## 2 Related Work
### Multi-Objective Optimization Approaches  
Compared to the single-objective optimization of police patrols, multi-objective optimization addresses the need to balance multiple conflicting criteria simultaneously, providing more comprehensive solutions to the PPOP. For instance, [PERSON] et al. employed Integer Programming (IP) to balance street coverage and minimize police deployment, introducing a decomposition approach to handle the NP-hard nature of the problem [25]. Subsequently, [PERSON] et al. extended their approach to a two-stage CSP framework, incorporating both offline and real-time patrol models to maximize incident service rates [26], later adapting it for traffic security patrols under frequency constraints [27]. [PERSON] et al. employed reinforcement learning (RL) to tackle dynamic patrol scheduling, focusing on two optimization objectives: maximizing the success rate of the incident response and minimizing patrol presence reduction [28]. They later extended this model to multi-agent cooperative scenarios, leveraging RL's strength in achieving global optimization [29]. [PERSON] et al. proposed a genetic algorithm-based method to optimize street patrol routes, aiming to minimize the response time and reduce the number of patrol inspectors [30].  
Despite their advancements, these multi-objective optimization methods exhibit significant limitations. [PERSON] et al.'s [25; 26; 27] approaches, while theoretically robust, facechallenges with scalability and real-time applicability. [PERSON] et al.'s [28,29] RL-based method, though effective in dual-objective optimization, essentially reduces the problem to a single-objective formulation through its reward function, limiting its ability to fully capture the trade-offs between competing goals. [PERSON] et al.'s [30] genetic algorithm assigns fixed weights to objectives, which fails to qualify as a true multi-objective optimization approach, and its reliance on static crime data constrains its effectiveness in dynamic environments. More broadly, existing methods struggle to achieve genuine constrained multi-objective optimization in intricate environments. They are often limited by scalability issues, the inability to balance competing objectives adequately, and reliance on static or simplistic models, which hinder their applicability in dynamic and multi-patrol scenarios. These limitations underscore the need for more efficient algorithms capable of addressing the complexities of patrol pathing and resource allocation in real-world settings.  
In recent years, Multi-Objective Evolutionary Algorithms (MOEAs) have demonstrated a remarkable performance in solving multi-objective optimization problems [31]. Numerous notable algorithms have emerged, such as NSGA-II [32], NSGA-III [33], SPEA2 [34], MOEA/D [35], IBEA [36], and HypE [37], which have significantly advanced the field. Notably, Constrained Multi-Objective Evolutionary Algorithms (CMOEAs) have effectively addressed optimization challenges by incorporating mechanisms specifically designed to handle constraints efficiently [38], including innovative algorithms like I-DBEA [39], ShiP [40], and BiCo [41]. Furthermore, the Push-Pull Search (PPS) algorithm offers distinct advantages, such as an enhanced convergence speed and adaptability to complex search landscapes, making it particularly suitable for dynamic and intricate optimization scenarios [42,43].  
Leveraging these advancements, this paper proposes the integration of the PPS-MOEA/D algorithm to tackle the complex and dynamic challenges inherent in PPOP, including resource allocation, route planning, and constraint satisfaction. This approach aims to provide a more robust and efficient solution to the multifaceted problems encountered in real-world patrol operations, ultimately improving the effectiveness and adaptability of patrol strategies.