Location of EMS Centers in Indiana CountyBy Paul Rossman

And Jason Smith

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

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

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

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.

Methodology

Downloaded Pennsylvania map from U.S. Census Bureau website

Used ArcGIS to extract map of Indiana County

Methodology

Used ArcGIS to find centers of Indiana County municipalities

Mapped and labeled municipalities in alphabetical order from 1 to 38

Methodology

Imported Indiana County municipality central points into Google Earth

Used Google Earth to find distances in minutes between municipality centers

Distance Matrix in Minutes

Distance Matrix in Miles

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

Converting Minutes to Miles

Miles = -1.9998789 + 0.56322676 Minutes

Sample size: 1444

R (correlation coefficient) = 0.9486

R-sq = 0.8999157

Model

= the set of all municipalities, 1 … 38 = the distance from node i to node j T = the maximum predetermined time

between municipalities

jXijd

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.

Sample Problem (Time <= 10 minutes)

38

1

mini

ix

1171 xx

1312 xx

13 x14 x

185 xx

1,0ix

Sample Problem (Time <= 10 minutes)

Technology

Generic Algebraic Modeling System

“a high-level modeling system for mathematical programming and optimization”

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

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;

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

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

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

Actual Placement of Centers

•According to model, it means a 23-minute response time.

•Much lower because of IRPs and other factors

Conclusions

• Model is sufficient, however leaves room for opportunities for improvement

• Provides a snapshot of needs and thresholds for building EMS centers

Any questions?