mathematical literacy in today’s world, 8th ed. finding...
TRANSCRIPT
Chapter 4: Linear Programming
Lesson Plan
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.
For All Practical Purposes
Chapter 4: Linear Programming
Mixture Problems
Mixture Problem
Limited resources are
combined into products
in such a way that the
profit from selling those
products is a maximum.
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.
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
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
How can we represent this information mathematically?
5x + 2y ≤ 60
This is called the resource constraint.
Chapter 4: Linear Programming
Mixture Problems
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
Write the minimum constraints inequalities, the resource
constraint inequality, and the profit equation for example 2.
x ≥ 0
y ≥ 0
5x + 2y ≤ 60
P = 1x + .55y
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
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
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
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.
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