The Theory of the
Chapter 5
Simplex Method
1. Geometric solution procedure ( for 2-dimensional problem)
2. Tableau form solution procedure
3. Matrix form solution procedure & avoid unnecessary computation
It streamlines the procedure considerably for computer
implementation.
Revised Simplex Method
5.1 Foundations of The simplex method
Geometric Concept
(Geometrical Interpretation)
2 - dimensional
dimensional
Algebraic Procedure
(Algebraic Interpretation)
n
- dimensional
dimensional
How thecorner feasible solutionsis identified algebraically?
Geometric Meaning and Algebraic Procedures:
Q: Why do we study thetheory of linearprogramming?
Starting at vertex ( 0, 0)
18
2
3
12
2
4
0
5
3
5
2
1
4
2
3
1
2
1
x
x
x
x
x
x
x
x
x
z
Move to the adjacent
vertex (0, 6).
Optimal? No
At vertex (0, 6)
1
4
1
3
2
4
1
4
5
5
3
30
2
4
1
6
2
3
6
z
x
x
x
x
x
x
x
x
x
Move to the adjacent
vertex (2, 6).
Optimal? No
At vertex (2, 6)
2
3
1
3
1
6
2
1
2
3
1
3
1
36
2
3
5
4
1
4
2
5
4
3
5
4
x
x
x
x
x
x
x
x
x
x
z
Stop
Optimal? Yes
constraint boundary equations
constraint boundary
corner-point solution
corner-point feasible solution
boundary of the feasible region
corner-point infeasible solution
For any LP with
n
decision variables, each
CPF solution
is a
simultaneous solution of
n
constraint boundary equations.
1
3
2
4
1
2
5
4
2
12
3
2
18
x
x
x
x
x
x
x
edge of a feasible region:
It lies at the intersection of
n
–
1 constraint equations.
Simplex Method:
Move from one
corner-point feasible
solution to an adjacent
corner-point feasible
solution.
Geometric interpretation
Algebraic Procedure
1
3
2
4
1
2
5
4
2
12
3
2
18
x
x
x
x
x
x
x
Two “adjacent” corner
-point feasible solutions:
a. Geometric meaning: e.g. (0, 0) and (0, 6), (0, 0) and (4, 0)
b. Algebraic meaning
: The defining equations. of two “adjacent”
corner-point
feasible
solutions.
1
2
0
(0,0) :
0
x
x
1
2
0
(0,6):
2
12
x
x
1
2
0
(0,0) :
0
x
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- Chen
- Optimization, CPF Solutions
