Applications of Mathematics
to
Real-World Problems
Michelle Dunbar
Maths Teachers Day @ UoW
SMAS/SMART
June 25, 2013
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 1/28
Outline
Classification and Pattern Recognition.
The Travelling Salesman Problem.
Vehicle Routing Problems.
Oceanography and Surfing.
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 2/28
Classification and Pattern Recognition
Electronic mail sorting Google ‘goggles’ Cancer detection
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 3/28
Classification and Pattern Recognition
Past Future
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 4/28
Classification and Pattern Recognition
The Problem:
How do we ‘teach’ a computer to read handwriting?
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 5/28
Classification and Pattern Recognition
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 6/28
Classification and Pattern Recognition
Convert the shades of grey into
numbers:
- white := 0
- black := 9
Read across the rows and store
the numbers in a list
(0, 0, 0, . . . , 0, 4, 3, 7, 7, 9, 8, 7, 8, 7, 5, 0, 0, . . . , 0, 0, 0)
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 7/28
Classification and Pattern Recognition
Example (2D):
Find a line
y = mx+ b
that separates the • from
the △.
ut
ut ut
utut ut
ut utut ut
utut
ut
bb
b
b
bb
bb
b b
b
b
b
A
B
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 8/28
Classification and Pattern Recognition
Example (2D):
Find a line
y = mx+ b
that separates the • from
the △.
ut
ut ut
utut ut
ut utut ut
utut
ut
bb
b
b
bb
bb
b b
b
b
b
A
B
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 8/28
Classification and Pattern Recognition
Example (2D):
Find a line
y = mx+ b
that separates the • from
the △.
ut
ut ut
utut ut
ut utut ut
utut
ut
bb
b
b
bb
bb
b b
b
b
b
A
B
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 8/28
Classification and Pattern Recognition
Example (2D):
Find a line
y = mx+ b
that separates the • from
the △.
ut
ut ut
utut ut
ut utut ut
utut
ut
bb
b
b
bb
bb
b b
b
b
b
A
B
ut
ut
b
b
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 8/28
Classification and Pattern Recognition
Example (2D):
Find a line
y = mx+ b
that separates the • from
the △.
The ‘best’ line has the
largest ‘buffer zone’.
ut
ut ut
utut ut
ut utut ut
utut
ut
bb
b
b
bb
bb
b b
b
b
b
A
B
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 8/28
Classification and Pattern Recognition
Example (2D):
Find a line
y = mx+ b
that separates the • from
the △.
The ‘best’ line has the
largest ‘buffer zone’.
ut
ut ut
utut ut
ut utut ut
utut
ut
bb
b
b
bb
bb
b b
b
b
b
A
B
ut
ut
b
b
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 8/28
Classification and Pattern Recognition
Example: Non-linear separator
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 9/28
Classification and Pattern Recognition
Example: Non-linear separator
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 10/28
Classification and Pattern Recognition
Text translation Identifying landmarks
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 11/28
The Travelling Salesman Problem (TSP)
Problem:
Visit every city
exactly once.
Find the minimal
cost tour
(return to start).
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 12/28
The Travelling Salesman Problem (TSP)
Problem:
Visit every city
exactly once.
Find the minimal
cost tour
(return to start).
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 13/28
The Travelling Salesman Problem (TSP)
Problem:
Visit every city
exactly once.
Find the minimal
cost tour
(return to start).
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 14/28
The Travelling Salesman Problem (TSP)
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 15/28
TSP for Efficient Circuit Board Production
Problem:
100s–1000s of holes to drill.
Want to do this efficiently.
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 16/28
TSP for Efficient Circuit Board Production
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 17/28
Aircraft Routing and Crew Pairing
Aircraft Crew
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 18/28
Aircraft Routing and Crew Pairing
A simple network:
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 19/28
Aircraft Routing and Crew Pairing
The objective is to design a schedule for both aircraft and crew.
Ensure every flight is covered exactly once by an aircraft and crew.
Determine the optimal aircraft routes and crew pairings.
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 20/28
Aircraft Routing and Crew Pairing
Time
Space
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 21/28
Aircraft Routing and Crew Pairing
Time
SpaceSYD
MEL
AKL
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 21/28
Aircraft Routing and Crew Pairing
Time
SpaceSYD
MEL
AKL0900 1030
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 21/28
Aircraft Routing and Crew Pairing
Time
SpaceSYD
MEL
AKL0900 1030 1145 1430
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 21/28
Aircraft Routing and Crew Pairing
Time
SpaceSYD
MEL
AKL0900 1030 1145 1430
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 21/28
Aircraft Routing and Crew Pairing
Time
SpaceSYD
MEL
AKL0900 1030 1145 1430
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 21/28
Aircraft Routing and Crew Pairing
200 400 600 800 1000 1200 1400 16000
2
4
6
8
10
12
14
16
18
20
22
Aircraft Number 1
Aircraft Number 2
Aircraft Number 3
Aircraft Number 4
Aircraft Number 5
Aircraft Number 6
Aircraft Number 7
Aircraft Number 8
Aircraft Number 9
Aircraft Number 10
Time (mins)
Plot of the flight durations for each flight and the corresponding aircraft route to which it belongs.
15
5
11
3
10
2
39
1
6
4
25
12
23
9
19
7
49
8
14
13
32
20
30
18
27
17
16
22
21
42
29
40
28
34
24
26
31
33
51
36
47
38
45
35
37
41
43
46
44
48
50
53
52
dold
= 17d
new = 7
dold
= 17d
new = 9
dold
= 7d
new = 3
dold
= 69d
new = 39
dold
= 9d
new = 14
dold
= 157d
new = 40
dold
= 13d
new = 6
dold
= 20d
new = 5
dold
= 32d
new = 10
dold
= 37d
new = 12
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 22/28
Oceanography
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 23/28
Surfing
Two concepts:
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 24/28
Surfing
The goal The optimal path
Brachistochrone (cycloid)
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 25/28
Surfing
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 26/28
Surfing
The cycloid
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 27/28
Surfing
Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 28/28
