Optimizing Police Facility Locations Based on Cluster Analysis and the Maximal Covering Location Problem
Abstract
:1. Introduction
2. Background
2.1. Facility Location Problem
2.1.1. MCLP
- i = index for demand points;
- j = index for candidate locations;
- I = the set of demand nodes;
- J = the set of potential operating locations;
- Ni = , the set of locations that can cover demand point i;
- dij = the shortest distance from i to j;
- S = maximum value of the distance between the demand node and the operation node (desired service distance);
- ai = coefficient reflecting the desirability of covering demand point i;
- P = number of operations (facilities) to be allocated;
- xj = 1 if a facility is located at potential location j—0 otherwise;
- yi = 1 if demand point i is covered at least one operation j—0 otherwise.
2.1.2. Clustering and k-Means Procedure
Algorithm 1: Pseudo-code for the k-means implementation |
Initialize k-means with random values |
While (a given number of iterations ‘i’ < = ‘I’) { |
Select ‘k’ cluster centers coinciding with ((k randomly chosen patterns) or (k randomly defined points)); |
Assign each pattern to the closest cluster center; |
For each ‘k’ new clusters: |
Recalculate (new centroids) = mean of all points assigned to that cluster; |
Recalculate new centroids; |
Iterate through items: |
Find the mean closest to the item; |
Assign item to mean; |
Update mean; |
} |
End when convergence criterion ‘i’ < = ‘I’ is met. |
- -
- If the centroid has been updated in the last step, for each data point included, the within-cluster sum of squares for each data point, if included in another cluster, is calculated.
- -
- If one of the cluster sum of squares is smaller than the current one, the case is assigned to this new cluster.
3. Materials and Methods
4. Results—MIS for Police Centers Location
4.1. Graphical User Interface (GUI)
4.2. The Preprint Reports Screen
5. Discussion
5.1. Clusters Analysis and Candidates
5.2. Optimal Locations
5.3. Comparing Scenarios
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Potential Locations | Latitude | Longitude |
---|---|---|
Candidate 1 | −7.99214 | −34.92050 |
Candidate 2 | −8.036942 | −34.88901 |
Candidate 3 | −8.058401 | −34.90614 |
Candidate 4 | −8.090015 | −34.89074 |
Candidate 5 | −8.086555 | −34.93611 |
Candidate 6 | −8.135032 | −34.91076 |
Candidate 7 | −8.035501 | −34.91289 |
Candidate 8 | −8.045003 | −34.94490 |
Candidate 9 | −8.115055 | −34.90337 |
Candidate 10 | −8.061584 | −34.88112 |
Coverage Radius (Km) | Number of Police Occurrences Covered | Optimal Coverage Rate |
---|---|---|
1.00 | 582 | 34.5% |
1.50 | 923 | 54.7% |
2.00 | 1143 | 67.7% |
2.50 | 1380 | 81.8% |
3.00 | 1521 | 90.1% |
3.50 | 1630 | 96.6% |
4.00 | 1664 | 98.6% |
4.50 | 1675 | 99.2% |
5.00 | 1683 | 99.7% |
N. of Candidates | Covered Occurrences | Cov. % | Non-Covered Occurrences | Non-Cov. % | Mean Distance | Std. Deviation |
---|---|---|---|---|---|---|
5 | 1558 | 0.92 | 130 | 0.08 | 1570.58 | 989.13 |
10 | 1521 | 0.90 | 167 | 0.10 | 1639.17 | 1045.29 |
15 | 1523 | 0.90 | 165 | 0.10 | 1714.41 | 1038.34 |
20 | 1568 | 0.93 | 120 | 0.07 | 1888.65 | 1026.29 |
25 | 1584 | 0.94 | 104 | 0.06 | 1723.50 | 969.96 |
30 | 1584 | 0.94 | 104 | 0.06 | 1723.50 | 969.96 |
35 | 1602 | 0.95 | 86 | 0.05 | 1844.77 | 958.51 |
40 | 1584 | 0.94 | 104 | 0.06 | 1718.15 | 943.99 |
45 | 1597 | 0.95 | 91 | 0.05 | 1715.32 | 930.77 |
50 | 1602 | 0.95 | 86 | 0.05 | 1819.48 | 964.39 |
100 | 1608 | 0.95 | 80 | 0.05 | 1803.64 | 917.07 |
1200 | 1625 | 0.96 | 63 | 0.04 | 1777.95 | 910.55 |
1500 | 1625 | 0.96 | 63 | 0.04 | 1.775.22 | 911.90 |
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Borba, B.F.d.C.; de Gusmão, A.P.H.; Clemente, T.R.N.; Nepomuceno, T.C.C. Optimizing Police Facility Locations Based on Cluster Analysis and the Maximal Covering Location Problem. Appl. Syst. Innov. 2022, 5, 74. https://doi.org/10.3390/asi5040074
Borba BFdC, de Gusmão APH, Clemente TRN, Nepomuceno TCC. Optimizing Police Facility Locations Based on Cluster Analysis and the Maximal Covering Location Problem. Applied System Innovation. 2022; 5(4):74. https://doi.org/10.3390/asi5040074
Chicago/Turabian StyleBorba, Bruno Ferreira da Costa, Ana Paula Henriques de Gusmão, Thárcylla Rebecca Negreiros Clemente, and Thyago Celso Cavalcante Nepomuceno. 2022. "Optimizing Police Facility Locations Based on Cluster Analysis and the Maximal Covering Location Problem" Applied System Innovation 5, no. 4: 74. https://doi.org/10.3390/asi5040074