Mr. Barker
Discrete math
Linear programming is a tool for maximizing or
minimizing a quantity, typically a profit or a
cost, subject to constraints.
It is an example of “New” mathematics. It came
about shortly after world war II.
Automobile requires
many complicated steps
and processes. Using
linear programming
techniques enables the
robots and humans to
carry out their tasks
faster and more
accurately.
May also be used in
making fuel, drinks,
baking bread, etc.
Linear programming is often used to solve
special problems known as mixture problems.
Mixture problem
In a mixture problem, limited resources are
combined into products so that the profit from
selling those products is a maximum
 Resources: Definite resources are available in
limited, known quantities.
 Products: Definite products can be made by
combining, or mixing, the resources
 Recipes: A recipe for each product specifies how
many units of each resource are needed to
make on unit of that product.
 Profits: Each product earns a known profit per
unit.
 Objective: The objective is to find how much of
each product to make so as to maximize profit
without exceeding resources
A toy manufacturer can manufacture only
skateboards, only dolls, or some mixture of
skateboards and dolls. Skateboards require five
units of plastic and can be sold for a profit of
$1.00, while dolls require two units of plastic and
can be sold for a profit of $0.55. If 60 units of
plastic are available, what number of skateboards
and/or dolls should be manufactured for the
company to maximize its profit?
Make a table to sort out all the information. Display the
products you want to make, the materials available, and
the profit of each product. This is called a mixture chart.
Resource(s)
Containers of
plastic
60
profit
Skateboards
(x-unit)
5
$1.00
Dolls
(y-unit)
2
$0.55
A clothing manufacturer has 60 yards of cloth
available to make shirts and decorated vests. Each
shirt requires 3 yards of cloth and provides a profit of
$5. Each vest requires 2 yards of cloth and provides a
profit of $3. Make a mixture table to show this.
Resource(s)
Yards of cloth
60
Profit
Shirts
(x-unit)
3
$5
Vests
(y-unit)
2
$3
Now we need to translate
the data into
mathematical form to
produce constraints
Equation for resources
5𝑥 + 2𝑦 ≤ 60
Equation for profit
𝑃 = 1𝑥 + 0.55𝑦
Resource(s)
Containers of
plastic
60
profit
Skateboards
(x-unit)
5
$1.00
Dolls
(y-unit)
2
$0.55
Write an equation for our
clothing manufacturer
Equation for resources as
a constraint.
3𝑥 + 2𝑦 ≤ 60
Equation for profit
Resource(s)
Yards of cloth
60
Profit
Shirts
(x-unit)
3
$5
Vests
(y-unit)
2
$3
𝑃 = 5𝑥 + 3𝑦
Find the intercepts (x)
5𝑥 + 2𝑦 = 60
5𝑥 + 2 0 = 60
5𝑥 = 60
5𝑥
5
=
60
5
𝑥 = 12
Find y
5𝑥 + 2𝑦 = 60
5(0) + 2𝑦 = 60
2𝑦 = 60
2𝑦
2
=
60
2
𝑦 = 30
Write the equation
Set 𝑦 = 0
Solve
The feasible set, also called the feasible region,
for a linear-programming problem is the
collection of all physically possible solution
choices that can be made.
5x+2y=60
5x+2y≤60
40
35
30
25
20
15
10
5
0
30
5x+2y=60
20
5x+2y≤60
10
0 2 4 6 8 10 12
Feasible region
0
0 2 4 6 8 10 12
Pg. 139 1,3, 7-13 odd, 17, 19
5x+2y=60
5x+2y≤60
40
35
30
25
20
15
10
5
0
30
5x+2y=60
20
5x+2y≤60
10
0 2 4 6 8 10 12
Feasible region
0
0 2 4 6 8 10 12