1. 11/20/2015 6:37 AM1 1 LINEAR PROGRAMMING Section 3.4, ©2008

2. 11/20/2015 6:37 AM2 3.5 Linear Programming2 Example 1 (-5, 3) (4, 3) (4, -1) (-5, -1)

3. 11/20/2015 6:37 AM3 3.4 Linear Programming3 Definitions  Optimization is finding the minimum and maximum value  For the most part, optimization involves point, P  Steps in Linear Programming 1. Find the vertices by graphing 2. Plug the vertices into the P equation, which is given 3. Find the minimum and maximum optimization values of P

4.  Linear Programming is a method of finding a maximum or minimum value of a function that satisfies a set of conditions called constraints  A constraint is one of the inequalities in a linear programming problem.  The solution to the set of constraints can be graphed as a feasible region.

5. 11/20/2015 6:37 AM5 3.5 Linear Programming5 Optimization  A Haunted House is opened from 7pm to 4am. Look at this graph and determine the maximization and minimization of this business. 7p 8p 9p 10p 11p 12a 1a 2a 3a 4a MAXIMIZATION MINIMIZATION

6. 11/20/2015 6:37 AM6 3.5 Linear Programming6 Example 1  Given Find the minimum and maximum for equation, Step 1: Find the vertices by graphing (-5, 3) (4, 3) (4, -1) (-5, -1)

7. 11/20/2015 6:37 AM7 3.5 Linear Programming7 Example 1  Given Find the minimum and maximum for equation, Step 2: Plug the vertices into the P equation, which is given verticesP = –2x + yprofit (-5, 3) (4, 3) (4, -1) (-5, -1) P = -2(-5) + (3) P = -2(4) + (3) P = -2(4) + (-1) P = -2(-5) + (-1) P = 13 P = –5 P = –9 P = 9P = 9

8. 11/20/2015 6:37 AM8 3.5 Linear Programming Example 1  Given Find the minimum and maximum for equation, Step 3: Find the minimum and maximum optimization values of P verticesP = -2x + yProfit (-5, 3) (4, 3) (4, -1) (-5, -1) P = -2(-5) + (3) P = -2(4) + (3) P = -2(4) + (-1) P = -2(-5) + (-1) P = –5 P = 9P = 9 Minimum: –9 @ (4,-1) Maximum: 13 @ (-5,3) P = –9 P = 13

9. 11/20/2015 6:37 AM9 3.5 Linear Programming9 Example 2 Given Find the minimum and maximum optimization for equation,

10. 11/20/2015 6:37 AM1011/20/2015 6:37 AM3.5 Linear Programming10 Example 2  Given Find the minimum and maximum for equation, VerticesP = 3x+4yProfit (2, 6)P = 3(2) + 4(6) 30 (5, 6)P = 3(5) + 4(6) 39 (2, 1)P = 3(2) + 4(1) 10 (5, 1)P = 3(5) + 4(1) 19 (2, 6) (5, 6) (2, 1) (5, 1) Minimum: 10 @ (2,1) Maximum: 39 @ (5,6)

11. 11/20/2015 6:37 AM1111/20/2015 6:37 AM3.5 Linear Programming Example 3  Given Find the minimum and maximum for equation, x ≥ 0 y ≥ 1.5 2.5x + 5y ≤ 20 3x + 2y ≤ 12 11 Vertices: (0, 4), (0, 1.5), (2, 3), and (3, 1.5) (2, 3) (3, 1.5) (0, 1.5) (0, 4)

12. 11/20/2015 6:37 AM12  Given Find the minimum and maximum for equation, (x, y)25x + 30yP($) (0, 4)25(0) + 30(4)120 (0, 1.5)25(0) + 30(1.5) 45 (2, 3)25(2) + 30(3) 140 (3, 1.5)25(3) + 30(1.5)120 3.5 Linear Programming Example 3 x ≥ 0 y ≥ 1.5 2.5x + 5y ≤ 20 3x + 2y ≤ 12 12 (2, 3) (3, 3/2) (0, 3/2) (0, 4)

13. 11/20/2015 6:37 AM1311/20/2015 6:37 AM3.5 Linear Programming13 Your Turn  Given Find the minimum and maximum for equation, (0, 2) (2, 0) (0, 0) Step 1: Find the vertices by graphing

14. 11/20/2015 6:37 AM1411/20/2015 6:37 AM3.5 Linear Programming14 Your Turn  Given Find the minimum and maximum for equation, Step 2: Plug the vertices into the P equation, which is given verticesP = x + 2yprofit (0, 2) (0, 0) (2, 0) P = (0) + 2(2)P = 4P = 4 P = (0) + 2(0)P = 0P = 0 P = (2) + 2(0)P = 2P = 2

15. 11/20/2015 6:37 AM1511/20/2015 6:37 AM3.5 Linear Programming15 Example 4  Given Find the minimum and maximum for equation, (0, 8) (4, 0) (0, 2) (2, 0) verticesP = 2x + 3yprofit (0, 8) P = 2(0) + 3(8) 24 (0, 2) P = 2(0) + 3(2) 6 (2, 0) P = 2(2) + 3(0) 4 (4, 0) P = 2(4) + 3(0) 8

16. Example 5 AA charity is selling T-shirts in order to raise money. The cost of a T-shirt is $15 for adults and $10 for students. The charity needs to raise at least $3000 and has only 250 T-shirts. Write and graph a system of inequalities that can be used to determine the number of adult and student T-shirts the charity must sell. Let a = adult t-shirts Let b = student t-shirts

17. 11/20/2015 6:37 AM17 Sue manages a soccer club and must decide how many members to send to soccer camp. It costs $75 for each advanced player and $50 for each intermediate player. Sue can spend no more than $13,250. Sue must send at least 60 more advanced than intermediate players and a minimum of 80 advanced players. Find the number of each type of player Sue can send to camp to maximize the number of players at camp. Warm-up 10-23-13

18. 11/20/2015 6:37 AM18 x = the number of advanced players, y = the number of intermediate players. x ≥ 80 y ≥ 0 75x + 50y ≤ 13,250 x – y ≥ 60 The number of advanced players is at least 80. The number of intermediate players cannot be negative. There are at least 60 more advanced players than intermediate players. The total cost must be no more than $13,250. Let P = the number of players sent to camp. The objective function is P = x + y. Example 6 MAKE a TABLE to show your work for the objective function

19. 11/20/2015 6:37 AM19 Graph the feasible region, and identify the vertices. Evaluate the objective function at each vertex. P(80, 0) = (80) + (0) = 80 P(80, 20) = (80) + (20) = 100 P(176, 0) = (176) + (0) = 176 P(130,70) = (130) + (70) = 200 Example 6

20. 11/20/2015 6:37 AM20 Check the values (130, 70) in the constraints. x ≥ 80 130 ≥ 80 y ≥ 0 70 ≥ 0 x – y ≥ 60 (130) – (70) ≥ 60 60 ≥ 60 75x + 50y ≤ 13,250 75(130) + 50(70) ≤ 13,250 13,250 ≤ 13,250 Example 6

21. 11/20/2015 6:37 AM2111/20/2015 6:37 AM3.5 Linear Programming21 Assignment  Pg 202: 11-19 odd, 20, 29, 31 (no need to identify the shape from 16-19)  Pg 209: 9-21 odd