9.2 LINEAR PROGRAMMING INVOLVING TWO VARIABLES
Many applications in business and economics involve a process called
optimization,
in
which it is required to find the minimum cost, the maximum profit, or the minimum use of
resources. In this section, one type of optimization problem called
linear programming
is
discussed.
A twodimensional linear programming problem consists of a linear
objective function
and a system of linear inequalities called
constraints.
The objective function gives the
quantity that is to be maximized (or minimized), and the constraints determine the set of
feasible solutions.
For example, consider a linear programming problem in which you are asked to
maximize the value of
Objective function
subject to a set of constraints that determine the region indicated in Figure 9.10. Because
every point in the region satisfies each constraint, it is not clear how to go about finding the
point that yields a maximum value of
z
. Fortunately, it can be shown that if there is an op
timal solution, it must occur at one of the vertices of the region. In other words,
you can find the maximum value by testing z at each of the vertices
, as illustrated in
Example 1.
EXAMPLE 1
Solving a Linear Programming Problem
Find the maximum value of
Objective function
subject to the following constraints.
Solution
The constraints form the region shown in Figure 9.11. At the four vertices of this region,
the objective function has the following values.
x
2
y
≤
1
x
1
2
y
≤
4
y
≥
0
x
≥
0
z
5
3
x
1
2
y
z
5
ax
1
by
522
CHAPTER 9
LINEAR PROGRAMMING
Figure 9.10
x
y
The objective function has its
optimal value at one of the vertices
of the region determined by the
constraints.
Feasible
solutions
Theorem 9.1
Optimal Solution of a
Linear Programming Problem
If a linear programming problem has a solution, it must occur at a vertex of the set of
feasible solutions. If the problem has more than one solution, then at least one of them
must occur at a vertex of the set of feasible solutions. In either case, the value of the
objective function is unique.
Constraints
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At
At
(Maximum value of
z
)
At
So, the maximum value of
z
is 8, and this occurs when
and
REMARK
:
In Example 1, try testing some of the interior points in the region. You will
see that the corresponding values of
z
are less than 8.
To see why the maximum value of the objective function in Example 1 must occur at a
vertex, consider writing the objective function in the form
This equation represents a family of lines, each of slope
Of these infinitely many
lines, you want the one that has the largest
z
value, while still intersecting the region deter
mined by the constraints. In other words, of all the lines whose slope is
you want
the one that has the largest
y
intercept
and
intersects the given region, as shown in Figure
9.12. It should be clear that such a line will pass through one (or more) of the vertices of
the region.
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 Spring '08
 Staff
 Linear Algebra, Algebra, Linear Programming, Optimization, objective function

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