File: EnterVar.doc
Deciding What Variable Enters the Basis
(
x
B
)
The analysis of which variable should enter the basis (
x
B
) from the nonbasic set (
x
N
)
has been avoided until now.
We can select any variable from
x
N
as the entering variable
and find an adjacent basic feasible solution.
However, the most promising variable to
enter the basis would be one that yields the maximum increase in the value of the
objective function relative to the current solution.
Recall that we are developing the
methodology to solve linear programming problems.
So consider the linear objective
function
z
c x
i
i
n
i
T
T
T
=
=
=
+
=
∑
1
c x
c x
c x
B
B
N
N
that we add to the system of constraints
Ax
Bx
Nx
b
x
0
B
N
B
=
+
=
≥
,
.
Given a basis,
x
B
, we can select the most promising variable from
x
N
to enter the basis
by choosing the variable which yields the maximum change in z.
However, obtaining
these increases in z requires a considerable computation.
For the time being, we will
select the variable that yields the “best” incremental change in z; that is, has the “best”
slope.
The best is in quotes here since we may be maximizing in one case and
minimizing in another.
Obviously, the select should change with the sense of the
optimization.
For a maximization problem, the “best” select would be the variable with
the largest slope. For a minimization problem, the “best” select would be the variable
with the most negative slope.
How can we determine what the contribution to the objective function would be
for bringing a nonbasic variable into the basis?
This question is answered, as one might
deduce by now, by considering the general solution relative to the current basis.
The
objective function value associated with a the current basic feasible solution is given by
z
T
T
=
+
c x
c x
B
B
N
N
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 Curry
 Optimization, ax, basic feasible solution

Click to edit the document details