Name:___________________________

LINEAR PROGRAMMING

NOTES AND HOMEWORK

EXAMPLE 1: The Firm

It takes a tailoring firm 2 hours of cutting and 4 hours of sewing to make a knit suit. To make a wool suit, it takes 4 hours

of cutting and 2 hours of sewing. At most, 20 hours per day are available for cutting and 16 hours per day for sewing.

The profit on a knit suit is $34 and on a wool suit is $31. How many of each type of suit should be made to maximize

profit and what will that maximum profit be?

HELPFUL TOOL: The Box

Max or Min circle

Knit (x) Wool (y) Symbol Total

Cutting

Sewing

Profit

Objective Function:

The variables are at the top

of the box & fill down

Constraints:

*Hidden Constraints:

Knit (x)

Wo

ol

(y)

Answer:

They should make

_______ knit suits & ________ wool suits

For a maximum profit of $___________.

Feasible Region:

Vertices:

EXAMPLE 2: Papa Johns

John is hosting dinner for his large family. He has decided to buy pizzas. He needs at least 4 cheese pizzas and

6 pepperoni pizzas but needs at least 14 pizzas total. Each cheese pizza costs $8.00 and each pepperoni is $2 more. How

many pizzas of each type should he buy to minimize the cost of dinner?

Max or Min circle

Linear Programming Steps:

1. Identify Variables

2. Write constraints

(Remember the hidden ones)

and fill in box

3. Solve for y and graph

4. Shade feasible region

5. Find vertices of feasible

region

6. Substitute vertices into

objective function

7. Find max or min of objective

function

Cheese (x) Pepperoni (y) Symbol

Total Pizzas

Cost

Other Inequalities:

Cheese (x)

Pep

per

on

i (y

)

Answer:

John should order

_______ cheese &

________ pepperoni pizzas

For a minimum cost of

$___________.

PROBLEM 1: Its Snow Problem

A water ski manufacturing company makes two types of skis: trick and slalom. Trick skis require 6 hours in the

fabricating department and slalom skis require 4 hours. Each type of ski requires one hour in the finishing department.

There are a total of 108 hours available for fabricating and 24 hours for finishing. The profit on the trick skis is $40 and

on slalom is $30. How many of each type of ski should be produced to maximize profit?

Max or Min

circle

Trick Slalom Symbol Total

Trick Skis (x)

Answer:

They should make

_______ trick &

________ slalom skis

For a maximum profit of

$___________.

Sla

lom

Sk

is (

y)

Profit

Fabricate

Finish

PROBLEM 2: Keep Fishin

A private fishing resort has bass and trout in its lake. The owner provides two types of food, A and B, for these fish.

Each week, the bass require 2 units of food A and 4 units of food B. The trout require 5 units of food A and 2 units of

food B. The owner can only supply 800 units of each type of food on a weekly basis. What is the maximum number of

each type of fish the lake can support under these conditions?

Max or Min

circle

Bass Trout Symbol Total

Answer:

The lake can

support

_______ Bass

&

_______ Trout.

For _______ total

fish.

Tro

ut

# of Fish

Food A

Food B

Bass 20

20

400

400

PROBLEM 3: Dont Keep It Lowe

A tool company manufactures two types of drills, one cordless and the other corded. The cord-type drill requires 2 labor

hours to make and the cordless type requires 3. The company only has 600 labor hours available each day and the

packaging department can package no more than 250 drills each day. The cordless drill sells for a profit of $60 and the

corded for $45. How many of each type of drill should be made to maximize profit?

Max or Min

circle

Answer:

They should make

_______ corded

&

_______ cordless.

For a max profit of

$______________

Profit

Packaging

10

10

200

100

Corded Symbol Total

200

PROBLEM 4: Fly Away

An airline with two types of airplanes, X and Y, has contracted with a tour group to provide accommodations for a

minimum of 2000 first-class passengers, 1500 tourist-class passengers, and 2400 economy class passengers. Airplane X

costs $12,000 per plan to operate ad can accommodate 40 first-class, 40 tourist-class, and 120 economy class passengers.

Airplane Y can accommodate 80 first-class, 30 tourist-class and 40 economy-class passengers and costs $10,000 per plane

to operate. How many of each type of airplane should be used to minimize cost?

Max or Min

circle

Answer:

They should use

_______ type- X airplanes

&

_______ type Y- airplanes

For a minimum cost of

$_____________________

Cost

Economy

10

200

Symbol Total

Tourist

First-Class

5

5

Problem 5: Will You Choose the Red or Blue?

Neo is ill and decides to take vitamins to help speed up his recovery. Each day he must have at least 16 mg of vitamin A,

at least 5 mg of vitamin B, and at least 20 mg of vitamin C. He can choose between red pills that cost $0.10 each and

contain 8 mg of A, 1 mg of B, and 2 mg of C; or blue pills that contain 2 mg of A, 1 mg of B and 7 mg of C but cost $0.20

each. How many of each pill should Neo buy to satisfy the minimum daily requirements at minimum cost?

Max or Min

circle

10

Answer:

Robin should take

_______ Red vitamin pills

&

_______ Blue vitamin pills

For a minimum cost of

$_____________________

For a profit of

Cost

Vitamin C

10

200

Red Pill Blue Pill Symbol Total

Vitamin B

Vitamin A

1

1

Red Pill (x)

Blue Pill

Symbol

Total

Problem 6: So Need an Answer

Sony makes two types of television sets. It produces an LCD set that gives a profit of $100 or a plasma set

which sells for $150 profit. On the assembly line the LCD set requires 3 hours while the plasma set requires 5 hours. The

electronics department spends one hour on the LCD and 3 hours on the plasma. Both sets require 2 hours for testing and

packaging. On a particular production run, the company has 3900 work hours available for the assembly line, 2100 hours

in electronics and 2200 hours in the testing and packaging area. How many of each type of set should be made to

maximize profit?

Max or Min

circle

Answer:

They should produce

_______ LCD TVs

&

_______ Plasma TVs

For a maximum profit of

$____________________

For a profit of

$______________

Profit

10

200

50

50

LCD (x)

Blue Pill

Symbol

Total

Problem 7: McMe Some Money

A fast-food chain plans to expand by opening several new restaurants. The chain operates two types of

restaurants, drive thru and full-service. A drive-thru restaurant costs $100,000 to construct, requires 5 employees and has

an expected annual revenue of $200,000. A full-service restaurant costs $150,000 to construct, requires 15 employees and

has an expected annual revenue of $500,000. The chain has $2,400,000 available for construction costs. Labor contracts

require that they hire no more than 210 employees and licensing restrictions require that they open no more than 20 new

restaurants. How many restaurants of which type should the chain open in order to maximize their expected revenue?

Max or Min

circle

Answer:

They should open

_______ Drive-Thrus

&

_______ Full-Service Restaurants

For a maximum profit of

$_____________________.

They will use

$_______________ in capital and

Will hire ________ employees and

Open _________ restaurants.

For a profit of

Profit

10

200

Drive-Thru Full-Service Symbol Total

1

1

Problem 8: Soy Maize

A farmer has a 320 acre farm on which she plants two crops: corn and soybeans. For each acre of corn planted, her

expenses are $50 and for each acre of soybeans planted, her expenses are $100. Each acre of corn requires 100 bushels of

storage and yields a profit of $60; each acre of soybeans requires 40 bushels of storage and yields a profit of $90. If the

total amount of storage space available is 19,200 bushels and the farmer has only $20,000 on hand, how many acres of

each crop should she plant in order to maximize her profit? What will her profit be if she follows this strategy?

Max or Min

circle

Answer:

He should plant

_______ acres of corn

&

_______ acres of soybeans

For a maximum profit of

$____________________

Profit

200

Hint: You may need

to solve for the

intersection as at

least one wont be

easy to tell from

the graph.

Problem 9: Its Electric

A plant makes aluminum and copper wire. Each pound of aluminum wire requires 5 kwh of electricity and 1/4 hr. of

labor. Each pound of copper wire requires 2 kwh of electricity and hr. of labor. Production of copper wire is restricted

by the fact that raw materials are available to produce at most 60 lbs./day. Electricity is limited to 500 kwh/day and labor

to 40 personhrs./day. If the profit from aluminum wire is $.25/lb. and the profit from copper is $.40/lb., how much of

each should be produced to maximize profit and what is the maximum profit?

Max or Min

circle

10

Answer:

They should product

_______ lbs. of aluminum

&

_______ lbs. of copper

For a maximum profit of

$____________________

200

Other constraint:

Problem 10: Burning for an Answer

TAE Electronics manufactures portable tape players and CD players. The manufacturing plant has the capacity to

manufacture at most 750 tape players and 500 CD players in one month. Combined, they can only manufacture 900

products. It takes 2 hours to make a tape player and 5 hours to make a CD player. The company can spend no more than

3000 hours manufacturing these products. TAE Electronics makes $4 profit on tape players and $7 profit on CD players.

To maximize profits, how many tape players and how many CD players should they make? What is the maximum profit?

Max or Min

circle

Answer:

They should make

_______ tape players

&

_______ CD players

For a maximum profit of

$____________________

200

Other:

Problem 11: Soarin

Charles is chief mathematician for Fly-By-Night Aircraft Corp. He is responsible for mathematical analysis of

the manufacturing of the company's two models of planes, the Eagle and the Hippo. The production department can make

no more than 7 Hippos and 11 Eagles. The shipping department can move no more than 12 planes total per day. The sales

department can sell no more than twice the number of Hippos than the number of Eagles. The personnel department must

use more than 800 man-hours of labor per day (it takes 100 man-hours to manufacture each Eagle and 200 for each

Hippo.) If one Eagle makes $300 in profit and one Hippo makes $200 in profit, how many of each should they make to

maximize profit?

Max or Min

circle

Answer:

They should make

_______ Eagles

&

_______ Hippos

For a maximum profit of

$____________________

200

Note: 3 of the constraints may

not be able to be placed in the

box. Write an inequality for it

here.

Eagles (x)

Blue Pill

Symbol

B ONUS LINEAR PROGRAMMING PROBLEM

Problem 12: Dont Gas

The manufacturing process requires that oil refineries must manufacture at least 2 gal of gasoline for every gallon of fuel

oil. To meet the winter demand for fuel oil, at least 3 million gal a day must be produced. The demand for gasoline is no

more than 6.4 million gal per day. It takes .25 hour to ship each million gal of gasoline and 1 hour to ship each million gal

of fuel oil out of the warehouse. No more than 4.65 hours are available for shipping. If the refinery sells gasoline for $1.25

per gal and fuel oil for $1 per gal, how much of each should be produced to maximize revenue? Find the maximum

revenue.