## 3 Model Formulation
### Objective Function  
The PPOP primarily involves three key objectives.  
Objective 1: Maximization of Risk Coverage  
The first objective is to maximize the total risk coverage, defined as the sum of risk values across all patrolled streets during the day. For instance, if two officer units are assigned to complete patrol tasks over 16 h, the goal is to maximize the sum of risk values across the \(2\times 16=32\) patrolled streets. This objective ensures that limited police resources are allocated to the streets with the highest risk, thereby achieving optimal risk coverage. This approach aligns with the principles of hotspot policing, which emphasizes focusing resources on high-risk areas to enhance public safety [45; 46; 47].  
\[\textit{Maximize objective}1=\sum_{i=1}^{m}\sum_{t=1}^{T}x_{i}^{t}r_{i}^{t} \tag{1}\]  
Objective 2: Maximization of Total Patrolled Street Length  
The second objective is to maximize the total length of all patrolled streets during the day. For example, if two officer units are assigned to complete patrol tasks over 16 h, the goal is to maximize the sum of the lengths of the \(2\times 16=32\) patrolled streets. This objective aims to achieve the highest possible police visibility by ensuring that officerscover the greatest cumulative distance across the street network. Maximizing the total patrolled length enhances the perceived presence of law enforcement, which can deter potential criminal activities and improve public safety [48; 49].  
\[Maximize\ objective2=\sum_{i=1}^{m}\sum_{t=1}^{T}x_{i}^{t}\cdot l_{i} \tag{2}\]  
Objective 3: Minimization of Transfer Time Between Streets  
The third objective is to minimize the transfer time between streets during patrols, thereby reducing unnecessary inefficiencies in resource allocation. Intuitively, when an officer unit transitions from the current street to the next, the nearest street (i.e., the one requiring the least travel time) should be selected from the available options. This minimizes the time wasted on necessary transitions, optimizing the utilization of police resources. Let \(x_{i\to j}^{t}(p)\) denote the decision variable, indicating that officer unit \(p\) transitions from street \(i\) to street \(j\) at the end of time interval \(t\), with this process continuing until the final transition is completed at time interval \(T-1\).  
\[Minimize\ objective3=\sum_{p=1}^{n}\sum_{t=1}^{T-1}x_{i\to j}^{t}(p) \cdot d_{ij} \tag{3}\]  
Multi-Objective Optimization  
The three objectives described above may exhibit certain trade-offs or conflicts in specific scenarios. However, for police patrol operations, it is desirable to simultaneously satisfy all three objectives. To facilitate the optimization process, the first two objectives (the maximization of risk coverage and maximization of the total patrolled street length) can be transformed into minimization problems by taking their reciprocals, while the third objective remains unchanged.  
\[\begin{cases}f_{1}(x)=1/objective1\\ f_{2}(x)=1/objective2\\ f_{3}(x)=objective3\end{cases}\]  
Therefore, the three-objective optimization problem can be formulated as:  
\[Minimize\ F(x)=(f_{1}(x),f_{2}(x),f_{3}(x))^{T} \tag{4}\]