3.5 Linear Programming
Objectives: Write and graph a set of constraints for a
linear-programming problem.
Use linear programming to find the
maximum or minimum value of an objective
function.
Standard: 2.5.11.A. Use appropriate mathematical
techniques to solve non-routine problems.
A method called linear programming is used to
find optimal solutions.
Linear programming problems have the following
characteristics:
• The inequalities contained in the problem
are called constraints.
• The solution to the set of constraints is
called the feasible region.
• The function to be maximized or
minimized is called the objective
function.
Ex 1. Max Desmond is a farmer who plants corn and wheat. In making planting
decisions, he used the 1996 statistics at right from the United States Bureau of
the Census.
Crop
Yield Per Acre
Average Price
Corn
113.5 bu
$3.15 / bu
Soy Beans
34.9 bu
$6.80 / bu
Wheat
35.8 bu
$4.45 / bu
Cotton
540 lb
$.759 / lb
Rice
564 lb
$.0865 / lb
•Let x represent the number of acres of corn
•Let y represent the number of acres of wheat
• Mr. Desmond wants to plant according to the following
constraints:
• No more than 120 acres of corn and wheat
• At least 20 and no more than 80 acres of corn
• At least 30 acres of wheat
• How many acres of each crop should Mr. Desmond plant to
maximize the revenue from his harvest?
• OBJECTIVE FUNCTION R = 357.525x + 159.31y
B.
C.
The Corner-Point Principle confirms that you
need only the vertices of the feasible region to find
the maximum or minimum value of the objective
function.
• Corner-Point Principle:
• In linear programming, the maximum and
minimum values of the objective function
each occur at one of the vertices of the
feasible region.
Ex 2. Using the information in Example 1,
maximize the objective function. Then graph
the objective function that represents the
maximum revenues along with the feasible
region.
Ex 3. A small company produces knitted afghans
and sweaters and sells them through a chain of
specialty stores. The company is to supply the
stores with a total of no more than 100 afghans and
sweaters per day. The stores guarantee that they
will sell at least 10 and no more than 60 afghans per
day and at least 20 sweaters per day. The company
makes a profit of $10 on each afghan and a profit of
$12 on each sweater. Write a system of
inequalities to represent the constraints. Graph
the feasible region. Write an objective function
for the company’s total profit, P, from the sales of
afghans and sweater.
a.
10 ≤ x ≤ 60
y ≥ 20
x + y ≤ 100
* b. (graph)
c. P = 10x + 12y
Ex. 4
Ex 5. Find the maximum and minimum values, if they
exist, of the objective function T = 3x + 2y given the set
of constraints provided:
x + y ≤ 10
Vertex
Objective function Amount
x + 2y ≥ 12
1,9
21
4x + y ≥ 13
8, 2
Maximum
28
2,5
A. Y = - 4x + 13
Minimum
B. y = -x + 10
- 4y = 4x + 40
y= - 2/x + 6
-3y = - 27
-2y = x – 12
y=9
x=1
(1,9)
-1y = -2
y=2
x=8
(8, 2)
16
C. y = - 4x + 13
y = -x / 2 + 6
y = -4x + 13
-8y = 4x – 48
-7y = - 35
y = 5; x = 2
(2,5)
Summary Linear-Programming
Procedure
•
•
•
•
Write a system of inequalities, and graph
the feasible region.
Write the objective function to be
maximized or minimized.
Find the coordinates of the vertices of the
feasible region.
Evaluate the objective function for the
coordinates of the vertices of the feasible
region. Then identify the coordinates that
give the required maximum or minimum.
Multiple Choice Practice:
Lesson Quiz: Linear Programming