Download - Location of EMS Centers in Indiana County
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Location of EMS Centers in Indiana CountyBy Paul Rossman
And Jason Smith
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Introduction
Optimize placement and minimize the number of EMS (Emergency Medical Service) centers in Indiana County so that the entire population is reached
EMS Act of 1973: 95% of calls should be reached in 10 minutes
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Introduction
Journal of Emergency Medical Services: 2007 survey of the 200 largest U.S. cities stated that 72.78% of EMS agencies use a target of 8:59 minutes or less
Fitch & Associates, 2005: Urban transport ambulance response time benchmarks ranging from 8:59 to 12:59 minutes
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Research
Toregas et al., 1971, pioneered binary integer programming and set covering to determine optimal EMS station placement
Eaton et al., 1985, similarly placed ambulance stations
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Research
Batta and Mannur, 1990, similarly determined optimal placement of fire stations and ambulance stations
NUMB3RS season 4 opening episode “Trust Metric” (2007): Charlie Eppes uses set covering to optimize the position of police officers in downtown L.A.
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Methodology
Downloaded Pennsylvania map from U.S. Census Bureau website
Used ArcGIS to extract map of Indiana County
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Methodology
Used ArcGIS to find centers of Indiana County municipalities
Mapped and labeled municipalities in alphabetical order from 1 to 38
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Methodology
Imported Indiana County municipality central points into Google Earth
Used Google Earth to find distances in minutes between municipality centers
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Distance Matrix in Minutes
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Distance Matrix in Miles
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PopulationsMunicipality Population Armagh borough 131 Armstrong township 3090 Banks township 997 Black Lick township 1317 Blairsville borough 3607
Inverse proportion = Total Population/ Population of Municipalities
Municipality Inverse Proportion Armagh borough 684.0076336 Armstrong township 28.99838188 Banks township 89.87462387 Black Lick township 68.03720577 Blairsville borough 24.84197394
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Converting Minutes to Miles
Miles = -1.9998789 + 0.56322676 Minutes
Sample size: 1444
R (correlation coefficient) = 0.9486
R-sq = 0.8999157
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Model
= the set of all municipalities, 1 … 38 = the distance from node i to node j T = the maximum predetermined time
between municipalities
jXijd
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Model j jXZ1min
jX
iXa
j
j jij
allfor 1 ,0
allfor 1
Td
Tda
ij
ijij if 1
if 0
Z1 = the number of EMS centers required
s. t.
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Sample Problem (Time <= 10 minutes)
38
1
mini
ix
1171 xx
1312 xx
13 x14 x
185 xx
1,0ix
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Sample Problem (Time <= 10 minutes)
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Technology
Generic Algebraic Modeling System
“a high-level modeling system for mathematical programming and optimization”
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Technology
option limcol=0;set mun /1*38/;alias (mun,i,j);table travelTime(i,j) "travel time between municipalities in Indiana County" 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
1 0.00 42.00 84.00 26.00 20.00 12.00 13.00 11.00 73.00 29.00 35.00 53.00 42.00 38.00 47.00 59.00 5.00 43.00 71.00 68.00 55.00 25.00 32.00 52.00 68.00 63.00 30.00 37.00 48.00 40.00 42.00 71.00 59.00 52.00 75.00 14.00 32.00 47.00
2 42.00 0.00 73.00 34.00 37.00 28.00 37.00 32.00 58.00 21.00 27.00 56.00 34.00 35.00 20.00 44.00 45.00 25.00 73.00 54.00 51.00 20.00 21.00 37.00 61.00 48.00 36.00 17.00 33.00 48.00 10.00 47.00 39.00 24.00 51.00 46.00 21.00 20.00
3 84.00 73.00 0.00 82.00 84.00 76.00 80.00 79.00 18.00 69.00 61.00 34.00 55.00 93.00 62.00 40.00 87.00 61.00 22.00 33.00 47.00 68.00 60.00 41.00 26.00 31.00 67.00 63.00 51.00 104.00 74.00 46.00 51.00 62.00 44.00 94.00 60.00 89.00
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Technologyset cover(i,j) "Center at municipality i covers municipality j";cover(i,j) = yes$(travelTime(i,j) <= 8.9333);parameter popWeight(mun) population weight/1 684.0076336 2 28.998381883 89.874623874 68.03720577...Binary Variables buildStation(mun);Variables totalStations;
Equations coverConstraint(mun), objective;objective.. totalStations =e= sum(mun, buildStation(mun)*popWeight(mun));coverConstraint(mun).. sum(j$cover(j,mun),buildStation(j)) =g= 1;
model indiana /all/;
* any optimum within <1 of the true optimum must BE the true optimum!indiana.optca = 0.999;solve indiana using rmip minimizing totalStations;
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Analysis
• Fitch & Associates, 2005: Urban transport ambulance response time benchmarks ranging from 8:59 to 12:59 minutes
• Using Uniform distribution, we picked 28 random numbers between 8:59 and 12:59
• Similarly, we picked 28 random numbers between 3.01 and 5.31
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SolutionsMinutes Solution Solution With Population Weight People per Center 8.983333 29 2860.536 31 9.041318 28 2661.855316 34 9.051694 28 2661.855316 34
* Changes were at 9 minutes, 10 minutes and 12 minutes
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SolutionsMiles Solution
Solution With Population Weight
People per Center
3.01 29 2686.69729 33 3.032181 29 2686.69729 33 3.060117 29 2686.69729 33
* Changes are at 3.8 and at 4.0
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Actual Placement of Centers
•According to model, it means a 23-minute response time.
•Much lower because of IRPs and other factors
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Conclusions
• Model is sufficient, however leaves room for opportunities for improvement
• Provides a snapshot of needs and thresholds for building EMS centers
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Any questions?