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Linear Programming
Unit 2, Lesson 410/13
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 + 12. y > x – 4, y ≤ 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 + 4yConstraints: x ≥ 0, y ≥ 0, x + y ≤
8
Example
Obj. Function: C = 5x + 6yConstraints: x ≥ 0, y ≥ 0, x + y ≥ 5,
3x + 4y ≥ 18
Your Turn
Obj. Function: C = -2x + yConstraints: 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?