## 2 GTSP Solution for UAV Material Drop Paths
### Improved Whale Optimization Algorithm (IWOA) for Solving GTSP
#### 2.2.2 Improvements to the Whale Position Update Mechanism  
Equation (8) demonstrates that the WOA utilizes the current position of a randomly selected individual or an optimally positioned individual as the navigation target to maintain population diversity while updating an individual whale's position. This is achieved by gradually updating the position through a linear decrease of the convergence factor as the number of iterations increases, leading the whales to search for encirclement during the feeding process.  
The advantage of adopting this type of position updating method is that the whales gradually narrow the searched solution space while searching for food, which efficiently improves the local retrieval ability of the WOA. However, the disadvantages lie in the fact that the group is too centralized, the value of \(A_{1}\) is unstable, and the lack of the navigation coordinates of the good individuals makes it easy for the whales to deviate from the optimal direction of foraging, which affect the convergence speed and reduce the global proficiency of the algorithm.  
This work proposes a revised spiral position updating algorithm. It sorts the fitness of individual whales in ascending order, yielding the following results:  
\[X^{1},\cdots,X^{i-1},X^{i},\cdots,X^{popSize} \tag{9}\]  
Second, whale individuals were categorized into three classes based on fitness: optimal, intermediate, and deviant individuals, and the number of individuals in each class was \(0.05\times popSize\), \(0.9\times popSize\), and \(0.05\times popSize\). Individuals in each of the three classes were denoted as \(X^{p,best}\), \(X^{m}\) and \(X^{p,worst}\).  
The modified position update formula is as follows:  
\[X_{k}^{j+1}(t+1)=X_{k}^{p\_best}(t)+D_{k}\cdot\mathrm{e}^{bl}\cdot\cos(2\pi l) \tag{10}\]  
\[D_{k}=X_{k}^{p\_best}(t)-X_{k}^{p\_worst}(t)+\left|X_{k}^{m}(t)-X_{k}^{j}(t)\right| \tag{11}\]  
where \(X_{k}^{p\_best}(t)\) is the position of the optimal individual in the \(k\)th dimension in the current iteration and \(X_{k}^{p\_worst}(t)\) is the position of the worst individual in the \(k\)th dimension in the current iteration. \(X_{k}^{m}(t)\)is the position in the \(k\)th dimension of an intermediate-mass individual randomly selected in the current iteration. \(X_{k}^{j}(t)\) is the position of the \(j\)th whale individual examined in the current iteration in the kth dimension. \(D_{k}\) is the final distance metric, reflecting the difference in relative position between the current individual and the optimal-, worst-, and intermediate-quality individuals.  
In the standard WOA, the spiral shape parameter \(b\) is usually set as a constant, leading to a lack of diversity in the whales' postures during search and prey, often resulting in overly simplistic position update methods and making the algorithm prone to premature convergence. To improve the fixed posture search during the predation process, the enhanced WOA selects the parameter \(b\) as a dynamically changing sine function value, which varies with the number of iterations, allowing the whale individuals to adjust their spiral postures with the global iteration changes while searching and preying. The sine function value \(b\) effectively prevents the WOA from falling into local convergence and improves the global search capability of the algorithm, enhancing the convergence precision when seeking the optimal solution to the GTSP.  
The improved parameter \(b\) is shown as follows:  
\[b=-\lambda\sin(\omega\tau\cdot\sqrt{\frac{t_{\max}-t}{t_{\max}}}) \tag{12}\]  
where \(\lambda\) is the spiral update coefficient, setting \(\lambda=10\), \(\omega\) is the attitude influence constant, \(\omega=0.5\), and \(t_{\max}\) is the maximum number of iterations.