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3.5 Linear Programming Objectives: Write and graph a set of constraints for a linear-programming problem Use linear programming to find the maximum or minimum value of an objective function. Standard: A. Use appropriate mathematical techniques to solve non-routine problems.

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**The inequalities contained in the problem are called constraints. **

A method called linear programming is used to find optimal solutions. Linear programming problems have the following characteristics: The inequalities contained in the problem are called constraints. The solution to the set of constraints is called the feasible region. The function to be maximized or minimized is called the objective function.

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**Let x represent the number of acres of corn **

Ex 1. Max Desmond is a farmer who plants corn and wheat. In making planting decisions, he used the 1996 statistics at right from the United States Bureau of the Census. Crop Yield Per Acre Average Price Corn 113.5 bu $3.15 / bu Soy Beans 34.9 bu $6.80 / bu Wheat 35.8 bu $4.45 / bu Cotton 540 lb $.759 / lb Rice 564 lb $.0865 / lb Let x represent the number of acres of corn Let y represent the number of acres of wheat Mr. Desmond wants to plant according to the following constraints: No more than 120 acres of corn and wheat At least 20 and no more than 80 acres of corn At least 30 acres of wheat How many acres of each crop should Mr. Desmond plant to maximize the revenue from his harvest? OBJECTIVE FUNCTION R = x y

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B. C.

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**Corner-Point Principle: **

The Corner-Point Principle confirms that you need only the vertices of the feasible region to find the maximum or minimum value of the objective function. Corner-Point Principle: In linear programming, the maximum and minimum values of the objective function each occur at one of the vertices of the feasible region.

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Ex 2. Using the information in Example 1, maximize the objective function. Then graph the objective function that represents the maximum revenues along with the feasible region.

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Ex 3. A small company produces knitted afghans and sweaters and sells them through a chain of specialty stores. The company is to supply the stores with a total of no more than 100 afghans and sweaters per day. The stores guarantee that they will sell at least 10 and no more than 60 afghans per day and at least 20 sweaters per day. The company makes a profit of $10 on each afghan and a profit of $12 on each sweater. Write a system of inequalities to represent the constraints. Graph the feasible region. Write an objective function for the company’s total profit, P, from the sales of afghans and sweater.

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10 ≤ x ≤ 60 y ≥ 20 x + y ≤ 100 * b. (graph) c. P = 10x + 12y

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Ex. 4

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Ex 5. Find the maximum and minimum values, if they exist, of the objective function T = 3x + 2y given the set of constraints provided: x + y ≤ 10 x + 2y ≥ x + y ≥ 13 Vertex Objective function Amount 1,9 21 8, 2 Maximum 28 2,5 Minimum 16 B. y = -x + 10 y= - x/2 + 6 -2y = x – 12 -1y = -2 y = 2 x = 8 (8, 2) y = - 4x + 13 - 4y = 4x + 40 -3y = - 27 y = 9 x = 1 (1,9) C. y = - 4x + 13 y = -x / 2 + 6 y = -4x + 13 -8y = 4x – 48 -7y = - 35 y = 5; x = 2 (2,5)

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**Summary Linear-Programming Procedure**

Write a system of inequalities, and graph the feasible region. Write the objective function to be maximized or minimized. Find the coordinates of the vertices of the feasible region. Evaluate the objective function for the coordinates of the vertices of the feasible region. Then identify the coordinates that give the required maximum or minimum.

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**Multiple Choice Practice:**

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**Lesson Quiz: Linear Programming**

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Homework Integrated Algebra II- Section 3.5 Level A Honors Algebra II- Section 3.5 Level B

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