Matrix Notation
Max CX
(X is a vector with n elements)
s.t.
AX
≥
b
(there are m constraints)
and X
≥
0
Convert by adding slacks so that
S = b – AX
So the constraints are now
AX + IS = b
where I is an identity matrix of dimension m x m
Redefine the X vector to contain both the original X's and the slacks.
Redefine the C vector to contain the original C along with the zeros for the
slacks, and the new A matrix will contain the original A matrix along with
the identity matrix for the slacks.
Max CX
(X is a vector with n + m elements)
s.t.
AX
≥
b
(A has dimensions m, n+m)
The A matrix is not square, so it can't be inverted.
Basic and nonbasic variables
z
The solution to the LP problem will have a set of potentially nonzero
variables equal in number to the number of constraints.
z
Such a solution is called a
Basic Solution
and the associated variables
are commonly called
Basic Variables
.
z
The other variables are set to zero and are called the
nonbasic
variables
.
z
Note:
if there are 3 constraints, no more than 3 X variables can be
nonzero and so on.
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Partition the problem
MAX C
B
X
B
B
B
+ C
NB
X
NB
s.t.
A
B
X
B
B
B
+ A
NB
X
NB
= b
X
B
, X
B
NB
≥
0.
Subscript B represents basic variables and NB, nonbasic (0 valued)
variables and corresponding coefficients.
AB will be square because only as
many variables as there are constraints can be in the solution.
Your text
calls this matrix "B" for basis matrix.
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 Fall '08
 Duffy,P
 Linear Algebra, Operations Research, Linear Programming, basic variables, Xη

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