3-4 Linear Programming

1. What you’ll learn … To find maximum and minimum values. To solve problems with linear programming. 2.10 Use systems of two or more equations or inequalities to model and solve problems; justify results. Solve using tables, graphs, matrix operations, and algebraic properties. 3-4 Linear Programming

2. Linear Programming Businesses use linear programming to find out how to maximize profit or minimize costs. Most have constraints on what they can use or buy.

3. Linear programming is a technique that identifies the minimum or maximum value of some quantity. This quantity is modeled with an objective function. Limits on the variables in the objective function are constraints, written as linear inequalities. These constraints form the system of inequalities at the right. The blue region in the graph, the feasible region, contains all points that satisfy all the constraints.

4. Steps to Solve Linear Programming Problem • Define the variables. • Write a system of inequalities. • Graph the system of inequalities on graph paper. • Find the coordinates of the vertices of the feasible region. • Write a function to be maximized or minimized. • Substitute the coordinates of the vertices into the function. • Select the greatest or least result. Answer the problem.

5. Example 1 Find the minimum and maximum value of the function f(x, y) = 3x - 2y. We are given the constraints: • y ≥ 2 • 1 ≤ x ≤5 • y ≤ x + 3 • Find the minimum and maximum values by graphing the inequalities and finding the vertices of the polygon formed. • Substitute the vertices into the function and find the largest and smallest values.

6. 1 ≤ x ≤5 (5,8) 8 7 6 (1,4) 5 4 y ≥ 2 3 2 y ≤ x + 3 (1,2) (5,2) 1 3 5 4 1 2

7. Linear Programming • The vertices of the quadrilateral formed are: (1, 2) (1, 4) (5, 2) (5, 8) • Plug these points into the function f(x, y) = 3x - 2y

8. Linear Programming f(x, y) = 3x - 2y • f(1, 2) = 3(1) - 2(2) = 3 - 4 = -1 • f(1, 4) = 3(1) - 2(4) = 3 - 8 = -5 • f(5, 2) = 3(5) - 2(2) = 15 - 4 = 11 • f(5, 8) = 3(5) - 2(8) = 15 - 16 = -1

9. Linear Programming • f(1, 4) = -5 minimum • f(5, 2) = 11 maximum

10. Find the minimum and maximum value of the function f(x, y) = 4x + 3y We are given the constraints: • y≤x + 2 • y ≤ -x + 2 • y ≥ 2x -5

11. y ≥ 2x -5 6 5 (4,3) 4 (0,2) 3 y ≥ -x + 2 2 1 1 2 3 4 5 ( , - )

12. Vertices f(x, y) = 4x + 3y • f( , ) = 4( ) + 3( ) = • f( , ) = 4( ) + 3( ) = • f( , ) = 4( ) + 3( ) =

13. Linear Programming • f( , ) = ____ minimum • f( , ) = ____ maximum

14. Find the values of x and y that maximize or minimize the objective function for each graph. Evaluate at each vertex Example p. 138 Problem 2 Maximum for P= 7x + 4y D. (0, 500) C. (400, 300) B. (600, 0) A. (0, 0)

15. Example p.138 Problem 6 Graph each system of constraints. Name all vertices. Then find the values of x and y that maximize or minimize the objective function. x + y ≤ 8 2x + y ≤ 10 x ≥ 0 y ≥ 0 Maximize for N = 100x + 40y Evaluate at each vertex

16. Example p.138 Problem 9 Graph each system of constraints. Name all vertices. Then find the values of x and y that maximize or minimize the objective function. 2 ≤ x ≤ 6 1 ≤ y ≤ 5 x + y ≤ 8 Maximize for P = 3x + 2y Evaluate at each vertex

17. 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? Define Variables x= y = P = Example 2a Real World Connection 12 cans per case You pay … $24 per case Sell at … $3.50 per can $18 profit per case 20 packs per case You pay … $15 per case Sell at … $1.50 per pack $15 profit per case Write constraints and objective function. Evaluate at each vertex

18. Example 2b Real World Connection Teams chosen from 20 forest rangers and 8 trainees are planting trees. An experienced team consisting of two rangers can plant 300 trees per week. A training team consisting of one ranger and two trainees can plant 120 trees per week. Define Variables x= y = P = Write constraints and objective function.

19. Example 2c Real World Connection Trees in urban areas help keep air fresh by absorbing carbon dioxide. A city has $2100 to spend on planting spruce and maple trees. The land available for planting is 45,000 ft2. How many of each tree should the city plant to maximize carbon dioxide absorption? Define Variables x= y = P = Write constraints and objective function.

20. Example 2e Real World Connection Baking a tray of corn muffins takes 4 cups of milk and 3 cups of wheat flour. A tray of bran muffins takes 2 cups of milk and 3 cups of wheat flour. A baker has 16 cups of milk and 15 cups of wheat flour. He makes $3 profit per tray of corn muffins and $2 profit per tray of bran muffins. How many trays of each type of muffins should the baker make to maximize his profits? Define Variables x= y = P = Write constraints and objective function.

21. In Chapter 3, You Should Have… • Learned to solve systems of equations and inequalities in two variables algebraically and by graphing. • Learned to graph points and equations in three dimensions. • Learned to solve systems of equations in three variables.