This preview shows pages 1–3. Sign up to view the full content.
9.4 THE SIMPLEX METHOD: MINIMIZATION
In Section 9.3, the simplex method was applied only to linear programming problems in
standard form where the objective function was to be
maximized
. In this section, this pro
cedure will be extended to linear programming problems in which the objective function is
to be
minimized
.
A minimization problem is in
standard form
if the objective function
is to be minimized, subject to the constraints
where
and
The basic procedure used to solve such a problem is to convert
it to a
maximization problem
in standard form, and then apply the simplex method as
discussed in Section 9.3.
In Example 5 in Section 9.2, geometric methods were used to solve the following mini
mization problem.
Minimization Problem:
Find the minimum value of
Objective function
subject to the following constraints
where
and
The first step in converting this problem to a maximization prob
lem is to form the augmented matrix for this system of inequalities. To this augmented
matrix, add a last row that represents the coefficients of the objective function, as follows.
Next, form the
transpose
of this matrix by interchanging its rows and columns.
Note that the rows of this matrix are the columns of the first matrix, and vice versa. Finally,
interpret the new matrix as a
maximization
problem as follows. (To do this, introduce new
variables,
y
1
,
y
2
, and
y
3
.) This corresponding maximization problem is called the
dual
of the
original minimization problem.
3
60
60
. . .
300
12
6
. . .
36
10
30
. . .
90
:
:
. . .
:
0.12
0.15
. . .
0
4
3
60
12
10
. . .
0.12
60
6
30
. . .
0.15
:
:
:
. . .
:
300
36
90
. . .
0
4
x
2
≥
0.
x
1
≥
0
10
x
1
1
30
x
2
≥
90
12
x
1
1
6
x
2
≥
36
60
x
1
1
60
x
2
≥
300
w
5
0.12
x
1
1
0.15
x
2
b
i
≥
0.
x
i
≥
0
a
m
1
x
1
1
a
m
2
x
2
1
.
.
.
1
a
mn
x
n
≥
b
m
.
.
.
a
21
x
1
1
a
22
x
2
1
.
.
.
1
a
2
n
x
n
≥
b
2
a
11
x
1
a
12
x
2
1
.
.
.
1
a
1
n
x
n
≥
b
1
1
. . .
1
c
n
x
n
w
5
c
1
x
1
1
c
2
x
2
546
CHAPTER 9
LINEAR PROGRAMMING
Constraints
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentDual Maximization Problem:
Find the maximum value of
Dual objective function
subject to the constraints
where
and
As it turns out, the solution of the original minimization problem can be found by
applying the simplex method to the new dual problem, as follows.
Basic
y
1
y
2
y
3
s
1
s
2
b
Variables
60
12
10
1
0
0.12
s
1
←
Departing
60
6
30
0
1
0.15
s
2
00
0
↑
Entering
Basic
y
1
y
2
y
3
s
1
s
2
b
10
y
1
0
–
6
20
–
11
s
2
←
Departing
02
4
–
40
5
0
↑
Entering
Basic
y
1
y
2
y
3
s
1
s
2
b
y
1
01
y
3
2 03
2
↑↑
x
1
x
2
So, the solution of the dual maximization problem is
This is the same value
that was obtained in the minimization problem given in Example 5, Section 9.2. The
x
values corresponding to this optimal solution are obtained from the entries in the bottom
row corresponding to slack variable columns. In other words, the optimal solution occurs
when
The fact that a dual maximization problem has the same solution as its original
minimization problem is stated formally in a result called the
von Neumann Duality
Principle,
after the American mathematician John von Neumann (1903–1957).
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Linear Algebra, Algebra, Linear Programming

Click to edit the document details