## 3 Driver-Inspection Vehicle Automatic Scheduling Method
### Dynamic Automatic Scheduling Algorithm Based on Multi-Objective Optimization
The particle's update depends on the individual best position \(pbest_{i}\) and the global best position \(gbest_{i}\). The position and velocity update methods for each particle are given as follows:  
\[v_{i}(t+1)=\omega v_{i}(t)+c_{1}r_{1}(pbest_{i}-x_{i}(t))+c_{2}r_{2}(gbest_{i} -x_{i}(t)) \tag{19}\]  
\[x_{i}(t+1)=x_{i}(t)+v_{i}(t+1) \tag{20}\]  
where \(\omega\) is the inertia weight controlling the influence of the particle's historical velocity; \(c_{1}\) and \(c_{2}\) are acceleration constants, typically ranging from 1.5 to 2.5, controlling the speed of the particle approaching the individual best position and global best position; and \(r_{1}\) and \(r_{2}\) are random numbers between 0 and 1, used to introduce randomness [41]. In practical applications, multiple objective functions often need to be optimized, which can lead to conflicts among them. PSO can be adapted for multi-objective optimization by employing a non-dominated sorting mechanism to rank particles and select solutions on the Pareto front. The density of particles on this front is evaluated to maintain solution diversity. By integrating PSO with multi-objective optimization techniques, the algorithm effectively preserves a set of non-dominated solutions in each iteration, enhancing the robustness of the search process. Differential Evolution (DE) is utilized in this study to generate new solutions by leveraging the differences between individuals. This method can be effectively extended to multi-objective optimization problems, enabling the resolution of complex issues with multiple conflicting objectives and facilitating the identification of Pareto optimal solution sets to support decision-making. In DE, three different individuals are selected for differential operations,  
\[v_{i}(t+1)=x_{r1}+F(x_{r2}-x_{r3}) \tag{21}\]  
where \(F\) is the scaling factor, typically ranging from 0.5 to 1. After the crossover operation, the new individuals generated are compared with the original individuals, and those exhibiting better fitness are retained for the next generation. The crossover operation is defined as follows:  
\[u_{i}(t+1)=\left\{ \begin{aligned} v_{i}(t+1),r<CR\\ x_{i}(t),otherwise\end{aligned} \right. \tag{22}\]  
where \(CR\) is the crossover probability [42], controlling the mixing degree of the mutant vector and the current individual. By comparing the trial vector's fitness with that of the current individual, the algorithm selects the better-performing individuals for the subsequent generation. The DE algorithm concludes based on predefined termination criteria, which may include reaching a maximum number of iterations or achieving the convergence of the objective function. **Step 4: Multi-objective optimization and pareto selection. **  
Non-dominated sorting and crowding distance metrics are employed to select optimal solutions that form the Pareto front solution set [35]. In multi-objective optimization, each objective function is assigned a specific priority or weights; however, these objectives often conflict with one another. Thus, it is essential to evaluate the performance of all objectives rather than focusing solely on the minimum or maximum value of an individual objective function. To address multi-objective optimization problems, the Pareto selection method is utilized to assess the relative advantages and disadvantages of different schemes. Specifically, the Pareto front comprises a set of solutions that cannot be dominated by any other solution across all objectives. The selection strategy involves the following:  
Non-dominated sorting: For all individuals in the current population, the number of individuals dominated by each one is calculated, and the individuals are ranked based on dominance relationships. The parent population is merged with the new generation population to form a combined population, followed by another round of non-dominated sorting and crowding distance calculations.