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PPT – 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 presentation | free to download - id: 40ac1b-N2JmY

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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.

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.

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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.

10 x 60y 20x y 100

b. (graph)

c. P 10x 12y

Ex. 4

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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)

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

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Lesson Quiz Linear Programming

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Homework Pg. 191-192 10-36 even