3-5: Linear Programming

Vocabulary constraints: conditions given to variables, often

expressed as linear inequalities feasible region: the intersection of the graphs in

a system of constraints bounded: a region is bounded when the graph of

a system of constraints is a polygonal region vertices: the maximum or minimum value that a

linear function has for the points in a feasible region

unbounded: a system of inequalities that forms a region that is open

Linear Programming

Businesses use linear programming to find out how to maximize profit or minimize costs. Most have constraints on what they can use or buy.

Find the minimum and maximumvalue of the function f(x, y) = 3x - 2y.

We are given the constraints: y ≥ 2 1 ≤ x ≤5 y ≤ x + 3

Linear Programming

Find the minimum and maximum values by graphing the inequalities and finding the vertices of the polygon formed.

Substitute the vertices into the function and find the largest and smallest values.

6

4

2

2 3 4

3

1

1

5

5

7

8

y ≤ x + 3

y ≥ 2

1 ≤ x ≤5

Linear Programming

The vertices of the quadrilateral formed are:

(1, 2) (1, 4) (5, 2) (5, 8) Plug these points into the

function f(x, y) = 3x - 2y

Linear Programming

f(x, y) = 3x - 2y f(1, 2) = 3(1) - 2(2) = 3 - 4 = -1 f(1, 4) = 3(1) - 2(4) = 3 - 8 = -5 f(5, 2) = 3(5) - 2(2) = 15 - 4 = 11 f(5, 8) = 3(5) - 2(8) = 15 - 16 = -1

Linear Programming

f(1, 4) = -5 minimum f(5, 2) = 11 maximum

Find the minimum and maximum value of the function f(x, y) = 4x + 3y

We are given the constraints: y ≥ -x + 2

y ≤ x + 2

y ≥ 2x -5

1

4

6

4

2

53 4

5

1

1

2

3y ≥ -x + 2

y ≥ 2x -5

y x 1

24

Vertices

f(x, y) = 4x + 3y f(0, 2) = 4(0) + 3(2) = 6 f(4, 3) = 4(4) + 3(3) = 25 f( , - ) = 4( ) + 3(- ) = -1 = 7

3

1

3

1

3

7

3

28

3

25

3

Linear Programming

f(0, 2) = 6 minimum f(4, 3) = 25 maximum

Lisa has an online jewelry shop where she sells earrings and necklaces.  She sells earrings for $30 and necklaces for $40.  It takes 30 minutes to make a pair of earrings and 1 hour to make a necklace, and, since Lisa is a math tutor, she only has 10 hours a week to make jewelry.  In addition, she only has enough materials to make 15 total jewelry items per week.  She makes a profit of $15 on each pair of earrings and $20 on each necklace.  How many pairs of earrings and necklaces should Lisa make each week in order to maximize her profit, assuming she sells all her jewelry?

Define the variables, write an inequality for this situation, and graph the solutions to the inequality.

Then take the vertices (corner points) and plug in the x and y values into the objective function.

So, in order to maximize her profits, Lisa should make 10 pair of earrings and 5 necklaces per week, and her weekly profit is $250.

Identify the vertices of the feasible region

Using the vertices, find the minimum and maximum values of the function f(x, y) = 3x + y

Identify the vertices of the feasible region

Using the vertices, find the minimum and maximum values of the function f(x, y) = 5x + 4y

Veterinary medicine

As a receptionist for a veterinarian, one of Dolores Alvarez’s tasks is to schedule appointments. She allots 20 minutes for a routine office visit and 40 minutes for a surgery. The veterinarian cannot do more than 6 surgeries per day. The office has 7 hours available for appointments. If an office visit costs $55 and most surgeries cost $125, find a combination of office visits and surgeries that will maximize the income the veterinarian practice receives per day.

Veterinary medicine