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Skip to main content. Turn off Animations. Turn on Animations. Police Patrol. It looks like your browser does not have JavaScript enabled. Please turn on JavaScript and try again. Police Law Enforcement Police Patrol. Sites Covered. Shots Fired Assembly Video. Detrick: RML: 0. Therefore, a districting plan can be described and represented by a set of districting parameters.
Once a districting plan is generated, some measurements can be quickly calculated without detailed simulation evaluation, such as compactness of plans and the variation of CFS probability of all districts. These intermediate measurements of districting plans can be used to select top proportion of plans for further simulation evaluation. Without prior knowledge about how districting parameters affect final response variables, we randomize these parameters to generate some plans, quickly calculate the intermediate measurements, and take some time to get final responses through simulation evaluation.
Then we build statistical models to study the relation between them, especially how districting parameters and intermediate measurements affect the final performance variables. The districting parameters can be adjusted to generate more plans that may have better performances.
Since it is time intensive to use simulation to evaluate these plans, they can be ranked by the combined weighted score of the intermediate metrics. The weights are adjusted based on the relation between intermediate variables and final performance variables.
Because the assessment of response times and workloads requires the incorporation of multiple factors that interact in complex ways we cannot use closed form expressions.
Also, field experiments in the law enforcement and safety management are clearly not feasible because of the risks and costs, not to mention, the public relations problems [ 14 ].
This means that evaluation of the police patrol districting plans requires a high fidelity simulation. A feature key needed in this simulation is the ability to accurately represent behaviors of the police in response to calls-for-service.
Agent-based simulations afford the ability to effectively represent these behaviors. Agent-based simulations capture of the behaviors of objects in an environment, such as police patrols in city, through the use of decision rules.
These decision rules govern the interactions between objects in the simulation. For example, when a police car object interacts with a road object the rules specify the rate of transition to the next road object. These rules can also represent static properties of the object, for example the speed limit, and the dynamic properties of the environment, such as weather, construction, and traffic conditions.
Other example rules used in our simulation include:. If the nearest available police car is in a different district it will cross the district boundary and respond to the CFS.
The interactions between multiple objects governed by the rule sets in the simulation produce emergent behaviors or properties that cannot be predicted before running the simulations.
For our purposes the most important emergent properties are the response times to CFS and police workloads. These properties are the metrics that allow us to score the effectiveness of different patrol districting plans. Neither of these properties can be accurately anticipated a priori using only a districting plan and the numbers of CFS within the districts. Although graph-partitioning is NP-hard; nonetheless, there are available heuristics than can be applied.
In fact, when we use our high fidelity simulation to evaluate police districting plans that minimize the difference in CFS between districts they actually do worse than some other plans.
The same is true for workload. The ability to discover these emerging properties is an important feature of the agent-based simulation we built and a critical requirement in the assessment of competing districting plans. We implemented the simulation using Java Repast.
Java Repast is an open source, agent-based modeling and simulation platform [ 15 ]. It uses object-oriented model and has a source library of classes for creating and running agent-based simulations and for displaying and collecting data from these simulations.
Geographic data, such as the data expressed in shapefiles, can be imported into the Java Repast model. Using these geographic data the behaviors and movements of the agents can be controlled according to rule sets that exploit these data. For example, the shapefile data layer on secondary roads can have attributes that provide speed limits in different segments of this layer. The inputs to our simulation model consist of the shapefiles of the city, the patrol district plan, the police patrol allocation plan and a data set of CFS times, locations, and severity this last attribute determines the distribution of the service time for the CFS.
The shapefiles for the city include primary and secondary roads, major highways, and obstacles or impediments e. In order to simulate the time and spatial pattern of actual CFS and maintain the randomness of the city environment, the time between incidents and the locations are randomly chosen based on the distribution of actual CFS. Rather than use a bootstrap approach which would resample from actual CFS, we instead use an empirical fit of the distributions of the CFS in space and time.
We then chose CFS values based on random draws according to these distributions. Police cars are on patrol in their districts until they are dispatched by a CFS. Their patrol routes are randomly chosen from the network of roads in their assigned districts. An incoming CFS will generate the dispatch of the nearest police car and that car will follow the shortest route to the location of the CFS.
In following this route it will use the maximum safe speed for the route which is greater than or equal to the speed limit on the route. After the police car reaches the CFS location, the police car will remain at that location for the service time of that CFS.
We obtain this service time as random draw from the service time distribution for the type and severity of the particular CFS. The service time distributions are empirical distributions found from the data set of actual CFS. When the service time ends the police car returns to its patrol route and again becomes available for dispatch to the next CFS.
We run the simulation for chosen number of runs or for a selected amount of simulation time. For each CFS in a run we record. The time of departure from the CFS by the responding car i. Using these data we can calculate the average response time and workload for each run and, hence, for each districting plan. To illustrate the use of our approach to police patrol districting we used data from the Charlottesville, VA, USA police department.
Charlottesville is a city with a diameter of about 7 miles and a population of about 40, However, this population increases during most of the year by another 26, due the presence of a major university. The population lives in multi-dwelling buildings, as well as, detached townhouses, apartments, and homes.
There are more densely populated buildings near the university and the downtown. There are also commercial areas and some light industrial parks. The current police patrol district of Charlottesville was designed more than 20 years ago. There are 8 patrol districts and in most of cases, one police car is assigned to patrol each district. The police managers and commanders want to draw district boundaries to incorporate census block groups. There are 37 block groups in Charlottesville. To create more useful atomic geographic units we decomposed the city into grids.
Figure 3 shows the locations of historical CFS incidents for several years, including , events. Many incidents happened at same places so each red point may represent many CFS events. To have a better view of the CFS density distribution in the city region, these historical incidents were spatially projected into the grid network.
The CFS distribution across these grids is shown in Figure 4. Due to size of the city and number of districts, two circles are used to generate the seeds. For large cities, seeds can be located on several circles, depending on the size of city and the number police cars.
For cities with general shape of rectangular, ellipses can be used to generate seeds instead of circles. Then, these randomly generated plans were ranked by the weighted sum of two normalized intermediate measurements: standard deviation of CFS probabilities among districts and compactness score.
Without prior information about the relevant importance of the two intermediate measurements of districting plans, they were considered equally important with the weights [0. Then, the top proportion of the ranked plans was evaluated using simulation model and final performance measurements were obtained. Due to the randomness of the districting algorithm, some combinations of parameter settings cannot generate compact districting plans.
Furthermore, we cannot use simulation to evaluate too many districting plans due to the evaluation complexity of the problem, especially at the beginning phase of the experiment. More plans can be evaluated if more computational resources can be used. Based on result from statistical analysis, the center of the concentric circles, the radius of the outer circle are significant to average response time.
For workload variation, the significant districting parameters are number of seeds on each circle, the center of the concentric circles and the number of iterations of balancing the CFS probabilities of districts.
For the intermediate measurements, the variation of CFS probabilities is more important for both response variables. Its linear relationship to both responses can be seen in Figure 5. It can be seen that lower variation of cumulative CFS probabilities among districts leads to better performances for both responses.
So, in the next iteration of ranking randomly generated plans, more weight can be given to standard deviation of CFS probabilities.
Then, the relationship between responses and each districting parameter was analyzed individually and they were adjusted in the steepest decent direction to responses and another batch of thousands of plans was generated. The weights for two intermediate measurements became [0.
For each of the selected districting plans we ran the simulation for ticks simulation time units and 50 minutes of actual time. Under parallel running in current computing resources 13 batches on 3 PCs , the actual evaluation time for a districting plan can be reduced to 5 minutes. This length of time ensured the convergence of the simulation to a steady state, which can be seen in Figure 6.
While we put more effort into analysis of the performance metrics, average response time and workload, we also wanted to visualize the simulation. This allowed us and our police colleagues to visually assess and validate the behavior of the simulation. We showed the simulation to members of the police and they confirmed its behavior was consistent with that of their patrols.
A static view of the simulation is shown in Figure 7. The current districting plan was also evaluated by the simulation system. The simulation results show both performance measurements can be improved. The average response time of current districting plan was The standard deviation of workload proportion among 8 districts was reduced from 0. Due to the NP-completeness of the graph-partition problem, there are too many possibilities of districting plans. We cannot use exact method to evaluate each of them.
The evaluated districting plans in this case study are only a small proportion of the whole solution set and the solutions provided are preliminary. The global optimality cannot be guaranteed.
We only find some significant districting parameters and intermediate measurements that may lead to better plans. More rigorous experimental design and statistical analysis can be conducted to further study the relationship between these factors and responses.
With more powerful computational resources, more districting plans can be further generated and evaluated. It is possible to make improvements on both response variables. Because we have a multiple objective problem i.
Further no single plan was best in both average response time and workload variation. To provide a multiple objective solution we used Pareto analysis. This analysis shows the positioning of each of the alternative districting plans with respect to each other on the two dimensional plot of both metrics.
Using this plot we can trace the Pareto frontier which is the set of non-dominated districting plans. These plans are not dominated because no other plan is better than them in at least one of the performance metrics. Figure 8 shows the Pareto frontier of average response time and workload standard deviation for the Charlottesville case study. This figure shows that 2 out of districting plans are on the Pareto frontier.
They are No. Under any weighting of response time and workload variation, one of the two districting plans dominates the others. So, they are the best 2 districting plans.
The police department can choose one of them based on their needs and some practical considerations. The actual physical compositions of districting plan No. Clearly, the boundaries based on the grid network violate these boundaries. However, some existing geographical units such as police beats and census block groups consider these boundaries.
So, replacing the grid boundaries needs to consulting with police departments. If patrol boundaries must be drawn based on police beats or census blocks, conversion can be made between grid network and these units.
Example can be seen in Figure If not necessary, the grid boundaries can be replaced by the nearest roads. In this way, the performance of the districting plan may be close the optimal solution based on grid network. In this paper, we reviewed the characteristics of the police patrol district design problem from the perspective of past and current work.
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