## 6 Simulation Results  
### Comparative Analysis  
To evaluate the performance of various algorithms under different scenarios, simulations were conducted by introducing two key parameters defined in the model: the number of patrol units, \(n=\{2,4,6,8,10\}\), and the time parameter, \(\delta=\{1,2,3\}\).  
This resulted in a total of 15 combinatorial scenarios. The final simulation results are presented in Figures 6-8, which provide a detailed comparison of the algorithms based on the performance metrics. For a more detailed presentation of the simulation results, please refer to Appendices B.1-B.3.  
From the results presented in Figure 6, it is evident that the modified PPS-MOEA/D algorithm outperformed all other comparative algorithms in solving the proposed police patrol problem. Specifically, in all 15 scenarios, the modified PPS-MOEA/D achieved superior Hypervolume (HV) values, demonstrating its exceptional capability in terms of convergence and diversity for this specific optimization task.  
The algorithms ANSGA-III and NSGA-II ranked second and third, respectively, based on their HV performance. This indicates that these NSGA-based algorithms consistently exhibit a strong performance in practical optimization problems, reaffirming their robustness and effectiveness.  
The dominance of the modified PPS-MOEA/D highlights its adaptability and efficiency in addressing the unique challenges of the police patrol problem, while the consistent performance of ANSGA-III and NSGA-II underscores the reliability of NSGA-based approaches in multi-objective optimization.  
Figure 6: Comparison of HVs from simulation results of various algorithms.  
Building on the comparison of the Hypervolume (HV) metric, Figure 7 provides insights into the Spacing metric, which evaluates the distribution uniformity of solutions in the Pareto front. Meanwhile, the modified PPS-MOEA/D algorithm outperforms the other algorithms across most scenarios.  
Furthermore, although the CCMO algorithm demonstrates a slightly better Spacing performance compared to the modified PPS-MOEA/D, its performance is highly unstable, as observed in Figure 7, which illustrates this inconsistency. The CTAEA algorithm ranks third, as stable, demonstrating a moderate performance in terms of solution distribution uniformity.  
These observations underscore the importance of critically interpreting performance metrics and highlight the modified PPS-MOEA/D as a robust and reliable algorithm for achieving a well-distributed Pareto front in the PPOP.  
As illustrated in Figure 8, most algorithms, excluding MOEA/D-DAE, are capable of generating a significant number of feasible solutions. Among these, the PPS-MOEA/D,  
Figure 8: Comparison of feasible solutions from simulation results of various algorithms.  
Figure 7: Comparison of Spacings from simulation results of various algorithms.  
NSGA-II, and ANSGA-III algorithms exhibit the most stable and consistent performance, further reinforcing their efficacy in addressing the proposed police patrol problem.  
In summary, the PPS-MOEA/D algorithm demonstrates the best overall performance in solving the police patrol optimization problem, as evidenced by its superior results across multiple evaluation metrics.