3.4a: Linear Programming
Nature does nothing uselessly.
What is Linear Programming
Linear Programming is a technique that we use to
identify a minimum or maximum value of some quantity.
The objective function models that quantity.
Constraints are the limits on the variables in our
objective function.
The feasible region of the graph contains all of the points
that satisfy our constraints.
Vertex Principle: If there is a minimum or maximum
value of the linear objective function, it occurs at one or
more vertices of the feasible region.
Vertex Principle
Ex1) What values of x and y maximize P for the
objective function P = 3x + 2y with constraints:
1) Graph your constraints.
P  3x  2 y
10
9
 x  0, y  0

3
y  x 3
2

 y   x  7
2) What are the coordinates of
each vertex?
8
7
6
5
3) Evaluate P for each vertex.
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12
Vertex Principle
Ex2) Find the values of x and y that maximize  x  2 y  4

and minimize P if P = –5x + 4y.
5 x  2 y  16
 x  4 y  32
1) Graph your constraints.

10
2) What are the coordinates of
each vertex?
9
8
7
6
5
3) Evaluate P for each vertex.
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12
Linear Programming ~ Application
Ex3) A furniture manufacturer can make from 30 to 60 tables a day and
from 40 to 100 chairs a day. It can make at most 120 units in one day.
The profit on a table is $150, and the profit on a chair is $65. How many
tables and chairs should they make per day to maximize profit? How
much is the maximum profit?
1) Define the variables.
2) Write the constraints and
objective function.
Linear Programming ~ Application
30  x  60

40  y  100
 x  y  120

3) Graph the constraints.
P = 150x + 65y
4) Find the coordinates of
each vertex.
5) Evaluate the objective
function at each vertex.
3.4a: Linear Programming
Homework: 3.4 #1-9,
p. 137# 40, 43, 50
Nature does nothing uselessly.