Optimal ambulance routing for mass casualty events

Optimal Ambulance Routing

for Mass Casualty Events

Conference on Dynamics of Disasters Athens, Greece,

October 5–7, 2006

by Alkis Vazacopoulos

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Research Team

Dash Optimization

– Alkis Vazacopoulos, PHD

– Gabriel Tavares

– Horia Tipi

Weill Medical College of

Cornell University

– Nathaniel Hupert, MD, MPH

– Eric Hollingsworth

– Wei Xiong

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Imagine…

Katrina, Rita, Andrew, Allison…

London, Madrid, Beslan, 9/11…

Boca Raton, Brentwood, Hamilton…

1918, 1957, 1968…

What is an acceptable disaster response?

What medical and logistical systems are needed?

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…But

Most science of disasters is pointed at

estimating casualty numbers, not at what to

do once those casualties appear on your

doorstep

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Outline

• Dispatching Problem

• Methodology

– MIP

– Simulation

• An Example Scenario

– Description

– A MIP based Greedy-Heuristic

– Visualization and Analysis of Results

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Mass Casualty Incident (MCI)

• A MCI is an event which causes multiple casualties, and will strain the resources of the local healthcare system

• At the site of a MCI, victims await medical treatment and transportation to hospital facilities

• In a MCI, there is a limited number of ambulances available to transport patients to hospitals, requiring ambulances to make multiple trips from site to hospital

• Hospitals have different characteristics, including:

– number of beds; patient throughput; distances, etc

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The Dispatching Question

• Where do we send the patients in order

to minimize the amount of time it takes to

treat them all?

transportation + waiting + treatment

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Related Questions

• Which hospitals should be included in

the response?

• Where should ambulances transport the

casualties?

• How many casualties should be

transported to each hospital?

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Hospitals Decision Tradeoffs

• Taking patients to nearby hospitals reduces the demand on ambulance time, however, it increases the burden that nearby hospitals must bear

• Taking patients to remote hospitals increases demand for ambulance time, but spreads out demand for hospital services more evenly

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Why is the

Dispatching Question Important?

1. Improved patient outcomes

2. Effective use of resources

3. Difficult to correct dispatching errors

after the fact

4. Determine which hospitals should be

used in a response

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1. Improved Patient Outcomes

• Patient outcomes suffer when patients are transported to the incorrect hospital

– Patients should be treated to the facility most capable of providing the type of care they need

– Longer delay to treatment time leads to a higher mortality rate

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2. Effective Use of Resources

• Resources are wasted when patients are

transported to the incorrect hospital

– Available resources at other hospitals

wasted

– During 9/11, uptown hospitals had an

abundance of staff on call and available space,

while downtown hospitals were overwhelmed

– Patients must be re-transported, which

wastes ambulance and paramedic time

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3. After the Fact Difficulties

• It may be difficult to “fix” incorrect patient

routing decisions after the fact

– Once patients are transported to a hospital, it may

be difficult to re-transport them

– They may need immediate medical care, and cannot wait

for re-transport

– There may be legal restrictions requiring the hospital to

treat them

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4. Determine which Hospitals

should be Used in a Response

• Hospitals a large distance from the disaster site may have available capacity, but is the long transportation time worth it?

– Which hospitals should respond to a MCI of a given size?

– Which hospitals are too far away to help?

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Dispatching Today

• Currently, dispatching is handled manually

by trained dispatchers

• Dispatching skill comes from experience

• Personal connections between the

dispatcher and officials at local hospitals

are important

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Our Goals

• Develop quantitative models of

ambulance dispatching which can

determine optimal ambulance routings

• Distill heuristics from the results of the

model which can assist dispatch

decisions during an MCI

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Methodology

• Formulate and solve the routing problem as a

Mixed Integer Program (MIP)

– Using Xpress-Mosel

• Build a simulation model in order to verify the

results

– Using Arena

• Develop a visualization tool to analyze,

evaluate and propose “interesting” solutions

– Using Xpress-XAD

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Proposed MIP model

• Objective is to find the minimum completion time, which includes transportation + treatment times

• Underlying problem is a

Minimum Makespan Scheduling Problem:

– Assign a set of jobs (patients) to a set of machines (ambulances+hospitals)

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General Framework Considered

• The patients (jobs) have different treatment

(processing) times

• The patient treatment time also depends on

the hospital (machine)

• Every ambulance may return to the disaster

site a certain number of times (trips)

• Every hospital has a capacity (number of beds)

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Problem Structure

• Two phases:

– Transportation

– Treatment

• Decisions made in the transportation phase

affect the treatment phase

– The assignment of patients first needs to consider

transportation and only after their treatment

(sequential assignment)

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Transportation MIP Component

• Key Decision Variables:

– assign an ambulance to a patient:

assign(patient,ambulance,trip)

• Time Decision Variables:

waitTimeForTransportation(patient)

arrivalTimeToSite(ambulance,trip)

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Treatment MIP Component

• Key Decision Variables:

– assign a patient to a hospital bed:

treatmentOrder(patient,hospital,bed)

• Time Decision Variables:

– starting time of patient treatment

startTime(bed,hospital)

– completion time of treatment

completionTime(bed,hospital)

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Transportation and Treatment

MIP Components Integration

• Key Decision Variables:

– assign a hospital to the ambulance’s trip

flow(hospital,ambulance,trip)

• Time Decision Variables:

– time to go to the hospital in a given trip

timeToHospital(ambulance,trip)

– time to go from the hospital to the disaster site

timeFromHospital(ambulance,trip)

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Xpress-Mosel Implementation

• Xpress-Mosel is a Language for both

Mathematical Modeling and

Programming

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MIP Goal

• Minimize the time needed to transport

and treat all patients from the disaster

site

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MIP is hard

• In general MIP is NP-hard

• Standard Vehicle Routing MIPs are

usually very hard to optimize, and the

optimality gap of the relaxed problems is

traditionally large

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How can our Problem be Solved?

• Decompose the problem into smaller

MIPs

• Develop heuristic methods:

– Implement greedy, local-descent heuristics

– Implement meta-heuristics

– Use MIP to find feasible solutions

– Use MIP to improve solutions

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We use Xpress-Optimizer for MIP

• To find starting feasible solutions

• To improve existing solutions

• To produce a guaranteed optimality gap

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An Example Scenario

• A bombing at the New York Stock Exchange

(NYSE) in downtown Manhattan leaves 150

trauma victims in need of emergency medical care

• The city has the following resources to respond:

– 50 ambulances

– 10 available hospitals of different size located

throughout lower Manhattan

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Treatment Facilities Considered for

NYSE Attack Scenario

Hospital Distance

(minutes) Capacity

Throughput

(patients/hour)

NEW YORK DOWNTOWN HOSPITAL 2 48 1

BELLEVUE HOSPITAL CENTER 11 370 8

BETH ISRAEL MEDICAL CENTER 10 332 6

NY EYE AND EAR 9 31 1

HOSPITAL FOR JOINT DISEASES 10 50 2

NY UNIVERSITY MEDICAL CENTER 11 212 4

NEW YORK HOSPITAL-NEW YORK 15 644 6

ST VINCENTS HOSPITAL & MEDICAL CENTER 8 45 4

ST LUKES ROOSEVELT/ROOSEVELT 16 323 5

LENOX HILL HOSPITAL 18 196 5

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Scenario Framework

• 1 Disaster Site (NYSE)

• 150 Victims with treatment times

~normal(60,15) minutes / Hospital Throughput

• 50 Ambulances with Single Patient Capacities

– Assume that ambulances initially arrive to the disaster

site 8 min after the attack

• 10 Hospitals in Manhattan

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NYSE Scenario MIP Statistics

• 508 860 constraints

• 361 379 variables (197 300 are binary)

• 3 313 838 non-zero elements

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NYSE Scenario LP relaxation

• LP relaxation is 220 min

• Found using Xpress-2006B with the Barrier algorithm – CPU time was 3300 sec

– Found on – dual AMD Athlon 64bit, 4800+ 2.41 GHz CPU with 4GB RAM

Optimal completion time is greater than 3h 40m

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Ambulances Dispatcher

Xpress-XAD Visualization Tool

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A MIP feasible solution

• The overall completion time is 301 min

• The average pickup time is 96 min

• Not all 50 ambulances are assigned immediately after the attack

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MIP based Greedy-Heuristics

Implemented Using Xpress-Mosel

• Assign all patients to a given hospital – Assign first patients with smallest expected completion time

• Assign patients to a specified number of

ambulance trips – Assign first patients to hospitals which have smallest expected

hospital completion time, according to the previous patient

assignments

• Use Xpress-Optimizer to solve Reduced MIP

obtained with the previous assignments

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A MIP based

Greedy-Heuristic solution

• The overall completion time is 262 min

• The average pickup time is 47 min

• All 50 ambulances are assigned immediately after the attack

The greedy-heuristic

solution time was found in

2200 sec in a dual Xeon

3.0 GHz 4 GB RAM.

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Treatment Facilities Considered for

NYSE Attack Scenario

Hospital Throughput

(pt/hr)

Patients

Allocated

Waiting

Time/Patient

Completion

Time

NEW YORK DOWNTOWN HOSPITAL 1 3 51’ 207’

BELLEVUE HOSPITAL CENTER 8 30 62’ 254’

BETH ISRAEL MEDICAL CENTER 6 23 74’ 262’

NY EYE AND EAR 1 3 60’ 224’

HOSPITAL FOR JOINT DISEASES 2 7 56’ 253’

NY UNIVERSITY MEDICAL CENTER 4 14 56’ 240’

NEW YORK HOSPITAL-NEW YORK 6 21 62’ 249’

ST VINCENTS HOSPITAL & MEDICAL CENTER 4 15 74’ 253’

ST LUKES ROOSEVELT/ROOSEVELT 5 17 57’ 244’

LENOX HILL HOSPITAL 5 17 66’ 250’

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Improving the MIP heuristic

• Use Local-Search heuristics

• Use Reduced MIP Search

• The initial solution is probably within 5-to-10 min of being optimal

****************************** Beth Israel Medical Center ******************************

Patient 21 arrived at 28 waited 0 and finish at 38.9127(10.9127)

Patient 25 arrived at 28 waited 10.9127 and finish at 48.533(9.62032)

Patient 27 arrived at 28 waited 20.533 and finish at 64.8155(16.2825)

Patient 36 arrived at 28 waited 36.8155 and finish at 74.9802(10.1646)

Patient 40 arrived at 28 waited 46.9802 and finish at 86.4991(11.5189)

Patient 16 arrived at 28 waited 58.4991 and finish at 97.5059(11.0067)

Patient 43 arrived at 54 waited 43.5059 and finish at 109.702(12.1965)

Patient 85 arrived at 56 waited 53.7023 and finish at 118.856(9.15339)

Patient 71 arrived at 58 waited 60.8557 and finish at 131.901(13.0454)

Patient 86 arrived at 60 waited 71.9012 and finish at 141.893(9.99153)

Patient 77 arrived at 68 waited 73.8927 and finish at 150.167(8.27402)

Patient 60 arrived at 70 waited 80.1667 and finish at 163.536(13.3689)

Patient 47 arrived at 74 waited 89.5356 and finish at 173.336(9.79992)

Patient 137 arrived at 86 waited 87.3355 and finish at 177.917(4.58128)

Patient 90 arrived at 88 waited 89.9168 and finish at 186.698(8.78111)

Patient 129 arrived at 88 waited 98.6979 and finish at 193.457(6.75863)

Patient 111 arrived at 88 waited 105.457 and finish at 204.848(11.3916)

Patient 140 arrived at 96 waited 108.848 and finish at 212.415(7.56672)

Patient 132 arrived at 100 waited 112.415 and finish at 217.704(5.28914)

Patient 148 arrived at 104 waited 113.704 and finish at 228.242(10.5379)

Patient 92 arrived at 116 waited 112.242 and finish at 240.188(11.946)

Patient 101 arrived at 116 waited 124.188 and finish at 248.046(7.85816)

Patient 94 arrived at 146 waited 102.046 and finish at 262.156(14.1099)

Hospital 3 completion time is after 262.156 minutes

the total waiting time in the ER is 1702.15 minutes

and the number of patients allocated to it is 23

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Conclusions • MIP model is highly adaptable to different scenarios

– Simultaneous attacks; non-homogeneous transporters; introducing priority patients according to their condition, etc

• MIP based greedy-heuristics provide good starting feasible solutions – Alternative heuristics can easily be tested/programmed using

Xpress-Mosel

• The Ambulances Dispatcher Xpress-XAD Visualization Tool is very helpful in planning, detecting bottlenecks and possible improvements, and in demonstrating the value of the MIP model to the dispatchers and officials