Delineations for Police Patrolling on Street Network Segments with p-Median Location Models

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Abstract

It would be beneficial to incorporate specific results in the abstract to provide a more complete overview of the study. Additionally, the keywords should include the geographical scope of the study to enhance relevance and visibility.

Literature Review (State of the Art)

In the literature review, it would be useful to include a summary table that highlights the contributions of the studies analyzed and the aspects addressed in each case. This would provide a clearer and more structured view of the existing research.

Methodology

The software used for GIS analysis is not mentioned and should be included. IBM CPLEX 20.1 is used for modeling, and this should be specified in the methodology section. Additionally, the methodology section would benefit from a diagram that explains each of the developed processes in detail, along with the results obtained.

There are methodological aspects related to the formulations included in the results section that would be better placed in the methodology section. This adjustment would improve the organization and clarity of the paper.

Results and Discussion

The results section should focus more on the outcomes of the analysis. It would be interesting to know the current distribution of the police patrolling units and compare it with the optimized distribution obtained through the modeling process.

The authors should also discuss the practical implications of the results, particularly highlighting how the findings could improve police response capabilities.

Graphs

Figures that contain two layers should be properly cited in the captions for clarity. In Figure 4, the map legend is not clearly visible and should be improved for better readability.

Author Response

Comment 1: 

Abstract- It would be beneficial to incorporate specific results in the abstract to provide a more complete overview of the study. Additionally, the keywords should include the geographical scope of the study to enhance relevance and visibility.

Response 1:

We revised the last sentence in the abstract:

• “….The analysis results of this paper suggest that the p-median models provide effective specifications, including their capability to define patrol areas that encompass the entire study region while minimizing distance costs. The inclusion of balancing constraints ensures a more equitable distribution of workloads among patrol areas, improving overall efficiency. Additionally, the model with NKDCE results in an improved workload balance among delineated areas for police patrolling activities, thus supporting more informed spatial decision-making processes for public safety.”
• Added keywords “Texas”

Comment 2:

Literature Review (State of the Art)- In the literature review, it would be useful to include a summary table that highlights the contributions of the studies analyzed and the aspects addressed in each case. This would provide a clearer and more structured view of the existing research.

Response 2:

We added two tables

Table 1. Studies on police patrol area design and optimization models

Study	Model	Contribution
Larson (1971)	p-median based approach, maximal covering location problem	Minimized travel distance for police service calls in New York City.
Mitchell (1972)	p-median model	Designed patrol beats to reduce response time and balance workloads.
Curtin et al. (2005, 2010)	Maximal covering location problem	Optimized patrol centers in Dallas, focusing on crime and service call priorities.
D'Amico et al. (2002)	Heuristic approach (simulated annealing)	Partitioned police jurisdictions using a patrol car allocation model.
Vlćek et al. (2024)	Contiguous p-median problem	Introduced preceding adjacent basic areas, ensuring compactness and contiguity in police service districting.

 

Table 2. Studies on districting problem considerations

Criterion	Study	Methods for addressing
Contiguity	Kalcsics (2015)	Use of proximity grids like Gabriel graphs, minimum spanning trees, or Voronoi diagrams to define contiguity in models
Balanced workloads	Curtin et al. (2005), Camacho-Collados et al. (2015)	Use metrics like deviations of district sizes, ranges of district sizes, and service call volumes to define balanced workloads
Compactness	Garfinkel and Nemhauser (1970), Young (1988)	Measure compactness using the area-to-circle ratio, Schwarzberg compactness score, or elongation index to assess district shape

 

Comment 3:  Methodology

The software used for GIS analysis is not mentioned and should be included.

IBM CPLEX 20.1 is used for modeling, and this should be specified in the methodology section. Additionally, the methodology section would benefit from a diagram that explains each of the developed processes in detail, along with the results obtained.

Response 3:

Response:

• We added “Figure 3 illustrates the spatial distribution of vehicle crash counts estimated using NKDE with a bandwidth of 1,000m and a cell size of 200m, processed via SANET 4.1 (Okabe et al., 2006).
• We added Figure 1

Figure 1. Flowchart of the model construction process.

 

Comment 4: 

There are methodological aspects related to the formulations included in the results section that would be better placed in the methodology section. This adjustment would improve the organization and clarity of the paper.

Response 4: 

We moved paragraphs regarding methods to the new section “3.3 The extended p-median location model with workload balancing constraints, and   3.4. Model results efficiency measurement”

 

Comment 5: Results and Discussion

The results section should focus more on the outcomes of the analysis. It would be interesting to know the current distribution of the police patrolling units and compare it with the optimized distribution obtained through the modeling process. The authors should also discuss the practical implications of the results, particularly highlighting how the findings could improve police response capabilities.

Response 5:

We appreciate the reviewer’s point. However, this paper focuses on the model implementation and its performance for two key reasons: (1) to demonstrate the capability of the p-median location models in delineating police patrol areas, and (2) to enhance its practical applicability. As noted in the Discussion and Conclusions section, the proposed models are not able to incorporate the practical methods employed by the Plano Police Department, as they declined to provide detailed data regarding police patrolling (almost certainly for security and welfare reasons). We consider that this paper contributes, as a preliminary step, to exploring the capability of the p-median location models and their potential extensibility for public safety applications. Future studies will address your comments and further advance this research.

 

Comment 6: Graphs

Figures that contain two layers should be properly cited in the captions for clarity. In Figure 4, the map legend is not clearly visible and should be improved for better readability.

Response 6: Thank you. We have revised it.

 

Reviewer 2 Report

Comments and Suggestions for Authors

An extremely interesting and high-quality scientific article, which will be of great interest to the professional public. I suggest minor corrections:

- review of the method of reference to sources and

- that figures 4, 7, 9 be corrected so that they are fully legible.

 

I propose to publish an article.

Author Response

Comment 1:

An extremely interesting and high-quality scientific article, which will be of great interest to the professional public. I suggest minor corrections:

- review of the method of reference to sources and

- Figures 4, 7, 9 be corrected so that they are fully legible.

 

Response 1: we have revised these. Thank you for your kind words. We appreciate your valuable feedback and will gladly make adjustments to improve the quality and clarity of the article.

Reviewer 3 Report

Comments and Suggestions for Authors

Minor issues:

Please change the abstract - write something about the results and conclusion;

Figure 1: Are there no accidents on the roads? You should check the data.

Figure 4 and others: Should contain the legend

Major issues:

There is no methodology chapter - please describe your study with the scheme and other parts. After that part, you should include a case study

 

 

Author Response

Comment 1: 

Please change the abstract - write something about the results and conclusion;

Response 1: We revised the last sentence in the abstract. “…this paper suggests that the p-median models provide effective specifications, including their ability to define patrol areas that encompass the entire study region while minimizing distance costs. The inclusion of balancing constraints ensures a more equitable distribution of workloads among patrol areas, improving overall efficiency. Additionally, using NKDCE results in a more balanced workload distribution for police patrolling activities, thus supporting more informed spatial decision-making processes for public safety.”

 

Comment 2:

Figure 1: Are there no accidents on the roads? You should check the data.

Response 2: Thank you for suggesting a double-check of the data. To ensure that there are no segments with 0 accidents, we carefully set the segment size appropriately and we collected crash data over a lengthy time horizon in order to have a sufficient count of events. As a result, there are no road segments with 0 accidents in the dataset.

Comment 3: Figure 4 and others: Should contain the legend

Response 3: Instead of adding a legend, we included a note to assist with reading the map because the areas represent qualitative properties. The note states: 'Segments with the same colors represent a patrol area unit.'"

 

Comment 4: Major issues

There is no methodology chapter - please describe your study with the scheme and other parts. After that part, you should include a case study

Response 4: This is an important comment. Section 3 focuses on the study area and methods, but we recognized that it lacks a detailed scheme. To address this issue, we have added a flowchart of the study and consolidated paragraphs regarding the methods, moving that latter originally-dispersed by now a single discussion from other sections into this chapter.

• Added Figure 1

Figure 1. Flowchart of the model construction process.

• We added a new section “3.3 The extended p-median location model with workload balancing constraints, and  3.4. Model results efficiency measurement” to Section 3.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Good work

Author Response

I appreciate your review and feedback, which have helped improve our paper.

Reviewer 3 Report

Comments and Suggestions for Authors

Minor issues:

The literature review provides limited insights into recent advancements in network-based spatial optimization for law enforcement.

 Some mathematical formulations (e.g., NKDCE) lack detailed explanation, making it challenging for readers unfamiliar with these methods to follow.

Figures depicting patrol areas lack sufficient annotation, which could impair interpretability

change the range of the legend of fig 2 to show accidents on roads.

Major issues:

The paper assumes the p-median model as optimal without thoroughly discussing its limitations relative to alternative location models (e.g., maximal covering models).

The paper lacks diverse evaluation metrics.

 

Author Response

Minor issues:

1. The literature review provides limited insights into recent advancements in network-based spatial optimization for law enforcement.

We added more specific discussion about recent advancements in network-based police patrol districting problem as below in 2.2 section:

(p.6) … Moreover, the literature on network-based spatial optimization for police patrol districting has recently advanced with several new approaches in mathematical modeling. Integrating road network segments into models yields more realistic and pragmatic results. For example, Chen et al. (2019) incorporate street segments and factors such as crime risk rate and area size in a mixed-integer programming approach, and Kong et al. (2019) employ center-based clustering for mixed-integer linear programming to consider balance workload, demonstrate the effectiveness of road-focused modeling. These approaches are often combined with how a shift toward data-driven, network-aligned patrol optimization, enhancing police responsiveness and adaptability across urban environments.

2. Some mathematical formulations (e.g., NKDCE) lack detailed explanation, making it challenging for readers unfamiliar with these methods to follow.

We added a function description below the equations:

(This editor does not support equations so I have attached PDF with equations to this note.)

(p.10) … Equation (7) represents the main NKDE formula, where  is the overall density estimate at a point  on the network, calculated by averaging vehicle crashes across  kernel centers. Each kernel center, , contributes to calculating this density estimation through its own localized density function, , as defined in Equation (8). The formula for   is divided into three cases, based on distance from : 1) For , where the distance is within the bandwidth range . 2) For , where the density function adapts, accounting for proximity to the network boundaries, and 3) For , where the point  lies near or at the bandwidth edge, limiting further influence from the kernel center.

3. Figures depicting patrol areas lack sufficient annotation, which could impair interpretability

We included patrol distance values to help compare the size of each patrol area.

4. change the range of the legend of fig 2 to show accidents on roads.

We modified the breaks to have equal intervals, except for the final break.

 

Major issues:

1. The paper assumes the p-median model as optimal without thoroughly discussing its limitations relative to alternative location models (e.g., maximal covering models).
2. The paper lacks diverse evaluation metrics.

We appreciate your feedback and acknowledge the limitations of the p-median model. However, we chose the p-median models over other alternative models because the weaknesses of one location model do not necessarily translate into strengths for another, as each model’s perspectives in solving location problems vary based on geographic context and specific formulation. Our goal was to underscore the contributions of the p-median model, particularly when it is combined with supportive elements, such as balancing constraints and network kernel density estimates, which have not been adequately addressed in the current literature.

We also recognize that our paper has limitation to embrace the characteristics of alternative location models into discussion as well as a lack of diverse evaluation metrics due to the limited space for in-depth discussion on the matter. To address this comprehensively, future research will involve cross-comparisons with alternative models to assess the strengths and weaknesses of each. We agree that applying diverse evaluation metrics to the p-median model is essential. However, we also believe that the current form should be focused on strategically to share our specific findings to emphasize the contribution of the p-median model in a practical context rather than from a purely theoretical modeling perspective. In response to your valuable suggestions, we have now included a discussion on future research directions to address these limitations, as detailed below:

 

(p.19) … This study intends to underscore the contributions of the p-median model, particularly when it is combined with supportive elements, such as balancing constraints and network kernel density estimates, which have not been adequately addressed in the current literature. This study emphasizes the contribution of the p-median model in a practical context rather than from a purely theoretical modeling perspective.

However, this research is limited in that it lacks diverse evaluation metrics. Future research can address this issue with cross-comparisons with alternative models to assess the strengths and weaknesses of each. Also,

Author Response File: Author Response.pdf