Linear Programming is actually one of the most useful topics in math because it is a simple
model for problems with limited resources. Many large companies, even the military, use linear
programming to solve problems such as finding maximum profits when given certain restrictions on
There are 5 main steps to solving a Linear Programming problem:
1) Place all of the necessary information into a neat and organized table .
2) Find out what has to be maximized or minimized, and in most cases it is the profit or cost.
Then, write the profit/cost equation.
3) Find the variables of the problem and what parts can be controlled. These equations will
be known as the constraints. Depending on the problem there can be many or very few
4) Once the constraints have been established, plot them on a graph to find the vertices of
the linear function.
5) After the graph is drawn, the vertices of the graph will tell you the maximum or minimum of
the system by plugging them into the profit/cost equation.
Now lets try this problem together.
A magazine company sells two main types of magazines: Healthwise sells for $12 and
Superteen sells for $10. It costs the company $9 to produce Healthwise and $8 to produce
Superteen. In one week the publishing company can print 200 – 300 copies of Healthwise and from
100 – 250 copies of Superteen, but no more than 500 copies in total. How many of each type
should be printed in order for the company to make maximum profit.
The first step is to make a chart to make everything neat and organized. This step is used
to make things easier for you to understand.
Magazine Sales Amount Production Costs Number of Printable
Healthwise $12 $9 200 – 300
Superteen $10 $8 100 – 250
Now that the information is organized we can find the profit equation: X= Healthwise Y = Superteen
The Profit equation will be P = the Profit X makes + the Profit Y makes
Since production costs for Healthwise is $9, we have to subtract it from the Sales amount, which is
So $12 $9 = $3
The production costs for Superteen are $8 and the price is $10 so the profit will be the two
So $10 $8 = $2
The Profit Equation will be P
= 3x + 2y
Now we have to find the constraints. These constraints will explain how to draw the graph.
The constraints will be:
x + y ≤ 500 This constraint means that the number of Healthwise magazine (x) plus the number
of Superteen magazines (y) is less than or equal to 500, because all
together it said in the problem that no more than 500 copies can be made in one week. Therefore x
+ y has to be less than or equal to 500
200 ≤ x ≤ 300 This constraint explains that only 200 – 300 copies of Healthwise can be printed
during one week
100 ≤ y ≤ 250 The last constraint shows the amount of copies that Superteen can make
within a week.
To find out how to graph the line x + y ≤ 500, you have to give x a number to solve for y.
X + y = 500 Lets make x = 100
100 + y = 500
y = 500 – 100
y= 400 So the coordinates are (100, 400)
Lets get another set of values to draw the line correctly, but this time, lets plug in for y.
x + y = 500 Lets make y = 200
x + 200 = 500
x = 500 – 200 x = 300 So the coordinates are (300, 200)
As you know x will have two straight lines one on 200 and one on 300. Y will also have two straight
lines, one on 100 and the other on 250.
Now we plot the graph, which will look like this:
Notice that the area shaded is shared by all regions of the
As you can see there is a boxed in area that is shaded which has
five labelled intercepts:
A (200,100), B (300, 100) and E (200, 250). The intercepts D and E are a little different since you
can only see one precise intercept (y = 250). To find the other value plug in the x or y value that
you do know and solve for x or y (substitution).
D = y = 250 so x + y = 500
x + 250 = 500
x = 500 – 250
x = 250 The vertices D are (250, 250)
C = x = 300 so x + y = 500
300 + y = 500
y = 500 – 300
y = 200
The Vertices for C are (300, 200)
Now that we have all of the vertices we can plug them into our profit equation and solve for the
A) P = 3x +2y B) P = 3x +2y
P = 3(200) + 2(100) P = 3(300) + 2(100)
P = 600 + 200 P = 900 +200
P = $800 P = $1100
C) P = 3x + 2y D) P = 3x + 2y
P = 3(300) + 2(200) P = 3(250) +2(250) P = 900 + 400 P = 750 +500
P = $1300 P = $1250
E) P = 3x + 2y
P = 3(200) + 2(250)
P = 600 + 500
P = $1100
P = 3x + 2y
The company should make 300 copies of Healthwise and 200 copies of Superteen to make a
maximum profit of $1300.
Lets try another question together.
A lumber company converts logs into baseball bats. In a week, the company can turn out
400 bats, of which 100 deluxe bats and 150 regular bats are required on a regular basis. The profit
of a deluxe baseball bat is $20 and the profit on a regular baseball bat is $30. How many of each
type should the lumber company make to have maximum profit?
Remember to first make a chart to organize the information.
Type of Bat Demand Profit
Deluxe 100 $20
Regular 150 $30
X = Deluxe Y = Regular
The profit equation this time will be P = 20x + 30y because there are no production costs.
The constraints are:
x + y ≤ 400 This means that in total the company cannot make anymore than 400 bats in one
week. X ≥ 100 The x is greater than or equal to 100 because that is the minimum amount of deluxe
bats that the company can make.
Y ≥ 150 The y is greater than or equal to 150 because it is the minimum amount of regular bats
that can be made in one week since there is a demand for at least that amount.
To plot the graph we have to find the coordinates for x + y ≤ 400. Remember to pick a number for
the x or y value and solve for either x or y.
X + y = 400 x = 100
100 + y = 400
y = 400 –100
y = 300 So the coordinates are (100,300)
x + y =400
x +100 = 400
x = 400 –100
x = 300 So the coordinates are (300,100)
And of course we know that there will be a line at x = 100 and y = 150
Now you can plot the graph, which will look like this:
Notice that the area shaded is shared by all regions of the constraints.
To find the coordinates at A, you know it is on the xaxis at 100 and the yaxis is 0, so the A is
To find the coordinates of B, we know that it is 0 on the yaxis and intercepts with the line for x + y
= 400. So all we do is sub in 0 for y and solve for x.
y = 0
x + y = 400
x + 0 = 400 x = 400
The coordinates at B are (400, 0)
To find the coordinates at C, we know that the two lines intersecting are
y = 150, and x + y = 400. Again, sub in 150 for y and solve for x.
y = 150
x + y = 400
x + 150 = 400
x = 400 – 150
x = 250
The coordinates at C are (250, 150)
The coordinates of D are pretty straight forward sine all we need to do is look at the graph and see
that the two lines intersecting at D are x = 100 and y = 150. So the coordinates of D are (100, 150)
The vertices are: A (100, 0), B(400, 0), C(250, 150) and D(100, 150). Remember that we plug
these intercepts into the profit equation and find the maximum profit.
A) P = 20x + 30y B) P = 20x + 30y
(100, 0) (400, 0)
P = 20(100) + 30(0) P = 20(400) + 30(0)
P = $2000 P = $8000
C) P = 20x + 30y D) P = 20x + 30y
(250, 150) (100, 150)
P = 20(250) + 30(150) P = 20(100) + 30(150)
P = $9500 P = $6500
Vertices P = 20x + 30y
(100, 0) $2000
(400, 0) $8000
(250, 150) $9500 maximum
(100, 150) $6500
To make a maximum profit the lumber company must make 250 deluxe bats and 150 regular bats
to make a maximum profit of $9500.
Let’s try one last question together:
A window manufacturing company makes two types of windows, regular and heavy duty.
Each regular window takes approximately 3 hours to cut and 1 hour to finish. The heavyduty windows take 2 hours to cut and 4 hours to finish. Each regular window makes a net profit of $80
and the heavyduty window makes a net profit of $200. If 4 cutting and 6 finishing workers are used
12 hours per day how many of each window should be made for the company to make a maximum
Draw the organization table
Type of Window Cutting Hours Finishing Hours Profit
Regular 3 hours 1 hour $80
HeavyDuty 2 hours 4 hours $200
The profit equation will be P = 80x + 200y.
The constraints are:
3x +2y ≤ 48 It takes three hours to cut a regular window and two hours to cut a heavy duty
one. The fortyeight comes from the 4 cutting workers multiplied by how long they worked which
was 12 hours at maximum.
x+ 4y ≤ 72 It takes one hour to finish a regular and 4 hours to finish a heavyduty window.
The seventytwo is the 6 workers multiplied by the 12 hours because that is the maximum time that
the finishers can work.
To plot the graph we have to draw the two constraints so lets sub in the values for x and y.
3x +2y = 48 3x +2y =48
3(10) + 2y = 48 3x + 2(3)= 48
2y = 48 – 30 3x = 48 6
2y = 18 3x = 42
y = 9
The coordinates are (10,9) x = 14
The coordinates are (14,3)
x + 4y = 72 x + 4y = 72
4 + 4y = 72 x + 4(15) = 72
4y = 72 – 4 x = 72 60
4y = 68 x = 12
y = 17
The coordinates are (4, 17) The coordinates are (12, 15)
Here’s what the graph should look like: Notice that the area shaded is shared by all regions of the constraints.
A) (0, 0)
B) y = 0
3x + 2y = 48
3x + 0 = 48
3x = 48
x = 16
The coordinates of B are (16, 0)
C) 3x + 2y = 48
x + 4y = 72
For this particular intercept, we use the method of elimination to solve for x and y.
(3x + 2y = 48)(2)
6x – 4y = 96
+ x + 4y = 72
5x = 24
x = 4.8
x + 4y = 72
4.8 + 4y = 72
4y = 72 – 4.8
4y = 67.2
y = 16.8
So, the coordinates for C are (4.8, 16.8)
D) x = 0
x + 4y = 72
0 + 4y = 72
y = 18
The coordinates at D are (0, 18)
P = 80x + 200y
(16,0) $1280 (4.8, 16.8) $3744 maximum
To make a maximum profit the company would have to sell 4.8 regular windows and 16.8 heavy
duty windows to make a profit of $3744.
Now you can try some of the practice problems on your own.
Let’s Refresh Your Memory…
( A review of finding max. and mins., and y = mx + b)
1) Determine the maximum and minimum values for each of the following:
a) ƒ (x,y) = 3x + 5y vertices at (4,8), (2,4), (1,1), (5,2)
b) ƒ (x,y) = x + 4y vertices at (0,7), (0,0), (6,2), (5,4)
2) The size of shoe a person needs varies linearly with the length of his or her foot. The
smallest adult shoe size is Size 5, and it fits a 9 inch long foot. An 11inch foot requires a
Size 11 shoe.
a) Write the particular equation, which expresses shoe size in terms of foot length.
b) If your foot is a foot long, what size shoe do you need?
c) Bob Lanier of the Detroit Pistons wears a Size 22 shoe. How long is his foot?
d) Plot a graph of adult shoe size versus foot length. Use the given points and
calculated points. Be sure that the domain is consistent with the information in this
3) If you jump out of an airplane at high altitude but do not open your parachute, you will soon
fall at a constant velocity called your “terminal velocity.” Suppose that at t=0, you jump. When
t=15 seconds, your wrist altimeter shows that your distance from the ground, d, is 3600 meters.
When t=35 seconds, you have dropped to d=2400 meters. Assume that you are at your terminal
velocity by the time t=15.
a) Explain why d varies linearly with t after you have reached your terminal velocity.
b) Write the particular equation expressing d in terms of t.
c) If you neglect to open your parachute, when will you hit the ground?
d) According to your mathematical model, how high was the airplane when you
e) The plane was actually at 4200 meters when you jumped. How do you reconcile
this fact with your answer in part d? Now we’re ready!
4) A lumber company can convert logs into either lumber or plywood. In a given week the mill
can turn out 400 units in production, of which 100 units of lumber and 150 units of plywood
are required by regular customers. The profit on a unit of lumber is $20 and the profit on
plywood for one unit is $30. How many units of each type should the mill produce per week
in order for maximum profit?
5) An accountant’s analysis shows that you have $40 000 to invest in stocks and bonds. The
least that you are allowed to invest in stocks is $6 000 and you cannot invest more than
$22 000 in stocks. You may also invest no more than $30 000 in bonds. The interest in the
stocks is 8% taxfree and the interest on the bonds is percent taxfree. How much should
you invest in each type to maximize your profit? What is the income form the $40 000
6) A stationary company makes two types of notebooks: a deluxe notebook, with subject
dividers that sell for $1.25 and a regular notebook that sells for $.90. The production cost is
$1.00 for each deluxe notebook and $0.75 for each regular notebook. The company has
the facilities to manufacture between 2000 and 3000 deluxe and 3000 and 6000 regular,
but not more than 7000 altogether. How many notebooks of each type should be
manufactured to maximize the difference between the selling prices and the production
7) A coffee company purchases mixed lots of coffee beans and grades them into premium,
regular and unusable beans. The company needs at least 280 tons of premium grade and
200 tons of regular grade coffee beans. The company can purchase ungraded coffee
beans from two suppliers in any amount desired. Samples from the two suppliers contain
the following percentages of premium, regular and unusable beans;
Sample Premium Regular Unusable
A 20% 50% 30%
B 40% 20% 40%
If supplier A charges $125 per ton and B charges $200 per ton how much should the
company purchase from each supplier to fulfill its needs at minimum cost?
8) Gary Smelting Company receives a monthly order for at least 40 tons of iron, 60 tons of
copper and 40 tons of lead. It can fill the order by smelting either alloy A or alloy B. Each
railroad carload of A will produce 1 ton of iron, 3 tons of copper and 4 tons of lead after
smelting. Each railroad carload of B will produce 2 tons of iron, 2 tons of copper and 1 ton
of lead after smelting. If the cost of smelting one carload of the alloy A is $350 and the cost of smelting one carload of the alloy B is $200, how many carloads of each should be used
to fill the order at the minimum cost to Gary Smelting? What is the minimum cost?
9) A furniture company makes two types of desks, one plain and one fancy. Each plain desk
takes 3 hours of work to assemble and 1 hour to finish. Each fancy desk takes 2 hours of
work to assemble and 4 hours to finish. The 4 assembly workers and 6 sanding workers
are each used 12 hours per day. Each plain desk has a net profit of $80 and each fancy
desk has a net profit of $200. If the company can sell all the desks it makes, how many of
each kind should be produced each day in order to make maximum profit.
10) Almosttexas makes two types of calculators. Deluxe sells for $12 and Top of the Line sells
for $10. It costs Almosttexas $9 to produce a deluxe and $8 to produce a Top of the Line
calculator. In one week, Almottexas can produce 200 to 300 deluxe calculators and 100 to
250 Top of the Line calculators, but no more than 500 in total. How many of each type
should the company make to have the maximum profit?
11) A carpenter makes tables and chairs. Each table can be sold for a profit of $30 and each
chair for a profit of $10. The carpenter can afford to spend up to 40 hours working and it
takes six hours to make a table and three hours to make a chair. Customer demand
requires her to make at least three times as many chairs as tables. Tables take up four
times as much storage space as chairs and there is no room for more than 4 tables per
week. How many tables and chairs will she have to make, to make maximum profit?
12) A nutrition centre sells health food to mountain climbing teams. The Trailblazer mix
package contains one pound of corn cereal mixed with four pounds of wheat cereal and
sells for $9.75. The Frontier mix package contains two pounds of corn cereal with three
pounds of wheat cereal and sells for $9.50. The centre has 60 pounds of corn cereal and
120 pounds of wheat cereal available. How many packages of each mix should the centre
sell to maximize its income?
13) Angie and Nicole run very successful lemonade stand for most of the summer. They made
both lemonade and fruit punch. Although the ingredients for the fruit punch cost more, they
sold both drinks for the same price. Their profit for a cup of lemonade was $0.20, but only
$0.15 a cup for fruit punch. Every morning they made drinks for the day. One morning they
discovered that they were low on supplies. The lemonade recipe requires ½ a cup of sugar
per quart, whereas the fruit punch only required ¼ a cup per quart. They had about 4.5
cups of sugar on hand and in addition they only had enough oranges to make 8 quarts of
fruit punch. The friends want to make as much profit as they can, how much profit will they
make if they only made lemonade? Only fruit punch? What would the profit be if they made
14) A manufacturer of CB radios makes a profit of $20 on a deluxe model and $15 on a
standard model. The company wishes to produce at least 70 deluxe models and at least 100 standard models per day. To maintain high quality, the daily production should not
exceed 200 radios. How many of each type should be produced daily in order to maximize
15) A manufacturer of tennis rackets makes a profit of $15 on each oversized racket and $8 on
each standard racket. To meet dealer demand, daily production of standard rackets
should be between 30 and 80, and production of oversized rackets should be between 10
and 30. To maintain high quality, the total number of rackets produced should no exceed
80 per day. How many of each type of racket should be manufactured daily to maximize
16) Put your knowledge to the test! Create a scenario in