Lecture 3: Graphical Method Special Cases

1
Lecture 3 Graphical Method - Special Cases
2
Redundant Constraints Unbounded Infeasibility A
lternative Optima
3

• A constraint that does not affect the feasible
  region and, hence, cannot affect the optimal
  solution (because it will not be part of any
  corner point) is called a redundant constraint.
  You dont actually need it.

4
x2
10
Max 5x17x2
9
x1lt6
What happens when we add the following
constraint x1lt9?
8
x1 x2 lt 8
7
6
5
4
Feasible Region
3
2
2x1 3x2 lt 19
1
x1
4
1
2
3
4
5
6
7
8
9
10
5
x2
10
Max 5x17x2
9
x1lt9
x1lt6
8
There is no change to the feasible regionthus
x1lt9 is a REDUNDANT constraint
x1 x2 lt 8
7
6
5
4
Feasible Region
3
2
2x1 3x2 lt 19
1
x1
5
1
2
3
4
5
6
7
8
9
10
6
Redundant Constraints Unbounded Infeasibility A
lternative Optima
7
Redundant Constraints Unbounded Infeasibility A
lternative Optima
8

• The feasible region is unbounded and the
  objective function line can be moved parallel to
  itself without bound. So for a maximization
  problem, a solution value can be increased
  infinitely. For a minimization problem, a
  solution value can be decreased infinitely.

Something is missing!!
9
1
x2
Max 3x14x212
Feasible Region
A
The objective function line can be moved parallel
to itself without bound so that the solution
value can be increased infinitely.
B
x1x2gt5
x1
C
3x1x2gt8
10
1
x2
Max 3x14x224
Feasible Region
A
objective function line can be moved parallel to
itself without bound so that the solution value
can be increased infinitely.
B
x1x2gt5
x1
C
3x1x2gt8
11
1
x2
Max 3x14x232
Feasible Region
A
objective function line can be moved parallel to
itself without bound so that the solution value
can be increased infinitely.
B
x1x2gt5
x1
C
3x1x2gt8
12
1
x2
Max 3x14x248
Feasible Region
A
objective function line can be moved parallel to
itself without bound so that the solution value
can be increased infinitely.
B
x1x2gt5
x1
C
3x1x2gt8
13
1
x2
Max 3x14x260
Feasible Region
A
objective function line can be moved parallel to
itself without bound so that the solution value
can be increased infinitely.
B
x1x2gt5
x1
C
3x1x2gt8
14
1
x2
Max 3x14x28
Feasible Region
A
Objective Function line can be moved parallel to
itself without bound so that the solution value
can be increased infinitely.
B
x1x2gt5
x1
C
3x1x2gt8
15
x2
15 14 13 12 11 10 9 8 7 6 5 4
3 2 1

If this was a maximization problem, would the
feasible region be unbounded? If this was a
minimization would the feasible region be
unbounded?

Feasible Set
Obj. fun.
x1
1 2 3 4 5
6 7 8 9 10 11
12 13 14 15
16
x2
15 14 13 12 11 10 9 8 7 6 5 4
3 2 1

If this was a maximization problem, would the
feasible region be unbounded? If this was a
minimization would the feasible region be
unbounded?
Yes!
No

Feasible Set
Obj. fun.
x1
1 2 3 4 5
6 7 8 9 10 11
12 13 14 15
17
x2
15 14 13 12 11 10 9 8 7 6 5 4
3 2 1

If this was a minimization problem, would the
feasible region be unbounded?

Obj. fun.
Feasible Set
x1
1 2 3 4 5
6 7 8 9 10 11
12 13 14 15
18
x2
15 14 13 12 11 10 9 8 7 6 5 4
3 2 1

If this was a minimization problem, would the
feasible region be unbounded?
No!

Obj. fun.
Feasible Set
x1
1 2 3 4 5
6 7 8 9 10 11
12 13 14 15
19
Redundant Constraints Unbounded Infeasibility A
lternative Optima
20
Redundant Constraints Unbounded Infeasibility A
lternative Optima
21

• Also called an overconstrained problem
• No point satisfies all the constraints
• This problem has no feasible region and no
  optimal solution.

Expectations are too high!
22

• No feasible region
• No Corner Points
• No Solution

23
Redundant Constraints Unbounded Infeasibility A
lternative Optima
24
Redundant Constraints Unbounded Infeasibility A
lternative Optima
25

• When you move the objective function line to the
  edge of the feasible region, and it is parallel
  to a boundary constraint (that is, it lies on a
  constraint), then there
• are alternate optimal solutions, with all points
  on this line segment being optimal.

26
0.5
x2
A
x1lt4
x1x2lt5
Feasible Region
D
Max 2x12x22
C
x1
B
27
0.5
x2
A
x1lt4
x1x2lt5
Feasible Region
Max 2x12x24
D
C
x1
B
28
0.5
x2
A
x1lt4
x1x2lt5
Max 2x12x26
Feasible Region
D
C
x1
B
29
0.5
x2
A
x1lt4
x1x2lt5
Max 2x12x28
Feasible Region
D
C
x1
B
30
0.5
x2
A
x1lt4
x1x2lt5
Max 2x12x210
Feasible Region
D
C
x1
B
31
0.5
x2
A
x1lt4
x1x2lt5
Max 2x12x210

• All points between and including A and D will
  optimize the objective function value.
• The objective function value is 10. An optimal
  point is (1,4) Another is. (5,0) is NOT an
  optimal point! (In fact, it is not even feasible!

Feasible Region
D
C
x1
B
32
0.5
x2
A
x1lt4
x1x2lt5
Max 2x12x210

• x1 x2
• 0 5
• 1 4
• 2 3
• 3.5 1.5
• 4 1

Feasible Region
D
C
x1
B
33
End of Lecture 3
34

• Read 2.6
• Complete Practice Problems
• Complete Math Review

35
x2
15 14 13 12 11 10 9 8 7 6 5 4
3 2 1

Graph the following constraints, determine the
direction of feasibility, denote the Feasible
Region and Corner Points xlt8 xy12 ylt10

x1
1 2 3 4 5
6 7 8 9 10 11
12 13 14 15
36
x2
15 14 13 12 11 10 9 8 7 6 5 4
3 2 1

Graph the following constraints, determine the
direction of feasibility, denote the Feasible
Region and Corner Points xlt8 xy12 ylt10

What is the optimal solution if the objective
function is to Maximize xy? Solve using the
graphical method
x1
1 2 3 4 5
6 7 8 9 10 11
12 13 14 15
37
x2
15 14 13 12 11 10 9 8 7 6 5 4
3 2 1

Graph the following constraints, determine the
direction of feasibility, denote the Feasible
Region and Corner Points xlt8 xy12 ylt10

What is the optimal solution if the objective
function is to Maximize x-y? Solve using the
graphical method.
x1
1 2 3 4 5
6 7 8 9 10 11
12 13 14 15
38
Hungry Harrys Day camp would like to create a
menu for the children that meets nutritional
requirements. They would like to determine the
optimal mix of toast and sausage to meet the
nutritional requirements while minimizing the
cost to provide the meal. Toast costs .04/pound
while sausage costs .08/pound. The minimal
nutritional requirements are 20, 15 and 16, for
Vitamin A, Vitamin B, and Iron, respectively.
Toast provides 2, 1.5, and 2 units of nutrition
for Vitamin A, Vitamin B, and Iron, respectively
while, Sausage provides 20, 3, and 8 units of
nutrition for Vitamin A, Vitamin B, and Iron,
respectively. The maximum amount of sausage that
can be served is 4 pounds. Graph the following
constraints, determine the direction of
feasibility, denote the Feasible Region. What is
the optimal solution of the objective function.
39
Equation of a line can be written in two ways
1) General (standard) form Ax By C
ex 5x 8y 3
A 5 B 8 C 3 ex
-4x 2y .5 A B
C 2) Slope intercept form y
mx b mslope by-intercept
ex y 3x 8
m 3
b 8

ex y _ 1 x 4 3 m
b
40
Equation of a line can be written in two ways
Always put equations in slope-intercept form by
solving for y. ex Convert to slope intercept
form and then function notation 6x 2y 8
6x 2y 8 -6x -6x
2y -6x8 y -3x4 ex Convert to slope
intercept form and then function notation 3y
2x 12 ex Find
m and b given the following equation 7x 5y
10
41
If you need helpemail me, see me, or call me.
Have a great day!