## 4 Methodology
### Genetic Algorithm  
In this section, the GA is presented based on the specialty of the recovery scheduling problem. The GA begins by creating an original population based on the encoding method. Then the elite mechanism, the crossover operation, and the mutation operation are carried out to generate a new population. The execution process is terminated when reaching a maximum generation number \(N_{\text{g}}\) or achieving the optimal solution. Herein, the optimal solution in the last generation is the expected recovery sequence. The flow chart of the GA is demonstrated in Figure 3.  
The core components of the GA that is employed in the recovery scheduling problem are explained in detail as follows:  
1. Encoding. To accomplish the genetic operation, an effective representation of the chromosomes must be proposed. As discussed in the second section, each chromosome can be directly encoded by a recovery sequence \(S\). The encoding method is, to a certain extent, similar to the path representation in the traveling salesman problem.
2. Initialization. In the GA, the searching space, which is called the population, is the collection of \(N_{\text{p}}\) recovery sequences. Utilizing the encoding method, the recovery sequences in the original population are randomly created to guarantee the diversity.
3. Objective evaluation. During the aerial recovery process, the UAVs need to keep a safe distance between each other to avoid collisions. The fitness value of each recovery sequence can be obtained from Equation (10) if the corresponding trajectories will not cause collisions. Conversely, the recovery sequences that will cause collisions are penalized with the \"death penalty\", i.e., they are given a fitness value of \(-1\). Thus, these invalid recovery sequences are eliminated during the evolutionary process.
4. Elitism. In each generation, the best \(N_{\text{e}}\) recovery sequences form the elite group, and the elite group is copied to the next generation without any other genetic operations. By applying the elite mechanism, the good recovery sequences are reserved during the evolutionary process so that the optimal recovery sequence can be obtained faster.
5. Crossover. To perform the crossover operation, two recovery sequences are first selected based on the roulette wheel operator. The order crossover operator (OX1) is used to carry out the crossover operation due to its convenience [45] and the two recovery sequences are applied with the crossover operation, with a relatively high probability \(P_{\text{c}}\). Figure 5 shows an example of the crossover operation. During the crossover operation, a sub-sequence is selected from one of the recovery sequences and it is copied to the same location of the offspring. Then, the other recovery serial numbers are filled with the same order as they appear in another recovery sequence starting from the second crossover point. The other offspring is created through the same process.  
Figure 3: Flow chart of the genetic algorithm.  
Mutation. The exchange mutation operator (EM) is selected to perform the mutation operation and each recovery serial number is affected by the mutation operator with a low probability \(P_{\text{m}}\). As demonstrated in Figure 4, the mutated recovery serial number swaps its position with another randomly selected recovery serial number in the same recovery sequence.