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LINEAR PROGRAMMING

Learn the basics of linear programming and how to find minimum and maximum values for a given equation. Includes solved examples.

thomasfuller
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LINEAR PROGRAMMING

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  1. LINEARPROGRAMMING 12/20/2019 11:59 AM 1

  2. Example 1 (-5, 3) (4, 3) (-5, -1) (4, -1) 12/20/2019 11:59 AM 12/20/2019 11:59 AM 3.5 Linear Programming 2 2

  3. 3.4 Linear Programming 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 Pequation, which is given 3. Find the minimum and maximum optimization values of P 12/20/2019 11:59 AM 3

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

  5. 3.5 Linear Programming 7p 8p 9p 10p 11p 12a 1a 2a 3a 4a Optimization • A Haunted House is opened from 7pm to 4am. Look at this graph and determine the maximization and minimization of this business. MAXIMIZATION MINIMIZATION MINIMIZATION 12/20/2019 11:59 AM 5

  6. 3.5 Linear Programming Example 1 • Given Find the minimum and maximum for equation, Step 1: Find the vertices by graphing (-5, 3) (4, 3) (-5, -1) (4, -1) 12/20/2019 11:59 AM 6

  7. 3.5 Linear Programming Example 1 • Given Find the minimum and maximum for equation, Step 2: Plug the vertices into the P equation, which is given 12/20/2019 11:59 AM 7

  8. 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 P=13 P=–9 Minimum: –9 @ (4,-1) 13 @ (-5,3) Maximum: 12/20/2019 11:59 AM

  9. 3.5 Linear Programming Example 2 Given Find the minimum and maximum optimization for equation, 12/20/2019 11:59 AM 9

  10. 3.5 Linear Programming Example 2 10 @ (2,1) Minimum: • Given Find the minimum and maximum for equation, Maximum: 39 @ (5,6) (2, 6) (5, 6) (2, 1) (5, 1) 12/20/2019 11:59 AM 10

  11. 3.5 Linear Programming x ≥ 0 y ≥ 1.5 2.5x + 5y ≤ 20 3x + 2y ≤ 12 Example 3 • Given Find the minimum and maximum for equation, (0, 4) (2, 3) Vertices: (0, 4), (0, 1.5), (2, 3), and (3, 1.5) (0, 1.5) (3, 1.5) 12/20/2019 11:59 AM 11

  12. 3.5 Linear Programming x ≥ 0 y ≥ 1.5 2.5x + 5y ≤ 20 3x + 2y ≤ 12 Example 3 • Given Find the minimum and maximum for equation, (0, 4) (2, 3) (0, 3/2) (3, 3/2) 12

  13. 3.5 Linear Programming Your Turn • Given Find the minimum and maximum for equation, (0, 2) Step 1: Find the vertices by graphing (0, 0) (2, 0) 12/20/2019 11:59 AM 13

  14. 3.5 Linear Programming Your Turn • Given Find the minimum and maximum for equation, Step 2: Plug the vertices into the P equation, which is given 12/20/2019 11:59 AM 14

  15. 3.5 Linear Programming Example 4 • Given Find the minimum and maximum for equation, (0, 8) (0, 2) (2, 0) (4, 0) 12/20/2019 11:59 AM 15

  16. Example 5 • 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 only250 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. Warm-up 10-23-13 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 eachintermediate player. Sue can spend no more than$13,250. Sue must send at least 60 more advancedthan intermediate players and a minimum of 80advanced players. Find the number of each type of player Sue can send to camp to maximize the number of players at camp.

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

  19. Example 6 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

  20. Example 6 Check the values (130, 70) in the constraints. x ≥ 80 y ≥ 0   130 ≥ 80 70 ≥ 0 x – y ≥ 60 75x + 50y ≤ 13,250 (130) – (70) ≥ 60 75(130) + 50(70) ≤ 13,250  60 ≥ 60 13,250 ≤ 13,250 

  21. 3.5 Linear Programming Assignment • Pg 202: 11-19 odd, 20, 29, 31 (no need to identify the shape from 16-19) • Pg 209: 9-21 odd 12/20/2019 11:59 AM 21

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