Special Topics
Chapter 4.1
Mixture Problems
Chapter 4: Linear Programming
Lesson Plan
For All Practical
Purposes
 Mixture Problems
 Combining Resources to Maximize Profit
 Finding the Optimal Production Policy
 Why the Corner Point Principle Works
 Decreasing-Time-List Algorithm
 Linear Programming
 Life Is Complicated
 A Transportation Problem
 Delivering Perishables
 Improving on the Current Solution
Mathematical Literacy in
Today’s World, 8th ed.
Chapter 4: Linear Programming
Mixture Problems
 Linear Programming
 A management science technique that helps a business
allocate the resources it has on hand to make a particular mix of
products that will maximize profit.
 One of the most frequently used management science
techniques.
 Mixture Problem
 Limited resources are
combined into products
in such a way that the
profit from selling those
products is a maximum.
Chapter 4: Linear Programming
Mixture Problems
 Production Policy
 A solution to a linear-programming mixture problem is a
production policy that tells us how many units of each
product to make.
 Optimal Production Policy Has Two Properties
 First, it is possible; that is, it does not violate any of
the limitations under which the manufacturer
operates, such as availability of resources.
 Second, the optimal production policy gives the
maximum profit.
Chapter 4: Linear Programming
Mixture Problems
Common Features of Mixture Problems
Resources – Available in limited, known quantities
for time period.
Products – Made by combining, or mixing, the
resources.
Recipes – How many units of each resource are
needed.
Profits – Each product earns a known profit per unit.
Objectives – To find how much of each product to
make to maximize profit without
exceeding any of the resource limitations.
Chapter 4: Linear Programming
Mixture Problems
 Mixture Problem: Making Skateboards and Dolls
 Skateboards require five units of plastic and are sold
for $1 profit.
 Dolls require two units of plastic and are sold for $0.55
profit.
 If 60 units of plastic are available, what numbers of
skateboards and/or dolls should be manufactured to
maximize the profits?
Chapter 4: Linear Programming
Mixture Problems
Step 1
Mixture Chart – display the verbal information into a
chart that includes the unknown variables (―x‖ units
of Skateboards, and ―y‖ units of dolls).
Chapter 4: Linear Programming
Mixture Problems
Example 2
Make a mixture chart to display this situation: 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.
Let x =
Let y =
Chapter 4: Linear Programming
Mixture Problems
Chapter 4: Linear Programming
Mixture Problems
Translating Mixture Charts into
Mathematical Form
Chapter 4: Linear Programming
Mixture Problems
What must be true about the sign of the numbers we can
use for ―x‖ and ―y‖ in the skateboard/doll problem?
Both ―x‖ and ―y‖ cannot be negative numbers.
How can we write this information using an inequality sign,
like >, or ≥, or <, or ≤?
x≥0
y≥0
Chapter 4: Linear Programming
Mixture Problems
These inequalities are called minimum constraints.
Which means that one cannot manufacture negative
numbers of objects.
Chapter 4: Linear Programming
Mixture Problems
The next problem is that we only have so much plastic
available to make skateboards and dolls.
How can we represent this information?
Since we need five units of plastic for each skateboard, we
can write that information mathematically as needing 5x
units of plastic for each skateboard.
Since we need 2 units of plastic for each doll, we can write
that information mathematically as needing 2y units of
plastic.
Hence we will need 5x + 2y units of plastic for the mixture
of skateboards and dolls we make.
Chapter 4: Linear Programming
Mixture Problems
Reading from the table, we have only a limited number of
units of plastic available.
How can we represent this information mathematically?
5x + 2y ≤ 60
This is called the resource constraint.
Notice that all of the numbers in this inequality can be
obtained from a column of the mixture chart. One of the
reasons we construct a mixture chart is that it helps us
speed up the conversion into inequalities of the information
about the problem we wish to solve.
Chapter 4: Linear Programming
Mixture Problems
Use the mixture chart to write an equation for the amount of
profit that will be produced when we manufacture different
mixture of skateboards and dolls.
1x + 0.55y = P, where P = profit
Our goal is to find which values of ―x‖ and ―y‖ (skateboards
and dolls) make this profit as large as possible.
Chapter 4: Linear Programming
Mixture Problems
Example 3
Write the minimum constraints inequalities, the resource
constraint inequality, and the profit equation for example 2.
x≥0
y≥0
3x + 2y ≤ 60
P = 5x + 3y
Chapter 4: Linear Programming
Mixture Problems
Feasibility set (feasibility region) - A
collection of all physically possible
solutions, or choices, that can be
made.
Chapter 4: Linear Programming
Mixture Problems
 Feasibility Set or Feasibility Region
Our goal is to find the best mixture of ―x‖ and ―y‖
(skateboards and/or dolls) to produce the largest
profit — two phases:
1. Find the feasible set for the mixture problem
subject to limited resources.
Graph line below 5x + 2y 60 (plastic)
2. Determine the mixture that gives rise to the
largest profit.
Chapter 4: Linear Programming
Mixture Problems
To draw the graph of an inequality, let’s first review how to
draw the graph of the equation of a straight line.
Remember that two points can be used to uniquely
determine a straight line. Let’s use the equation associated
with the resource constraint in example 1.
The resource constraint is 5x + 2y ≤ 60. The equation
associated with this inequality is
5x + 2y = 60
Chapter 4: Linear Programming
Mixture Problems
There are two points that are easy to find on this line.
When x = 0, this gives rise to one point on the line, and
when y = 0, we can find another point. Find these two
points.
Chapter 4: Linear Programming
Mixture Problems
Let x = 0 so,
5(0) + 2y = 60
0 + 2y = 60
2y = 60
y = 30
The point (0, 30) is on the line.
Chapter 4: Linear Programming
Mixture Problems
Chapter 4: Linear Programming
Mixture Problems
Let y = 0 so,
5x+ 2(0) = 60
5x + 0 = 60
5x = 60
x = 12
The point (12, 0) is on the line.
Chapter 4: Linear Programming
Mixture Problems
Chapter 4: Linear Programming
Mixture Problems
Chapter 4: Linear Programming
Mixture Problems
Now that we know the graph of the equation 5x + 2y = 60
looks , we can think through where points (x, y) that satisfy
5x + 2y < 60 are located. The points that are either on the
line 5x + 2y = 60 or satisfy 5x + 2y < 60 will satisfy
5x + 2y ≤ 60.
Any line, for example, 5x + 2y = 60, divides the xy-plane
into three parts: those points on the line, and the points in
one of the two half-planes. In one of the half-planes we
have the points for which 5x + 2y < 60 and in the other we
have the points for which 5x + 2y > 60.
How can we tell which of the two half-planes is above the
line 5x + 2y = 60 and which is below?
Chapter 4: Linear Programming
Mixture Problems
The key is the use of a test point (x, y) that is not on the line
and whose half-planes we wish to distinguish.
We saw that (3, 10) is not on the line 5x + 2y = 60 and is
below the line. This enables us to see that the half-plane
for which 5x + 2y < 60 consists of the points below the line
5x + 2y = 60.
Chapter 4: Linear Programming
Mixture Problems
1. Graph of 5x + 2y = 60
2. Shade in the feasible region is where all equations
are true: 5x + 2y 60, and where x ≥ 0 , y ≥ 0
Chapter 4: Linear Programming
Mixture Problems
Example 4
In the earlier clothing manufacturer example, we developed
a resource constraint of 3x + 2y ≤ 60. Draw the feasible
region corresponding to that resource constraint, using the
reality minimums of x ≥ 0 and y ≥ 0.
1. Graph 3x + 2y = 60.
Let x = 0
0 + 2y = 60
2y = 60
y = 30
Graph (0, 30).
Chapter 4: Linear Programming
Mixture Problems
Let y = 0
3x + 0 = 60
3x = 60
x = 20
So graph (20, 0).
Which way are you going to shade?
Below the line 3x + 2y = 60.
Chapter 4: Linear Programming
Mixture Problems
Assignment
Textbook Pages 139-140 #2,4,6,12,16