Linear Programming Unit 2, Lesson 4 10/13

Warmup Have your homework problem(s) out on your desk. Pick up a sheet of graph paper from the front table. Graph the following systems of inequalities & STATE 2 SOLUTIONS: 1. y < -2x – 4 & y ≥ 3x + 1 2. y > 2 3 x – 4, y ≤ −1 5 x + 4, and x > 0

Key Terms Optimization – finding the maximum or minimum value of some quantity Linear Programming – the process of optimizing an objective function Objective function – the equation used to find the maximum or minimum value Constraints – the system of inequalities that defines where the max or min can occur Feasible region – the graph of the constraints Vertex (vertices) – the most important values of the feasible region

Solutions in Linear Programming If an objective function has a maximum or minimum value, it MUST occur at a vertex of the feasible region. If the feasible region is bounded, the objective function will have BOTH a maximum and a minimum value.

Feasible Regions Bounded Unbounded

Finding Max/Min Values Graph the constraints Identify the feasible region Find all the vertices of the feasible region Substitute the coordinates of each vertex into the objective function Determine the max and/or min values

Example Obj. Function: C = 3x + 4y Constraints: x ≥ 0, y ≥ 0, x + y ≤ 8

Example Obj. Function: C = 5x + 6y Constraints: x ≥ 0, y ≥ 0, x + y ≥ 5, 3x + 4y ≥ 18

Your Turn Obj. Function: C = -2x + y Constraints: x ≥ 0, y ≥ 0, x + y ≥ 7, 5x + 2y≥ 20

Problem 1: Porscha’s Cupcake Shop 1.) What are we trying to find? 3.) Equations by topic: 5.) Hidden constraints? 7.) Vertices of feasible region: 2.) Define variables: Let x = __________ Let y = __________ 4.) Constraints: 6.) Graph the feasible region: 8.) Test each vertex in both equations.

Problem 2: Taking a Test 1.) What are we trying to find? 3.) Equations by topic: 5.) Hidden constraints? 7.) Vertices of feasible region: 2.) Define variables: Let x = __________ Let y = __________ 4.) Constraints: 6.) Graph the feasible region: 8.) Test each vertex in both equations.

Exit Ticket A company produces packs of pencils and pens. The company produces at least 100 packs of pens each day, but no more than 240. The company produces at least 70 packs of pencils each day, but no more than 170. A total of less than 300 packs of pens and pencils are produced each day. Each pack of pens makes a profit of $1.25. Each pack of pencils makes a profit of $0.75. What is the maximum profit the company can make each day?