again with each individual centering approach using the expanded dataset described in Section 4.2.2. The results are depicted as graphs in Figures 13 and 14, where Figure 13i shows the original results from the initial 650 scenarios, and Figure 13 ii presents the results from the expanded 65,000 scenarios. Both graphs indicate a shared trend among approaches \(a\), \(b\), and \(c\), contrasting \(d\), \(e\), \(f\), and \(g\)'s behavior. This difference arises from the fact that the former group operates across all stations, while the latter focuses solely on the first and last stations within each route. The values on the Y-axis represent the improvement as a percentage over the non-optimized mTSP values. For example, if the standard mTSP for a particular flight plan yields a total distance of 2500 units, and employing a centering approach reduces it to 2100 units, the improvement is calculated as follows: \(1-\frac{2100}{2500}=0.16\to 16\%\). Figure 13 shows the performance of the centering approaches as the number of drones is adjusted. Both the original (Figure 13i) and expanded (Figure 13 ii) datasets demonstrate  
Figure 13: Centering approaches’ improvement for varying drones: (i) 650 unique scenarios, 5200 tests; (ii) 65,000 unique scenarios, 520,000 tests; (iii) percentage difference in geometric median and brute-force for 650 and 65k scenarios; (iv) data series legend. that the improvement escalates with the increase in drones, confirming the robustness of this trend across different sample sizes. Comparing the original dataset (650 scenarios) with the expanded dataset (65,000 scenarios) reveals several notable patterns. The expanded dataset shows remarkable consistency with the original findings, with most values remaining identical or showing only minor variations of \(\pm\)1-2% (see Figure 13 iii). This stability across the significantly larger sample size confirms the reliability of the geometric median approach and validates the initial experimental methodology. Figure 14 displays the centering approaches' performance relative to the number of stations. The expanded dataset reinforces the pattern observed in the original data, where increased station count results in a rapid decline in improvement rates. The comparison between the original and expanded datasets for station-based performance reveals, once more, interesting patterns. The station-based analysis shows even greater consistency between datasets, with most values remaining identical and only minor variations of \(\pm\)1-3% in a few cases (see Figure 14 iii). Notably, both datasets show the same general trend: highest improvements at lower station counts (22-25% for 5-10 stations) declining to moderate improvements at higher station counts (4-7% for 60-100 stations). This remarkable consistency across the expanded dataset with 100 times more scenarios demonstrates the robustness of the experimental methodology. Such stability validates that the initial experimental design was sound and that the patterns observed in the smaller dataset were genuine rather than statistical artifacts. Figure 14: Centering approaches’ improvement for varying stations: **(i)** 650 unique scenarios, 5200 tests; **(ii)** 65,000 unique scenarios, 520,000 tests; **(iii)** percentage difference in geometric median and brute force for 650 and 65k scenarios; **(iv)** data series legend. Building on this validation, the trend confirmation across both datasets becomes particularly significant. Both the drone-based and station-based performance patterns remain consistent regardless of dataset size, confirming that performance improvements increase with more drones and decrease with more stations and that the geometric median approach consistently outperforms other methods.