3-4: Linear Programming

1. 3-4: Linear Programming • Objectives: • Find the maximum and minimum values of a function over a region • Solve real-world problems using linear programming • Standards addressed: • 2.1 & 2.2

2. Definitions • Constraints: the inequalities • Feasible region: where the graphs intersect • Bounded: when the graphs of the inequalities intersect and form a “region” and it is bounded. • Vertices: the points where the bounded region come together

3. More definitions • Unbounded: when the inequalities overlap, but do not form a “shape” • Linear programming: the process of finding maximum or minimum values of a function for a region defined by inequalities

4. 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.

5. When solving linear programming problems, use the following steps: • Define variables. (i.e. Write “lets”.) • Write an expression (profit equation) to be maximized or minimized. • Write a system of inequalities. • Graph the system of inequalities. • Find the coordinates of the vertices of the feasible region (this will be where they overlap) • Substitute the coordinates of the vertices in the expression you wrote into the profit equation (#2). • Select the greatest or least result to answer the problem.

6. #10 Workbook pg. 16 • The drama club is selling tickets to its play. An adult ticket costs $15 and a student ticket costs $11. The auditorium will seat 300 ticket-holders. The drama club wants to collect at least $3630 from ticket sales. • Write and graph a system of four inequalities that describe how many of each type of ticket the club must sell to meet its goal. • Make a chart

7. Chart Now – write the four inequalities that go with this problem

8. …inequalities

9. Graph

10. Find the minimum and maximumvalue of the function f(x, y) = 3x - 2y. We are given the constraints: • y ≥ 2 • 1 ≤ x ≤5 • y ≤ x + 3

11. 1 ≤ x ≤5 8 7 6 5 4 y ≥ 2 3 2 y ≤ x + 3 1 3 5 4 1 2

12. 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

13. 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

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

15. 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

16. y ≥ 2x -5 6 5 4 3 y ≥ -x + 2 2 1 1 2 3 4 5

17. Vertices f(x, y) = 4x + 3y • f(0, 2) = 4(0) + 3(2) = 6 • f(4, 3) = 4(4) + 3(3) = 25 • f( , - ) = 4( ) + 3(- ) = -1 =

18. Linear Programming • f(0, 2) = 6 minimum • f(4, 3) = 25 maximum

19. Example • A landscaping company has crews who mow lawns and prune shrubbery. The company schedules 1 hour for mowing jobs and 3 hours for pruning jobs. Each crew is scheduled for no more than 2 pruning jobs per day. Each crew’s schedule is set up for a maximum of 9 hours per day. On the average, the charge for mowing a lawn is $40 and the charge for pruning shrubbery is $120. Find a combination of mowing lawns and pruning shrubs that will maximize the income the company receives per day for one of its crews.

21. Homework • Workbook pg. 17 (1-9)