Linear Programming

Objective:

I can solve problems using linear programming.

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

x

y

x

y

x

y

x

y

x

y

Constraints: Inequalities

Feasible region:

All points that satisfy the constraints

Objective Function:

• Quantity you want to maximize or minimize

• Vertices of feasible region will be maximum or minimum

Linear Programming:

Method for finding a minimum or maximum of some quantity given constraints { 𝒙 ≥𝟐

𝒚 ≥𝟑𝒚 ≤𝟔

𝒙+𝒚 ≤𝟏𝟎

2 𝑥+𝑦

Maximum

Minimum

Objective Function:

• Find the maximum or minimum points{𝒙+𝟐 𝒚 ≤𝟖𝒙− 𝒚 ≤𝟐𝒙≥𝟎𝒚 ≥𝟎

2 𝑥+𝑦

Vertices:

-5 5

-5

5

x

y

x

y

x

y

x

y

x

y

(0, 0)

(2, 0)

(4, 2)

(0, 4)

Test Vertices

2(0) + 0 = 0

2(2) + 0 = 4

2(4) + 2 = 10

2(0) + 4 = 4

Total

10xMinutes to work

Number

Cost

Profitx

4x

6x

30y

y

20y

22y50

20(60) = 1200

600

≤

≤≥

Maximize

T-Shirts• 10 minutes to make• Supplies cost $4• Profit $6

Sweatshirts• 30 minutes to make• Supplies cost $20• Profit $22

Sweatshirts (y)T-Shirts (x)

You are printing T-shirts and sweatshirts to sell before homecoming.You have at most 20 hours to work.You can spend no more than $600 and you must sell at least 50 items.

+

++

+

Maximum

T-Shirts (x) Sweatshirts (y) Total

Minutes to work 10x 30y ≤ 1200

Cost 4x 20y ≤ 600

Number x y ≥ 50

Profit 6x 22y maximize

300 6(50)+22(0) =( 50, 0)

7006(25)+22(25) =(25, 25)

780 6(75)+22(15) =(75, 15)

720 6(120)+22(0) =(120, 0)

Pg. 160#10-13

75 T-shirts, 15 Sweatshirts

20 40 60 80 100 120 140 160

5

10

15

20

25

30

35

40

45

50

x

y

x

y

x

y

x

y

Vertices: Test Vertices

+

++

+

10000 20000 30000 40000 50000

10000

20000

30000

40000

50000

0

x

y

Paying for CollegeYou have been given $40,000 to invest for a college scholarship. You must invest in both stocks and bonds.

Let x represent the dollars invested in stocks.

Let y represent the dollars invested in bonds.

What inequality can you write to represent the amount of money invested in stocks and bonds?

10000 20000 30000 40000 50000

10000

20000

30000

40000

50000

0

x

y

000,40 yx StocksB

ond

s

10000 20000 30000 40000 50000

10000

20000

30000

40000

50000

0

x

y

Paying for CollegeEach investment requires a minimum purchase of $5,000

What are your new inequalities?

Since stocks are more risky you want to at least twice as many bonds as stocks.

What is the inequality?

000,40 yx

Stocks10000 20000 30000 40000 50000

10000

20000

30000

40000

50000

0

x

y

000,5

000,5

y

x

xy 2

Feasible Region

Bon

ds

Constraints

Pg 339

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