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Locating and dispatching ambulances
using discrete optimization methodologies
Laura Albert McLay
Industrial & Systems Engineering
University of Wisconsin-Madison
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