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4.6 – Linear Programming Day 1

4.6 – Linear Programming Day 1. Linear programming. maximum. minimum. is a technique that identifies the and values of some quantity. This quantity is modeled with an . on the variables in the objective function are , written as linear. objective function. conditions.

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4.6 – Linear Programming Day 1

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  1. 4.6 – Linear Programming Day 1

  2. Linear programming maximum minimum • is a technique that identifies the and values of some quantity. • This quantity is modeled with an . • on the variables in the objective function are , written as linear . objective function conditions constraints inequalities

  3. feasibility region • The contains all of the points that satisfy all the constraints. • The points of the graph where the lines that form the boundary intersect are called of the feasibility region. corner points

  4. 1. Suppose you are selling cases of mixed nuts and roasted peanuts. You can order no more than a total of 500 cans and packages and spend no more than $600. How can you maximize your profit? How much is the maximum profit? • Mixed Nuts: 12 cans per case, you pay $24 per case, and you make $18 profit per case • Roasted Peanuts: 20 packages per case, you pay $15 per case, you make $15 profit per case • Let x = number of cases of mixed nuts ordered • Let y = number of cases of roasted peanuts ordered • Let P = total profit

  5. Complete the table above from the given information.

  6. What is the objective function? (Hint: what is the profit?)

  7. What are the constraints?

  8. Graph the constraints:

  9. Graph the constraints:

  10. Find the coordinates of each vertex:(the corner points)

  11. Evaluate P at each vertex.

  12. When will you have your maximum profit? • The maximum profit of $510 will occur when 15 cases of mixed nuts and 16 cases of roasted peanuts are sold.

  13. Steps for Linear Programming • 1. Clearly identify the variables in the problem. • 2. Write down the objective function which represents the quantity to be maximized or minimized. • 3. Write down your constraints which are restrictions or limits set in the problem. These constraints are expressed as a system of linear inequalities. • 4. Graph your constraints to create your feasible region. The feasible region is the shaded part of the graph. Accurately label all vertices of the feasible region. • 5. Substitute all vertices into the objective function found in step 2. Your final answer will either maximize or minimize the objective function. It will include values for both variables and the maximum or minimum of the objective function.

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