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.

• 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

•Let x represent the number of acres of corn•Let y represent the number of acres of wheat

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

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 ≤ 60y ≥ 20x + 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 ≤ 10x + 2y ≥ 12 4x + y ≥ 13

A.y = - 4x + 13

- 4y = 4x + 40

-3y = - 27

y = 9

x = 1

(1,9)

B. y = -x + 10

y= - x/2 + 6

-2y = x – 12

-1y = -2

y = 2

x = 8

(8, 2)

C. y = - 4x + 13

y = -x / 2 + 6

y = -4x + 13

-8y = 4x – 48

-7y = - 35

y = 5; x = 2

(2,5)

Vertex Objective function Amount

1,9 21

8, 2 Maximum 28

2,5 Minimum 16

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

HomeworkPg. 191-192 #10-36 even