Lecture 1: linear optimization: introduction · Lecture 1: linear optimization: introduction...
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1 • Definition of cost / objective function • Example of cost functions, affine functions, linear functions • Definition of constraints • Example of constraints, linear constraints • Linear programs • General form of a linear program • Sigma notation • Extended example 1: the transportation problem • Google maps • Extended example 2: the shortest path problem Lecture 1: linear optimization: introduction Minimize a cost or objective function (for ex. cost of production) or Maximize a cost or objective function (for ex. profit) with respect to constraints - Employee cannot work more than x hours a day - Only three people can use the same machine at a time - The pipeline’s maximal fuel throughput is y i.e. find a solution that is optimal within limits given What is optimization?
Transcript of Lecture 1: linear optimization: introduction · Lecture 1: linear optimization: introduction...
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• Definition of cost / objective function• Example of cost functions, affine functions, linear functions • Definition of constraints• Example of constraints, linear constraints• Linear programs• General form of a linear program• Sigma notation• Extended example 1: the transportation problem• Google maps• Extended example 2: the shortest path problem
Lecture 1: linear optimization: introduction
Minimize a cost or objective function (for ex. cost of production)or
Maximize a cost or objective function (for ex. profit)
with respect to constraints
- Employee cannot work more than x hours a day- Only three people can use the same machine at a time- The pipeline’s maximal fuel throughput is y
i.e. find a solution that is optimal within limits given
What is optimization?
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Example: cost of a mile as a function of the distan ce
What is a cost function?
[http://www.skyaid.org]
For some application, some cost functions look almo st like “lines”, i.e. are linear or affine . Example here: cost of building a dam as a function of the size of the pon d
Linear or affine cost functions
[http://home.hetnet.nl/~krekelberg/chapter3.htm]
cost
approximation
pond size
cost
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Linar functions
Price of gas ($):
1 2 3 4 5 6 7 8 9 10 11 12
1
23456789
10
gallon
0 gallon = $ 01 gallon = $ 2.252 gallons = $ 4.50
Affine functions
Price of renting a U-haul ($):
1 2 3 4 5 6 7 8 9 10 11 12
10
2030405060708090
X 10 (mile)
$19.95
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Linear or affine cost functions: formal definition
is the same as minimizing the linear cost function
Minimizing the affine cost function
A more general expression of the cost function:
Minimizing affine or linear function is the same
Minimizing a function f(x)
f
x
f(x)
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Minimizing affine or linear function is the same
Minimizing a function f(x) or f(x)+c is the same
f f(x)
x
f(x)+CC<0
Minimum obtained at the same x
Example: cost of building a wall
Cost of a pound of cement ($ per lb)Cost of a feet of steel beam ($ per ft)Weight of cement (lb)Length of steel beam (ft)
Total cost ($)
Note that none of the variables above has the same unit!
Note however that and and ha ve the same unit
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What is a constraint?
Your maximal budget for cement is
Your minimal budget for steel is
You want to spend twice as much for steel as for ce ment
You want to spend a given minimum amount for the wa ll
A constraint is a condition on variables which rest ricts the values they can take
Summary
Your optimization program incorporating all your co nstraints can be formulated as follows.
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Constraints in the form of equalities (I)
Sometimes, constraints are given in the form of equ alities
Example: you want to spend exactly twice as much fo r steel as for cement:
This is exactly the same as
and
Constraints in the form of equalities (II)
So you could rewrite the program in the following f orm:
One can thus assume that all constraints are always given in the form of inequalities.
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General form for a linear program
So you could rewrite the program in the following f orm:
Sigma notation
So you could rewrite the program in the following f orm:
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Example: the transportation problem (I)Paul’s farm produces 4 tons of apples per dayRon’s farm produces 2 tons of apples per dayMax’s factory needs 1 ton of apples per dayBob’s factory needs 5 tons of apples per day
George owns both farms and factories. He is paying the cost of shipping all the apples from the farms to the fa ctories.
The shipping costs for George are: Paul ����Max: 1000$ per tonRon ����Max: 1350$ per tonPaul ����Bob: 1250$ per tonRon ����Bob: 1450$ per ton
What is the best way to ship the apples?
Example: the transportation problem (II)
Paul
Ron Bob
Max
George pays for the shipping
1000$
1250$
1350$
1450$
4
2
1
5
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Example: the transportation problem (III)
Paul
Ron Bob
Max1000$
1250$
1350$
1450$
4
2
1
5
Example: the transportation problem (IV)
Paul
Ron Bob
Max1000$
1250$
1350$
1450$
4
2
1
5
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General form of the transportation problem
Paul
Ron Bob
Max1000$
1250$
1350$
1450$
4
2
1
5
Please, be lazy, do not write pages of equations…
Use summations, they leave you more time to go to t he movies
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Example: a « small » network (air traffic control)
[Robelin, Sun, Bayen, tech. rep., 2005]
Example: a « large » network (the internet)
[http://research.lumeta.com/ches/map/]
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Example: shortest path
Alice
Beatrix
Example: shortest path
Alice
Beatrix
A
2
3
4
5
6
7
8
9
10
B
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Example: shortest path: length of the shortest path
Alice
Beatrix
A
2
3
4
5
6
7
8
9
10
B
Example: shortest path: length of the shortest path
Alice
Beatrix
A
2
3
4
5
6
78
9
10
B
For every (i,j) on the shortest path
For every (i,j) not on the shortest path
Define
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Example: shortest path: length of the shortest path
Alice
Beatrix
A
2
3
4
5
6
78
9
10
B
For every (i,j) on the shortest path
For every (i,j) not on the shortest path
Define
Example: shortest path: choice of path
AliceBeatrix
For every (i,j) on the shortest path
For every (i,j) not on the shortest path
Define
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Example: shortest path
Define a graph (road network)
Call the cost to go from i to j (for exmple fu el burned)
For example is the cost to go from node 3 to node 4
AliceBeatrix
Example: shortest path
Define a graph (road network)
Take if Alice decides to go throug h link (i,j), zero otherwise
For example if Alice decides to use route (3,4)
AliceBeatrix
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Example: shortest path
All other are zero
Total length of this path:
AliceBeatrix
Example: shortest path
Total length:
AliceBeatrix
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Example: shortest path
Minimize:
Total length
1 if link (i,j) chosen0 otherwise
Cost of arc (i,j)
set of nodes j with direct connections to node i
Example: shortest path
minimize:
such that:
All links arriving at node i
All links leaving node i
set of nodes j with direct connections to node i
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Example: shortest path
minimize:
such that:
set of nodes j with direct connections to node A
A
Starting from A, Alice can only take one path
Example: shortest path
minimize:
such that:
set of nodes j with direct connections to node B
Arriving at B, one can only take one pathB
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Example: shortest path
minimize:
such that:
set of nodes j with direct connections to node i
Example: a « small » network (air traffic control)
[Robelin, Sun, Bayen, tech. rep., 2005]
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Example: a « large » network (the internet)
[http://research.lumeta.com/ches/map/]
