## 2 GTSP Solution for UAV Material Drop Paths
### Based on a Hybrid Whale-Swarm Solution  
Although the above IWOA has made progress in global search capability, it still suffers from the problem of insufficient local search refinement. When dealing with complex optimization problems, it is difficult for a single algorithm to balance global and local searches. Therefore, in this paper, a Whale-Swarm Hybrid Algorithm is explicitly proposed to solve path planning in the GTSP. The hybrid algorithm combines the careful optimization ability of the ABC algorithm in local search and the extensive exploration ability of the IWOA in global search. With this combination, the algorithm can quickly identify potential high-quality solution regions and perform fine-grained optimization and refinement, leading to efficient exploration and exploitation of the solution space. This approach offers the advantages of balancing global and local search, enhancing adaptability, accelerating the convergence process, and providing a new solution strategy for complex optimization problems. Step 1: Initialization variables. Initialize the set of paths \(X=\{X_{1},X_{2},\ldots,X_{N}\}\). Each path \(X_{i}\) represents a possible path for material placement. Initialize each parameter of the ABC algorithm and IWOA algorithm. Initialize the weight \(w_{IWOA}=w_{ABC}=0.5\) of the ABC Algorithm and IWOA strategies. Step 2: Division of population size. The overall population size is divided based on weights through Equation (14):  
\[\left\{ \begin{array}{l}N_{IWOA}=\left[N\times w_{IWOA}\right]\\ N_{ABC}=N-N_{IWOA}\end{array} \right. \tag{14}\]  
where \(\left[\ \right]\) denotes an upward rounding function that rounds numbers to the nearest integer, making sure that \(N_{IWOA}\) and \(N_{ABC}\) are integers and that the entire population is distributed between the two strategies without overlap. Step 3: Phase I--Global exploration phase of the IWOA algorithm. For each solution \(X_{i}\), update its position is as follows:  
\[\left\{ \begin{array}{l}X_{i\_\text{new}}\left(t+1\right)=X_{\text{best}} \left(t\right)+D_{k}\cdot e^{bl}\cdot\cos(2\pi l)\\ D=X_{best}(t)-X_{worst}(t)+\left|X_{m}(t)-X_{k}(t)\right|\end{array} \right. \tag{15}\]  
where the dynamically changing spiral shape parameter b is adapted to be calculated according to Equation (12), which makes the postures of individual whales more diversified during the search and predation process; meanwhile, the nonlinearly decreasing convergence factor \(\alpha\) is calculated according to Equation (13), which enhances the search performance in the initial iteration stage of the WOA while improving the local contraction ability of the algorithm and further optimizing the global search performance. Step 4: Phase II--Localized search phase of the ABC algorithm. For the ABC algorithm, update the path formula as follows:  
\[Y_{i}=X_{i}+\phi\cdot\left(X_{i}-X_{k}\right) \tag{16}\]  
where \(X_{i}\) represents the current path, \(Y_{i}\) represents the updated path, \(X_{k}\) is other paths in the same category, and \(\phi\) is a random number in the interval [\(-\)1, 1] that controls the update magnitude. Calculate the fitness value of the current solution as follows:  
\[f(X)=\sum_{i=1}^{n-1}d(X_{i},X_{i+1})+d(X_{n},X_{1}) \tag{17}\]  
Based on the fitness value of the current solution, the list of quality paths is updated. Paths with low fitness values will be eliminated, and paths with high fitness values will be retained and optimized for the next iteration. Step 5: Perform 2-OPT optimization every t rounds. Let a material placement path be denoted as a vertex sequence \(V_{1},V_{2},\ldots,V_{nt}\), where \(V_{i}\) denotes the \(i\)th material placement point. The optimization is performed using the following steps: (1) Choose two paths to exchange: two pairs of vertices \((V_{a},V_{a+1})\) and \((V_{b},V_{b+1})\) of which \(1\leq a<b<n\) are chosen at random from the paths.