## 3 Model Formulation
### Constraints  
The optimization problem is subject to the following constraints:  
Patrol Assignment Constraint:  
Ensure that in each time interval \(t\), all \(n\) officer units are assigned to \(n\) distinct streets for patrol:  
\[\sum_{i=1}^{m}x_{i}^{t}=n\ \forall t\in\{1,2,...,T\} \tag{5}\]  
Transition Constraint:  
Ensure that each officer unit transitions to a different street for patrol in the next time interval, preventing units from remaining on the same street continuously. This constraint enhances the efficiency and motivation of police patrol work:  
\[\sum_{p=1}^{n}\sum_{t}x_{i\to j}^{t}(p)=\sum_{p=1}^{n}\sum_{t+1}x_{i \to j}^{t}(p)\ i\
eq j \tag{6}\]Starting Street Constraint:  
Ensure that all officer units start their patrol from a designated starting street \(s_{0}\) :  
\[x_{i}^{0}=s_{0} \tag{7}\]  
Ending Street Constraint:  
Ensure that all officer teams return to the designated starting street \(s_{0}\) at the end of the patrol period:  
\[x_{i}^{T+1}=s_{0} \tag{8}\]  
Risk Coverage Constraint:  
Ensure that the total risk coverage across all patrolled streets is at least equal to the average risk value:  
\[\sum_{i=1}^{m}\sum_{t=1}^{T}x_{i}^{t}.t_{i}^{t}\geq\frac{\sum_{i}\sum_{t}R_{i }^{t}}{m\times T}\times n \tag{9}\]  
Street Length Coverage Constraint:  
Ensure that the total length of all patrolled streets is at least equal to the average length of all streets:  
\[\sum_{i=1}^{m}\sum_{t=1}^{T}x_{i}^{t}.t_{i}\geq\frac{\sum_{i}L_{i}}{m}\times n \tag{10}\]  
Transfer Time Constraint:  
Ensure that the average time required for all necessary street transitions is less than or equal to 10 min. This ensures that even in extreme cases (e.g., \(\delta=3\)) each street is patrolled for at least 10 min. According to statistical data from the Amp platform, the average driving distance within a 10 min interval on urban road networks, such as those in Beijing, is approximately 3-5 km. This empirical observation provides a practical basis for setting the transfer time constraint in the police patrol optimization model.  
\[\sum_{p=1}^{n}\sum_{t=1}^{T-1}x_{i\to j}^{t}(p)\cdot d_{ij}\leq n\times(T-1) \times 3000 \tag{11}\]  
Exploration of Diverse Streets  
To ensure that the patrol routes cover a diverse range of streets within the limited patrol time and enhance the exploratory nature of the model, we introduce a binary variable \(z_{i}\). When \(z_{i}=1\), street \(i\) is visited more than once during the whole patrol period, otherwise \(z_{i}=0\).  
\[\frac{\sum_{i}(1-z_{i})}{\sum_{i}\sum_{t}x_{i}^{t}}\geq\theta \tag{12}\]  
where \(\theta\) is a predefined threshold that limits the proportion of streets that can be revisited. This constraint encourages the patrol routes to explore new streets rather than repeatedly visiting the same ones, thereby maximizing the coverage of the street network. The parameter \(\theta\) is set to a default value of 0.5, which signifies that a minimum of 50% of the streets should remain non-repeatedly visited during the patrol cycles, ensuring sufficient coverage and diversity in the patrol routes.  
Coverage of Historically High-Risk Streets  
To ensure that the patrol routes prioritize streets with a history of incidents, we introduce a binary variable \(h_{i}\). When \(h_{i}=1\), street \(i\) has a history of real incidents, otherwise \(h_{i}=0\).  
\[\frac{\sum_{i}\sum_{t}x_{i}^{t}.h_{i}}{\sum_{i}\sum_{t}x_{i}^{t}}\geq\mu \tag{13}\]where \(\mu\) is a predefined threshold that ensures the proportion of historically high-risk streets among all patrolled streets is above a certain level. The parameter \(\mu\) is set to a default value of 0.3. This threshold is determined based on statistical analysis, as out of the total 390 streets, only 129 have historically recorded incidents, accounting for less than 30% of the total street count. This constraint guarantees that the patrol routes focus on areas with a higher likelihood of incidents, aligning with the principles of proactive policing.