## 3 Algorithm  
The bus scheduling problem belongs to the classical mixed-integer nonlinear programming problem, which contains a large number of linear and nonlinear constraints, as well as integer and continuous variables in the model, and is generally regarded as an NP-hard problem in existing studies. Heuristic algorithms are usually used to solve this problem. Common heuristic algorithms used to solve bus scheduling problems include genetic algorithms (GAs), ant colony algorithms (ACAs), simulated annealing algorithms (SAAs), tabu search (TS), and particle swarm optimization (PSO). SAAs avoid falling into the local optimal solution by setting a certain probability to accept the worse solution and realize global search in the search space, which also requires a large number of random perturbations and acceptance and rejection decisions. The GA is based on the basic idea of \"survival of the fittest\", and after cross mutation and other operations, each generation of individuals gradually evolves in the direction of the optimal solution, but it is easy to deviate from the direction of evolution due to early maturity and gradually fall into the local minimum. Therefore, the two algorithms have complementary advantages, so this study combines the ideas of the two algorithms and proposes a new hybrid heuristic algorithm. This new hybrid heuristic algorithm is based on the GA and utilizes the principles of SAAs to perturb and search each individual to some degree to obtain a better solution. The main steps and processes of the algorithm include:  
(1) Chromosome coding  
Bus paths are encoded using integer coding. Each gene position corresponds to each stop on the route, and each chromosome contains the complete path of the vehicle during the corresponding time period. As shown in Figure 4, when the bus operates uni-directionally on the initial route and does not skip any bus stop, the path is represented as \(\overrightarrow{n_{1}}=[1,2,3,\cdots,26,27]\). When the bus skips at one of the stops (e.g., stop 3), which has a gene locus value of 0, the path can be represented as \(\overrightarrow{n_{2}}=[1,2,0,4,5,\cdots,26,27]\). Then, the chromosome code of the initial population can be obtained as \(\left\{[1,2,3,\cdots,26,27],[1,2,0,4,5,\cdots,26,27],\cdots,\overrightarrow{n }\right\}\), where n is the total number of chromosomes in the initial population. (2)  
The population initialization of the GA can be carried out in a variety of ways, and a common practice is to generate a number of random points comparable to the population size. Due to the complexity of the optimization model, the distribution of the initial points in the search space obtained by this random generation method has great uncertainty, which can easily make the algorithm prematurely fall into the local optimum. When the quality of the initial solution is average, it also leads to a greatly extended solution time. Therefore, the distribution of initial values is homogenized using the Tent-based chaotic mapping method. Tent chaotic mapping is a one-dimensional discrete segmented mapping method based on chaos theory, and the function image presents a \"tent\" shape with a relatively uniform distribution function and good correlation. The expression of this method is:  
\[x_{n+1}=f(\Delta x_{n})=\left\{ \begin{matrix}x_{n}/\alpha,x_{n}\in[0,\alpha) \\ (1-x_{n})/(1-\alpha),\ x_{n}\in[\alpha,1]\end{matrix} \right. \tag{19}\]  
Generally, \(0<\alpha<1\), but when \(\alpha<0.5\), the periodic state of the system will converge quickly, leading to unsatisfactory results. Therefore, this study takes \(\alpha\) as \(0.75\). The initialization of the population is carried out after obtaining the chaotic components of each optimization parameter according to the above equation.