# Enhanced Path Planning by Repositioning the Starting Point +
## 3 Proposed Method
### Adjusting the Starting Point
 \tag{11}\]  
where Grid\({}_{SPD}\) represents all discrete points within the MBR formed by the first and last stations of each route, \((x_{j,first},y_{j,first})\) are the coordinates of the first station in route \(j\), and \((x_{j,last},y_{j,last})\) are coordinates of the last station in route \(j\). The coordinates of the top-left and bottom-right corners of the MBR shown in Figure 12 are (224, 107) and (1326, 518), respectively. The width of the rectangle is calculated as the difference between the x-coordinates (\(1326-224=1102\)), and the height is the difference between the y-coordinates (\(518-107=411\)). With a width of 1102 units and a height of 411 units, there are a total of 1102 \(\times\) 411 = 452,922 possible starting locations within the MBR. This means that employing the mTSP to find the shortest possible routes (minimum total distance) among these points would require evaluating each of the 452,922 locations as a starting point. Given this massive search space, the computational implications become apparent when comparing different optimization strategies. The brute-force approach requires significantly more computational resources to execute but serves an essential purpose as it consistently identifies the truly optimal point. This makes it invaluable for verifying the accuracy of the other approaches. Methodological Rationale and Strategic Progression  
The selection of these seven approaches follows a deliberate strategic progression that moves from simple geometric intuition to mathematically rigorous optimization. The methodology begins with basic geometric centers such as the field center, MBR center, and centroid of all stations, which provide straightforward computational solutions based on fundamental spatial relationships. These initial approaches test the spatial uniformity hypothesis, where optimal positioning depends primarily on the overall geometric distribution of the operational area. Building upon this foundation, this research incorporates domain-specific knowledge by developing SPD-focused variants that concentrate exclusively on the endpoint stations of each route. This refinement acknowledges that only the first and last stations in each drone's path directly connect to the starting point, representing a relevance filtering hypothesis  
Figure 12: Brute force in SPD. that eliminates unnecessary computational complexity while targeting the distances that actually influence the objective function. The progression then advances to theoretically grounded optimization through the geometric median approach, which applies established mathematical principles for minimizing the sum of Euclidean distances. This distance minimization hypothesis represents the most sophisticated analytical solution, leveraging proven geometric optimization theory to identify the point that mathematically minimizes the total starting point distances. Finally, the methodology incorporates exhaustive verification through the brute-force approach, which serves as the computational ground truth by testing every possible position within the relevant search space. This exhaustive optimization hypothesis provides definitive validation for the other approaches, ensuring that theoretical optimality translates to practical performance. Each approach thus represents a distinct hypothesis about the fundamental characteristics that define an optimal starting point, ranging from simple spatial considerations to complex mathematical optimization principles.