Outline
1
Review
2
How to check the optimality of a fbp?
3
How to check the unboundedness of an LP?
4
How to find a better fbp?
Lizhi Wang ([email protected])
IE 534 Linear Programming
September 24, 2012
23 / 30
The Simplex diagram
Lizhi Wang ([email protected])
IE 534 Linear Programming
September 24, 2012
24 / 30
Improving a fbp
Suppose the current fbp is
(
B
,
N
)
.
The way we improve this fbp is by selecting an
j
*
from
N
and a
i
*
from
B
and then updating the fbp as
(
B
1
=
B\{
i
*
} ∪ {
j
*
}
,
N
1
=
N\{
j
*
} ∪ {
i
*
}
)
.
The variable
x
j
*
is called the
entering variable
, because it will
enter the basis. Similarly,
x
i
*
is called the
leaving variable
.
Geometrically, such an update means a move from a fbs to an
adjacent one.
Lizhi Wang ([email protected])
IE 534 Linear Programming
September 24, 2012
25 / 30
Bland’s rule
Suppose the current fbp is
(
B
,
N
)
.
The rule for selecting the entering and leaving variables is called a
pivoting rule
.
Bland’s rule is a commonly used pivoting rule.
The Bland’s rule
j
*
=
min
{
j
∈ N
: (
c
>
N

c
>
B
A

1
B
A
N
)
j
>
0
}
.
i
*
=
min
(
argmin
i
∈B
(
(
A

1
B
b
)
i
(
A

1
B
A
N
)
i,j
*
: (
A

1
B
A
N
)
i,j
*
>
0
))
.
Lizhi Wang ([email protected])
IE 534 Linear Programming
September 24, 2012
26 / 30
Bland’s rule
According to Bland’s rule
The entering variable has the
smallest index
with a positive
reduced cost.
The leaving variable is the winner of the
ratio test
:
min
(
(
A

1
B
b
)
i
(
A

1
B
A
N
)
i,j
*
)
for all
i
such that
(
A

1
B
A
N
)
i,j
*
>
0
.
In case of a tie, choose the smallest index as the winner of the
ratio test.
Lizhi Wang ([email protected])
IE 534 Linear Programming
September 24, 2012
27 / 30
Example 5
Consider the following dictionary.
ζ
=
0 +
5
4
3
x
1
x
2
x
3
>
w
4
w
5
w
6
=
5
11
8

2
3
1
4
1
2
3
4
2
x
1
x
2
x
3
.
The current fbp is
(
B
0
=
{
4
,
5
,
6
}
,
N
0
=
{
1
,
2
,
3
}
)
.
The entering variable is
x
1
.
The ratio test for
w
4
w
5
w
6
is
5
/
2
11
/
4
8
/
3
=
2
.
5
2
.
75
2
.
67
.
The leaving variable is
w
4
, because it’s the winner of the ratio test.
The updated fbp is
(
B
1
=
{
1
,
5
,
6
}
,
N
1
=
{
4
,
2
,
3
}
)
.
Lizhi Wang ([email protected])
IE 534 Linear Programming
September 24, 2012
28 / 30
Exercise
Solve the following LP using the Simplex algorithm and the Matlab
programs
dictionaryL1.m
and
dictionaryL0.m
, which can be
downloaded from the course web.
max
x
ζ
= 13
x
1
+ 7
x
2

12
x
3
s
.
t
.
2
x
1
+ 3
x
2

x
3
≤
5

4
x
1

7
x
2
+ 2
x
3
≤ 
11
3
x
1

4
x
2

2
x
3
≤ 
8
x
1
, x
2
, x
3
≥
0
.
Lizhi Wang ([email protected])
IE 534 Linear Programming
September 24, 2012
29 / 30
Exercise solution
A=[2 3 1; 4 7 2; 3 4 2];
b=[5; 11; 8];
c=[13; 7; 12];
Phase I
I
dictionaryL1(A,b,c,[4 7 8],[1 2 3 5 6]);
I
dictionaryL1(A,b,c,[1 7 8],[4 2 3 5 6]);
I
dictionaryL1(A,b,c,[1 2 8],[4 7 3 5 6]);
I
dictionaryL1(A,b,c,[1 2 3],[4 7 8 5 6]);
Phase II
I
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 Spring '12
 lizhiwang
 Operations Research, Linear Programming, Optimization, Simplex algorithm, Lizhi Wang