The Simplex Algorithm

140 Pages·2017·0.41 MB·English

Math 407: Linear Optimization
Lecture 5: Simplex Algorithm I
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1 Dictionaries, Augmented Matrices, the Simplex Tableau
Dictionaries
The Simplex Tableau
2 Basic Feasible Solutions (BFS)
3 The Grand Strategy: Pivoting
4 The Simplex Algorithm in Matrix Form
5 The Simplex Algorithm in Matrix Form
Lecture5:SimplexAlgorithmI Math407:LinearOptimization 2/35
max5x +4x +3x
1 2 3
s.t. 2x +3x +x ≤ 5
1 2 3
4x +x +2x ≤ 11
1 2 3
3x +4x +2x ≤ 8
1 2 3
0 ≤ x ,x ,x
1 2 3
At this point we only have one tool for attacking linear systems.
Gaussian elimination, a method for solving linear systems of equations.
Let’s try to use it to solve LPs.
The Simplex Algorithm
We develop a method for solving standard form LPs.
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At this point we only have one tool for attacking linear systems.
Gaussian elimination, a method for solving linear systems of equations.
Let’s try to use it to solve LPs.
The Simplex Algorithm
We develop a method for solving standard form LPs.
max5x +4x +3x
1 2 3
s.t. 2x +3x +x ≤ 5
1 2 3
4x +x +2x ≤ 11
1 2 3
3x +4x +2x ≤ 8
1 2 3
0 ≤ x ,x ,x
1 2 3
Lecture5:SimplexAlgorithmI Math407:LinearOptimization 3/35
Gaussian elimination, a method for solving linear systems of equations.
Let’s try to use it to solve LPs.
The Simplex Algorithm
We develop a method for solving standard form LPs.
max5x +4x +3x
1 2 3
s.t. 2x +3x +x ≤ 5
1 2 3
4x +x +2x ≤ 11
1 2 3
3x +4x +2x ≤ 8
1 2 3
0 ≤ x ,x ,x
1 2 3
At this point we only have one tool for attacking linear systems.
Lecture5:SimplexAlgorithmI Math407:LinearOptimization 3/35
Let’s try to use it to solve LPs.
The Simplex Algorithm
We develop a method for solving standard form LPs.
max5x +4x +3x
1 2 3
s.t. 2x +3x +x ≤ 5
1 2 3
4x +x +2x ≤ 11
1 2 3
3x +4x +2x ≤ 8
1 2 3
0 ≤ x ,x ,x
1 2 3
At this point we only have one tool for attacking linear systems.
Gaussian elimination, a method for solving linear systems of equations.
Lecture5:SimplexAlgorithmI Math407:LinearOptimization 3/35
The Simplex Algorithm
We develop a method for solving standard form LPs.
max5x +4x +3x
1 2 3
s.t. 2x +3x +x ≤ 5
1 2 3
4x +x +2x ≤ 11
1 2 3
3x +4x +2x ≤ 8
1 2 3
0 ≤ x ,x ,x
1 2 3
At this point we only have one tool for attacking linear systems.
Gaussian elimination, a method for solving linear systems of equations.
Let’s try to use it to solve LPs.
Lecture5:SimplexAlgorithmI Math407:LinearOptimization 3/35
max5x +4x +3x
1 2 3
s.t. 2x +3x +x ≤ 5
1 2 3
4x +x +2x ≤ 11
1 2 3
3x +4x +2x ≤ 8
1 2 3
0 ≤ x ,x ,x
1 2 3
Dictionaries
We must first build a linear system of equations that encodes all of the
information associated with the LP.
Lecture5:SimplexAlgorithmI Math407:LinearOptimization 4/35
Dictionaries
We must first build a linear system of equations that encodes all of the
information associated with the LP.
max5x +4x +3x
1 2 3
s.t. 2x +3x +x ≤ 5
1 2 3
4x +x +2x ≤ 11
1 2 3
3x +4x +2x ≤ 8
1 2 3
0 ≤ x ,x ,x
1 2 3
Lecture5:SimplexAlgorithmI Math407:LinearOptimization 4/35
x = 5 − [2x +3x +x ] ≥ 0,
4 1 2 3
x = 11 − [4x +x +2x ] ≥ 0,
5 1 2 3
x = 8 − [3x +4x +2x ] ≥ 0.
6 1 2 3
Slack Variables: x ,x ,x
4 5 6
Next we introduce a variable to represent the objective.
z = 5x +4x +3x .
1 2 3
This system of equations is called a dictionary for the the LP.
The slack variables and the objective are defined by the original decision
variables.
Slack Variables and Dictionaries
For each linear inequality we introduce a new variable, called a slack
variable, so that we can write each linear inequality as an equation.
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