Enhanced Path Planning by Repositioning the Starting Point ^†

Abstract

Drone power management poses ongoing challenges that significantly impact operational effectiveness across various applications. This research examines path planning optimization, particularly focusing on distance minimization to enhance efficiency and performance. When drones must visit static ground stations, analyzing the constituent elements of flight paths reveals that segments connecting the launch pad to initial and final stations emerge as a distinct area for further path optimization. Given scenarios where launch pad relocation remains feasible, this study proposes several alternative methodologies for adjusting launch positions to minimize total flight distances across multiple drone operations. The investigation employed extensive experimentation involving diverse configurations with varying station counts and available drone units. Results demonstrate that repositioning the launch pad to serve as an optimal center point for all drone routes yields substantial improvements in total distance minimization, ranging from 4% to 22% across different operational scenarios. The geometric median approach consistently outperformed alternative positioning strategies, achieving these improvements while maintaining computational efficiency. These findings contribute to sustainable drone operations by reducing energy consumption through optimized flight planning. The methodology proves particularly valuable for applications requiring flexible launch point positioning, offering practical solutions for enhancing operational efficiency in environmental monitoring, precision agriculture, and infrastructure inspection tasks where energy conservation directly impacts mission success and operational viability.

1. Introduction

Unmanned aerial vehicles (UAVs), commonly known as drones, initially captured widespread attention through their military applications. However, these versatile platforms have rapidly expanded into civilian domains, transforming numerous industries by offering innovative solutions across diverse fields, including film production [1] and photography [2], agriculture [3], search and rescue operations [4], and infrastructure inspection [5].

Beyond their commercial applications, drones have emerged as powerful tools for environmental sustainability, offering innovative solutions across multiple domains. In wildlife protection and conservation, high-resolution cameras and advanced sensors enable comprehensive monitoring of wildlife populations and protected areas without disturbing natural habitats [6]. These systems not only safeguard endangered species but also detect and deter illegal activities such as poaching [7].

The agricultural sector has particularly benefited from drone integration, where precision agriculture applications utilize sensors and artificial intelligence to provide farmers with detailed data on crop health, irrigation requirements, and pesticide usage. This technological advancement enhances farming practices while simultaneously minimizing environmental impact [8,9,10]. Similarly, infrastructure maintenance has increasingly adopted drone technology, especially for inspecting and maintaining renewable energy installations, resulting in improved functionality and reduced downtime for critical structures such as wind turbines [11] and solar panels [12].

During natural disasters, drones prove invaluable for rapid damage assessment, search and rescue coordination, and identification of areas requiring immediate assistance [13]. Forest management applications leverage this technology for comprehensive health assessments, disease outbreak identification, deforestation monitoring, and wildfire impact analysis [14,15]. Environmental monitoring capabilities extend to ecosystem data collection, providing insights into vegetation health [16,17], biodiversity patterns [18], and environmental changes in previously inaccessible locations [19,20]. Additionally, drones serve crucial roles in assessing river and lake conditions [21], coastal zone monitoring [22], and water quality assessment [23].

The operational advantages of drone deployment become evident when considering the challenges posed by their absence. Traditional approaches requiring human intervention face significant limitations, including resource dispersion, increased energy consumption, task monotony, and extended completion times. Remote or difficult-to-access operations necessitate deploying teams of personnel to challenging terrains [6,24], resulting in elevated transportation costs, increased carbon emissions, and potential disturbance to fragile ecosystems.

Manual operations inherently demand greater energy expenditure compared to drone-based alternatives [10,25]. Human-dependent approaches require more extensive effort and time, leading to increased fatigue and potential safety risks. The financial implications of deploying large workforces significantly exceed drone operational costs, potentially limiting investment in other sustainable technologies and initiatives.

The efficiency advantages of drone operations become particularly apparent in large-area coverage capabilities [10,25]. Drones dramatically reduce time requirements for inspection, monitoring, and data collection tasks. Delays in traditional approaches can severely impact environmental response times, affecting disaster response effectiveness and conservation efforts.

Motivation and Contribution

Achieving optimal balance between drone technology and traditional methodologies represents a critical factor in addressing these operational challenges. Advancing drone sustainability requires technological improvements encompassing enhanced battery life [26], improved operational efficiency [27], and exploration of alternative power sources, including solar energy and hydrogen fuel cells [28].

Path planning emerges as a fundamental component for effective drone operations [29]. Enhancing drone efficiency necessitates careful trajectory planning to compensate for inherent energy capacity constraints. Path planning encompasses the process of determining optimal routes from initial to target points, with primary objectives focused on minimizing distance and time requirements. Optimal path generation should achieve minimum energy consumption while reducing operational duration [30].

Power source limitations represent the primary constraint in drone operations, often characterized as the Achilles’ heel of UAV systems [31,32]. Limited flight duration due to battery constraints restricts application scope and creates operational challenges. Ongoing research efforts focus on advancing battery technology through lighter, more efficient systems such as solid-state batteries [26] and alternative power sources, including solar energy and hydrogen fuel cells [28]. Complementary advancements in drone design emphasize power consumption optimization through lightweight materials, aerodynamic improvements, efficient motor systems [27], and energy-efficient path planning algorithms to mitigate current battery technology limitations.

These constraints become particularly significant when considering large-area monitoring requirements. Data collection across extensive areas typically necessitates multiple drone deployment, as individual drones can only cover limited areas per flight due to power resource constraints [31,32]. Multi-drone systems enable comprehensive area coverage within reduced time frames.

Given these considerations, our research focuses on implementing multiple drones for data collection from ground stations distributed across large areas, emphasizing optimal path planning to minimize travel distance and consequently reduce energy consumption. Specifically, this work implements and enhances the multiple Traveling Salesman Problem (mTSP) [33,34] optimization algorithm through strategic starting point adjustments.

The proposed optimization applies specifically to scenarios where starting point (depot) repositioning is feasible. While certain applications such as parcel delivery systems, airport security patrols, and hospital medical supply delivery require fixed depot locations due to regulatory, infrastructure, or operational constraints, many operational scenarios benefit from adjustable starting points. These include wildlife surveillance where drones monitoring animal populations across protected areas can relocate launch points to optimize coverage of different habitat zones, forest health monitoring where mobile command vehicles can reposition to minimize flight distances when inspecting trees for disease or fire damage, infrastructure inspection teams that can reposition their base stations strategically across wind farms or solar installations, and many others.

Our research demonstrates that strategic starting point adjustment yields significantly improved optimization results. The key contributions of this work include the following:

• Novel Distance Decomposition Framework: Introduction of a systematic approach that decomposes drone flight paths into constituent parts, identifying starting point distances (SPDs) as a distinct optimization target separate from inter-station distances.
• Multiple Starting Point Repositioning Methodologies: Development and evaluation of seven different approaches for optimal launch pad positioning, ranging from simple geometric centers to mathematically rigorous optimization techniques.
• Geometric Median Optimization Discovery: Identification of the geometric median approach as the optimal solution for starting point repositioning, achieving identical results to brute-force methods while maintaining computational efficiency.
• Comprehensive Experimental Validation: Execution of extensive testing across 520,000 scenarios (65,000 unique configurations with multiple approaches), demonstrating statistically significant improvements ranging from 4% to 22% in total distance reduction.
• Performance Metric Development: Creation of starting point distance (SPD) and Starting Point Factor (SPF) metrics to quantify the impact of starting point optimization on overall mission efficiency.
• Scalability Analysis: Systematic evaluation of optimization effectiveness across varying numbers of drones (2–10) and stations (5–100), revealing predictable performance patterns for mission planning applications.
• Simulation Framework: Development of a graph-based framework modeling flight planning using mTSP to identify distance-minimizing solutions.

The remainder of this paper is structured as follows: Section 2 examines related work on the mTSP and its variants, the local search algorithm and the search operators used in the current implementation. Section 3 presents the proposed framework, while Section 4 discusses the experimental setup, results, and evaluation methodologies. Finally, Section 5 provides concluding remarks and future research directions.

2. Background and Related Work

To address the multi-drone path optimization challenges outlined in the previous section, our research builds upon the multiple Traveling Salesman Problem (mTSP) framework. The mTSP extends the classical Traveling Salesman Problem (TSP) [35] by incorporating m concurrent salesmen. While the standard TSP requires a single salesman to visit every city in a territory exactly once before returning home via the shortest possible route, the mTSP involves multiple salesmen working collaboratively to minimize their combined travel distance. Each salesman visits a subset of cities exactly once, with all salesmen departing from and returning to a common location known as the Depot [33]. The computational complexity of the mTSP exceeds that of the already challenging TSP, establishing it as an NP-hard optimization problem [36].

The scholarly landscape surrounding mTSP solutions has been systematically analyzed by several researchers. Rostami et al. [37] provide a comprehensive taxonomic overview of solution methodologies, organizing approaches into three primary categories based on their computational strategy. Their classification encompasses six exact solution methods, ten heuristic approaches, and three transformation-based techniques, offering researchers a structured framework for understanding the diverse solution landscape.

Building upon this foundation, Cheikhrouhou and Khoufi [34] present an even more detailed categorization of mTSP variants, organizing them into five comprehensive groups based on fundamental problem characteristics. Their first category examines salesman properties, including type, quantity, and cooperative behavior patterns. The second classification focuses on depot characteristics, encompassing variations such as single versus multiple depots, fixed versus mobile configurations, open versus closed path requirements, and refueling considerations. The third group addresses city distribution patterns, distinguishing between the standard mTSP where all salesmen share the same city set and the colored mTSP where cities are partitioned between salesman groups. Their fourth category differentiates problems based on objective function complexity, separating single-objective from multi-objective formulations. Finally, the fifth group categorizes constraint types, examining energy limitations, capacity restrictions, and time window requirements. This comprehensive framework illuminates the rich diversity within the mTSP family.

For our experimental investigation, we employ a specific mTSP variant with carefully defined characteristics. All drones operate from a single mobile depot, departing together and returning to the same location upon mission completion, creating closed-path trajectories. The optimization objective focuses exclusively on minimizing total distance traveled, which under simplified operational assumptions (where energy consumption occurs only during flight) directly corresponds to energy minimization. Following the formulation presented in [34], we can express this mathematically using Equations (1) and (2). For a drone $U_i$ beginning its mission at starting point H, visiting an assigned sequence of ground stations $S_{i_1},\dots ,S_{i_n}$ and returning to H, the mission cost C is calculated as follows:

$$\begin{array}{c}\hfill C\left(Missio{n}_{U_i}\right)=C(H,S_{i_1})+\sum _{k=1}^{n-1}C(S_{i_k},S_{i_{k+1}})\\ \hfill +C(S_{i_n},H)\end{array}$$

(1)

The total operational cost for d drones is then defined as follows:

$$\mathit{Total}\mathit{Cost}=\sum _{i=1}^{d}C\left(Missio{n}_{U_i}\right)$$

(2)

Our proposed methodology employs the local search [38] variant of mTSP optimization. Local search techniques have demonstrated significant effectiveness in addressing mTSP challenges [37,39,40], establishing their importance within AI-driven optimization frameworks [41].

The local search methodology applies to optimization problems requiring the identification of superior solutions among multiple candidates. The algorithmic approach navigates the solution space through incremental, localized modifications, transitioning from one solution to another until discovering an optimal result or satisfying predetermined stopping criteria (such as maximum iteration limits or convergence thresholds indicating no further improvement).

The operational mechanism as described in Algorithm 1 involves deploying various local search operators to systematically modify current solutions, generating new candidates for evaluation. Each newly generated solution undergoes comparison with its predecessor; superior solutions replace existing ones, while inferior solutions are discarded. This iterative refinement continues until achieving an optimal solution set or reaching convergence conditions where no further improvements are detected.    

Algorithm 1: mTSP using local search operators

Our implementation incorporates three specialized local search operators: Or-Optimization (Or-Opt), Two-Optimization (2-Opt), and Cross-Exchange-Optimization (Cross-Opt). The Or-Opt operator functions by selecting consecutive station sequences within individual routes and repositioning them to alternative locations within the same route, optimizing local arrangements [42].

The 2-Opt operator focuses on intra-route refinement, systematically examining station pair exchanges within individual routes to minimize total travel distances [43]. In contrast, Cross-Exchange operates at the inter-route level, facilitating sub-route exchanges between different drones to enhance overall solution quality through strategic station reassignment [40]. This inter-drone exchange capability extends the search beyond single-route modifications, enabling exploration of diverse route combinations that can escape local optima [44].

The diversification introduced by Cross-Exchange significantly enriches the search process, revealing improved solutions that remain inaccessible through purely localized operators like Or-Opt and 2-Opt [38]. The synergistic interaction among all three operators drives continuous solution improvement through systematic incremental refinements. Strategic integration of these operators, potentially combined with additional heuristic methods, can further enhance solution quality for complex mTSP instances [37,39,40].

3. Proposed Method

While the mTSP framework provides a robust foundation for multi-drone optimization, our research identifies opportunities for enhancement through a novel perspective on distance calculation. Distance plays a fundamental role in drone path planning and represents a crucial factor in determining efficient flight paths. Given the limited battery capacity of drones, minimizing traveled distance becomes essential for energy conservation and operational success.

Existing research in this field [27,29,30] has traditionally treated distance as a single, unified factor. However, our approach considers distance as a composite element that can be decomposed into constituent parts. This decomposition enables a more detailed examination of each segment, facilitating the development and enhancement of algorithms that are more effective at minimizing overall distance. To the best of our knowledge, no existing work addresses this particular combination of challenges. This publication builds upon the research of Gasteratos & Karydis [45].

3.1. Distance Analysis

To illustrate our approach, we begin with a simple scenario shown in Figure 1. This scenario involves a single drone that must visit static ground stations with predefined coordinates in a specific order, collect data, and return to its starting point.

The conventional approach involves finding the shortest path by applying optimization algorithms that consider all possible combinations of station-to-station flights. Using the TSP formulation, which forms the foundation of this study, we obtain the result depicted in Figure 2. The total distance of 1419.3 units, calculated as $d\left(1\right)+d\left(2\right)+d\left(3\right)+d\left(4\right)+d\left(5\right)$, represents the optimal flight path under traditional TSP constraints.

While the TSP produces an optimal solution with a fixed station visiting order, further distance minimization remains possible through strategic starting point adjustment. Consider the sum of distances $d\left(1\right)+d\left(5\right)=470.9$, which can be reduced by repositioning the starting point along the straight line between stations $S3$ and $S4$, as demonstrated in Figure 3. This repositioning yields an improved value of $d\left(1\right)+d\left(5\right)=398.2$, representing the minimum possible distance between $S3$ and $S4$.

This analysis reveals a fundamental distinction in flight path distance calculations. Two distinct distance types emerge: inter-station distances $d\left(2\right),d\left(3\right),d\left(4\right)$ (shown as solid lines) and starting point distances $d\left(1\right),d\left(5\right)$ (shown as dotted lines). The key insight is that starting point repositioning can significantly influence path optimization, opening new avenues for improvement beyond traditional TSP solutions.

This principle extends naturally to multi-drone missions where all drones share a common starting point, as illustrated in Figure 4. The scalability of this approach demonstrates its practical relevance for complex operational scenarios.

3.1.1. Starting Point Distance

To quantify the impact of starting point optimization, we establish appropriate performance indicators. The most direct measure involves calculating the sum of all distances connected to the starting point. We define starting point distance (SPD) in Equation (3) as follows:

$$\mathit{Starting\; Point\; Distance}\left(\mathit{SPD}\right)=\sum _{i=1}^n(d_{i,<first>}+d_{i,<last>})$$

(3)

where n represents the number of paths (routes) in a mission, $d_{first}$ denotes the distance from the starting point to the first station in each path, and $d_{last}$ represents the distance from the last station back to the starting point.

Applying this definition to our examples, the SPD in Figure 3 equals the sum of dotted line distances: $d\left(1\right)+d\left(5\right)=398.2$. For the multi-drone scenario in Figure 4, which comprises three distinct paths, the SPD calculation becomes $d(1,1)+d(1,6)+d(2,1)+d(2,6)+d(3,1)+d(3,6)=1953$, representing the sum of all starting point connections.

3.1.2. Starting Point Factor

To provide additional analytical perspective, we can view SPD as a proportion of the total mission distance. This leads us to define the Starting Point Factor (SPF) in Equation (4) as follows:

$$\mathit{Starting\; Point\; Factor}\left(\mathit{SPF}\right)={\displaystyle \frac{\mathit{Starting}\mathit{Point}\mathit{Distance}\left(\mathit{SPD}\right)}{\mathit{Total}\mathit{Distance}}}$$

(4)

Since SPD represents a subset of the total distance, SPF values fall within the range $[0,1]$. The lower bound $SPF=0$ occurs in the theoretical scenario where starting, ending, and depot points coincide. Lower SPF values indicate that starting point distances constitute a small proportion of total distance, offering limited optimization potential. Conversely, higher SPF values suggest that starting point distances represent a significant portion of total distance, providing substantial opportunities for improvement through SPD minimization, as demonstrated in Figure 5.

3.2. Adjusting the Starting Point

Thus far, it appears that the starting point needs to be adjusted to a better position that serves as the optimal position for a drone to travel to its connecting stations in the shortest possible distance. To achieve this, the following approaches were chosen in order to determine the best position for the task at hand.

(a)

Center of the field. This is the point where the diagonals of the area under investigation cross (see Figure 6), and is thus selected irrespective of the stations’ locations. It is defined in Equation (5) as

$${\mathrm{StartingPoint}}_{field}=\left({\displaystyle \frac{x_{min}+x_{max}}{2}},{\displaystyle \frac{y_{min}+y_{max}}{2}}\right)$$

(5)

where $(x_{min},y_{min})$ and $(x_{max},y_{max})$ are the coordinates of the field boundaries.

(b)

Center of minimum bounding rectangle (MBR) [46]. The point where the diagonals of the MBR intersect (see Figure 7). It is defined in Equation (6) as

$${\mathrm{StartingPoint}}_{MBR}=\left({\displaystyle \frac{x_{min}^{all}+x_{max}^{all}}{2}},{\displaystyle \frac{y_{min}^{all}+y_{max}^{all}}{2}}\right)$$

(6)

where $(x_{min}^{all},y_{min}^{all})$ and $(x_{max}^{all},y_{max}^{all})$ are the coordinates of the minimum bounding rectangle containing all stations.

(c)

Centroid of all stations [47,48]. The x and y coordinates of the starting points are obtained by averaging the x and the y coordinates of the n stations in the field (see Figure 8). It is defined in Equation (7) as

$${\mathrm{StartingPoint}}_{centroid}=\left({\displaystyle \frac{{\sum }_{i=1}^n{x}_i}{n}},{\displaystyle \frac{{\sum }_{i=1}^n{y}_i}{n}}\right)$$

(7)

where $(x_i,y_i)$ are the coordinates of the n stations in the field.

(d)

MBR in SPD. Similar to approach (b), but this time focusing exclusively on the starting and ending stations (endpoints) of each route (see Figure 9). It is defined in Equation (8) as

$${\mathrm{StartingPoint}}_{MBR\_SPD}=\left({\displaystyle \frac{x_{min}^{SPD}+x_{max}^{SPD}}{2}},{\displaystyle \frac{y_{min}^{SPD}+y_{max}^{SPD}}{2}}\right)$$

(8)

where $(x_{min}^{SPD},y_{min}^{SPD})$ and $(x_{max}^{SPD},y_{max}^{SPD})$ are the coordinates of the minimum bounding rectangle containing only the endpoints of each route.

(e)

Centroid in SPD. Similar to approach (c), but this time focusing exclusively on the endpoints of each route (see Figure 10). It is defined in Equation (9) as

$${\mathrm{StartingPoint}}_{centroid\_SPD}=\left({\displaystyle \frac{{\sum }_{j=1}^m(x_{j,first}+x_{j,last})}{2m}},{\displaystyle \frac{{\sum }_{j=1}^m(y_{j,first}+y_{j,last})}{2m}}\right)$$

(9)

where $(x_{j,first},y_{j,first})$ and $(x_{j,last},y_{j,last})$ are the coordinates of the first and last stations of route j, m is the number of routes, and $2m$ is the number of points (each route contributes 2 points: first station and last station).

(f)

Geometric median in SPD [49]. The point y that minimizes the sum of distances to all endpoint coordinates of each route (see Figure 11). It is defined in Equation (10) as

$${\mathrm{StartingPoint}}_{geometric\_median}=\underset{y}{argmin}\sum _{k=1}^{2m}{\parallel x_k-y\parallel }_2$$

(10)

where

• $\underset{y}{argmin}$—“Argument of the minimum” means “find the point y that minimizes the following expression” and y represents a candidate starting point with coordinates $(x_y,y_y)$. We are searching over all possible points y in the 2D plane.
• ${\sum }_{k=1}^{2m}$—“Sum over all endpoints”, m = number of routes (drones) and $2m$ = total number of endpoint coordinates (each route has 2 endpoints: start and end). We sum over all these endpoint coordinates.
• $x_k$—“The k-th endpoint coordinate”. $x_1,x_2,\dots ,x_{2m}$ are the coordinates of all first and last stations. For example looking at Figure 11,we have 3 routes: $x_1$ = $S4$ of orange route, $x_2$ = $S11$ of orange route, $x_3$ = $S9$ of blue route, $x_4$ = $S6$ of blue route, $x_5$ = $S10$ of green route, and $x_6$ = $S3$ of green route.
• ${|x_k-y|}_2$—“Euclidean distance”. This is the straight-line distance between point $x_k$ and candidate point y. If $x_k=(x_{k,x},x_{k,y})$ and $y=(y_x,y_y)$, then ${|x_k-y|}_2=\sqrt{{(x_{k,x}-y_x)}^2+{(x_{k,y}-y_y)}^2}$.

Figure 11. Geometric median in SPD.

Figure 11. Geometric median in SPD.

(g)

Brute force in SPD. Test every point in the MBR area formed by the endpoints of each route and see which one is the optimal point to act as the starting point (see Figure 12). It is defined in Equation (11) as

$$\begin{array}{cc}\hfill {\mathrm{StartingPoint}}_{brute\_force}=\underset{(x,y)\in {\mathrm{Grid}}_{SPD}}{argmin}\sum _{j=1}^m(& \sqrt{{(x-x_{j,first})}^2+{(y-y_{j,first})}^2}\hfill \\ \hfill & +\sqrt{{(x-x_{j,last})}^2+{(y-y_{j,last})}^2})\hfill \end{array}$$

(11)

where ${\mathrm{Grid}}_{SPD}$ 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 × 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 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.

4. Experimental Evaluation

4.1. Experimental Setup

A simulation model was developed in order to test various scenarios using the mTSP Algorithm 1 as described by Equations (1) and (2).

The simulation model was implemented in the .NET framework using C#. The machine specifications used for conducting the tests include an Intel Quad Core i7-6820HK CPU @ 2.70GHz with 16 GB of RAM (Intel, Santa Clara, CA, USA).

4.2. Evaluation Results

4.2.1. The Experiment

In every trial, we varied the flight plan, using 2 to 10 drones and between 5 and 100 stations (specifically, 5, 10, 15, 20, 25, 45, 60, or 100). Each of these configurations was then tested across 10 different, randomly generated station deployment scenarios. Deliberately commencing with a starting number of drones at two, rather than one, is intentional, as this work primarily focuses on the mTSP rather than the TSP. Cases (70 in total, resulting from 7 × 10 scenarios) where the number of drones was equal to or greater than the number of stations were excluded from the analysis, as it makes no sense to deploy more drones than the available stations.

Table 1. Excluded test scenarios.

Drones	Stations
5	5
6	5
7	5
8	5
9	5
10	5
10	10

shows the excluded cases.

This resulted in 650 unique scenarios (9 drones × 8 station groupings × 10 scenarios − 70).

4.2.2. Expanded Experimental Validation

To strengthen our initial results (in [45]) and ensure the statistical significance of our findings, we expanded the test scenarios significantly. The experimental parameters were scaled up to include one thousand random scenarios per configuration, rather than the initial ten scenarios. This expansion resulted in 65,000 unique scenarios (9 drones × 8 station groupings × 1000 scenarios − 7000 exclusions, as per Table 1). Consequently, the total number of tests increased to 520,000 (the initial run + 7 runs for the 7 approaches = 8 × 65,000 = 520,000), providing a much more robust statistical foundation for our analysis.

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 Figure 13 and Figure 14, where Figure 13i shows the original results from the initial 650 scenarios, and Figure 13ii 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: $1-{\displaystyle \frac{2100}{2500}}=0.16\to 16\%$.

Figure 13 shows the performance of the centering approaches as the number of drones is adjusted. Both the original (Figure 13i) and expanded (Figure 13ii) datasets demonstrate 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 ±1–2% (see Figure 13iii). 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 ±1–3% in a few cases (see Figure 14iii). 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.

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.

The statistical reliability of these findings becomes evident when examining the minimal variations between datasets, which typically range from ±1–2%. While this suggests that the original 650-scenario dataset was sufficient to capture the essential patterns, the expanded dataset offers researchers additional confidence in the statistical significance of the results.

This enhanced confidence is reflected in the superior performance of the geometric median approach across both datasets. When compared to other methods for centering the starting point in the mTSP, the geometric median consistently achieves optimized total distances. The brute-force approach produces identical results, further affirming the geometric median’s effectiveness as an optimal solution.

The practical implications of these findings become clear when considering algorithm efficiency. The geometric median approach maintains its position as the optimal centering method across both datasets, with results consistently matching the computationally expensive brute-force approach. This consistency across different sample sizes reinforces the practical value of the geometric median method as an efficient alternative to exhaustive search approaches.

To provide additional statistical validation, we applied the Wilcoxon Signed-Rank Test [50] to compare the total distance produced by the standard mTSP against the total distance after repositioning the starting point using the geometric median approach. The test yielded a statistic value of 2145.0 and a p-value of $1.203\times {10}^{-12}$, indicating very strong evidence against the null hypothesis. This suggests a significant difference between the paired samples, confirming that the observed improvements are not due to random chance but represent a meaningful enhancement in performance.

To further validate these findings and ensure the reproducibility of our results, we conducted an additional round of testing using a completely new set of 65,000 randomly generated scenarios (another 520,000). This independent validation dataset, generated with different random station configurations, produced virtually identical results to our original expanded dataset (Figure 13 and Figure 14). The consistency between these two independent large-scale datasets reinforces the validity and reliability of our findings, demonstrating that the observed performance patterns are robust and not dependent on specific random configurations. This additional validation strengthens the statistical evidence supporting the superiority of the geometric median approach.

4.2.4. Employing the Best Centering Approach in the mTSP

As previously discussed, the geometric median approach emerged as the optimal choice for centering the starting point in the mTSP. To further examine its effectiveness, we conducted a thorough analysis of how adjusting the starting point impacts performance when the number of drones or stations varies. This analysis follows the same experimental procedure described in Section 4.2.2.

The experimental design involved applying the mTSP to each flight plan using the default starting point and then employing the geometric median approach to refine the starting location for subsequent mTSP optimization. This two-step process resulted in 130,000 tests across the 65,000 scenarios. The simulation model generated comprehensive data, which is presented in Figure 15 and Figure 16.

The Y-axis values represent the percentage improvement over the vanilla mTSP for three key metrics: SPD (as defined in Equation (3)), total flight distance, and SPF (as defined in Equation (4)). The X-axis shows the variation in the number of drones and stations for each respective analysis.

• Analysis of Drone Variation Effects

When examining the impact of varying drone numbers in Figure 15, several clear patterns emerge. For SPD, there is a consistent upward trend in improvement, ranging from 20% to 24% as the number of drones increases. This pattern is mirrored in the distance analysis, which shows a positive correlation between drone count and improvement, with gains ranging from 7% to 18%.

An increase in improvement is anticipated for both SPD and distance as more drones are engaged in the flight plan. Each additional drone introduces new starting point distances (as detailed in Section 3.1.1), which increases the total distance and consequently provides greater opportunities for optimization through strategic starting point repositioning.

The SPF analysis reveals a contrasting pattern, showing a consistent decrease in improvement as drone numbers grow. This decline is initially more pronounced but gradually levels off with additional drones. However, this apparent contradiction can be explained by examining the relationship between the components. Considering the almost constant change in Diff (Figure 15d) when comparing “Vanilla mTSP” and “After Centering”, along with the observation that SPF increases with the addition of drones, the apparent decrease in SPF improvement is anticipated, as the improvement is proportional to the evolution of SPF.

• Analysis of Station Variation Effects

The station-based analysis, presented in Figure 16, reveals different but equally important patterns. SPD improvement shows a clear inverse relationship with station count, achieving the highest gains when fewer stations are present and declining as station numbers increase. The average SPD improvement across all station configurations reaches 22%.

Distance improvements follow a similar declining trend as station numbers increase, with higher gains observed at lower station counts. The average improvement for total distance across all station configurations is 13%. This behavior aligns with expectations, as additional stations create longer flight paths, making the relative impact of starting point optimization less significant compared to the overall mission distance.

Interestingly, SPF shows a different behavior pattern, displaying a gradual but modest increase in improvement from 9% to 11%, with an average improvement of 9%. This trend occurs because adding stations to a flight plan does not change the number of drones executing the mission, and drone count is the primary factor influencing SPF values.

The comprehensive analysis reveals that both SPD and distance follow predictable declining improvement trends as station numbers increase. This behavior stems from the fundamental nature of the optimization approach: since improvements are achieved by optimizing distances between the first and last stations of each drone route while leaving intermediate station sequences unchanged, the relative benefit naturally diminishes as mission complexity grows. These findings provide valuable insights for mission planners seeking to maximize the benefits of geometric median-based starting point optimization in various operational scenarios.

• Computational Efficiency Analysis

While the comprehensive analysis of drone and station variations provides valuable insights into optimization effectiveness, practical implementation requires careful consideration of computational requirements. The time complexity differences between approaches become particularly significant when evaluating real-world applicability, as operational scenarios often demand rapid decision-making capabilities.

To thoroughly evaluate computational performance, an additional 130,000 tests were conducted across the 65,000 scenarios, comparing the geometric median approach against brute force’s timings. This extensive testing was necessary to establish statistically reliable performance benchmarks and validate the efficiency claims across diverse operational configurations. The results are depicted in in Figure 17.

Examining the computational performance data presented in Figure 17, the brute-force methodology demonstrates exponentially increasing processing demands, requiring approximately 12.6 s for configurations with 5 stations and extending beyond 92 s when handling 100 stations. These extended computation periods present significant limitations for time-critical applications where rapid deployment decisions are essential. In contrast, the geometric median approach maintains consistently efficient performance, completing optimization calculations within one second across all tested scenario complexities. This dramatic efficiency advantage occurs without compromising solution quality, as the geometric median consistently identifies identical optimal positions to those discovered through exhaustive search methods.

4.2.5. Scaled-Up Experimental Analysis

Building upon the comprehensive evaluation presented in Section 4.2.4, the significant computational advantages of the geometric median approach enabled us to undertake a more ambitious experimental investigation. While the original 65,000-scenario dataset provided robust statistical validation, the growing complexity of operational requirements in real-world drone applications demanded evaluation across broader parameter ranges, particularly for large-scale missions involving extensive station networks and substantial drone fleets.

Given the substantial computational time requirements for processing larger scenario sets—with complete test cycles extending to multiple days—we implemented a strategic adjustment to maintain experimental feasibility while expanding coverage. The iteration count was reduced from 1000 to 500 scenarios per configuration, allowing for a significant expansion in the parameter space without compromising statistical reliability.

The scaled experimental framework encompassed station configurations ranging from 5 to 800 stations (specifically 5, 10, 15, 20, 25, 45, 60, 100, 200, 400, and 800 stations) and drone fleet sizes extending from 2 to 60 drones (2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, 30, and 60 drones).

This expanded configuration resulted in another set of 58,500 randomly generated scenarios, representing a completely new, independently generated dataset that enables analysis of optimization performance across industrially relevant scales, including large infrastructure monitoring operations, extensive agricultural surveys, and comprehensive environmental assessment missions.

The scaled experimental results, presented in Figure 18 and Figure 19, reveal several critical insights regarding optimization performance at increased operational scales, building upon the patterns initially observed in Figure 15 and Figure 16.

Figure 18 illustrates performance patterns for larger drone fleet configurations, extending the insights originally presented in Figure 15.

The extended drone fleet analysis, as illustrated in Figure 18, encompasses operational scenarios relevant to major commercial and research applications, extending well beyond the original 2–10 drone range examined in Figure 15. Notably, the extended drone analysis reveals that performance improvements actually increase with larger fleet sizes, contrary to the declining pattern observed with increased station counts. This behavior aligns with theoretical expectations, as each additional drone introduces two new starting point connections (departure and return paths), providing additional optimization opportunities while maintaining the same fundamental operational constraints.

Similarly, Figure 19 demonstrates the behavior of the geometric median approach across the extended station range, providing direct comparison with the original station-based analysis from Figure 16.

The extended station analysis confirms and extends the declining performance trend clearly established in the original experimental range shown in Figure 16. While the original findings demonstrated the predictable declining trend as station counts increase from 5 to 100 stations, the extended range in Figure 19 reveals the continuation of this pattern at much larger scales and identifies specific performance transition zones. Beyond 200 stations, improvements stabilize at lower but still meaningful levels, indicating that even for very large missions, geometric median optimization provides consistent benefits.

• Comparative Analysis: Original vs. Scaled Results

Direct comparison between the original results (Figure 15 and Figure 16) and scaled experimental results (Figure 18 and Figure 19) provides valuable insights into the robustness and scalability of the optimization approach. The overlapping parameter ranges involving 5 to 100 stations and 2 to 10 drones demonstrate remarkable consistency between datasets. Station-based consistency analysis reveals minimal variation, with 5 stations showing original 32.7% versus scaled 32.3% SPD improvement, 25 stations demonstrating original 21.8% versus scaled 23.0% SPD improvement, and 100 stations exhibiting original 13.1% versus scaled 16.8% SPD improvement. Drone-based consistency analysis similarly shows acceptable variation ranges, with 2 drones producing original 19.6% versus scaled 17.3% SPD improvement, 5 drones yielding original 22.9% versus scaled 17.7% SPD improvement, and 10 drones achieving original 24.1% versus scaled 18.7% SPD improvement.

The minor variations between datasets (typically ±2–4%) fall well within expected statistical ranges given the reduced iteration count in the scaled experiment. This consistency validates both the robustness of the geometric median approach and the reliability of the experimental methodology across different sampling intensities.

• Operational Implications and Scaling Insights

The scaled experimental analysis provides several critical insights for practical implementation. Performance predictability emerges as a key finding, with optimization benefits following predictable patterns across all tested scales, enabling mission planners to estimate potential improvements based on operational parameters. Threshold identification represents another significant discovery, as the 25 to 100 station range represents a critical transition zone where optimization benefits decline most rapidly, suggesting focused analysis for missions in this range. Large fleet viability becomes evident through the increasing effectiveness with larger drone fleets, confirming the approach’s suitability for major commercial operations requiring substantial autonomous vehicle coordination. Scalability validation demonstrates that the maintenance of optimization benefits even at the largest tested scales involving 800 stations and 60 drones establishes the approach’s practical applicability across industry-relevant operational ranges.

These findings establish the geometric median optimization approach as a robust, scalable solution for drone fleet path planning across the full spectrum of anticipated commercial and research applications, from small-scale monitoring missions to large-scale infrastructure and environmental assessment operations.

5. Conclusions

The widespread adoption of drone technology across numerous sectors continues to face fundamental challenges, with power management and energy efficiency serving as critical bottlenecks that limit operational effectiveness and application scope. This research tackles these constraints through innovative path planning optimization, focusing specifically on distance minimization to enhance operational efficiency while contributing to broader environmental sustainability goals.

5.1. Primary Contributions and Research Outcomes

This study presents a novel decomposition methodology for drone path planning that fundamentally challenges conventional approaches by examining individual path components rather than treating flight distance as an indivisible whole. The central innovation involves identifying and optimizing the connection segments between launch pads and terminal stations, components that prove to be prime candidates for enhancement when launch pad repositioning remains feasible.

This research develops a comprehensive graph-based framework that models multi-drone operations using enhanced multiple Traveling Salesman Problem formulations. Two new performance metrics, starting point distance and Starting Point Factor, were introduced to quantify how launch pad positioning affects overall mission efficiency. Seven distinct approaches for optimal launch pad positioning were proposed and systematically evaluated, spanning from basic geometric centroid calculations to sophisticated optimization algorithms.

Extensive experimentation (≈1,320,000 tests) across 130,000 unique operational scenarios validated the proposed methodology. These scenarios encompassed variable drone configurations ranging from two to ten units and station arrangements from five to one hundred. The geometric median approach consistently demonstrated superior performance, achieving distance improvements between 4% and 22% across different operational configurations. Average improvements reached 22% for starting point distance calculations and 13% for total flight distance measurements. The approach also delivered computational efficiency gains of several orders of magnitude compared to exhaustive search methods while maintaining solution quality.

5.2. Operational and Economic Implications

The research findings reveal substantial practical benefits for drone operations across multiple industrial sectors. The optimization approach enables considerable energy savings through reduced flight distances, directly addressing the primary limitation of restricted battery capacity in current drone systems. Clear performance patterns emerged from the analysis, showing that improvements increase with drone count while decreasing with station density, providing mission planners with predictive capabilities for various operational scenarios.

These distance reductions translate into tangible economic advantages, including extended operational range, reduced battery consumption, and improved mission success rates for commercial drone applications. From an environmental perspective, minimizing energy consumption through optimized flight paths contributes meaningfully to sustainable drone operations and supports broader conservation initiatives across industries.

5.3. Research Limitations and Scope Boundaries

While this research demonstrates significant improvements in path optimization, several limitations warrant acknowledgment. The optimization methodology applies specifically to scenarios where launch pad repositioning remains feasible, restricting applicability in fixed-depot operations such as commercial package delivery systems. The current analysis focuses primarily on distance-based energy consumption without accounting for variable factors including payload weight, atmospheric conditions, or altitude variations that influence real-world energy usage. Additionally, the methodology assumes static ground stations and predictable flight environments, which may not accurately reflect dynamic operational conditions encountered in practice.

5.4. Future Research Trajectories

Several promising research directions emerge from these findings. Future investigations should incorporate environmental variables including wind patterns, temperature fluctuations, and atmospheric conditions into the optimization framework to enhance real-world applicability. The development of adaptive algorithms capable of adjusting launch pad positioning and route assignments in real time based on changing mission requirements and environmental conditions presents another valuable research avenue.

Extending the current distance-focused approach to encompass multiple objectives such as flight duration, energy consumption, and mission completion probability would provide more comprehensive optimization capabilities. Investigation of optimization strategies for heterogeneous drone fleets with varying capabilities, battery capacities, and operational constraints could significantly expand the methodology’s practical utility. Comprehensive field testing across diverse operational scenarios and geographical conditions remains essential for validating laboratory findings and refining optimization algorithms.

5.5. Broader Scientific and Technological Impact

This research contributes substantially to the advancement of autonomous systems and sustainable technology applications. The geometric median-based optimization approach demonstrates how fundamental mathematical principles can be effectively applied to complex engineering challenges, providing a template for similar optimizations in other autonomous vehicle domains.

The findings support growing evidence that intelligent path planning can dramatically enhance the viability and sustainability of drone operations across sectors, including precision agriculture, environmental monitoring, infrastructure assessment, and emergency response coordination. As drone technology continues evolving, optimization approaches like those presented here become increasingly vital for maximizing operational efficiency and environmental responsibility.

The demonstrated improvements of 4–22% in flight distance efficiency, while appearing incremental, represent substantial gains when scaled across the expanding global drone fleet. These improvements could contribute to significant energy savings and extended operational capabilities that may accelerate drone adoption across multiple industries. This research establishes a foundation for future developments in autonomous vehicle optimization while addressing pressing needs for more sustainable and efficient robotic systems in an increasingly technology-dependent world.