OR320/520
9/18/07
Prof. Bland
The Simplex Method: Parts III and IV
Part III: Tableaus
In our illustration of the simplex method in Part II on the example from Part I we used
the term
Tableau
to describe a system of equations that gets updated at each iteration. To
simplify the presentation of the tableau we usually write down in each row the coefficients
of the variables without repeating the variable names.
Return to the example from Parts I and II. The linear programming problem we were solving
there has a vector
x
= (
x
1
, x
2
, x
3
, x
4
, x
5
)
T
of five nonnegative variables, an objective function
cx
with coefficients (
c
1
, c
2
, c
3
, c
4
, c
5
) = (20
,
30
,
0
,
0
,
0) , and a 3
×
5 system of equations
Ax
=
b
with:
A
=
2
2
1
0
0
4
2
0
1
0
3
6
0
0
1
b
=
80
120
210
The initial tableau in this example is particularly simple: three of its four rows are just the
system
Ax
=
b
, but we present it in tabular form, leaving out the
x
j
’s and simply writing
down the entries from
A
and
b
. These three rows comprise what is called the
body
of the
tableau. The other row of the tableau is its
objective function row
. First we add a variable
z
; for any choice of the vector
x
the value of
z
is the objective function value
cx
. So we think
of an equation like
z
=
cx
being appended to the
Ax
=
b
system. However,
z
is a variable,
and we like to keep all of our variables on the lefthandside of our equations, so we rewrite
this as

z
+
cx
= 0 and append it to the
Ax
=
b
system to get an extended system with one
additional row, the objective function row, and one additional column, the (

z
) column. (It
turns out that it will be convenient if we think of (

z
) as the added variable, rather than
z
).
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 Fall '07
 BLAND
 Linear Programming, Optimization, Simplex algorithm

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