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Simplex Method and the Tableau Procedure
The solution procedure presented in these notes is the
simplex method
for solving linear
programming problems.
A key observation about the linear programming solution method (using Basic Feasible Solutions)
is that when we select a basis, the set
B
x
, and develop the coefficient matrix of
( )
,
T
B
zx
, call it
M
,
then
1
M
−
times the original data coefficients always yields an identity matrix.
Consider the data
10
0
T
zc
x
zA
x
b
−=
+=
(
1
)
partitioned by
( )
,,
TT
BN
zx x
then
0
BB
NN
zcx cx
zB
x N
x b
−−
=
++
=
(
2
)
Now,
1
0
T
B
c
M
B
⎡⎤
−
=
⎢⎥
⎣⎦
,
1
1
1
0
T
B
cB
M
B
−
−
=
and
1
1
1
0
T
B
B
−
−
0
x
c
x
x
N
x
b
=
++=
yields
()
11
0
TTT
T
N
N
B
zxc
B
N
c
x
c
B
b
zI
x
B
N
x
B
b
−+
−
=
=
(3)
The key observation here is that the
mm
+
×
+
identity matrix comprises the
( )
,
T
B
coefficients.
Thus, by setting
0
N
x
=
, we have the basic solution
1
1
T
B
B
zcBb
x
Bb
−
−
=
=
.
The matrix multiplication method for creating (3) from (2) is not the only method for obtaining this
solution.
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View Full Document A computationally convenient method is to use Gaussian reduction to diagonalize the
( )
,
T
B
zx
coefficients.
This procedure is particularly useful for moving from one BFS to an adjacent BFS.
To
illustrate, consider the three tableaux used to solve the example problem:
Max
z
=
1
x
+ 5
2
x
s.t.
2
1
x
+ 3
2
x
≤
12

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This note was uploaded on 10/26/2011 for the course ISEN 620 taught by Professor Curry during the Fall '10 term at Texas A&M.
 Fall '10
 Curry

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