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9
Degeneracy
In Section 5, we introduced the idea of a “degenerate” tableau, by which we
mean at least one of the numbers
¯
b
i
on the righthand side of the body of the
tableau is zero. Thus a tableau and the corresponding basis are degenerate
when one of the basic variables takes the value zero at the corresponding
basic feasible solution.
In the course of the simplex method, if we have to choose between sev
eral possible leaving indices, then the next tableau will be degenerate. To
illustrate, consider the following simple linear program:
maximize
2
x
1
+
x
2
subject to
x
1

x
2
≤
1
x
1
≤
1
x
2
≤
1
x
1
,
x
2
≥
0
.
After introducing slack variables, the initial tableau is
z

2
x
1

x
2
= 0
x
1

x
2
+
x
3
= 1
x
1
x
4
= 1
x
2
+
x
5
= 1
.
The basic variables are
x
3
, x
4
, x
5
, the corresponding basic feasible solution
is [0
,
0
,
1
,
1
,
1]
T
, and the corresponding value is 0. The current values of the
basic variables are all nonzero, so this tableau is nondegenerate.
Starting the simplex method, we could choose
x
1
as the entering variable,
and compute the ratios in the usual way.
basic variable
x
3
x
4
x
5
ratio
1
1
1
1
We now have a choice of the leaving variable: either
x
3
or
x
4
. This poses no
diﬃculty in itself. We could choose
x
3
, for example, in which case after the
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 Fall '07
 BLAND

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