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USE OF MATRIX FORMULAS
An Example: Consider the LP and its standard form:
max
z
=
x
1
+
4
x
2
x
1
+
2
x
2
≤
6
2
x
1
+
x
2
≤
8
x
1
≥
0
x
2
≥
0
max
z
=
x
1
+
4
x
2
x
1
+
2
x
2
+
s
1
= 6
2
x
1
+
x
2
+
s
2
= 8
x
1
≥
0
x
2
≥
0
s
1
≥
0
s
2
≥
0
QUESTION
Given that the optimal
basis
is
BV
=
{
x
2
,
s
2
}
, ﬁnd the
optimal tableau.
Saigal (U of M)
IOE 310
1 / 20
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View Full Document AN EXAMPLE
First: Compute
B

1
.
B
=
±
2 0
1 1
²
,
B

1
=
±
1
2
0

1
2
1
²
,
b
=
±
6
8
²
.
Second: Note that
B

1
Ax
=
B

1
b
gives
1
2
x
1
+
x
2
+
s
1
= 3
3
2
x
1

1
2
s
1
+
s
2
= 5
Saigal (U of M)
IOE 310
2 / 20
Using
c
T
BV
= (4
,
0) and
c
T
= (1
,
4
,
0
,
0) the objective row of the
optimal tableau is:
c
T
BV
B

1
A

c
T
which is obtained by ﬁrst computing
c
T
BV
B

1
= (2
,
0)
and then
c
T
BV
B

1
A

c
T
= (1
,
0
,
2
,
0)
with the objective value:
c
T
BV
B

1
b
= 12
which gives the ﬁnal tableau as
z
+
x
1
+2
s
1
= 12
1
2
x
1
+
x
2
+
s
1
= 3
3
2
x
1

1
2
s
1
+
s
2
= 5
Saigal (U of M)
IOE 310
3 / 20
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View Full Document SENSITIVITY ANALYSIS
Changing the obj coeﬀ of a nonbasic variable.
Changing the obj coeﬀ of a basic variable.
Changing the RHS of a constraint.
Changing the column of a nonbasic variable.
Adding a new activity.
Adding a new constraint.
Shadow Price.
Saigal (U of M)
IOE 310
4 / 20
AN EXAMPLE
I
A furniture manufacturer has the following data:
Desk
Table
Chair
Available
Lumber in Board ft
8
6
1
48
Finishing Hours
4
2
1.5
20
Carpentry Hours
2
1.5
0.5
8
Selling Price
$60
$30
$20
Decision variables
x
1
x
2
x
3
number produced
Saigal (U of M)
IOE 310
5 / 20
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View Full Document AN EXAMPLE
II
The LP Model:
max
z
= 60
x
1
+
30
x
2
+
20
x
3
8
x
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This note was uploaded on 04/02/2008 for the course IOE 310 taught by Professor Saigal during the Spring '08 term at University of Michigan.
 Spring '08
 Saigal

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