Linear Programming Sample Problems
Problem 1.
Complete the following tableau, where the objective function is
z
= 3
x
1

2
x
2
+ 2
x
4

x
5
.
c
*
x
1
x
2
x
3
x
4
x
5
b
3
1
1
0
0
1

1
2
0
1
0
5
4
5
0
0
1
9
Next, maximize the objective function using the simplex method.
Solution.
First read off the basic solution:
x
3
= 1
, x
4
= 5
, x
5
= 9
, x
1
=
x
2
= 0. Then
compute the “truncated” vector
c
*
by picking the components of
c
corresponding to the
basic variables
x
3
, x
4
, x
5
: This gives
c
*
= (0
,
2
,

1).
Next, find the value of
z
for this solution by computing
z
=
c
*
b
= 0
·
1 + 2
·
5 + (

1)
·
9 = 1.
(You can also find this by plugging in the values for
x
1
, . . . , x
5
into the formula for
z
.
Finally we compute the simplex indicators using the formula
c
*
A

c
.
The first simplex
indicator
s
1
is computed as
s
1
=
0
2

1
3

1
4

3 = (0
·
2 + 1
·
(

1) + (

1)
·
4)

3 =

9
.
The other simplex indicators are found similarly.
Putting all of this together gives the
following tableau:
c
*
x
1
x
2
x
3
x
4
x
5
b
0
3
1
1
0
0
1
2

1
2
0
1
0
5

1
4
5
0
0
1
9

9
1
0
0
0
1
Next, we note that the first simplex indicator is negative. Because not all entries of the
corresponding column are negative, we can improve the solution. This is done by using row
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reduction, pivoting on one of the elements in the first column. We pivot on the first element
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 Spring '08
 Hofstra
 Linear Algebra, Algebra, Linear Programming, Optimization, BMW Sports Activity Series, Simplex algorithm

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