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Integer programming for locating ambulances
1. Integer programming for locating
ambulances
Laura Albert McLay
The Industrial & Systems Engineering Department
University of Wisconsin-Madison
laura@engr.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. CMMI -1054148, 1444219.
2. The problem
โข We want to locate ๐ ambulances at stations in a
geographic region to โcoverโ the most calls in 9
minutes
โข What we need to include:
1. Different call volumes at different locations
2. Non-deterministic travel times
3. Each ambulance responds to roughly the
same number of calls
4. Ambulances that are not always available
(backup coverage is important)
2
4. Anatomy of a 911 call
Response time
Service provider:
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
4
5. Objective functions
โข NFPA standard yields a coverage objective function
for response time threshold (RTT)
โข Most common RTT: nine minutes for 80% of calls
โข A call with response time of 8:59 is covered
โข A call with response time of 9:00 is not covered
Why RTTs?
โข Easy to measure
โข Intuitive
โข Unambiguous
5
6. Coverage models for EMS
โข Expected coverage objective
โข Maximize expected number of calls covered by a 9
minute response time interval
โข Coverage issues:
โข Ambulance unavailability: Ambulances not available
when servicing a patient (spatial queuing)
โข Fractional coverage: coverage is not binary due to
uncertain travel times
โข Other issues:
โข Which ambulance to send? As backup?
โข Side constraints:
โข Balanced workload
6
8. Why use optimization models?
Because it helps you identify solutions that are not
intuitive. This adds value!
8
MODEL
9. Model 1: covering location models
Adjusts for different call volumes at different locations (#1),
but does not include our other needs
9
10. Model 1 formulation
Parameters
โข ๐ = set of demand locations
โข ๐ = set of stations
โข ๐๐ = demand at ๐ โ ๐
โข ๐๐๐= fraction of calls at location ๐ โ
๐ that can be reached by 9
minutes from an ambulance from
station ๐ โ ๐.
โข When travel times are
deterministic, then ๐๐๐ โ {0, 1}
โข ๐ฝ๐ โ ๐ = subset of stations that can
respond to calls at ๐ within 9
minutes, ๐ โ ๐:
โข ๐ฝ๐ = ๐: ๐๐๐ = 1
โข ๐ฝ๐ = all stations that encircle ๐
Decision variables
โข ๐ฆ๐ = 1 if we locate an ambulance
at station ๐ โ ๐ (and 0 otherwise)
โข ๐ฅ๐ = 1 if calls at ๐ โ ๐ are covered
(and 0 otherwise)
โข We must locate all ๐ ambulances
at stations
โข Linking constraint: a location ๐ โ
๐ is covered only if one of the
stations in ๐ฝ๐ has an ambulance
โข Integrality constraints on the
variables
10
Constraints (in words)
11. Maximal Covering Location Problem #1
max
๐โ๐
๐๐ ๐ฅ๐
Subject to:
๐ฅ๐ โค
๐โ๐ฝ ๐
๐ฆ๐
๐โ๐
๐ฆ๐ = ๐
๐ฅ๐ โ 0, 1 , ๐ โ ๐
๐ฆ๐ โ 0, 1 , ๐ โ ๐
11
Church, Richard, and Charles R. Velle. "The maximal covering location problem." Papers in
regional science 32, no. 1 (1974): 101-118.
13. Example solution
Model 1 solutions
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
Limitations
13
โข Does not look at backup coverage
โข Assumes all calls in circles are 100%
โcoveredโ
โข Does not assign calls to stations
โข Each ambulance does not respond to
same number of calls
14. Model 2: p-median models to
maximize expected coverage
Addresses:
1. Different call volumes at different locations
2. Non-deterministic travel times
3. Each ambulance responds to roughly the same number of calls
Does not address #4: backup coverage
14
17. Model formulation
Parameters
โข ๐ = set of demand locations
โข ๐ = set of stations
โข ๐๐ = demand at ๐ โ ๐
โข ๐๐๐= fraction of calls at location
๐ โ ๐ that can be reached by 9
minutes from an ambulance
from station ๐ โ ๐.
โข ๐๐๐ โ ๐, ๐ (fractional!)
โข ๐ = lower bound on number of
calls assigned to each open
station
โข ๐ = capacity of each station (max
number of ambulances, ๐ = 1)
Decision variables
โข ๐ฆ๐ = 1 if we locate an ambulance at
station ๐ โ ๐ (and 0 otherwise)
โข ๐๐๐ = 1 if calls at ๐ โ ๐ต are assigned
to station ๐ (and 0 otherwise)
โข We must locate all ๐ ambulances at
stations
โข Each open station must have at least
๐ calls assigned to it
โข Linking constraint: a location ๐ โ ๐ต
can be assigned to station ๐ only if ๐
has an ambulance
โข Each location must be assigned to at
most one (open) station
โข Integrality constraints on the
variables
17
Constraints (in words)
18. Integer programming bag of tricks
โข ๐ฟ = 1 โ ๐โ๐ ๐๐ ๐ฅ๐ โค ๐
โข ๐โ๐ ๐๐ ๐ฅ๐ + ๐ ๐ฟ โค ๐ + ๐
โข ๐โ๐ ๐๐ ๐ฅ๐ โค ๐ โ ๐ฟ = 1
โข ๐โ๐ ๐๐ ๐ฅ๐ โ ๐ โ ๐ ๐ฟ โค ๐ + ๐
โข ๐ฟ = 1 โ ๐โ๐ ๐๐ ๐ฅ๐ โฅ ๐
โข ๐โ๐ ๐๐ ๐ฅ๐ + ๐ ๐ฟ โฅ ๐ + ๐
โข ๐โ๐ ๐๐ ๐ฅ๐ โฅ ๐ โ ๐ฟ = 1
โข ๐โ๐ ๐๐ ๐ฅ๐ โ ๐ + ๐ ๐ฟ โค ๐ โ ๐
โข ๐ฟ is a binary variable
โข ๐โ๐ ๐๐ ๐ฅ๐ constraint
LHS
โข ๐: constraint RHS
โข ๐: upper bound on
๐โ๐ ๐๐ ๐ฅ๐ โ ๐
โข ๐: lower bound on
๐โ๐ ๐๐ ๐ฅ๐ โ ๐
โข ๐: constraint violation
amount (0.01 or 1)
18
Each open station must have at least ๐ calls assigned to it
20. Bag of tricks
โข We want to use this one:
โข ๐ฟ = 1 โ ๐โ๐ ๐๐ ๐ฅ๐ โฅ ๐
โข ๐โ๐ ๐๐ ๐ฅ๐ + ๐ ๐ฟ โฅ ๐ + ๐
For this:
โข ๐ฆ๐ = 1 โ ๐โ๐ ๐๐ ๐ฅ๐๐ โฅ ๐
โข Step 1:
โข Find ๐: lower bound on ๐โ๐ ๐๐ ๐ฅ๐๐ โ ๐
โข This is โ๐ since we could assign no calls to ๐
โข Step 2: Put it together and simplify
โข ๐โ๐ ๐๐ ๐ฅ๐๐ โ ๐ ๐ฆ๐ โฅ โ๐ + ๐ simplifies to
โข ๐โ๐ ๐๐ ๐ฅ๐๐ โฅ ๐ ๐ฆ๐
โข Note: this will also work when we let up to ๐
ambulances located at a station
โข ๐ฆ๐ โ 0, 1, โฆ , ๐
20
23. Model 3: p-median models to maximize
expected (backup) coverage
Addresses:
1. Different call volumes at different locations
2. Non-deterministic travel times
3. Each ambulance responds to roughly the same number of calls
4. Ambulances that are not always available (backup coverage is
important)
23
24. Ambulances that are not always available
Letโs model this as follows:
โข Letโs pick the top 3 ambulances that should respond to each call
โข Ambulance 1, 2, 3 responds to a call with probability ๐1, ๐2, ๐3
with ๐1 + ๐2 + ๐3 < 1.
Ambulances are busy with probability ๐
1. First choice ambulance response with probability ฯ1 โ 1 โ ๐
2. Second choice ambulance response with probability ฯ2 โ
๐(1 โ ๐)
3. Third choice ambulance response with probability ฯ3 โ
๐2 1 โ ๐
โข If ๐ = 0.3 then ๐1 = 0.7, ๐2 = 0.21, ๐3 = 0.063 (sum to 0.973)
โข If ๐ = 0.5 then ๐1 = 0.5, ๐2 = 0.25, ๐3 = 0.125 (sum to 0.875)
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25. New variables
We need to change this variable:
โข ๐ฅ๐๐ = 1 if calls at ๐ โ ๐ are assigned to station ๐ (and
0 otherwise)
to this:
โข ๐ฅ๐๐๐ = 1 if calls at ๐ โ ๐ are assigned to station ๐ at
the ๐ ๐กโ priority, ๐ = 1, 2, 3.
Note: this is cool!
This tells us which ambulance to send, not just where
to locate the ambulances.
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26. Model formulation
Parameters
โข ๐ = set of demand locations
โข ๐ = set of stations
โข ๐๐ = demand at ๐ โ ๐
โข ๐๐๐= fraction of calls at location
๐ โ ๐ that can be reached by 9
minutes from an ambulance
from station ๐ โ ๐.
โข ๐๐๐ โ 0,1
โข ๐ = lower bound on number of
calls assigned to each open
station
โข ๐ ๐, ๐ ๐, ๐ ๐ = proportion of calls
when the 1st, 2nd, and 3rd
preferred ambulance responds
Decision variables
โข ๐ฆ๐ = 1 if we locate an ambulance at
station ๐ โ ๐ (and 0 otherwise)
โข ๐๐๐๐ = 1 if station ๐ is the ๐th preferred
ambulance for calls at ๐ โ ๐ต, ๐ = ๐, ๐, ๐.
โข We must locate all ๐ ambulances at
stations (at most one per station)
โข Each open station must have at least
๐ calls assigned to it
โข Linking constraint: a location ๐ โ ๐ can be
assigned to station ๐ only if ๐ has an
ambulance
โข Each location must be assigned to 3
(open) stations
โข 3 different stations
โข Stations must be assigned in a specified
order
โข Integrality constraints on the variables26
Constraints (in words)
29. Related blog posts
โข A YouTube video about my research
โข In defense of model simplicity
โข Operations research, disasters, and science
communication
โข Domino optimization art
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