LINEARPROGRAMMING 5/23/2018 11:13 AM 5/23/2018 11:13 AM 1

Example 1 (-5, 3) (4, 3) (-5, -1) (4, -1) 5/23/2018 11:13 AM 3.5 Linear Programming 2 2 2

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 5/23/2018 11:13 AM 5/23/2018 11:13 AM 3.4 Linear Programming 3

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.

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 7p 8p 9p 10p 11p 12a 1a 2a 3a 4a 5/23/2018 11:13 AM 5/23/2018 11:13 AM 3.5 Linear Programming 5

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) 5/23/2018 11:13 AM 5/23/2018 11:13 AM 3.5 Linear Programming 6

Example 1 vertices P = –2x + y profit (-5, 3) (4, 3) (4, -1) (-5, -1) Given Find the minimum and maximum for equation, Step 2: Plug the vertices into the P equation, which is given vertices P = –2x + y profit (-5, 3) (4, 3) (4, -1) (-5, -1) P = -2(-5) + (3) P = 13 P = -2(4) + (3) P = –5 P = -2(4) + (-1) P = –9 P = -2(-5) + (-1) P = 9 5/23/2018 11:13 AM 5/23/2018 11:13 AM 3.5 Linear Programming 7

Example 1 Minimum: –9 @ (4,-1) 13 @ (-5,3) Maximum: Given Find the minimum and maximum for equation, Step 3: Find the minimum and maximum optimization values of P vertices P = -2x + y Profit (-5, 3) (4, 3) (4, -1) (-5, -1) P = -2(-5) + (3) P = 13 P = -2(4) + (3) P = –5 P = -2(4) + (-1) P = –9 P = -2(-5) + (-1) P = 9 Minimum: –9 @ (4,-1) Maximum: 13 @ (-5,3) 5/23/2018 11:13 AM 5/23/2018 11:13 AM 3.5 Linear Programming

Given Find the minimum and maximum optimization for equation, Example 2 Given Find the minimum and maximum optimization for equation, 5/23/2018 11:13 AM 5/23/2018 11:13 AM 3.5 Linear Programming 9

Example 2 39 10 Minimum: 10 @ (2,1) Maximum: 39 @ (5,6) Given Find the minimum and maximum for equation, Maximum: 39 @ (5,6) Vertices P = 3x+4y Profit (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) 5/23/2018 11:13 AM 5/23/2018 11:13 AM 3.5 Linear Programming 10

Example 3 Given Find the minimum and maximum for equation, Vertices: y ≥ 1.5 2.5x + 5y ≤ 20 3x + 2y ≤ 12 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) 5/23/2018 11:13 AM 5/23/2018 11:13 AM 3.5 Linear Programming 11

Example 3 45 140 Given Find the minimum and maximum for equation, y ≥ 1.5 2.5x + 5y ≤ 20 3x + 2y ≤ 12 Given Find the minimum and maximum for equation, (x, y) 25x + 30y P($) (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) (0, 4) (2, 3) (0, 3/2) (3, 3/2) 5/23/2018 11:13 AM 3.5 Linear Programming 12

Your Turn Given Find the minimum and maximum for equation, Step 1: (0, 2) Step 1: Find the vertices by graphing (0, 0) (2, 0) 5/23/2018 11:13 AM 5/23/2018 11:13 AM 3.5 Linear Programming 13

Your Turn vertices P = x + 2y profit (0, 2) (0, 0) (2, 0) Given Find the minimum and maximum for equation, Step 2: Plug the vertices into the P equation, which is given vertices P = x + 2y profit (0, 2) (0, 0) (2, 0) P = (0) + 2(2) P = 4 P = (0) + 2(0) P = 0 P = (2) + 2(0) P = 2 5/23/2018 11:13 AM 5/23/2018 11:13 AM 3.5 Linear Programming 14

Example 4 24 4 (0, 8) (0, 2) 6 (2, 0) (4, 0) 8 (0, 8) P = 2(0) + 3(8) Given Find the minimum and maximum for equation, (0, 8) vertices P = 2x + 3y profit (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 (0, 2) (2, 0) (4, 0) 5/23/2018 11:13 AM 5/23/2018 11:13 AM 3.5 Linear Programming 15

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 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

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 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. 5/23/2018 11:13 AM

Example 6 MAKE a TABLE to show your work for the objective function 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 is P = x + y. MAKE a TABLE to show your work for the objective function 5/23/2018 11:13 AM

Example 6 P(80, 0) = (80) + (0) = 80 P(80, 20) = (80) + (20) = 100 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 5/23/2018 11:13 AM

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  5/23/2018 11:13 AM

Assignment Pg 202: 11-19 odd, 20, 29, 31 (no need to identify the shape from 16-19) Pg 209: 9-21 odd 5/23/2018 11:13 AM 5/23/2018 11:13 AM 3.5 Linear Programming 21