## 3 Driver-Inspection Vehicle Automatic Scheduling Method
### Dynamic Automatic Scheduling Algorithm Based on Multi-Objective Optimization
The computational complexity of the dynamic automatic scheduling algorithm based on multi-objective optimization in this study is illustrated through two key performance metrics: the algorithm's runtime change curve and its memory usage change curve as the dataset size varies. Runtime curve: This curve represents the change in the algorithm's runtime across different dataset scales (input size). The horizontal axis represents the scale of input data (i.e., number of vehicles waiting for inspection), while the vertical axis indicates the algorithm's runtime performance scales with an increasing problem size, highlighting its efficiency in handling larger datasets. Memory usage curve: This curve illustrates the change in the algorithm's memory consumption as the dataset size varies. The horizontal axis represents input data scale, while the vertical axis denotes memory usage. The memory usage curve provides insights into the algorithm's space complexity, allowing for a comparison of resource requirements between different algorithms, and the memory usage curve is shown in Figure 5.

In terms of time complexity, the algorithm's runtime was tested on different dataset scales (e.g., 100, 500, 1000, 5000 vehicle data). The results indicate that the proposed algorithm exhibits low computational time for small-scale datasets (100 to 500 vehicles), with an average time complexity of approximately \(O(n\log n)\). As the dataset scale expands to 1000 vehicles or more, the algorithm's runtime exhibits polynomial growth, reflecting its ability to handle larger problem sizes effectively. In contrast, the time complexity of benchmark algorithms such as NSGA-II and MOEA/D is higher, demonstrating an \(O\big{(}n^{2}\big{)}\) growth trend. The proposed algorithm converges faster on large datasets, attributed to its efficient Pareto front solution update and population management strategy. Specifically, on a dataset with 5000 vehicles, the runtime of the proposed algorithm is approximately 30% lower than that of NSGA-II and about 25% lower than MOEA/D. In terms of space complexity, the experiment measured the algorithm's memory usage under various dataset scales. The results indicate that the proposed algorithm's space complexity is \(O(n)\), exhibiting a nearly linear growth trend, which is related to its efficient storage management mechanism. In contrast, NSGA-II and MOEA/D show a significantly increased memory usage when handling large-scale data, demonstrating an \(O\big{(}n^{2}\big{)}\) space complexity. This indicates that the proposed algorithm can better control memory usage as the data scale increases, making it particularly suitable for environments with limited computing resources. These findings underscore the advantages of the proposed multi-objective optimization algorithm in both time and space efficiency, enhancing its applicability in real-time scheduling scenarios involving customs inspection vehicles. To further present the experimental results of the automatic scheduling method for customs inspection vehicles based on Vehicle Electronic Identification and biometric recognition, this study focuses particularly on the performance of various indicators and analyzes their effectiveness in multi-objective optimization. The same test dataset is used to evaluate the performance of the proposed method and the comparative algorithms. Figure 5: (**a**) Runtime curve. (**b**) Memory usage curve. Table 2 shows the experimental results of each algorithm on eight performance indicators, with all results representing the average values and standard deviations from 20 experiments conducted along the path from Parking Lot 6 to Inspection Laboratory 1. In the vehicle waiting time objective function, the proposed method significantly outperforms the comparative algorithms, achieving an average waiting time of 14.30 min, which is the shortest among the three algorithms.