## 3 Model Formulation and Scheme Design
### Path Optimization
#### 3.3.2 Description of the TGA Algorithm  
The genetic algorithm (GA) is a heuristic search algorithm based on the process of biological evolution, which generates the next generation of solutions through selection, crossover, and mutation. In this process, the solution with a low fitness function value is gradually eliminated, and the solution with a high fitness function value is increased, so as to approach the optimal solution of the problem gradually through evolution. The algorithm has the advantages of being simple and general, strong robustness, and suitable for parallel processing. However, the local search ability of the genetic algorithm is poor, and the initial solution is uncertain. At this stage, the algorithm is used for blind search and makes the convergence rate slower. Similar to the GA algorithm, the Teaching-Learning-Based Optimization algorithm (TLBO) is also a population-based heuristic stochastic optimization algorithm, which is inspired by the classroom teaching process and imitates the teacher's influence on students. On the one hand, the number of hyperparameters in TLBO is lower; thus, the search speed is fast. On the other hand, the optimal solution obtained is not as good as the GA. Furthermore, novel state transition rules are utilized to avoid the local optimal and improve the quality of solution. The TGA (improved genetic algorithm based on TLBO) algorithm designed combines the advantages of the two algorithms, which considers the convergence rate while obtaining a better solution. First, the UAV path planning model is solved by TLBO with fast convergence, and the sub-optimal solution is used as the initial solution of the GA. In the TLBO algorithm, the higher the individual's learning performance, the better the individual. The goal of UAV flight path optimization is to obtain the shortest flight distance, so the fitness function of TLBO is Equation (6), where \(M\) is the number of students. \[f(m)=\frac{1}{L_{m}}\ m\in\{1,2,\ldots,M\} \tag{6}\]  
In the teaching phase, students improve their own level by learning the difference between the teacher and the average level of the class. For the \(j\)th learner \(m_{j}\) in the group, the update mechanism is expressed as follows:  
\[m_{j}^{\prime}=m_{j}+rand\cdot(m_{t}-T_{F}\cdot m_{mean}) \tag{7}\]  
\[m_{mean}=\frac{\sum_{j=1}^{M}m_{j}}{M} \tag{8}\]  
\[T_{F}=round[1+rand(0,1)] \tag{9}\]  
where \(m_{j}^{\prime}\) is the known solution \(m_{j}\); \(m_{t}\) is the current optimal UAV flight path; \(m_{mean}\) is the average state of the population; \(T_{F}\) is a teaching factor, which determines the degree to which the average value is changed, typically 1 or 2, which is an important step in the heuristic; \(rand\) is a random vector, each element of which is a random number in the range of \([0,1]\). In the learning phase, learning occurs by simulating the way students communicate with each other in order to improve their own level. For \(m_{j}\), the updated formula is:  
\[m_{j}^{\prime}=\left\{ \begin{array}{l}m_{j}+rand\cdot(m_{k}-m_{j}),\ if\ f(m_{j})<f(m_{k})\\ m_{j}+rand\cdot(m_{j}-m_{k}),\ otherwise\end{array} \right. \tag{10}\]  
where \(m_{j}^{\prime}\) is the new value of flight path \(m_{j}\), \(m_{k}\) is another randomly chosen flight path from the population that is different from \(m_{j}\). \(f(m_{j})\) and \(f(m_{k})\) are the fitness of paths \(m_{j}\) and \(m_{k}\), respectively. \(rand\) is a random vector, each element of which is a random number in the range of \([0,1]\). Similar to the teaching phase, the value of \(m_{j}\) will adopt the larger one between its original fitness and the new fitness, and then enter the next round of teaching. The generated suboptimal solution is used as the initial solution of the TGA algorithm, when the TLBO reaches the set number of iterations and terminates. A local search update mechanism is added to enhance the local search capability of the TGA algorithm.