## 2 GTSP Solution for UAV Material Drop Paths
### Improved Whale Optimization Algorithm (IWOA) for Solving GTSP  
#### 2.2.1 Principles and Applications of the WOA  
The WOA mimics the feeding behavior of humpback whales by iteratively updating their positions to improve spatial solutions [26]. This section is divided into three parts to elaborate on the WOA solution for the material placement GTSP. Step 1: Encircling the prey (positioning of material drop-off points). In this step, the algorithm considers the optimal path as the \"prey.\" In a \(d\)--dimensional space, let the position of the current best whale be \(X^{*}=\left(X_{1}^{*},X_{2}^{*},\cdots,X_{d}^{*}\right)\), and the position of a random whale be \(X^{j}=\left(X_{1}^{j},X_{2}^{j},\cdots,X_{d}^{j}\right)\). Then, the latest position coordinates of the individual whale \(X^{j}\) are constantly updated \(X^{j\_new}=\left(X^{j\_new1},X^{j\_new2},\cdots,X^{j\_newd}\right)\) as they approach the optimal whale individual \(X^{*}\) as described in the following equation:  
\[X_{k}^{j\_new}\left(t+1\right)=X_{k}^{*}(t)-A_{1}\cdot D_{k} \tag{2}\]  
\[D_{k}=\left|C_{1}\cdot X_{k}^{*}(t)-X_{k}^{j}(t)\right| \tag{3}\]  
where \(X_{k}^{j\_new}\) is the new position of the \(j\)th material drop path at the \(k\)th drop point; \(X_{k}^{*}\) is the position of the optimal path at the \(k\)th drop point; and \(t\) is the current iteration number. \(D_{k}\) is the distance between the \(k\)th whale individual and the optimal whale individual in the \(k\)th dimension, representing the difference between the two material placement sites. \(A_{1}\) is the coefficient that controls the magnitude of the path update, and \(C_{1}\) enhances the randomness of the path search. \(C_{1}\) is determined by the random number \(r_{2}\) and calculated by Equation (4):  
\[\left\{ \begin{array}{l}A_{1}=2\alpha\cdot r_{1}-\alpha\\ C_{1}=2r_{2}\end{array} \right. \tag{4}\]  
where both \(r_{1}\) and \(r_{2}\) are taken randomly from among \([0,1]\) and \(\alpha\) is a linear convergence factor, converging linearly from \(2\sim 0\). Step 2: Bubble net attack (Spiral Path Optimization). In this step, the study employs a spiral search strategy to refine and optimize material drop patterns in a path-planning optimization problem. The algorithm alternates between linear and spiral updates to explore the solution space flexibly and find shorter or less costly material drop paths. Its position change rule is as follows:  
\[X_{k}^{j\_new}(t+1)=X_{k}^{*}(t)+D_{k}\cdot\mathrm{e}^{bl}\cdot\mathrm{cos}(2 \pi l) \tag{5}\]\[D_{k}=\left|X_{k}^{*}(t)-X_{k}^{j}(t)\right| \tag{6}\]  
where \(e^{bl}\cos(2\pi l)\) determines the spiral path's shape and direction to simulate the material delivery strategy's spiral adjustment; \(b\) is the spiral shape parameter that influences the whale's position update; and \(l\) is randomized across the range \([-1,1]\). The probability of both the whale's contraction encirclement behavior and the selection of a helical swimming update position was set to allow the whale to contract encirclement while swimming towards its prey, as shown in Equation (7):  
\[X_{k}^{j,\text{new}}\left(t+1\right)= \begin{cases}X_{k}^{*}(t)-A_{1}\cdot D_{k }&p<0.5\\ X_{k}^{*}(t)+D_{k}\cdot\mathrm{e}^{bl}\cdot\cos(2\pi l)&p\geqslant 0.5\end{cases}  \tag{7}\]  
where \(p\) is the probability. Step 3: Search predation (global search path update). In this step, mutual information sharing and position updates help the algorithm to escape local optima and explore a more comprehensive solution space to find the optimal material delivery solution, as calculated by Equation (8):  
\[\left\{ \begin{array}{l}X_{k}^{j,new}\left(t+1\right)=X_{k}^{\text{rand}}-D_{ k}\cdot A_{1}\\ D_{k}=\left|C_{1}\cdot X_{k}^{\text{rand}}-X_{k}^{j}(t)\right|\\ C_{1}=2r_{2}\\ A_{1}=2\alpha\cdot r_{1}-\alpha\end{array} \right. \tag{8}\]  
where \(X_{k}^{\text{rand}}\) is the \(k\)th component of the position vector of a random whale individual in the solution space.