2015 Fuzzy Vance Lecture in Mathematics at Oberlin College:

1. 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
1
This 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.

2. 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

3. Collaborators
Maria Mayorga
North Carolina State University
Students
Sardar Ansari
Ben Grannan
Philip Leclerc
3

4. 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)
1972
1972

5. Early urban operations research models
5
Set cover / maximum cover models
How 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 models
How 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.
Analytics
How 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 models
What 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.

6. 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 call
Unit
dispatched
Unit is en
route
Unit arrives
at scene
Service/care
provided
Unit leaves
scene
Unit arrives
at hospital
Patient
transferred
Unit returns
to service
6

7. 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. EMS
Paid 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

8. Operationalizing recommendations
Priority dispatch:
… but which ambulance when there is a choice?
8
Type Capability Response Time
Priority 1
Advanced Life Support (ALS) Emergency
Send ALS and a fire engine/BLS
E.g., 9 minutes
(first unit)
Priority 2
Basic Life Support (BLS) Emergency
Send BLS and a fire engine if available
E.g., 13 minutes
Priority 3
Not an emergency
Send BLS
E.g., 16 minutes

9. 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

10. Response times vs. cardiac arrest survival
10
CDF of
calls for
service
covered
Response time (minutes) 9
80%

11. 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”

12. 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

13. 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

14. 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

15. Dispatching models
15

16. Optimal dispatching policies
using Markov decision process models
911 call
Unit
dispatched
Unit is en
route
Unit arrives
at scene
Service/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
priority
HT or LT
16
Information changes over the course of a call
Decisions made based on classified priority.
Performance metrics based on true priority.
Classified customer risk
Map Priority 1, 2, 3 call types to high-priority (𝐻) or low-priority (𝐿)
Calls of the same type treated the same
True customer risk
Map all call types to high-priority (𝐻 𝑇) or low-priority (𝐿 𝑇)

17. Optimal dispatching policies
using Markov decision process models
Optimality equations:
𝑉𝑘 𝑆 𝑘 = max
𝑥 𝑘∈𝑋(𝑆 𝑘)
𝐸 𝑢𝑖𝑗
𝜔
𝑥 𝑘 + 𝑉𝑘+1 𝑆 𝑘+1 𝑆 𝑘, 𝑥 𝑘, 𝜔
Formulate problem as an undiscounted, infinite-horizon,
average reward Markov decision process (MDP) model
• The state 𝒔 𝒌  𝑆 describes the combinations of busy and free ambulances.
• 𝑋(𝒔 𝑘) denotes the set of actions (ambulances to dispatch) available in state 𝒔 𝒌.
• Reward 𝑢𝑖𝑗
𝜔 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

18. 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 Pr3
HT
HT
Pr1 Pr2 Pr3
HT
Pr1 Pr2 Pr3
HT
Informational
accuracy captured by:
𝛼 =
𝑃 𝐻 𝑇 𝐻
𝑃(𝐻 𝑇|𝐿)
18
Classified high-priority
Classified low-priority
Improved accuracy

19. Structural properties
RESULT
It is more beneficial for a server to be idle than busy.
RESULT
It is more beneficial for a server to be serving closer customers.
RESULT
It is not always optimal to send the closest ambulance, even for
high priority calls.

20. Coverage
0 10 20 30 40 50
0.405
0.41
0.415
0.42
0.425
0.43
0.435
0.44
0.445

Expectedcoverage
Optimal Policy, Case 1
Optimal Policy, Case 2
Closest Ambulance
20
Better accuracy

21. Low and high priority calls
Conditional probability that the closest unit is dispatched given
initial classification
High-priority calls Low-priority calls0 10 20 30 40 50
0.98
0.985
0.99
0.995
1
1.005

Proportionclosestambulanceisdispatched
Closest Ambulance
Optimal Policy, Case 1
Optimal Policy, Case 2
0 10 20 30 40 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

Proportionclosestambulanceisdispatched
Closest Ambulance
Optimal Policy, Case 1
Optimal Policy, Case 2
Classified high-priority Classified low-priority
21

22. Case 1 (𝛼 = ∞), Case 2 policies
High-priority calls
Case 2: First to send to high-priority calls
Station
1
2
3
4
Case 2: Second to send to high-priority calls
Station
1
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

23. Districting and location models
23

24. Early location models
24
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
Maximum coverage model
How can we “cover” the maximum
number of locations with 𝑝
ambulances?
𝒑-median model
How can we locate 𝑝 ambulances such
that we minimize the average distance
an ambulance must travel to a call?

25. 0 1 2 3 4 5 6 7 8 9 10
0
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.

26. Districting model
Mixed Integer Linear Program
max 𝑤∈𝑊 𝑗∈𝐽 𝑝=1
𝑠
𝑚=1
min(𝑐 𝑤,𝑠−𝑝+1)
𝑞 𝑗𝑝𝑚 1 − 𝑟 𝑚
𝑟 𝑝−1
𝜆𝑗
𝐻
𝑅 𝑤𝑗 𝑧 𝑤𝑗𝑝𝑚
subject to
𝑝=1
𝑠
𝑚=1
𝜅 𝑤𝑝
𝑧 𝑤𝑗𝑝𝑚 ≤ 1, 𝑗 ∈ 𝐽, 𝑤 ∈ 𝑊
𝑝=1
𝑠
𝑚=1
𝜅 𝑤𝑝
𝑧 𝑤𝑗𝑝𝑚 ≤ 𝑦 𝑤, 𝑗 ∈ 𝐽, 𝑤 ∈ 𝑊
𝑤∈𝑊 𝑥 𝑤𝑗𝑝 = 1, 𝑗 ∈ 𝐽, 𝑝 = 1, … , 𝑠
𝑝=1
𝑠
𝑥 𝑤𝑗𝑝 = 𝑦 𝑤 , 𝑗 ∈ 𝐽, 𝑤 ∈ 𝑊
𝑥 𝑤𝑗𝑝′ = 𝑝=max 1,𝑝′−𝑐 𝑤+1
𝑝′
𝑚=𝑝′−𝑝+1
𝜅 𝑤𝑝
𝑧 𝑤𝑗𝑝𝑚 ,
𝑗 ∈ 𝐽, 𝑤 ∈ 𝑊, 𝑝′
= 1, … , 𝑠
𝑤∈𝑊 𝑦 𝑤 = 𝑠
𝑦 𝑤 ≤ 𝑐 𝑤, 𝑤 ∈ 𝑊
𝑟 − 𝛿 𝑦 𝑤 ≤
𝑗∈𝐽 𝑝=1
𝑠
𝑚=1
𝜅 𝑤𝑝
𝜆𝑗
𝐻
+ 𝜆𝑗
𝐿
𝑞𝑗𝑝𝑚 1 − 𝑟 𝑚
𝑟 𝑝−1
𝜏 𝑤𝑗 𝑧 𝑤𝑗𝑝𝑚
≤ 𝑟 + 𝛿 𝑦 𝑤, 𝑤 ∈ 𝑊
𝑥 𝑤𝑗′1 ≥ 𝑥 𝑤𝑗1, 𝑗 ∈ 𝐽, 𝑤 ∈ 𝑊, 𝑗′
∈ 𝑁 𝑤𝑗
𝑦 𝑤 ∈ 𝑍0
+
, 𝑤 ∈ 𝑊
𝑥 𝑤𝑗𝑝 ∈ 0,1 , 𝑤 ∈ 𝑊, 𝑗 ∈ 𝐽, 𝑝 = 1, … , 𝑠
𝑧 𝑤𝑗𝑝𝑚 ∈ 0,1 , 𝑤 ∈ 𝑊, 𝑗 ∈ 𝐽, 𝑝 = 1, … , 𝑠, 𝑚 = 1, … , 𝑐 𝑤
26
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 𝑐 𝑤 per
station
Expected coverage
Contiguous first priority districts
Binary and integrality
constraints on the variables

27. Parameters
• 𝐽: set of all customer (demand) nodes
• 𝑊: set of all potential station locations
• 𝑠: total number of servers in the system
• 𝜆𝑗
𝐻
(𝜆𝑗
𝐿
): mean high-priority (low-priority)
call arrival rates from node 𝑗
• 𝜆: system-wide total call arrival rate
• 𝜏 𝑤𝑗: 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
• 𝑐 𝑤: capacity of station 𝑤
27
• 𝑟: system-wide average server utilization
• 𝑃𝑠: loss probability (probability that all 𝑠
servers are busy)
• 𝑅 𝑤𝑗: expected proportion of calls from 𝑗
that are reached by servers from station 𝑤
in nine minutes
• 𝑞 𝑗𝑝𝑚: correction factor for customer 𝑗's 𝑝th
priority server at which there are 𝑚 servers
located.
• 𝑁 𝑤𝑗: set of demand nodes that are
neighbors to 𝑗 and are closer to station 𝑤
than 𝑗.
Decision variables
• 𝑦 𝑤 = number of servers located at station 𝑤, 𝑤 ∈ 𝑊.
• 𝑧 𝑤𝑗𝑝𝑚= 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, … , 𝑐 𝑤 and 0 otherwise.
• 𝑥 𝑤𝑗𝑝= 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.

28. Issue
28
Queuing model
Spatial queuing
outputs
Spatial queuing
inputs
Mixed integer
programming
model
How it usually works:
What we need:

29. The solution: iterate
29
Spatial
queuing
model
Mixed
integer
programming
model

30. Results
RESULT
The 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

31. Hanover County: example

32. First Priority Districts
• Base Model: no workload balancing

33. LBM Model
First Priority Districts for WD12am6am
• LBM (2 servers at Ashland and 1 at every other)

34. First Priority Districts
• Workload balancing
• Contiguous first priority districts
Two ambulances

35. 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

36. 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

37. Results quantify impact of using QRVs
37

38. 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.

39. Where do EMS systems need to go?
39

40. 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?

41. 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

42. Disasters and Homeland security
42

43. 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?

44. 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.edu
punkrockOR.wordpress.com
bracketology.engr.wisc.edu
@lauramclay