ISyE 323 Lecture #7
Prof. Jeﬀ Linderoth
September 24, 2009
ISyE Lecture #7
Outline
Outline
The Algebra of the Simplex Method (Section 4.3)
The Simplex Method in Tabular Form (Section 4.4)
Tie Breaking in the Simplex Method (Section 4.5)
Adapting to Other Model Forms (Section 4.6)
Postoptimality Analysis (Section 4.7)
ISyE Lecture #7
2
ISyE Lecture #7
The Algebra of the Simplex Method (Section 4.3)
Outline
The Algebra of the Simplex Method (Section 4.3)
Introduction
Initialization
Step 1
Step 2
Step 3
Optimality Test
Simplex Iteration: Summary
Iteration 2
The Simplex Method in Tabular Form (Section 4.4)
Tie Breaking in the Simplex Method (Section 4.5)
ISyE Lecture #7
3
ISyE Lecture #7
The Algebra of the Simplex Method (Section 4.3)
Introduction
Before We Begin.
..
±
In this section, we will discuss the algebra of the simplex
method.
±
We’ll discuss the
algebraic
equivalent to each of the
geometric
steps in our geometric explanation.
±
We’ll keep the geometry in mind when discussing the algebra.
±
But, one important note:
±
The geometry is based on the
original
form of the LP (2
variables).
±
The algebra is based on the
augmented
form of the LP (5
variables).
ISyE Lecture #7
4
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentISyE Lecture #7
The Algebra of the Simplex Method (Section 4.3)
Initialization
Initialization (Step 0)
±
In Step 0, we chose
(0
,
0)
as our initial BF solution (SC #3).
±
In other words, we chose
x
1
and
x
2
as our initial nonbasic
variables.
±
We need to solve the following system for the basic variables
(in
red
):
x
1
+
x
3
= 4
(1)
2
x
2
+
x
4
= 12
(2)
3
x
1
+ 2
x
2
+
x
5
= 18
(3)
±
This is easy, since the nonbasic variables equal 0!
x
3
= 4
x
4
= 12
x
5
= 18
±
Our BF solution is
(0
,
0
,
4
,
12
,
18)
.
ISyE Lecture #7
5
ISyE Lecture #7
The Algebra of the Simplex Method (Section 4.3)
Initialization
Optimality Test
±
The objective function is
Z
= 3
x
1
+ 5
x
2
so
Z
= 0
.
±
How do we know whether this is optimal?
±
If we increased
either
nonbasic variable (
x
1
or
x
2
),
Z
would
increase.
±
Because the coeﬃcients of both
x
1
and
x
2
are positive in the
expression for
Z
.
±
Therefore,
(0
,
0
,
4
,
12
,
18)
is not optimal.
±
For our initial solution, all of the basic variables have
coeﬃcients equal to 0 in the expression for
Z
.
±
This will not be true after this iteration.
±
So we’ll have to ﬁnd a way of writing
Z
in terms of just the
nonbasic variables.
ISyE Lecture #7
6
ISyE Lecture #7
The Algebra of the Simplex Method (Section 4.3)
Initialization
Summary
Summary of Initialization Step
±
Choose
(0
,
0)
as the initial CPF solution if possible.
±
Find the corresponding BF solution by solving the constraint
system for the basic variables.
±
Test for optimality.
ISyE Lecture #7
7
ISyE Lecture #7
The Algebra of the Simplex Method (Section 4.3)
Step 1
Step 1: Determining the Direction of Movement
±
Increasing a nonbasic variable from 0 corresponds to moving
along an edge emanating from the current CPF solution.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 JEFF
 Optimization, Optimal Solutions, Wyndor Glass, ISyE Lecture

Click to edit the document details