Chapter 3 Linear Programs

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Chapter 3 Linear Programs

• Section 3.1 Linear Inequalities in Two Variables
• Section 3.2 Solutions of Systems of Inequalities
  A Geometric Picture
• Section 3.3 Linear Programming A Geometric
  Approach
• Section 3.4 Applications

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Graph the linear inequality.
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Graph the linear inequality.
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Graph the linear inequality.
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Graph the linear inequality.
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Graphing a Linear Inequality
Section 3.1

• Graph the inequality of the form ax by lt c.
  (The procedure also applies if the
  inequality symbols are lt, gt or gt.)
• Select a point that is not on the line from one
  half plane. The point (0,0) is usually a good
  choice when it is not on the line. If (0,0) is
  on the line. If (0,0) is on the line, use a
  point that is not on the line.
• Continued on next slide

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Continued

• Substitute the coordinates of the point for x and
  y in the inequality.
• If the selected point satisfies the inequality,
  then shade the half plane where the point lies.
  These points are on the graph.
• If the selected point does not satisfy the
  inequality, shade the half plane opposite the
  point.
• If the inequality symbol is lt or gt, use a dotted
  line for the graph of ax by c. This
  indicates that the points on the line are not a
  part of the graph.
• If the inequality symbol is lt or gt, use a solid
  line for the graph of ax by c. This
  indicates that the line is a part of the graph.

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Example
An automobile assembly plant has an assembly line
that produces the Hatchback Special and the
Sportster. Each Hatchback requires 2.5 hours of
assembly line time, and each Sportster requires
3.5 hours. The assembly line has a maximum
operating time of 140 hours per week. Graph the
number of cars of each type that can be produced
in one week.
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A bakery is making whole-wheat bread and apple
bran muffins. The bread takes 4 hours to prepare.
The muffins take 0.5 hour to prepare. The maximum
preparation time available is 16 hours. Graph the
number of of each type that can be prepared in
one day.
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Acme Manufacturing has two product lines. Line A
can produce 200 gadgets per hour and line B can
produce 350 widgets per hour. Because of
warehouse limitations, the total number of
gadgets and widgets produced must not exceed
75,000. Write an inequality that describes the
number of each that can produced and graph it.
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A service club agrees to donate at least 500
hours of community service. A full member is to
give 4 hours and a pledge is to give 6 hours.
Write an inequality that expresses this
information and graph it.
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On a typical long distance call you talk for 30
minutes. On a typical local call you talk for 10
minutes. Your phone company offers a special low
rate of 0.08 per minute for long distance calls
and 0.03 per minute for local calls, for
customers who spend at least 240 minutes on the
phone per month. Your parents have set a limit of
no more than 15 long distance calls per month and
30 local calls per month. Write some
inequalities that describe this situation.
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Two manufacturing plants make the same kind of
bicycle. The table gives the hours of general
labor, machine time, and technical labor required
to make one bicycle in each plant. For the two
plants combined, the manufacturer can afford to
use up to 4000 hours of general labor, up to 1500
hours of machine time, and up to 2300 hours of
technical labor per week. Write some linear
inequalities that describe this situation.
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HW 3.1

• Pg 202-204 1-35

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3.2 Systems of Linear Inequalities
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Graph the system of linear inequalities.
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Graph the system of linear inequalities.
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Graph the system of linear inequalities.
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Graph the system of linear inequalities.
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Graph the system of linear inequalities.
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Graph the system of linear inequalities.
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Graph the system of linear inequalities.
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Graph the system of linear inequalities.
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Your Turn
Graph the system of linear inequalities.
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A Geometric Picture
Section 3.2
_______________ problems are described by systems
of linear inequalities. The ________________ is
the region of intersection on a graph of a system
of inequalities and is the ___________ to the
system of linear inequalities.
_____________________ problems generally have a
_____________________ on the variables that
states that some or all of the variables can
never be ____________ because the quantities they
measure (number of items, weight of materials)
can never be negative.
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Example
Graph the solutions (feasible region) to the
following system
SOLUTION The lines x y 4, 3x 2y 3 and x
0 determine _____________ of the solution set.
The intersection of these half plane solutions of
the boundaries forms the _________________________
_.
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Example continued
The points A and B on the graph are called the
_______________________________ because they are
the points in the feasible region where the
boundaries ___________.
Corners are the _______________ solution to a
linear programming problem. Corners can be found
by solving pairs of simultaneous equations of the
lines forming the ________.
The corner A is found by finding the intersection
of x 0 and 3x 2y 3.
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Types of Solutions
The system of inequalities has
__________________________ if the feasible region
can be enclosed in a region where all points are
a finite distance apart.
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Types of Solutions
A system of inequalities has _____________________
______ because some of the points in the feasible
region are infinitely apart.
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Example
Find the solution set (feasible region) of the
system
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Summary

• Graphing a System of Inequalities
• Replace each inequality symbol with an equals
  sign to obtain a linear equation.
• Graph each line. Use a solid line if it is a
  part of the solution. Use a dotted line if it is
  not a part of the solution. The line is a part
  of the solution when lt or gt is used. The line is
  not a part of the solution when lt or gt is used.
• Select a test point not on the line.

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continued

1. If the test point satisfies the original
   inequality, it is in the correct half plane. If
   it does not satisfy the inequality, the other
   half plane is the correct one.
2. Shade the correct half plane.
3. When the above steps are completed for each
   inequality, determine where the shaded half
   planes overlap. This region is the graph of the
   system of inequalities.

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Write a system of linear inequalities that has
the given graph
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Write a system of linear inequalities that has
the given graph
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Write a system of linear inequalities that has
the given graph
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Write a system of linear inequalities that has
the given graph
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A retired couple has up to 50,000 to invest. As
their financial adviser, you recommend that they
place at least 35,000 in Treasury bills yielding
7 and at most 10,000 in corporate bonds
yielding 10.

• Using x to denote the amount of money invested in
  Treasury bills and y the amount invested in
  corporate bonds, write a system of linear
  inequalities that describes the possible amounts
  of each investment.
• Graph the system and label the corner points.

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Mikes Toy Truck Company manufacturers two models
of toy trucks, a standard model and a deluxe
model. Each standard model requires 2 hours for
painting and 3 hours for detail work each deluxe
model requires 3 hours for painting and 4 hours
for detail work. Two painters and three detail
workers are employed by the company, and each
works 40 hours per week.

• Using x to denote the number of standard model
  trucks and y to denote the number of deluxe
  model trucks, write a system of linear
  inequalities that describes the possible number
  of each model of truck that can be manufactured
  in a week.
• Graph the system and label the corner points.

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Bills Coffee House, a store that specializes in
coffee, has available 75 pounds of A grade coffee
and 120 pounds of B grade coffee. These will be
blended into 1 pound packages as follows An
economy blend that contains 4 ounces of A grade
coffee and 12 ounces of B grade coffee and a
superior blend that contains 8 ounces of A grade
coffee and 8 ounces of B grade coffee.

• Using x to denote the number of packages of the
  economy blend and y to denote the number of
  packages of the superior blend, write a system of
  linear inequalities that describes the possible
  number of packages of each kind of blend.
• Graph the system and label the corner points.

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Nolas Nuts, a store that specializes in selling
nuts, has available 90 pounds of cashews and 120
pounds of peanuts. These are to be mixed in
12-ounce packages as follows a lower-priced
package containing 8 ounces of peanuts and 4
ounces of cashews and a quality package
containing6 ounces of peanuts and 6 ounces of
cashews.

• Using x to denote the number of ounces of cashews
  and y to denote the number of peanuts, write a
  system of linear inequalities that describes the
  possible number of packages of each kind of
  blend.
• Graph the system and label the corner points.

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A small truck can carry no more than 1600 pounds
of cargo nor more than 150 cubic feet of cargo. A
printer weighs 20 pounds and occupies 3 cubic
feet of space. A microwave oven weighs 30 pounds
and occupies 2 cubic feet of space.

• Using x to represent the number of microwave
  ovens and y to represent the number of printers,
  write a system of linear inequalities that
  describes the number of ovens and printers that
  can be hauled by the truck.
• Graph the system and label the corner points.

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Your Turn

• A theater wishes to book a musical group that
  requires a guarantee 7000. Tickets prices are
  10 for students and 15 for adults, and the
  theaters maximum capacity is 550 seats.
• State the inequalities that represent this
  information.
• Graph the system of linear inequalities
• Find the corner points

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Hw 3.2

• Pg 210-212 1-40

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Linear Programming
Section 3.3
Constraints and Objection Function
A linear inequality of the form a1x a2y lt b
or a1x a2y gt b is called a __________ of a
linear programming problem. The restrictions x gt
0 and y gt 0 are __________________.
THEOREM Given a linear _____________________
subject to linear inequality constraints, if the
objection function has an ____________________
(maximum or minimum), it must occur at a corner
point of the feasible region.
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Example
Find the maximum value of the objective
function z 10x 15y subject to the constraints
SOLUTION First, ______________________ of the
system of inequalities and locate the
__________________. The corner points can be
found by solving the system of equations of the
lines that intersect at the point.
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Example continued
The corner points of the feasible region are
________, _________, __________, and ____________.
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Example continued
Now, find the value of z at each corner point.
Corner z 10x 15y
(0,90)
(0,0)
(150,0)
(120,60)
The maximum value of z is _______________ and
occurs at the corner ________________.
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Find the minimum and maximum values of the
objective function subject to the given
constraints.
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Your Turn
Find the minimum and maximum values of the
objective function subject to the given
constraints.
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Theorems
Bounded Feasible Region When the feasible region
is not __________ and is __________, the
objective function has both a ______________ and
a _______________ value, which must occur at
corner points.
Unbounded Feasible Region When a feasible region
with ______________ conditions is
_________________, an objective function assumes
a ___________ at a corner point of the feasible
region. However, the objective function can be
arbitrarily large for points in the feasible
region, so no optimal ______________ solution
exists.
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Example
Find the maximum value of the objective
function z 4x 6y subject to the constraints
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Example continued
Now, find the value of ______ at each corner
_________.
Corner z 4x 6y





The maximum value of z is ___ and occurs at the
corner _____.
The minimum value of z is ____ and occurs at the
corner ______.
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Maximizing Profit A manufacturer of skis produces
two types downhill and cross-country. Use the
following table to determine how many of each
kind of ski should be produced to achieve a
maximum profit. What is the maximum profit? What
would the maximum profit be if the maximum time
available for manufacturing is increased to 48
hours?
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Farm Management A farmer has 70 acres of land
available or planting either soybeans or wheat.
The cost of preparing the soil, the workdays
required, and the expected profit per acre
planted for each type of crop are given in the
following table
The farmer cannot spend more than 1800 in
preparation costs nor use more than a total of
120 workdays. How many acres of each crop should
be planted to maximize the profit? What is the
maximum profit? What is the maximum profit if the
farmer is willing to spend no more than 2400 on
preparation?
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Farm Management A small farm in Illinois has 100
acres of land available on which to grow corn and
soybeans. The following table shows the
cultivation cost per acre, the labor cost per
acre, and the expected profit per acre. The
column on the right shows the amount of money
available for each of these expenses. Find the
number of acres of each crop that should be
planted to maximize profit.
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Dietary Requirements A certain diet requires at
least 60 units of carbohydrates, 45 units of
protein, and 30 units of fat each day. Each ounce
of Supplement A provides 5 units of
carbohydrates, 3 units of protein, and 4 units of
fat. Each ounce of Supplement B provides 2 units
of carbohydrates, 2 units of protein, and 1 unit
of fat. If Supplement A costs 1.50 per ounce and
Supplement B costs 1.00 per ounce, how many
ounces of each supplement should be taken daily
to minimize the cost of the diet?
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Production Scheduling In a factory, machine 1
produces 8-inch pliers at the rate of 60 units
per hour and 6-inch pliers at the rate of 70
units per hour. Machine 2 produces 8-inch pliers
at the rate of 40 units per hour and 6-inch
pliers at the rate of 20 units per hour. It costs
50 per hour to operate machine 1, and machine 2
costs 30 per hour to operate. The production
schedule requires that at least 240 units of
8-inch pliers and at least 140 units of 6-inch
pliers be produced during each 10-hour day. Which
combination of machines will cost the least money
to operate?
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Farm Management An owner of a fruit orchard hires
a crew of workers to prune at least 25 of his 50
fruit trees. Each newer tree requires one hour to
prune, while eacholder tree needs one-and-a-half
hours. The crew contracts to work for at least 30
hours and charge 15 for each newer tree and 20
for each older tree. To minimize his cost, how
many of each kind of tree will the orchard owner
have pruned? What will be the cost?
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Groups

• On your way into class you picked up one of 8
  different problems.
• Find the other people in class who have your
  problem and form a group.
• When you have a solution to your problem let me
  know and I will check your solution.
• One person from your group will need to explain
  your problem to the class
• Everyone is responsible to know how to do every
  problem presented. Quiz tomorrow

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Discussion and Writing

• Explain in your own words what a linear
  programming problem is and how it can be solved.

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HW 3.3

• Pg 231-239 1-75 odd

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Applications
Section 3.4

• Solving linear programming problems geometrically
  works well when there are only two variables and
  a few constraints.
• Typically though, linear programming problems
  will require dozens of variables with several
  constraints.
• A correct analysis and description of the problem
  is essential before applying any method. An
  erroneous constraint will yield an erroneous
  solution.
• This section works on correctly setting up linear
  programming problems in more than two variables
  so that the methods in Chapter 4 can be utilized
  to solve these larger systems.

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Example
Adventure Time offers one-week summer vacations
during the month of August. The package includes
round-trip transportation and a weeks
accommodations at the Lodge. The Lodge gives a
discount to Adventure Time if they rent two- or
three-week blocks of condos, rent of 1000 per
condo for a two-week period and 1300 per condo
for a 3-week period. Adventure Time expects to
need the number of condos shown.
Week Number of Condos needed
First 30
Second 42
Third 21
Fourth 32
How many condos should Adventure Time rent for
two weeks, and how many should be rented for
three weeks, to meet the needed number and to
minimize Adventure Times rental costs?
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Example continued
SOLUTION First, determine all possible ways to
schedule two- and three-week blocks in August.
The table below helps to visualize the possible
ways to schedule the blocks.
Two-week periods Three-week periods Number of condos needed
Week 1
Week 2
Week 3
Week 4
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Example continued
Since Adventure Time wants to minimize their
rental costs, we need to minimize
The relationship between the number of condos
needed and the five time periods can be written as
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HW 3.4

• Pg 243-246 1-20