3.4 Linear Programming
p. 163
Optimization • Finding the minimum or maximum value of
some quantity.
• Linear programming is a form of
optimization where you optimize an
objective function with a system of linear
inequalities called constraints.
• The overlapped shaded region is called
the feasible region.
Solving a linear programming problem
1. Graph the constraints.
2. Locate the ordered pairs of the vertices of
the feasible region.
3. If the feasible region is bounded (or closed),
it will have a minimum & a maximum.
If the region is unbounded (or open), it will
have only one (a minimum OR a
maximum).
4. Plug the vertices into the linear equation
(C=) to find the min. and/or max.
A note about: Unbounded Feasible
Regions
• If the region is
unbounded, but has a
top on it, there will be
a maximum only.
• If the region is
unbounded, but has a
bottom, there will be a
minimum only.
Find the min. & max. values of C=-x+3y
subject to the following constraints.
x2
x5
y0
y  -2x+12
• Vertices of feasible
region:
(2,8)
C= -2+3(8)= 22
Max. of 22
at (2,8)
(2,0)
C= -2+3(0)= -2
(5,0)
C= -5+3(0)= -5
Min. of -5
(5,2)
at (5,0)
C= -5+3(2)= 1
Ex: C=x+5y Find the max. & min.
subject to the following constraints
x0
y2x+2
5x+y
• Vertices?
(0,2)
C=0+5(2)=10
(1,4)
C=1+5(4)=21
Maximum only!
Max of 21 at (1,4)
Linear Programming
• You want to buy
tickets for some
concerts; rock and
jazz.
• You want at least 5
rock tickets and at
least 8 jazz tickets.
• No more than 20 total
tickets.
• Rock tickets cost $3
and jazz tickets cost
$5.
• You cannot spend
more than $80.
• Find the number of
each type of ticket
you can buy at the
lowest price total.