11/20/2015 6:37 AM1 1 LINEAR PROGRAMMING Section 3.4, ©2008

04/20/23 19:03 104/20/23 19:03 1

LINEARPROGRAMMING

Section 3.4, ©2008

04/20/23 19:03 204/20/23 19:03 3.5 Linear Programming 2

Example 15

4

1

3

x

x

y

y

(-5, 3) (4, 3)

(4, -1)(-5, -1)


Definitions

Optimization is finding the minimum and maximum value

For the most part, optimization involves point, P

Steps in Linear Programming1. Find the vertices by graphing2. Plug the vertices into the P equation, which is

given3. Find the minimum and maximum optimization

values of P

Linear Programming is a method of finding a maximum or minimum value of a function that satisfies a set of conditions called constraints

A constraint is one of the inequalities in a linear programming problem.

The solution to the set of constraints can be graphed as a feasible region.


Optimization

A Haunted House is opened from 7pm to 4am. Look at this graph and determine the maximization and minimization of this business.

7p 8p 9p 10p 11p 12a 1a 2a 3a 4a

MAXIMIZATION

MINIMIZATION MINIMIZATION


Example 1

Given Find the minimum and maximum

for equation,

5

4

1

3

x

x

y

y

2 .P x y

Step 1:

Find the vertices by graphing

(-5, 3) (4, 3)

(4, -1)(-5, -1)


Example 1

Given Find the minimum and maximum for equation,

5

4

1

3

x

x

y

y

2 .P x y

Step 2: Plug the vertices into the P equation, which is given

( 5,3),(4,3),(4, 1),( 5, 1) vertices P = –2x + y profit

(-5, 3)

(4, 3)

(4, -1)

(-5, -1)

P = -2(-5) + (3)

P = -2(4) + (3)

P = -2(4) + (-1)

P = -2(-5) + (-1)

P = 13

P = –5

P = –9

P = 9

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Example 1


5

4

1

3

x

x

y

y

2 .P x y

Step 3: Find the minimum and maximum optimization values of P

( 5,3),(4,3),(4, 1),( 5, 1) vertices P = -2x + y Profit

(-5, 3)

(4, 3)

(4, -1)

(-5, -1)

P = -2(-5) + (3)

P = -2(4) + (3)

P = -2(4) + (-1)

P = -2(-5) + (-1)

P = –5

P = 9

Minimum: –9 @ (4,-1) Maximum:13 @ (-5,3)

P = –9

P = 13


Example 2

Given Find the minimum and

maximum optimization for equation,

3 4 .P x y

2

5

1

6

x

x

y

y


Example 2

Given Find the minimum and maximum for

equation,

2

5

1

6

x

x

y

y

3 4 .P x y

Vertices P = 3x+4y Profit

(2, 6) P = 3(2) + 4(6)

30

(5, 6) P = 3(5) + 4(6) 39

(2, 1) P = 3(2) + 4(1) 10

(5, 1) P = 3(5) + 4(1)

19

(2, 6) (5, 6)

(2, 1) (5, 1)

Minimum: 10 @ (2,1)Maximum: 39 @ (5,6)

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Example 3


for equation, 25 30 .P x y

x ≥ 0y ≥ 1.5

2.5x + 5y ≤ 203x + 2y ≤ 12

11

Vertices:(0, 4), (0, 1.5), (2, 3), and (3, 1.5)

(2, 3)

(3, 1.5)(0, 1.5)

(0, 4)

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for equation,

(x, y) 25x + 30y P($)

(0, 4) 25(0) + 30(4) 120

(0, 1.5) 25(0) + 30(1.5) 45(2, 3) 25(2) + 30(3) 140

(3, 1.5) 25(3) + 30(1.5) 120

3.5 Linear Programming

Example 3

25 30 .P x y

x ≥ 0y ≥ 1.5

2.5x + 5y ≤ 203x + 2y ≤ 12

12

(2, 3)

(3, 3/2)(0, 3/2)

(0, 4)


Your Turn


for equation,

0

0

2

x

y

x y

2 .P x y

2y x (0, 2)

(2, 0)(0, 0)Step 1:

Find the vertices by graphing


Your Turn

Given Find the minimum and maximum for equation, 0

0

2

x

y

x y

2 .P x y

Step 2: Plug the vertices into the P equation, which is given

(0,0),(0,2),(2,0)vertices P = x + 2y profit

(0, 2)

(0, 0)

(2, 0)

P = (0) + 2(2) P = 4P = (0) + 2(0) P = 0

P = (2) + 2(0) P = 2


Example 4


0

0

2 8

2 2 4

x

y

x y

x y

2 3 .P x y

(0, 8)

(4, 0)

(0, 2)

(2, 0)

vertices P = 2x + 3y profit

(0, 8) P = 2(0) + 3(8) 24

(0, 2) P = 2(0) + 3(2)

6

(2, 0) P = 2(2) + 3(0) 4

(4, 0) P = 2(4) + 3(0)

8

Example 5 A charity is selling T-shirts in order to raise money. The

cost of a T-shirt is $15 for adults and $10 for students. The charity needs to raise at least $3000 and has only 250 T-shirts. Write and graph a system of inequalities that can be used to determine the number of adult and student T-shirts the charity must sell.

Let a = adult t-shirts

Let b = student t-shirts

250

a b15 10 3000 a b

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Sue manages a soccer club and must decide how many members to send to soccer camp.

It costs $75 for each advanced player and $50 for each intermediate player.

Sue can spend no more than $13,250.

Sue must send at least 60 more advanced than intermediate players and a minimum of 80 advanced players.

Find the number of each type of player Sue can send to camp to maximize the number of players at camp.

Warm-up 10-23-13

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x = the number of advanced players, y = the number of intermediate players.

x ≥ 80

y ≥ 0

75x + 50y ≤ 13,250

x – y ≥ 60

The number of advanced players is at least 80.

The number of intermediate players cannot be negative.There are at least 60 more advanced players than intermediate players.

The total cost must be no more than $13,250.

Let P = the number of players sent to camp. The objective function is P = x + y.

Example 6

MAKE a TABLE to show your work for the objective function

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Graph the feasible region, and identify the vertices. Evaluate the objective function at each vertex.

P(80, 0) = (80) + (0) = 80

P(80, 20) = (80) + (20) = 100

P(176, 0) = (176) + (0) = 176

P(130,70) = (130) + (70) = 200

Example 6

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Check the values (130, 70) in the constraints.

x ≥ 80

130 ≥ 80

y ≥ 0

70 ≥ 0

x – y ≥ 60

(130) – (70) ≥ 60

60 ≥ 60

75x + 50y ≤ 13,250

75(130) + 50(70) ≤ 13,250

13,250 ≤ 13,250

Example 6


Assignment

Pg 202: 11-19 odd, 20, 29, 31 (no need to identify the shape from 16-19)

Pg 209: 9-21 odd

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11/20/2015 6:37 AM1 1 LINEAR PROGRAMMING Section 3.4, ©2008