1. Linear Programming

2. 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.

3. Solving a linear programming problem
Graph the constraints.
Locate the ordered pairs of the vertices of the feasible region.
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.

4. 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.

5. Find the min. & max. values of C=-x+3y subject to the following constraints.
Vertices of feasible region:
(2,8)
C= -2+3(8)= 22
(2,0)
C= -2+3(0)= -2
(5,0)
C= -5+3(0)= -5
(5,2)
C= -5+3(2)= 1
x  2
x  5
y  0
y  -2x+12
Max. of 22 at (2,8)
Min. of -5 at (5,0)

6. Ex: C=x+5y Find the max. & min. subject to the following constraints
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)