p. 112
• I can solve linear programing problem.
• 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.
• Infeasible – when the constraints of a linear
programing application do not overlap.
• Alternate optimal solutions- when there are two or
more possible linear programing application (usually
the graph is parallel to one side
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.
• 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.
x
x
y
y
• Vertices of feasible region:
(2,8)
2
C= -2+3(8)= 22
5
(2,0)
0
Max.
C= -2+3(0)=
-2 of 22
-2x+12
at (2,8)
(5,0)
C= -5+3(0)= -5
(5,2)
C= -5+3(2)= 1
Min. of -5
at (5,0)
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)
Here is a plan of the steps used to solve word problems
using linear programming:
1. After reading the question, make a chart to see the
information more clearly.
2. Assign variables to the unknowns.
3. Form expressions to represent the restrictions.
4. Graph the inequalities.
5. Find the coordinates of the corner points of the feasible
region.
6. Find the vertex point that maximizes or minimizes what we
are looking for.
7. State the solution in a sentence.
Theory – Solving Problems
Using Linear Programming
Example – Seven Steps
Example – Seven Steps cont’d
Example – Seven Steps cont’d
Example 2 – Seven Steps
Example 2 – Seven Steps cont’d
Example 2 – Seven Steps cont’d
Groupwork
Every table will answer one problem on page 116 to 118.
Remember to assign an equipment manager, recorder, presentor and
facilitator. Write legibly your answer on a huge graphing paper.
Equiptment manager get the materilas (3 colors of markers,
Huge graphing paper and ruler.)