3.4 Linear Programming

Rita Korsunsky

Example: Maximizing a Profit

A small TV manufacturing company produces console and portable TV’s using three different machines, A, B, and C. The table below shows how many hours are required on each machine per day in order to produce a console TV or a portable TV. Company makes a $60 profit on each console TV and a $40 on each portable TV. How many of each should be produced every day to maximize the profit?

Machine Console Portable Hours available

A 1 h 2 h 16

B 1 h 1 h 9

C 4 h 1 h 24

162 yx

0

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8

9 yx

244 yx x 0 6

y 8 5

162 yx

x 3 4

y 6 5

9 yx

x 4 5

y 8 4

244 yx

Machine Console Portable Hours available

A 1 h 2 h 16

B 1 h 1 h 9

C 4 h 1 h 24

0

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8

Plot several profit lines:

dollars 4060 yxP yx 4060120

03y

20x

x 0 6

y 9 0

2004060 yx

As profit increases, the corresponding profit lines move farther away from the origin.

A company makes a $60 profit on each console TV and a $40 on each portable TV. How many of each should be produced every day to maximize the profit?

3604060 yx

4604060 yx

0

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8

Exclude any profit value > 460 because these profit lines don’t have points in the region.

“greatest-profit line” w/points in region: 4604060 yx

maximum profit: $460 (5 console TV’s and 4 portable TV’s)

“least-profit line”: through corner point (0,0).In general, the greatest or least value of a linear expression will always occur at a corner point of the region. So…

4604060 yx

320840060

460440560 400740260

360040660

2nd method of finding Max

Example: Minimizing a CostEvery day Rhonda Miller needs 4 mg of vitamin A, 11 mg of vitamin B, and 100 mg of vitamin C. Either of two brands of vitamins can be used: Brand X at 6¢ a pill or Brand Y at 8¢ a pill. A Brand X pill supplies 2 mg of vitamin A, 3 mg of vitamin B, and 25 mg of vitamin C. A Brand Y pill supplies 1 mg of vitamin A, 4 mg of vitamin B, and 50 mg of vitamin C. How many pills of each brand should she take each day in order to satisfy the minimum daily need most economically?

Brand X Brand Y Minimum Daily Need

Vitamin A 2 mg 1 mg 4 mg

Vitamin B 3 mg 4 mg 11 mg

Vitamin C 25 mg 50 mg 100 mg

Cost per pill 6¢ 8¢

Brand X Brand Y Minimum Daily Need

Vitamin A 2 mg 1 mg 4 mg

Vitamin B 3 mg 4 mg 11 mg

Vitamin C 25 mg 50 mg 100 mg

Cost per pill 6¢ 8¢

0

1

2

3

4

1 2 3 4

x 0 2

y 4 0

42 yx

x 1 3

y 2 0.5

1143 yx

x 0 4

y 2 0

42 yx

0

1

2

3

4

1 2 3 4

yxC 86

Method 2: Evaluate the cost at each corner point:

(0,4)

(1,2)

(3,0.5)

(4,0)

324806 ¢

222816 ¢

225.0836 ¢

240846 ¢

2486Graph yxx 0 4

y 3 0

Method 1: Continue graphing parallel lines until you reach the last corner point in the region.

The solutions are (1,2) and (3,0.5), but it is inconvenient to take half a pill.

The best choice is 1 Brand X pill and 2 Brand Y pills.

Brand X is at 6¢ a pill and Brand Y at 8¢ a pill.How many of each should she take to minimize the cost?