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 - PowerPoint PPT Presentation

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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

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Title: 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


1
3.5 Linear ProgrammingObjectives 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.
2
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.

3
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

4
B.
C.
5
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.

6
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.
7
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8
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 companys total
profit, P, from the sales of afghans and sweater.
9
10 x 60y 20x y 100
b. (graph)
c. P 10x 12y
10
Ex. 4
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12
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 10x 2y 12


4x y 13
Vertex Objective function Amount
1,9 21
8, 2 Maximum 28
2,5 Minimum 16
B. y -x 10 y - x/2 6 -2y x 12
-1y -2 y 2 x 8 (8, 2)
  • y - 4x 13
  • - 4y 4x 40
  • -3y - 27
  • y 9
  • x 1
  • (1,9)

C. y - 4x 13 y -x / 2 6 y -4x
13 -8y 4x 48 -7y - 35 y 5 x 2
(2,5)
13
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.

14
Multiple Choice Practice
15
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16
Lesson Quiz Linear Programming
17
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18
Homework Pg. 191-192 10-36 even
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