9.3
THE SIMPLEX METHOD: MAXIMIZATION
For linear programming problems involving two variables, the graphical solution method
introduced in Section 9.2 is convenient. However, for problems involving more than two
variables or problems involving a large number of constraints, it is better to use solution
methods that are adaptable to computers. One such method is called the
simplex method,
developed by George Dantzig in 1946. It provides a systematic way of examining the ver
tices of the feasible region to determine the optimal value of the objective function. This
method is introduced with the following example.
Suppose you want to find the maximum value of
where
and
subject to the following constraints.
Because the lefthand side of each
inequality
is less than or equal to the righthand side,
there must exist nonnegative numbers
and
that can be added to the left side of each
equation to produce the following system of linear
equations
.
The numbers
and
are called
slack variables
because they take up the “slack” in
each inequality.
s
3
s
1
,
s
2
,
x
1
x
1
2
x
1
x
2
x
2
5
x
2
s
1
s
2
s
3
11
27
90
s
3
s
1
,
s
2
,
2
x
1
5
x
2
≤
90
x
1
x
2
≤
27
x
1
x
2
≤
11
x
2
≥
0,
x
1
≥
0
z
4
x
1
6
x
2
,
530
CHAPTER 9
LINEAR PROGRAMMING
Standard Form of a
Linear Programming
Problem
A linear programming problem is in
standard form
if it seeks to
maximize
the objec
tive function
subject to the constraints
where
and
After adding slack variables, the corresponding system of
constraint equations
is
where
s
i
≥
0.
a
m
1
x
1
a
m
2
x
2
.
.
.
a
mn
x
n
s
m
b
m
.
.
.
a
21
x
1
a
22
x
2
.
.
.
a
2
n
x
n
s
2
b
2
a
11
x
1
a
12
x
2
.
.
.
a
1
n
x
n
s
1
b
1
b
i
≥
0.
x
i
≥
0
a
m
1
x
1
a
m
2
x
2
.
.
.
a
mn
x
n
≤
b
m
.
.
.
a
21
x
1
a
22
x
2
.
.
.
a
2
n
x
n
≤
b
2
a
11
x
1
a
12
x
2
.
.
.
a
1
n
x
n
≤
b
1
z
c
1
x
1
c
2
x
2
.
.
.
c
n
x
n
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R E M A R K
:
Note that for a linear programming problem in standard form, the objective
function is to be maximized, not minimized. (Minimization problems will be discussed in
Sections 9.4 and 9.5.)
A
basic solution
of a linear programming problem in standard form is a solution
of the constraint equations in which
at most m
variables are
nonzero––the variables that are nonzero are called
basic variables.
A basic solution for
which all variables are nonnegative is called a
basic feasible solution.
The Simplex Tableau
The simplex method is carried out by performing elementary row operations on a matrix
called the
simplex tableau.
This tableau consists of the augmented matrix corresponding to
the constraint equations together with the coefficients of the objective function written in
the form
In the tableau, it is customary to omit the coefficient of
z
. For instance, the simplex tableau
for the linear programming problem
Objective function
is as follows.
Basic
x
1
x
2
s
1
s
2
s
3
b
Variables
1
1
0
0
11
s
1
1
1
0
1
0
27
s
2
2
5
0
0
1
90
s
3
0
0
0
0
↑
Current z–value
For this
initial simplex tableau,
the
basic variables
are
and
and the
nonbasic
variables
are
and
The nonbasic variables have a value of zero, yielding a current
z
value of zero. From the columns that are farthest to the right, you can see that the basic
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 Spring '08
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 Linear Algebra, Algebra, Linear Programming, Optimization, Tax Returns, objective function, George Dantzig

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