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Linear Programming

Linear Programming. Mr. Barker Discrete 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.

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Linear Programming

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  1. Linear Programming Mr. Barker Discrete math

  2. 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.

  3. 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.

  4. What else? • 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

  5. 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

  6. 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?

  7. 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.

  8. 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.

  9. Example 1 continued… • Now we need to translate the data into mathematical form to produce constraints • Equation for resources • Equation for profit

  10. You try • Write an equation for our clothing manufacturer • Equation for resources as a constraint. • Equation for profit

  11. Graph the function • Find the intercepts (x) • Write the equation • Set • Solve • Find y

  12. 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.

  13. Graph the line of the function Feasible region

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

  15. Graph the line of the function Feasible region

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