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Math 141-copyright Joe Kahlig, 10B
Section 3.3: Graphical Solution of Linear Programming Problems
Example: Find the point(s) that maximize the objective function P = x + 5y subject to these constraints.
2x + y ≤ 12
2x + 5y ≤ 20
12
11
x, y ≥ 0
10
9
8
7
6
5
4
3
2
F.R.
1
1
2
3
4
5
6
7
8
Theorem: Given a feasible region R and the objective function f = ax + by.
A) If f has a maximum or a minimum then it will happen at a corner point.
B) If R is bounded (and includes all of the corner points) then f will have both a maximum and a
minimum.
C) If R is unbounded, you might have a max or a min (or neither). More work is needed to decide.
9
10
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Math 141-copyright Joe Kahlig, 10B
Example: Maximize P = x + y subjected to
x+y ≤8
3x + y ≥ 12
12
11
−2x + 3y ≥ 3
10
9
What is the maximum value of P ?
8
7
(2,6)
6
Where does P have a maximum?
5
F.R.
4
(4.2, 3.8)
(3, 3)
3
2
1
1
Example: Minimize f = 3x − y for this feasible region.
(2, 4)
4
Feasible
Region
(5, 0)
2
3
4
5
6
7
8
9
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Math 141-copyright Joe Kahlig, 10B
Example: Minimize f = 3x − 2y for this feasible region.
Points
A (0,1)
B (0,12)
C (3,9)
D (3,2)
Feasible
B
E (3,0)
Region
C
F (8.25, 3.75)
H
G (12,0)
H (12,5)
D
F
A
E
G