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 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 Resource(s) Yards of cloth 60 Profit Shirts (x-unit) 3$5 Vests (y-unit) 2$3

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.

Feasible region

Pg ,3, 7-13 odd, 17, 19

Feasible region