## 1 Introduction  
Policate patrol strategies have been extensively studied and empirically demonstrated to significantly reduce crime rates while enhancing public perceptions of security, thereby constituting a critical operational focus for frontline law enforcement agencies [1]. In China, police deployment predominantly relies on the \(110\) call for the police service platform, which allocates the nearest available units based on reported incidents [2]. However, this reactive policing model falls short of meeting the evolving societal demands for security and the heightened expectations for proactive police services. Consequently, transitioning to an active policing paradigm has emerged as an imperative reform, aiming to optimize police resource allocation and enable timely interventions at potential risk sites [3]. A central challenge in this transition lies in determining the optimal spatial distribution of patrol units to ensure rapid response capabilities while minimizing resource inefficiencies. Fundamentally, this challenge represents a quintessential multi-objective optimization problem, balancing competing priorities such as the response time, resource utilization, and crime prevention efficacy.  
Currently, there are still significant challenges in applying multi-objective optimization methods to solve the Police Patrol Optimization Problem (PPOP). As illustrated in Figure 1, PPOP is supposed to determine optimal patrol routes for police units. The solution aims to achieve three key objectives: (1) ensuring a rapid response to incidents by prioritizing the coverage of nearby streets [4; 5], (2) maximizing the probability of police presence on main streets to enhance visibility [6; 7], and (3) minimizing the time spent transitioning between streets to improve the operational efficiency and allocate more time to active patrolling or incident management [8; 9]. These objectives often present inherent trade-offs, necessitating a delicate balance to formulate an effective and practical solution. Moreover, real-world constraints--such as geographical limitations, resource availability, and dynamic conditions like traffic congestion [10; 11]--introduce significant layers of complexity. These factors not only complicate the optimization process, but also amplify the challenge of developing strategies that are both efficient and adaptable to changing environments.  
Traditional multi-objective optimization algorithms often underperform in handling constraint conditions, failing to fully meet the diverse requirements of actual patrol tasks, particularly in dynamically changing environments where their adaptability is limited. Additionally, many existing methods exhibit deficiencies in global search efficiency, leading to the suboptimal convergence and robustness of solutions. These shortcomings become especially evident when tackling large-scale optimization problems, defined here as scenarios involving hundreds of patrol streets, and high-dimensional optimization problems, characterized by three or more conflicting objectives and constraints.  
To address these issues, this paper proposes a method based on the PPS-MOEA/D (Push and Pull Search + Multi-Objective Evolutionary Algorithm based on Decomposition). This approach leverages the efficient convergence of the PPS algorithm [12] and the decomposition strategy advantages of MOEA/D [13], enabling the effective balancing of multi-objective optimization under complex constraint conditions. Simulation validation was carried out using real-world data from a subdistrict in Chaoyang, Beijing.