Locating and dispatching ambulances

using discrete optimization methodologies

Laura Albert McLay

Industrial & Systems Engineering

University of Wisconsin-Madison

lmclay@wisc.edu

punkrockOR.wordpress.com

@lauramclay

1This work was in part supported by the U.S. Department of the Army under Grant Award Number W911NF-10-1-0176 and by the National Science Foundation under Award No. 1054148, 1444219, 1541165.

The road map

• How do emergency medical service (EMS) systems work?

• How do we know when EMS systems work well?

• How can we improve how well EMS systems work?

• Where is EMS OR research going?

• Where do they need to go?

2

Collaborators

Maria MayorgaNorth Carolina State University

Students

Sardar Ansari

Ben Grannan

Philip Leclerc

3

OR in EMS, fire & policing

4

The President’s

Commission on Law

Enforcement and the

Administration of Justice

(1965)

Al Blumstein chaired the

Commission’s Science

and Technology Task

Force (CMU)

Richard Larson did

much of the early

work (MIT)

19721972

Early urban operations research models

5

Set cover / maximum cover modelsHow can we “cover” the maximum number of locations with ambulances?Church, R., & ReVelle, C. (1974). The maximal covering location problem. Papers in regional science, 32(1), 101-118.

Markov modelsHow many fire engines should we send?Swersey, A. J. (1982). A Markovian decision model for deciding how many fire companies to dispatch. Management Science, 28(4), 352-365.

AnalyticsHow far will a fire engine travel to a call?Kolesar, P., & Blum, E. H.(1973). Square root laws

for fire engine response distances. Management Science, 19(12), 1368-1378.

Hypercube queuing modelsWhat is the probability that our first choice ambulance is unavailable for this call?Larson, R. C. (1974). A hypercube queuing model for facility location and redistricting in urban emergency services. Computers & Operations Research, 1(1), 67-95.

Anatomy of a 911 call

Call arrives to call center

queue

Call answered by call taker

Triage / data entry

Call sent to dispatcher

Information collected from

caller

Instructions to caller

Call taker ends call

Dispatcher answers call

First unit assigned

Additional units assigned

Pre-arrival instructions to

service providers

Dispatcher ends call

Response time

Service provider:

Dispatcher:

Call taker:

Dispatch time

Dispatch time

Emergency 911 callUnit

dispatchedUnit is en

routeUnit arrives

at sceneService/care

provided

Unit leaves scene

Unit arrives at hospital

Patient transferred

Unit returns to service

6

EMS design varies by community:One size does not fit all

7McLay, L.A., 2011. Emergency Medical Service Systems that Improve Patient Survivability. Encyclopedia of Operations Research in the area of “Applications with Societal Impact,” John Wiley & Sons, Inc., Hoboken, NJ (published online: DOI: 10.1002/9780470400531.eorms0296)

Fire and EMS vs. EMSPaid staff vs. volunteers

Publicly run vs. privately run

Emergency medical technician (EMT) vs. Paramedic (EMTp)

Mix of vehicles

Ambulance location, relocation, and relocation

on-the-fly

Mutual aid

Operationalizing recommendations

Priority dispatch:

… but which ambulance when there is a choice?

8

Type Capability Response Time

Priority 1Advanced Life Support (ALS) Emergency

Send ALS and a fire engine/BLSE.g., 9 minutes

(first unit)

Priority 2Basic Life Support (BLS) Emergency

Send BLS and a fire engine if availableE.g., 13 minutes

Priority 3Not an emergency

Send BLSE.g., 16 minutes

Performance standards

National Fire Protection Agency (NFPA) standard yields a coverage objective function for response times

Most common response time threshold (RTT): 9 minutes for 80% of calls

• Easy to measure

• Intuitive

• Unambiguous

9

Response times vs. cardiac arrest survival

10

CDF of calls for service covered

Response time (minutes) 9

80%

What is the best response time threshold?

• Guidelines suggest 9 minutes

• Medical research suggests ~5 minutes• But this would disincentive 5-9 minute responses

11

Responses no longer “count”

What is the best response time threshold?

• Guidelines suggest 9 minutes

• Medical research suggests ~5 minutes• But this would disincentive 5-9 minute responses

• Which RTT is best for design of the system?

12

What is the best response time threshold based on retrospective survival rates?

Decision context is locating and dispatching ALS ambulances

• Discrete optimization model to locate ambulances *

• Markov decision process model to dispatch ambulances

13* McLay, L.A. and M.E. Mayorga, 2010. Evaluating Emergency Medical Service Performance Measures. Health Care Management Science 13(2), 124 - 136

Survival and dispatch decisions

14

Across different ambulance configurations

Across different call volumes

McLay, L.A., Mayorga, M.E., 2011. Evaluating the Impact of Performance Goals on Dispatching Decisions in Emergency Medical Service. IIE Transactions on Healthcare Service Engineering 1, 185 – 196

Minimize un-survivability when altering dispatch decisions

Dispatching models

15

Optimal dispatching policiesusing Markov decision process models

911 callUnit

dispatchedUnit is en

routeUnit arrives

at sceneService/care

provided

Unit leaves scene

Unit arrives at hospital

Patient transferred

Unit returns to service

Determine which ambulance to send based

on classified priority

Classified priority(H or L)

True priorityHT or LT

16

Information changes over the course of a callDecisions made based on classified priority.Performance metrics based on true priority.

Classified customer riskMap Priority 1, 2, 3 call types to high-priority (𝐻) or low-priority (𝐿)Calls of the same type treated the same

True customer riskMap all call types to high-priority (𝐻𝑇) or low-priority (𝐿𝑇)

Optimal dispatching policiesusing Markov decision process models

Optimality equations:

𝑉𝑘 𝑆𝑘 = max𝑥𝑘∈𝑋(𝑆𝑘)

𝐸 𝑢𝑖𝑗𝜔 𝑥𝑘 + 𝑉𝑘+1 𝑆𝑘+1 𝑆𝑘 , 𝑥𝑘 , 𝜔

Formulate problem as an undiscounted, infinite-horizon, average reward Markov decision process (MDP) model

• The state 𝒔𝒌 𝑆 describes the combinations of busy and free ambulances.

• 𝑋(𝒔𝑘) denotes the set of actions (ambulances to dispatch) available in state 𝒔𝒌.

• Reward 𝑢𝑖𝑗𝜔 depend on true priority (random).

• Transition probabilities: the state changes when (1) one of the busy servers completes service or (2) a server is assigned to a new call.

Select best

ambulance to send

Value in current

state

Values in (possible)

next states

(Random) reward based

on true patient priority

Under- or over-prioritize

• Assumption: No priority 3 calls are truly high-priority

Case 1: Under-prioritize with different classification accuracy

Case 2: Over-prioritize

Pr1 Pr2 Pr3

Pr1 Pr2 Pr3HT

HT

Pr1 Pr2 Pr3HT

Pr1 Pr2 Pr3HT

Informational accuracy captured by:

𝛼 =𝑃 𝐻𝑇 𝐻

𝑃(𝐻𝑇|𝐿)

18

Classified high-priorityClassified low-priority

Improved accuracy

Structural properties

RESULTIt is more beneficial for a server to be idle than busy.

RESULTIt is more beneficial for a server to be serving closer customers.

RESULTIt is not always optimal to send the closest ambulance, even for high priority calls.

Coverage

0 10 20 30 40 500.405

0.41

0.415

0.42

0.425

0.43

0.435

0.44

0.445

Ex

pe

cte

d c

ov

era

ge

Optimal Policy, Case 1

Optimal Policy, Case 2

Closest Ambulance

20

Better accuracy

Low and high priority callsConditional probability that the closest unit is dispatched given initial classification

High-priority calls Low-priority calls0 10 20 30 40 500.98

0.985

0.99

0.995

1

1.005

Pro

po

rtio

n c

losest

am

bu

lan

ce is d

isp

atc

hed

Closest Ambulance

Optimal Policy, Case 1

Optimal Policy, Case 2

0 10 20 30 40 500.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

po

rtio

n c

losest

am

bu

lan

ce is d

isp

atc

hed

Closest Ambulance

Optimal Policy, Case 1

Optimal Policy, Case 2

Classified high-priority Classified low-priority

21

Case 1 (𝛼 = ∞), Case 2 policiesHigh-priority calls

Case 2: First to send to high-priority calls

Station1

2

3

4

Case 2: Second to send to high-priority calls

Station1

2

3

4

Service can be improved via optimization of backup service and response to low-priority patients

Rationed for high-priority calls

Rationed for low-priority calls

22

Districting and location models

23

Early location models

24

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Maximum coverage modelHow can we “cover” the maximum number of locations with 𝑝ambulances?

𝒑-median modelHow can we locate 𝑝 ambulances such that we minimize the average distance an ambulance must travel to a call?

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Ambulance response districts

How should we locate ambulances?

How should we design response districts around each ambulance?

• Ambulance unavailability (spatial queueing)

• Uncertain travel times / Fractional coverage

• Workload balancing: all ambulances do the same amount of work

25

Fractional coverage / UnavailabilityPerfect coverage / Availability

Ansari, S., McLay, L.A., Mayorga, M.E., 2015. A maximum expected covering problem for locating and dispatching servers. To appear in Transportation Science.

Districting modelMixed Integer Linear Program

max 𝑤∈𝑊 𝑗∈𝐽 𝑝=1𝑠 𝑚=1

min(𝑐𝑤,𝑠−𝑝+1) 𝑞𝑗𝑝𝑚 1 − 𝑟𝑚 𝑟𝑝−1𝜆𝑗𝐻𝑅𝑤𝑗𝑧𝑤𝑗𝑝𝑚

subject to

𝑝=1𝑠

𝑚=1

𝜅𝑤𝑝 𝑧𝑤𝑗𝑝𝑚 ≤ 1, 𝑗 ∈ 𝐽, 𝑤 ∈ 𝑊

𝑝=1𝑠

𝑚=1

𝜅𝑤𝑝 𝑧𝑤𝑗𝑝𝑚 ≤ 𝑦𝑤 , 𝑗 ∈ 𝐽, 𝑤 ∈ 𝑊

𝑤∈𝑊 𝑥𝑤𝑗𝑝 = 1, 𝑗 ∈ 𝐽, 𝑝 = 1,… , 𝑠

𝑝=1𝑠 𝑥𝑤𝑗𝑝 = 𝑦𝑤 , 𝑗 ∈ 𝐽, 𝑤 ∈ 𝑊

𝑥𝑤𝑗𝑝′ = 𝑝=max 1,𝑝′−𝑐𝑤+1𝑝′

𝑚=𝑝′−𝑝+1

𝜅𝑤𝑝 𝑧𝑤𝑗𝑝𝑚 ,𝑗 ∈ 𝐽, 𝑤 ∈ 𝑊, 𝑝′ = 1,… , 𝑠

𝑤∈𝑊 𝑦𝑤 = 𝑠

𝑦𝑤 ≤ 𝑐𝑤, 𝑤 ∈ 𝑊

𝑟 − 𝛿 𝑦𝑤 ≤ 𝑗∈𝐽 𝑝=1

𝑠 𝑚=1

𝜅𝑤𝑝 𝜆𝑗𝐻 + 𝜆𝑗

𝐿 𝑞𝑗𝑝𝑚 1 − 𝑟𝑚 𝑟𝑝−1𝜏𝑤𝑗𝑧𝑤𝑗𝑝𝑚≤ 𝑟 + 𝛿 𝑦𝑤 , 𝑤 ∈ 𝑊

𝑥𝑤𝑗′1 ≥ 𝑥𝑤𝑗1, 𝑗 ∈ 𝐽, 𝑤 ∈ 𝑊, 𝑗′ ∈ 𝑁𝑤𝑗

𝑦𝑤 ∈ 𝑍0+, 𝑤 ∈ 𝑊

𝑥𝑤𝑗𝑝 ∈ 0,1 , 𝑤 ∈ 𝑊, 𝑗 ∈ 𝐽, 𝑝 = 1,… , 𝑠

𝑧𝑤𝑗𝑝𝑚 ∈ 0,1 , 𝑤 ∈ 𝑊, 𝑗 ∈ 𝐽, 𝑝 = 1,… , 𝑠,𝑚 = 1,… , 𝑐𝑤26

Every customer has all the priorities and the

number of assignments to a station is equal to

the number of servers located at that station

A customer location is not assigned to

a station more than once and no call

location is assigned to a closed station

Linking constraints

Balance the load amongst the

servers

Locate 𝑠 servers with no more than 𝑐𝑤 per

station

Expected coverage

Contiguous first priority districts

Binary and integrality

constraints on the variables

Parameters

• 𝐽: set of all customer (demand) nodes

• 𝑊: set of all potential station locations

• 𝑠: total number of servers in the system

• 𝜆𝑗𝐻 (𝜆𝑗

𝐿): mean high-priority (low-priority)

call arrival rates from node 𝑗

• 𝜆: system-wide total call arrival rate

• 𝜏𝑤𝑗: mean service time for calls originated

from node 𝑗 and served by a server from a potential station 𝑤.

• 𝜏: system-wide mean service time

• 𝑐𝑤: capacity of station 𝑤

27

• 𝑟: system-wide average server utilization

• 𝑃𝑠: loss probability (probability that all 𝑠servers are busy)

• 𝑅𝑤𝑗: expected proportion of calls from 𝑗

that are reached by servers from station 𝑤in nine minutes

• 𝑞𝑗𝑝𝑚: correction factor for customer 𝑗's 𝑝th

priority server at which there are 𝑚 servers

located.

• 𝑁𝑤𝑗: set of demand nodes that are

neighbors to 𝑗 and are closer to station 𝑤than 𝑗.

Decision variables• 𝑦𝑤 = number of servers located at station 𝑤,𝑤 ∈ 𝑊.• 𝑧𝑤𝑗𝑝𝑚= 1 if there are 𝑝 − 1 servers located at stations that node 𝑗 prefers over 𝑤 and there

are 𝑚 servers located at station 𝑤,𝑤 ∈ 𝑊, 𝑗 ∈ 𝐽, 𝑝 = 1,… , 𝑠,𝑚 = 1,… , 𝑐𝑤 and 0 otherwise.• 𝑥𝑤𝑗𝑝= 1 if 𝑝′ < 𝑝 < 𝑠 − 𝑝′′ where are 𝑝′ is the number of servers located at stations that

node 𝑗 prefers over 𝑤, and 𝑝′′ is the number of servers located at stations that node 𝑗 prefers less than 𝑤,𝑤 ∈ 𝑊, 𝑗 ∈ 𝐽, 𝑝 = 1,… , 𝑠, and 0 otherwise.

Issue

28

Queuing modelSpatial queuing

outputs

Spatial queuing inputs

Mixed integer programming

model

How it usually works:

What we need:

The solution: iterate

29

Spatial queuing model

Mixed integer

programming model

Results

RESULTThe Base model that does not maintain contiguity or a balanced load amongst the ambulances is NP-complete.

• reduction from k-median

RESULT

The first priority response districts for the Base model are contiguous if there is no more than one server per station.

RESULT

Identifying districts that balance the workload is NP-complete.• reduction from bin packing

RESULT

Reduced model to assign only the top 𝑠′ ≤ 𝑠 servers• Not trivial, allows model to scale up to have many servers

30

Hanover County: example

First Priority Districts

• Base Model: no workload balancing

LBM ModelFirst Priority Districts for WD12am6am

• LBM (2 servers at Ashland and 1 at every other)

First Priority Districts

• Workload balancing

• Contiguous first priority districts

Two ambulances

Weekday solutions: first priority districts

• (a) 2 servers at Ashland (b) 1 server at every station (c) 1 server at every station (d) 2 servers at Ashland

Coordinating multiple types of vehicles

• Not intuitive how to use multiple types of vehicles• ALS ambulances / BLS ambulances (2 EMTp/EMT)• ALS quick response vehicles (QRVs) (1 EMTp)

• Double response = both ALS and BLS units dispatched

• Downgrades / upgrades for Priority 1 / 2 calls• Who transports the patient to the hospital?

• Research goal: operationalize guidelines for sending vehicle types to prioritized patients

• (Linear) integer programming model for a two vehicle-type system: ALS Non-transport QRVs and BLS ambulances

36

Results quantify impact of using QRVs

37

Application in a real setting

38

Achievement Award Winner for Next-Generation Emergency Medical Response Through Data Analysis & Planning (Best in Category winner), National Association of Counties, 2010.

McLay, L.A., Moore, H. 2012. Hanover County Improves Its Response to Emergency Medical 911 Calls. Interfaces 42(4), 380-394.

Where do EMS systems need to go?

39

EMS = Prehospital care

Operations Research

• Efficiency

• Optimality

• Utilization

• System-wide performance

Healthcare

• Efficacy

• Access

• Resources/costs

• “Patient centered outcomes”

40

Healthcare

Transportation

Public sector

Common ground?

More thoughts on patient centered outcomes

Operational measures used to evaluate emergency departments

• Length of stay

• Throughput

Increasing push for more health metrics

• Disease progression

• Recidivism

Many challenges for EMS modeling

• Health metrics needed

• Information collected at scene

• Equity models a good vehicle for examining health measures (access, cost, efficacy)

41

Healthcare

Transportation

Public sector

Disasters and Homeland security

42

EMS response during/after extreme events

43

EMS service largely dependent on other interdependent systems and networks

E.g., Health risks during/after hurricanes:• Increased mortality, traumatic injuries, low-priority calls• Carbon monoxide poisoning, Electronic health devices

* Caused by power failures

Decisions may be very different during disasters• Ask patients to wait for service• Patient priorities may be dynamic (not static)• Evacuate patients from hospitals• Massive coordination with other agencies (mutual aid)

Data needs are real: what is going on?• Descriptive analytics: what is happening?• Predictive analytics: what will happen?• Prescriptive analytics: what do we do about it?

Thank you!

44

1. McLay, L.A., Mayorga, M.E., 2013. A model for optimally dispatching ambulances to emergency calls with classification errors in patient priorities. IIE Transactions 45(1), 1—24.

2. McLay, L.A., Mayorga, M.E., 2011. Evaluating the Impact of Performance Goals on Dispatching Decisions in Emergency Medical Service. IIE Transactions on Healthcare Service Engineering 1, 185 – 196

3. McLay, L.A., Mayorga, M.E., 2014. A dispatching model for server-to-customer systems that balances efficiency and equity. To appear in Manufacturing & Service Operations Management, doi:10.1287/msom.1120.0411

4. Ansari, S., McLay, L.A., Mayorga, M.E., 2015. A maximum expected covering problem for locating and dispatching servers. To appear in Transportation Science.

5. Kunkel, A., McLay, L.A. 2013. Determining minimum staffing levels during snowstorms using an integrated simulation, regression, and reliability model. Health Care Management Science 16(1), 14 – 26.

6. McLay, L.A., Moore, H. 2012. Hanover County Improves Its Response to Emergency Medical 911 Calls. Interfaces 42(4), 380-394.7. Leclerc, P.D., L.A. McLay, M.E. Mayorga, 2011. Modeling equity for allocating public resources. Community-Based Operations Research: Decision

Modeling for Local Impact and Diverse Populations, Springer, p. 97 – 118.8. McLay, L.A., Brooks, J.P., Boone, E.L., 2012. Analyzing the Volume and Nature of Emergency Medical Calls during Severe Weather Events using

Regression Methodologies. Socio-Economic Planning Sciences 46, 55 – 66.9. McLay, L.A., 2011. Emergency Medical Service Systems that Improve Patient Survivability. Encyclopedia of Operations Research in the area of

“Applications with Societal Impact,” John Wiley & Sons, Inc., Hoboken, NJ (published online: DOI: 10.1002/9780470400531.eorms0296)10. McLay, L.A. and M.E. Mayorga, 2010. Evaluating Emergency Medical Service Performance Measures. Health Care Management Science 13(2),

124 - 136

laura@engr.wisc.edupunkrockOR.wordpress.combracketology.engr.wisc.edu

@lauramclay