WarmUp

1
Warm-Up
5 minutes
1) Solve the system.
2) Graph the solution.
2
3.5.1 Linear Programming

• Objectives
• Write and graph a set of constraints for a
  linear-programming problem

3
Example 1
A small company produces knitted afghans and
sweaters and sells them through a chain of
specialty stores. The company is to supply the
stores with a total of no more than 100 afghans
and sweaters per day. The stores guarantee that
they will sell at least 10 and no more than 60
afghans per day and at least 20 sweaters per day.
The company makes a profit of 10 on each afghan
and a profit of 12 on each sweater.
the inequalities contained in the problem are
called the constraints
Let x the of afghans
Let y the of sweaters
the solution to the set of constraints is called
the feasible region
4
Example 1
A small company produces knitted afghans and
sweaters and sells them through a chain of
specialty stores. The company is to supply the
stores with a total of no more than 100 afghans
and sweaters per day. The stores guarantee that
they will sell at least 10 and no more than 60
afghans per day and at least 20 sweaters per day.
The company makes a profit of 10 on each afghan
and a profit of 12 on each sweater.
the function to be maximized or minimized is
called the objective function
Write an objective function for the companys
total profit, P, from the sales of afghans and
sweaters.
P
10x
12y
5
Homework
p.191 11-17 odds
6
Warm-Up
6 minutes
1) Graph the feasible region for the set of
constraints below.
2) Identify the vertices of the feasible region.
7
3.5.2 Linear Programming

• Objectives
• Use linear programming to find the maximum or
  minimum value of an objective function

8
Example 1
A small company produces knitted afghans and
sweaters and sells them through a chain of
specialty stores. The company is to supply the
stores with a total of no more than 100 afghans
and sweaters per day. The stores guarantee that
they will sell at least 10 and no more than 60
afghans per day and at least 20 sweaters per day.
The company makes a profit of 10 on each afghan
and a profit of 12 on each sweater.
a) Maximize the objective function.
P 10x 12y
Vertex
Objective function
(10,20)
P 10(10) 12(20)
340
(60,20)
P 10(60) 12(20)
840
(60,40)
P 10(60) 12(40)
1080
(10,90)
P 10(10) 12(90)
1180
9
Corner-Point Principle
In linear programming, the maximum and minimum
values of the objective function each occur at
one of the vertices of the feasible region.
10
Example 1
A small company produces knitted afghans and
sweaters and sells them through a chain of
specialty stores. The company is to supply the
stores with a total of no more than 100 afghans
and sweaters per day. The stores guarantee that
they will sell at least 10 and no more than 60
afghans per day and at least 20 sweaters per day.
The company makes a profit of 10 on each afghan
and a profit of 12 on each sweater.
b) Graph the objective function, indicating the
maximum profit and the feasible region.
P 10x 12y
1180 10x 12y
12y 10x 1180
11
Example 2
Find the maximum and minimum values, if they
exist, of the objective function T 3x 2y
given the set of constraints below.
12
Homework
p.192 19,21,23,27,29