Integer programming for locating ambulances

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Integer programming for locating
ambulances
Laura Albert McLay
The Industrial & Systems Engineering Department
University ...

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

The solution:
3

Anatomy of a 911 call
Response time
Service provider:
Emergency 911 call
Unit
dispatched
Unit is en
route
Unit arrives
at ...

Objective functions
• NFPA standard yields a coverage objective function
for response time threshold (RTT)
• Most common R...

Coverage models for EMS
• Expected coverage objective
• Maximize expected number of calls covered by a 9
minute response t...

$Ambulance unavailability /fractional coverage 7 Fractional coverage / Facility unavailability 0 1 2 3 4 5 6 7 8 9 10 0 1 2...$

Why use optimization models?
Because it helps you identify solutions that are not
intuitive. This adds value!
8
MODEL

Model 1: covering location models
Adjusts for different call volumes at different locations (#1),
but does not include our...

$Model 1 formulation Parameters • 𝑁 = set of demand locations • 𝑆 = set of stations • 𝑑𝑖 = demand at 𝑖 ∈ 𝑁 • 𝑟𝑖𝑗= fraction ...$

Maximal Covering Location Problem #1
max
𝑖∈𝑁
𝑑𝑖 𝑥𝑖
Subject to:
𝑥𝑖 ≤
𝑗∈𝐽 𝑖
𝑦𝑗
𝑗∈𝑆
𝑦𝑗 = 𝑠
𝑥𝑖 ∈ 0, 1 , 𝑖 ∈ 𝑁
𝑦𝑖 ∈ 0, 1 , 𝑗 ∈ ...

Example inputs
Input parameters Coverage
12
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
Covering all locations
is imposs...

Example solution
Model 1 solutions
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0
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9
10
Limitations
13
• Does not look at backup ...

Model 2: p-median models to
maximize expected coverage
Addresses:
1. Different call volumes at different locations
2. Non-...

Non-deterministic travel times
15
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8 9 10
Coverage
Distance (miles)
Partial coverag...

Let’s extend our formulation
16

$Model formulation Parameters • 𝑁 = set of demand locations • 𝑆 = set of stations • 𝑑𝑖 = demand at 𝑖 ∈ 𝑁 • 𝑟𝑖𝑗= fraction of...$

Integer programming bag of tricks
• 𝛿 = 1 → 𝑗∈𝑁 𝑎𝑗 𝑥𝑗 ≤ 𝑏
• 𝑗∈𝑁 𝑎𝑗 𝑥𝑗 + 𝑀 𝛿 ≤ 𝑀 + 𝑏
• 𝑗∈𝑁 𝑎𝑗 𝑥𝑗 ≤ 𝑏 → 𝛿 = 1
• 𝑗∈𝑁 𝑎𝑗 𝑥𝑗 − ...

Maximal Covering Location Problem #2
max
𝑖∈𝑁 𝑗∈𝑆
𝑑𝑖 𝑟𝑖𝑗 𝑥𝑖𝑗
Subject to:
𝑥𝑖𝑗 ≤ 𝑦𝑗, 𝑖 ∈ 𝑁, 𝑗 ∈ 𝑆
𝑗∈𝑆
𝑥𝑖𝑗 = 1, 𝑖 ∈ 𝑁
𝑗∈𝑆
𝑦𝑗 =...

Bag of tricks
• We want to use this one:
• 𝛿 = 1 → 𝑗∈𝑁 𝑎𝑗 𝑥𝑗 ≥ 𝑏
• 𝑗∈𝑁 𝑎𝑗 𝑥𝑗 + 𝑚 𝛿 ≥ 𝑚 + 𝑏
For this:
• 𝑦𝑗 = 1 → 𝑖∈𝑁 𝑑𝑖 𝑥𝑖𝑗...

0 1 2 3 4 5 6 7 8 9 10
0
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10
Example solution
No minimum per station
obj = 194.6
0 1 2 3 4 5 6 7 8 9 10
0...

Example solution
Model 1
0 1 2 3 4 5 6 7 8 9 10
0
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2
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7
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9
10
Model2
0 1 2 3 4 5 6 7 8 9 10
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1...

Model 3: p-median models to maximize
expected (backup) coverage
Addresses:
1. Different call volumes at different location...

Ambulances that are not always available
Let’s model this as follows:
• Let’s pick the top 3 ambulances that should respon...

New variables
We need to change this variable:
• 𝑥𝑖𝑗 = 1 if calls at 𝑖 ∈ 𝑁 are assigned to station 𝑗 (and
0 otherwise)
to ...

$Model formulation Parameters • 𝑁 = set of demand locations • 𝑆 = set of stations • 𝑑𝑖 = demand at 𝑖 ∈ 𝑁 • 𝑟𝑖𝑗= fraction of...$

Maximal Covering Location Problem #3
max
𝑖∈𝑁 𝑗∈𝑆 𝑘=1
3
𝑑𝑖 𝜋 𝑘 𝑟𝑖𝑗 𝑥𝑖𝑗𝑘
Subject to:
𝑥𝑖𝑗𝑘 ≤ 𝑦𝑗, 𝑖 ∈ 𝑁, 𝑗 ∈ 𝑆, 𝑘 = 1,2,3 [no ...

Example solution
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
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5
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7
8
9
10
28
0 1 2 3 4 5 6 7 8 9 10
0
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0 1 2 3 4 ...

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