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.

The problem

• We want to locate 𝑠 ambulances at stations in a geographic region to “cover” the most calls in 9 minutes

• 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)

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The solution:

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Anatomy of a 911 call

Response time

Service provider:

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

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Objective functions

• NFPA standard yields a coverage objective function for response time threshold (RTT)• Most common RTT: nine minutes for 80% of calls

• A call with response time of 8:59 is covered

• A call with response time of 9:00 is not covered

Why RTTs?

• Easy to measure

• Intuitive

• Unambiguous

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Coverage models for EMS

• Expected coverage objective• Maximize expected number of calls covered by a 9

minute response time interval

• Coverage issues:• Ambulance unavailability: Ambulances not available

when servicing a patient (spatial queuing)

• Fractional coverage: coverage is not binary due to uncertain travel times

• Other issues:• Which ambulance to send? As backup?

• Side constraints:• Balanced workload

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Ambulance unavailability /fractional coverage

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Fractional coverage /

Facility unavailability

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Perfect coverage /

Availability

Why use optimization models?

Because it helps you identify solutions that are not intuitive. This adds value!

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MODEL

Model 1: covering location modelsAdjusts for different call volumes at different locations (#1), but does not include our other needs

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Model 1 formulation

Parameters• 𝑁 = set of demand locations

• 𝑆 = set of stations

• 𝑑𝑖 = demand at 𝑖 ∈ 𝑁

• 𝑟𝑖𝑗= fraction of calls at location 𝑖 ∈𝑁 that can be reached by 9 minutes from an ambulance from station 𝑗 ∈ 𝑆.

• When travel times are deterministic, then 𝑟𝑖𝑗 ∈ {0, 1}

• 𝐽𝑖 ⊂ 𝑆 = subset of stations that can respond to calls at 𝑖 within 9 minutes, 𝑖 ∈ 𝑁:

• 𝐽𝑖 = 𝑗: 𝑟𝑖𝑗 = 1

• 𝐽𝑖 = all stations that encircle 𝑖

Decision variables

• 𝑦𝑗 = 1 if we locate an ambulance at station 𝑗 ∈ 𝑆 (and 0 otherwise)

• 𝑥𝑖 = 1 if calls at 𝑖 ∈ 𝑁 are covered (and 0 otherwise)

• We must locate all 𝑆 ambulances at stations

• Linking constraint: a location 𝑖 ∈𝑁 is covered only if one of the stations in 𝐽𝑖 has an ambulance

• Integrality constraints on the variables

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Constraints (in words)

Maximal Covering Location Problem #1

max

𝑖∈𝑁

𝑑𝑖𝑥𝑖

Subject to:

𝑥𝑖 ≤

𝑗∈𝐽𝑖

𝑦𝑗

𝑗∈𝑆

𝑦𝑗 = 𝑠

𝑥𝑖 ∈ 0, 1 , 𝑖 ∈ 𝑁𝑦𝑖 ∈ 0, 1 , 𝑗 ∈ 𝑆

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Church, Richard, and Charles R. Velle. "The maximal covering location problem." Papers in

regional science 32, no. 1 (1974): 101-118.

Example inputsInput parameters Coverage

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Covering all locations is impossible!

Example solutionModel 1 solutions

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Limitations

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• Does not look at backup coverage

• Assumes all calls in circles are 100% “covered”

• Does not assign calls to stations

• Each ambulance does not respond to same number of calls

Model 2: p-median models to maximize expected coverageAddresses:

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 coverage14

Non-deterministic travel times

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0

0.2

0.4

0.6

0.8

1

1.2

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Co

vera

ge

Distance (miles)

Partial coverage from data 0-1 coverage

Expon. (Partial coverage from data)

Let’s extend our formulation

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Model formulation

Parameters

• 𝑁 = set of demand locations

• 𝑆 = set of stations

• 𝑑𝑖 = demand at 𝑖 ∈ 𝑁

• 𝑟𝑖𝑗= fraction of calls at location 𝑖 ∈ 𝑁 that can be reached by 9 minutes from an ambulance from station 𝑗 ∈ 𝑆.

• 𝒓𝒊𝒋 ∈ 𝟎, 𝟏 (fractional!)

• 𝒍 = lower bound on number of calls assigned to each open station

• 𝑐 = capacity of each station (max number of ambulances, 𝑐 = 1)

Decision variables• 𝑦𝑗 = 1 if we locate an ambulance at

station 𝑗 ∈ 𝑆 (and 0 otherwise)

• 𝒙𝒊𝒋 = 1 if calls at 𝒊 ∈ 𝑵 are assigned to station 𝒋 (and 0 otherwise)

• We must locate all 𝑠 ambulances at stations

• Each open station must have at least 𝒍 calls assigned to it

• Linking constraint: a location 𝒊 ∈ 𝑵can be assigned to station 𝒋 only if 𝒋has an ambulance

• Each location must be assigned to at most one (open) station

• Integrality constraints on the variables

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Constraints (in words)

Integer programming bag of tricks

• 𝛿 = 1 → 𝑗∈𝑁 𝑎𝑗𝑥𝑗 ≤ 𝑏

• 𝑗∈𝑁 𝑎𝑗𝑥𝑗 +𝑀 𝛿 ≤ 𝑀 + 𝑏

• 𝑗∈𝑁 𝑎𝑗𝑥𝑗 ≤ 𝑏 → 𝛿 = 1• 𝑗∈𝑁 𝑎𝑗𝑥𝑗 − 𝑚 − 𝜖 𝛿 ≤ 𝑏 + 𝜖

• 𝛿 = 1 → 𝑗∈𝑁 𝑎𝑗𝑥𝑗 ≥ 𝑏

• 𝑗∈𝑁 𝑎𝑗𝑥𝑗 +𝑚 𝛿 ≥ 𝑚 + 𝑏

• 𝑗∈𝑁 𝑎𝑗𝑥𝑗 ≥ 𝑏 → 𝛿 = 1• 𝑗∈𝑁 𝑎𝑗𝑥𝑗 − 𝑀 + 𝜖 𝛿 ≤ 𝑏 − 𝜖

• 𝛿 is a binary variable

• 𝑗∈𝑁 𝑎𝑗𝑥𝑗 constraint LHS

• 𝑏: constraint RHS

• 𝑀: upper bound on 𝑗∈𝑁 𝑎𝑗𝑥𝑗 − 𝑏

• 𝑚: lower bound on 𝑗∈𝑁 𝑎𝑗𝑥𝑗 − 𝑏

• 𝜖: constraint violation amount (0.01 or 1)

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Each open station must have at least 𝒍 calls assigned to it

Maximal Covering Location Problem #2

max

𝑖∈𝑁

𝑗∈𝑆

𝑑𝑖𝑟𝑖𝑗𝑥𝑖𝑗

Subject to:𝑥𝑖𝑗 ≤ 𝑦𝑗 , 𝑖 ∈ 𝑁, 𝑗 ∈ 𝑆

𝑗∈𝑆

𝑥𝑖𝑗 = 1, 𝑖 ∈ 𝑁

𝑗∈𝑆

𝑦𝑗 = 𝑠

𝑖∈𝑁

𝑑𝑖𝑥𝑖𝑗 ≥ 𝑙 𝑦𝑗 , 𝑗 ∈ 𝑆

𝑥𝑖𝑗 ∈ 0, 1 , 𝑖 ∈ 𝑁, 𝑗 ∈ 𝑆𝑦𝑖 ∈ 0, 1,… , 𝑐 , 𝑗 ∈ 𝑆

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Bag of tricks

• We want to use this one:• 𝛿 = 1 → 𝑗∈𝑁 𝑎𝑗𝑥𝑗 ≥ 𝑏

• 𝑗∈𝑁 𝑎𝑗𝑥𝑗 +𝑚 𝛿 ≥ 𝑚 + 𝑏

For this:

• 𝑦𝑗 = 1 → 𝑖∈𝑁 𝑑𝑖𝑥𝑖𝑗 ≥ 𝑙

• Step 1:• Find 𝑚: lower bound on 𝑖∈𝑁 𝑑𝑖𝑥𝑖𝑗 − 𝑙• This is −𝑙 since we could assign no calls to 𝑗

• Step 2: Put it together and simplify• 𝑖∈𝑁 𝑑𝑖𝑥𝑖𝑗 − 𝑙 𝑦𝑗 ≥ −𝑙 + 𝑙 simplifies to

• 𝑖∈𝑁 𝑑𝑖𝑥𝑖𝑗 ≥ 𝑙 𝑦𝑗

• Note: this will also work when we let up to 𝑐ambulances located at a station• 𝑦𝑖 ∈ 0, 1, … , 𝑐

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Example solution

No minimum per stationobj = 194.6

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Minimum of 60 per stationobj = 191.4

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60

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Example solution

Model 1

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Model2

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Model 3: p-median models to maximize expected (backup) coverageAddresses:

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)

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Ambulances that are not always available

Let’s model this as follows:

• Let’s pick the top 3 ambulances that should respond to each call

• Ambulance 1, 2, 3 responds to a call with probability 𝜋1, 𝜋2, 𝜋3with 𝜋1 + 𝜋2 + 𝜋3 < 1.

Ambulances are busy with probability 𝜌

1. First choice ambulance response with probability π1 ≈ 1 − 𝜌

2. Second choice ambulance response with probability π2 ≈𝜌(1 − 𝜌)

3. Third choice ambulance response with probability π3 ≈𝜌2 1 − 𝜌

• If 𝜌 = 0.3 then 𝜋1 = 0.7, 𝜋2 = 0.21, 𝜋3 = 0.063 (sum to 0.973)

• If 𝜌 = 0.5 then 𝜋1 = 0.5, 𝜋2 = 0.25, 𝜋3 = 0.125 (sum to 0.875)

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New variables

We need to change this variable:

• 𝑥𝑖𝑗 = 1 if calls at 𝑖 ∈ 𝑁 are assigned to station 𝑗 (and 0 otherwise)

to this:

• 𝑥𝑖𝑗𝑘 = 1 if calls at 𝑖 ∈ 𝑁 are assigned to station 𝑗 at the 𝑘𝑡ℎ priority, 𝑘 = 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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Model formulation

Parameters• 𝑁 = set of demand locations

• 𝑆 = set of stations

• 𝑑𝑖 = demand at 𝑖 ∈ 𝑁

• 𝑟𝑖𝑗= fraction of calls at location 𝑖 ∈ 𝑁 that can be reached by 9 minutes from an ambulance from station 𝑗 ∈ 𝑆.

• 𝑟𝑖𝑗 ∈ 0,1

• 𝑙 = lower bound on number of calls assigned to each open station

• 𝝅𝟏,𝝅𝟐, 𝝅𝟑 = proportion of calls when the 1st, 2nd, and 3rd

preferred ambulance responds

Decision variables• 𝑦𝑗 = 1 if we locate an ambulance at

station 𝑗 ∈ 𝑆 (and 0 otherwise)

• 𝒙𝒊𝒋𝒌 = 1 if station 𝒋 is the 𝒌th preferred ambulance for calls at 𝒊 ∈ 𝑵, 𝒌 = 𝟏, 𝟐, 𝟑.

• We must locate all 𝑆 ambulances at stations (at most one per station)

• Each open station must have at least 𝑙 calls assigned to it

• Linking constraint: a location 𝑖 ∈ 𝑁 can be assigned to station 𝑗 only if 𝑗 has an ambulance

• Each location must be assigned to 3 (open) stations

• 3 different stations• Stations must be assigned in a specified

order

• Integrality constraints on the variables26

Constraints (in words)

Maximal Covering Location Problem #3

max

𝑖∈𝑁

𝑗∈𝑆

𝑘=1

3

𝑑𝑖𝜋𝑘𝑟𝑖𝑗𝑥𝑖𝑗𝑘

Subject to:

𝑥𝑖𝑗𝑘 ≤ 𝑦𝑗 , 𝑖 ∈ 𝑁, 𝑗 ∈ 𝑆, 𝑘 = 1,2,3 [no longer needed]

𝑗∈𝑆

𝑥𝑖𝑗𝑘 = 1, 𝑖 ∈ 𝑁, 𝑘 = 1,2,3

𝑥𝑖𝑗1 + 𝑥𝑖𝑗2 + 𝑥𝑖𝑗3 ≤ 𝑦𝑗 , 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑆

𝑗∈𝑆

𝑦𝑗 = 𝑠

𝑖∈𝑁

𝑘=1

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𝑑𝑖𝜋𝑘𝑥𝑖𝑗𝑘 ≥ 𝑙 𝑦𝑗 , 𝑗 ∈ 𝑆

𝑥𝑖𝑗𝑘 ∈ 0, 1 , 𝑖 ∈ 𝑁, 𝑗 ∈ 𝑆, 𝑘 = 1,2,3𝑦𝑖 ∈ 0, 1 , 𝑗 ∈ 𝑆

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Example solution

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First priority assignments Second priority assignments Third priority assignments

𝜋1 = 0.65𝜋2 = 0.23𝜋3 = 0.08

Related blog posts

• A YouTube video about my research

• In defense of model simplicity

• Operations research, disasters, and science communication

• Domino optimization art

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