OR3300/5300
Fall 2011
Prof. Bland
Review of the Simplex Method
Part I: Standard Form Systems and Ordered Bases
Let
A
be an
m
×
n
matrix and let
b
be an
m
×
1 vector. We will be interested in linear
programming problems in which the constraints take the form
Ax
=
b, x
≥
0. Note that the
set
S
=
{
x
∈
R
I
n
:
Ax
=
b, x
≥
0
}
of feasible solutions to such l.p. problems is a polyhedron.
This polyhedron may not be bounded (for example if
m
= 0), but if it is nonempty, it has
at least one extreme point. (We will give an intuitive argument in class.)
For a little while we are going to ignore the nonnegativity constraints and focus on the linear
system
Ax
=
b
. For each index 1
≤
j
≤
n
let
A
j
denote the
jth
column of
A
. Now suppose
that for the list
B
= (
B
1
, ..., B
m
) of indices in
{
1
, ..., n
}
the
m
×
m
matrix
A
B
= [
A
B
1
, ..., A
B
m
]
is nonsingular. Then we say that
B
is
basic
for
A
, and that
B
is an
ordered basis
correspond
ing to the linear system
¯
Ax
=
¯
b
, where (
¯
A,
¯
b
) :=
A

1
B
(
A, b
) The variables
x
B
1
,
· · ·
, x
B
m
are
basic variables
, and the other variables
x
j
, for
j
not in
B
, are called
nonbasic
. It is convenient
to denote by
N
the set of indices of the nonbasic variables.
For example let
A
=
2
2
1
0
0
4
2
0
1
0
3
6
0
0
1
b
=
80
120
210
Then
B
= (3
,
4
,
5) is basic for
A
, and, since
A
B
=
I
3
,
¯
A
=
A,
¯
b
=
b
, so this choice of
B
is an
ordered basis for the system
Ax
=
b
. Another
B
that is basic for
A
is
B
= (2
,
1
,
4). For this
choice of
B
we have:
A
B
= [
A
2
, A
1
, A
4
] =
2
2
0
2
4
1
6
3
0
A

1
B
=

1
2
0
1
3
1
0

1
3

3
1
2
3
and
¯
A
=
0
1

1
2
0
1
3
1
0
1
0

1
3
0
0

3
1
2
3
¯
b
=
30
10
20
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For any
m
×
n
matrix
A
,
m
×
1 vector
b
and
B
that is basic for
A
, the matrix
¯
A
=
A

1
B
A
has a very special form: specifically
¯
A
B
=
I
m
. We say that such a matrix
¯
A
is in
standard
form
, and we also say that the linear system
¯
Ax
=
¯
b
is in standard form.
In each row
i
= 1
,
· · ·
, m
the basic variable
x
B
i
has a coefficient of 1 but each of the other basic variables
have coefficient 0.
We can think of
x
B
i
as the basic
or dependent variable in row
i
of
¯
A
.
The nonbasic variables
x
j
play the role of independent variables in this system.
For any
arbitrary assignment of values to
{
x
j
:
j
∈
N
}
setting
x
B
i
=
¯
b
i

∑
j
∈
N
¯
a
ij
x
j
(
i
= 1
, ..., m
)
assures that
¯
Ax
=
¯
b
and, therefore, that
Ax
=
b
. Of special interest is the
unique solution
of
Ax
=
b
in which every nonbasic variable
x
j
= 0
; it has
x
B
=
¯
b
.
We call this a basic
solution of
Ax
=
b
.
The time has come to bring the nonnegativity constraints back into
the picture.
If
¯
b
is nonnegative, then we say that the solution given by
x
B
=
¯
b, x
N
= 0
is a
basic feasible solution
of our linear programming problem. In our 3
×
5 example there
are 5 basic feasible solutions and five other basic solutions that are not feasible. For exam
ple, if we take
B
= (1
,
2
,
3), then the corresponding basic solution has
x
3
<
0.
It would
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 TODD
 Linear Programming, Optimization, objective function, Simplex algorithm

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