Linear Programming

Mr. BarkerDiscrete math

What is Linear Programming?

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.

What is it used for

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.

What else?

Linear programming is often used to solve special problems known as mixture problems.

Mixture problemIn a mixture problem, limited resources are combined into products so that the profit from selling those products is a maximum

Features of a mixture problem 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

Example problem 1

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?

Example 1 continued…

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

plastic60

profit

Skateboards (x-unit)

5 $1.00

Dolls(y-unit)

2 $0.55

You Try!

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

Example 1 continued…

Now we need to translate the data into mathematical form to produce constraints

Equation for resources

Equation for profit

Resource(s)Containers of

plastic60

profit

Skateboards (x-unit)

5 $1.00

Dolls(y-unit)

2 $0.55

You try

Write an equation for our clothing manufacturer

Equation for resources as a constraint.

Equation for profitResource(s)

Yards of cloth60

Profit

Shirts(x-unit)

3 $5

Vests(y-unit)

2 $3

Graph the functionFind the intercepts (x)

Write the equationSet Solve

Find y

The Feasible set

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.

Graph the line of the function

0 2 4 6 8 10 120

10

20

30

40

5x+2y=60

5x+2y=60

0 2 4 6 8 10 1205

101520253035

5x+2y≤60

5x+2y≤60

Feasible region

Assignment

Pg. 139 1,3, 7-13 odd, 17, 19

Graph the line of the function

0 2 4 6 8 10 120

10

20

30

40

5x+2y=60

5x+2y=60

0 2 4 6 8 10 1205

101520253035

5x+2y≤60

5x+2y≤60

Feasible region