## 3 Model Formulation and Scheme Design
### Path Optimization
#### 3.3.2 Description of the TGA Algorithm
In order to avoid premature convergence of the algorithm during the calculation process, the result of the non-global optimal solution appears. The solution generated by each iteration is broken and then repaired. Specifically, firstly a DCP is randomly removed from the current solution \(\sigma\) to set \(S\), as the first element of set \(S\), \(p\) represents the number of stay points removed, and there are still \(p-1\) elements left. Then, a DCP \(z\) is randomly selected from the set \(S\) each time, and the remaining DCPs\(z\) in \(\sigma\) are arranged in ascending order of correlation. The DCP\(c\) is selected from \(\sigma\) that has the greatest correlation with \(z\), \(c\) is removed from \(\sigma\), and added to \(S\). This process is repeated \(p-1\) times until the remaining \(p-1\) elements are selected. \(c_{ij}\) is the Euclidean distance between \(i\) and \(j\). The correlation formula is:  
\[R(i,j)=\frac{\max c_{ij}}{c_{ij}} \tag{11}\]  
In actual situations, there is no perfect correlation function. If over-reliance on the correlation function selects the DCP to be moved, it may be limited to local correlation. To avoid this situation, a random element \(\gamma\) needs to be added to the algorithm. \(\sigma\) is the number of DCPs remaining in \(\sigma\). \[\gamma=(Random([0,1]))^{D}\times\sigma\cdot \tag{12}\]  
The remaining DCPs in \(\sigma\) are arranged in descending order of correlation with \(z\), and the sorted sequence is \(l_{st}\), so the result of the above process is \(l_{st}[index]\). \(D\) is a random number not less than 1. From that we can conclude, when \(D\) is 1, the DCP that is removed is completely selected at random; when \(D\) is equal to positive infinity, it is close to selecting the DCP with the greatest correlation. In other words, the higher the value of \(D\) is, the more favorable it is to the DCP with high correlation. Finally, the damaged solution is repaired, and the elements in the removed DCPs set \(S\) are reinserted back into the partial solution \(\sigma^{\prime}\) to produce a better solution. First, the optimal insertion position of each DCP in \(S\) is calculated, and the DCP \(c\) in \(S\) is inserted back into the partial solution \(\sigma^{\prime}\). In the process of selecting the DCP \(c\) from \(S\) and inserting it back to \(\sigma^{\prime}\), the increment of the objective function value is calculated where each DCP in \(S\) is inserted into its optimal insertion position, and the DCP with the largest increment is chosen as the first insertion point. The method is repeated until all the DCPs are reinserted into the partial solution. The steps of the TGA algorithm are described as shown in Algorithm 1. ```
0: Maximum number of iterations, all cluster head nodes \((x_{i},y_{j})\), UAV take-off point \((x_{0},y_{0})\), population size, crossover probability \(P_{c}\), probability of variation \(P_{m}\) Output: The optimal transportation route. 1. Perform TLBO algorithm to randomly initialize population
2. Evaluate population \(m_{mean}\)
3. While (number of iterations)
4. for all port do
5. TLBO algorithm produce \(m_{j}^{\prime}\)
6. end for
7. end while
8. Execute the GA algorithm, using solution \(m_{j}\) generated by TLBO as the initial solution of the improved GA algorithm
9. While (number of iterations)
10. for each individual do
11. Calculate the objective function for each individual based on (4)
12. Crossing and mutation to produce a new generation of population
13. New-generation populations perform local search operations with the concept of destruction and repair
14. Delete duplicate individuals from the population and complete the deleted individuals
15. end for
16. Update parameter \(R(i,j)\), \(\gamma\) according to Equations (11) and (12)
17. end while
```  
**Algorithm 1** Implementation of the TGA algorithm