Economics 146 – Fall, 2007
Linear Programming – Part III
Last revision: 10/22/07.
Artificial Basis Vectors.
If an LP problem in computational form has no obvious
initial basis, one method to use to start the simplex method is to introduce
artificial basis
vectors.
We give an example first.
Consider the following LP problem in computational
form (which we did not get from a max problem in standard form):
minimize

x
1

2
x
2

3
x
3
+
x
4
subject to
x
1
+2
x
2
+3
x
3
= 15
2
x
1
+
x
2
+5
x
3
= 20
x
1
+
2
x
2
+
x
3
+
x
4
= 10
and
x
1
, . . . , x
4
≥
0
(1)
The simplex tableau (with no initial feasible basis chosen yet) is:

1

2

3
1
a
1
a
2
a
3
a
4
c
1
2
3
0
15
2
1
5
0
20
1
2
1
1
10
(2)
Note that there is no obvious choice for an initial feasible basis.
(We did not get this
problem by adding slack variables to a problem in standard maximum form.) There is a
good choice for one basis vector, viz.
a
4
. We will augment the problem by adding two new
artificial variables
x
5
, x
6
and two new
artificial basis vectors
a
5
= [1
,
0
,
0]
T
,
a
6
= [0
,
1
,
0]
T
.
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 Fall '07
 farmer
 Economics, Optimization, initial feasible basis

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