## 3 Materials  
### Operations of the GA  
The GA is an optimization technique inspired by the principles of natural selection and genetics. It seeks approximate or optimal solutions by simulating evolutionary processes such as selection, crossover, and mutation. GAs offer several key advantages. First, they have strong global optimization capabilities, allowing them to avoid local optima and explore a broader solution space. Second, GAs are highly adaptable, capable of solving complex, nonlinear, and multi-modal optimization problems without the need for extensive prior knowledge. They are also well-suited for parallel processing, enabling the simultaneous exploration of multiple solutions. Additionally, GAs are flexible and can be combined with other optimization techniques to address more complex challenges. These attributes make GAs highly effective for a wide range of optimization tasks.  
The operations of the GA primarily include the following steps [21]. (1) Initialization: A set of possible solutions is randomly generated as the initial population. Each individual (solution) is typically represented by a code, such as a binary string. (2) Selection: Individuals are selected based on their fitness. The goal of selection is to retain the best genes and improve the search efficiency of the algorithm. (3) Crossover: One or more offspring are generated from two parent individuals, typically by exchanging parts of their genes. This operation simulates biological reproduction and helps explore the solution space. (4) Mutation: A small, random change is made to an individual's genes, such as flipping certain bits in a binary encoding. Mutation simulates genetic mutations in nature and helps explore new areas of the solution space. (5) Fitness Evaluation: Each individual's fitness value is calculated based on the objective function. The fitness value reflects how well the individual solves the problem, often corresponding to the objective function value in optimization problems. (6) Replacement: The newly generated offspring are replaced by individuals in the current population. Replacement strategies affect the population's diversity and the algorithm's convergence speed. (7) Termination Condition: The algorithm terminates when certain conditions are met, such as reaching the maximum number of iterations, finding a solution that satisfies the desired accuracy, or when the population's fitness changes very little.  
GA uses selection, crossover, and mutation to filter individuals, retaining those with high fitness values to form new generations. The new generation inherits information from the previous generation but is improved. As the process continues, the fitness of individuals in the population increases until a stopping condition is satisfied. The fitness function serves as the basis for selection in GA, and its formulation directly influences the convergence speed and the quality of the solution.  
The selection, crossover, and mutation processes for the MATC system thrust allocation using the GA are as follows:(1) Selection: A random sampling selection operator is used to select individuals with higher fitness values from a randomly generated thrust allocation population, forming a new thrust allocation population with a higher probability.  
(2) Crossover: A single-point crossover operator is used. A crossover point is randomly selected from the binary-encoded thrust allocation population obtained through the selection process, and the binary encodings are exchanged to form the next generation of thrust allocation individuals.  
(3) Mutation: A bit-flip mutation operator is used. Random mutation positions are selected from the binary-encoded thrust allocation population obtained after crossover, and the selected bit is flipped to generate the next generation of thrust allocation individuals.