In a linear programming problem, a linear function is to be optimized subject to linear inequality constraints. The corner principle says to solve such a problem all we have to do is look at the corners of the feasibility set.

1. Lesson 29 (Section 19.1)
Linear Programming: The Corner Principle
Math 20
November 30, 2007
Announcements
Problem Set 11 on the WS. Due December 5.
next OH: Monday 1–2 (SC 323)
next PS: Sunday 6–7 (SC B-10)
Midterm II review: Tuesday 12/4, 7:30-9:00pm in Hall E
Midterm II: Thursday, 12/6, 7-8:30pm in Hall A

2. Outline
Motivation: LP Problems
The Corner Principle
Solving 2D or 3D LP Problems
More Rolls and Cookies
The Duality Theorem

3. Motivational Example
Suppose a baker makes rolls and cookies. Rolls contain two ounces
of flour and one ounce of sugar per each. Cookies contain two
ounces of sugar and one ounce of flour per each. The profit on
each roll sold is 8 cents, while the profit on each cookie sold is 10
cents. The baker has 50 ounces of flour and 70 ounces of sugar on
hand. How much of each bakery product should he make that day?

4. Motivational Example
Suppose a baker makes rolls and cookies. Rolls contain two ounces
of flour and one ounce of sugar per each. Cookies contain two
ounces of sugar and one ounce of flour per each. The profit on
each roll sold is 8 cents, while the profit on each cookie sold is 10
cents. The baker has 50 ounces of flour and 70 ounces of sugar on
hand. How much of each bakery product should he make that day?
This is an optimization problem with inequalities. We have to
maximize
f (x, y ) = 8x + 10y
subject to the constraints
2x + y ≤ 50 x ≥0
x + 2y ≤ 70 y ≥ 0.

5. When the objective function and constraint functions are linear
(possibly with constants), we say the optimization problem is a
linear programming problem.

6. The feasibility set
These are the points representing combinations of rolls and cookies
that can be produced.
y
35
30
25
20
15
10
5
x
5 10 15 20 25

7. The feasibility set
These are the points representing combinations of rolls and cookies
that can be produced.
y
35
30
x ≥0
25
20
15
10
5
x
5 10 15 20 25

8. The feasibility set
These are the points representing combinations of rolls and cookies
that can be produced.
y
35
30
x ≥0
25
y ≥0
20
15
10
5
x
5 10 15 20 25

9. The feasibility set
These are the points representing combinations of rolls and cookies
that can be produced.
y
35
30
x ≥0
25
y ≥0
20 2x + y ≤ 50
15
10
5
x
5 10 15 20 25

10. The feasibility set
These are the points representing combinations of rolls and cookies
that can be produced.
y
35
30
x ≥0
25
y ≥0
20 2x + y ≤ 50
x + 2y ≤ 70
15
10
5
x
5 10 15 20 25

11. The feasibility set
These are the points representing combinations of rolls and cookies
that can be produced.
y
35
30
x ≥0
25
y ≥0
20 2x + y ≤ 50
x + 2y ≤ 70
15
The complete
10
feasiblity set
5
x
5 10 15 20 25

12. Outline
Motivation: LP Problems
The Corner Principle
Solving 2D or 3D LP Problems
More Rolls and Cookies
The Duality Theorem

13. The Corner Principle
y
35
Look at the feasible
30 set as a subset of a
contour plot of the
25
objective function
20 f (x, y ):
15
10
5
x
5 10 15 20 25

14. The Corner Principle
y
35
Look at the feasible
30 set as a subset of a
contour plot of the
25
objective function
20 f (x, y ):
15
10
5
x
5 10 15 20 25

15. The Corner Principle
y
35
Look at the feasible
30 set as a subset of a
contour plot of the
25
objective function
20 f (x, y ):
15
10
5
x
5 10 15 20 25

16. The Corner Principle
y
35
Look at the feasible
30 set as a subset of a
contour plot of the
25
objective function
20 f (x, y ):
isoquants are
15
lines
10
5
x
5 10 15 20 25

17. The Corner Principle
y
35
Look at the feasible
30 set as a subset of a
contour plot of the
25
objective function
20 f (x, y ):
isoquants are
15
lines
10
5
x
5 10 15 20 25

18. The Corner Principle
y
35
Look at the feasible
30 set as a subset of a
contour plot of the
25
objective function
20 f (x, y ):
isoquants are
15
lines
10
5
x
5 10 15 20 25

19. The Corner Principle
y
35
Look at the feasible
30 set as a subset of a
contour plot of the
25
objective function
20 f (x, y ):
isoquants are
15
lines
10
5
x
5 10 15 20 25

20. The Corner Principle
y
35
Look at the feasible
30 set as a subset of a
contour plot of the
25
objective function
20 f (x, y ):
isoquants are
15
lines
10
5
x
5 10 15 20 25

21. The Corner Principle
y
35
Look at the feasible
30 set as a subset of a
contour plot of the
25
objective function
20 f (x, y ):
isoquants are
15
lines
10
5
x
5 10 15 20 25

22. The Corner Principle
y
35
Look at the feasible
30 set as a subset of a
contour plot of the
25
objective function
20 f (x, y ):
isoquants are
15
lines
10
5
x
5 10 15 20 25

23. The Corner Principle
y
35
Look at the feasible
30 set as a subset of a
contour plot of the
25
objective function
20 f (x, y ):
isoquants are
15
lines
10
Where the
critical points?
5
x
5 10 15 20 25

24. The Corner Principle
y
(0, 35)
35
(10, 30) Look at the feasible
30 set as a subset of a
contour plot of the
25
objective function
20 f (x, y ):
isoquants are
15
lines
10
Where the
critical points?
5
x
(0, 0) 10 15 20 25(25, 0)
5

25. Theorem of the Day
Theorem (The Corner Principle)
In any linear programming problem, the extreme values of the
objective function, if achieved, will be achieved on a corner of the
feasibility set.

26. A Diet Problem
Example
A nutritionist is planning a menu that includes foods A and B as
its main staples. Suppose that each ounce of food A contains 2
units of protein, 1 unit of iron, and 1 unit of thiamine; each ounce
of food B contains 1 unit of protein, 1 unit of iron, and 3 units of
thiamine. Suppose that each ounce of A costs 30 cents, while each
ounce of B costs 40 cents. The nutritionist wants the meal to
provide at least 12 units of protein, at least 9 units of iron, and at
least 15 units of thiamine. How many ounces of each of the foods
should be used to minimize the cost of the meal?

27. A Diet Problem
Example
A nutritionist is planning a menu that includes foods A and B as
its main staples. Suppose that each ounce of food A contains 2
units of protein, 1 unit of iron, and 1 unit of thiamine; each ounce
of food B contains 1 unit of protein, 1 unit of iron, and 3 units of
thiamine. Suppose that each ounce of A costs 30 cents, while each
ounce of B costs 40 cents. The nutritionist wants the meal to
provide at least 12 units of protein, at least 9 units of iron, and at
least 15 units of thiamine. How many ounces of each of the foods
should be used to minimize the cost of the meal?
We want to minimize the cost
c = 30x + 40y
subject to the constraints
2x + y ≥ 12 x +y ≥9 x + 3y ≥ 15 x ≥0 y ≥0

28. Feasibility Set for the diet problem
14
12
10
8
6
4
2
2 4 6 8 10 12 14

29. Feasibility Set for the diet problem
14
12
10
x ≥0
y ≥0
8
6
4
2
2 4 6 8 10 12 14

30. Feasibility Set for the diet problem
14
12
10
x ≥0
y ≥0
8
2x + y ≥ 12
6
4
2
2 4 6 8 10 12 14

31. Feasibility Set for the diet problem
14
12
10
x ≥0
y ≥0
8
2x + y ≥ 12
6
x +y ≥9
4
2
2 4 6 8 10 12 14

32. Feasibility Set for the diet problem
14
12
10
x ≥0
y ≥0
8
2x + y ≥ 12
6
x +y ≥9
4 x + 3y ≥ 15
2
2 4 6 8 10 12 14

33. Feasibility Set for the diet problem
14
12
10
x ≥0
y ≥0
8
2x + y ≥ 12
6
x +y ≥9
4 x + 3y ≥ 15
The complete
2
feasiblity set. Notice
it’s unbounded.
2 4 6 8 10 12 14

34. Outline
Motivation: LP Problems
The Corner Principle
Solving 2D or 3D LP Problems
More Rolls and Cookies
The Duality Theorem

35. Go back to the baker.
f (x, y ) = 8x + 10y
We need only check the corners:

36. Go back to the baker.
f (x, y ) = 8x + 10y
We need only check the corners:
f (0, 0) =

37. Go back to the baker.
f (x, y ) = 8x + 10y
We need only check the corners:
f (0, 0) = 0

38. Go back to the baker.
f (x, y ) = 8x + 10y
We need only check the corners:
f (0, 0) = 0
f (25, 0) =

39. Go back to the baker.
f (x, y ) = 8x + 10y
We need only check the corners:
f (0, 0) = 0
f (25, 0) = 200

40. Go back to the baker.
f (x, y ) = 8x + 10y
We need only check the corners:
f (0, 0) = 0
f (25, 0) = 200
f (0, 35) =

41. Go back to the baker.
f (x, y ) = 8x + 10y
We need only check the corners:
f (0, 0) = 0
f (25, 0) = 200
f (0, 35) = 350

42. Go back to the baker.
f (x, y ) = 8x + 10y
We need only check the corners:
f (0, 0) = 0
f (25, 0) = 200
f (0, 35) = 350
f (10, 30) =

43. Go back to the baker.
f (x, y ) = 8x + 10y
We need only check the corners:
f (0, 0) = 0
f (25, 0) = 200
f (0, 35) = 350
f (10, 30) = 380

44. Go back to the baker.
f (x, y ) = 8x + 10y
We need only check the corners:
f (0, 0) = 0
f (25, 0) = 200
f (0, 35) = 350
f (10, 30) = 380
So the baker should make 10 rolls and 30 cookies.

45. Go back to the baker.
f (x, y ) = 8x + 10y
We need only check the corners:
f (0, 0) = 0
f (25, 0) = 200
f (0, 35) = 350
f (10, 30) = 380
So the baker should make 10 rolls and 30 cookies. Mmmm . . .
cookies.

46. For the diet problem, we have to evaluate the cost function at each
of the corner points.
c = 30x + 40y

47. For the diet problem, we have to evaluate the cost function at each
of the corner points.
c = 30x + 40y
c(15, 0) =

48. For the diet problem, we have to evaluate the cost function at each
of the corner points.
c = 30x + 40y
c(15, 0) = 450

49. For the diet problem, we have to evaluate the cost function at each
of the corner points.
c = 30x + 40y
c(15, 0) = 450
c(0, 12) =

50. For the diet problem, we have to evaluate the cost function at each
of the corner points.
c = 30x + 40y
c(15, 0) = 450
c(0, 12) = 480

51. For the diet problem, we have to evaluate the cost function at each
of the corner points.
c = 30x + 40y
c(15, 0) = 450
c(0, 12) = 480
c(6, 3) =

52. For the diet problem, we have to evaluate the cost function at each
of the corner points.
c = 30x + 40y
c(15, 0) = 450
c(0, 12) = 480
c(6, 3) = 300

53. For the diet problem, we have to evaluate the cost function at each
of the corner points.
c = 30x + 40y
c(15, 0) = 450
c(0, 12) = 480
c(6, 3) = 300
c(3, 6) =

54. For the diet problem, we have to evaluate the cost function at each
of the corner points.
c = 30x + 40y
c(15, 0) = 450
c(0, 12) = 480
c(6, 3) = 300
c(3, 6) = 330

55. For the diet problem, we have to evaluate the cost function at each
of the corner points.
c = 30x + 40y
c(15, 0) = 450
c(0, 12) = 480
c(6, 3) = 300
c(3, 6) = 330
So we should use 6 units of ingredient A and 3 units of ingredient
B.

56. Outline
Motivation: LP Problems
The Corner Principle
Solving 2D or 3D LP Problems
More Rolls and Cookies
The Duality Theorem

57. Let’s go back to the baker. Suppose a factory wants to buy
ingredients from the baker and use them to make their own baked
goods. What prices should they quote to the baker?

58. Let’s go back to the baker. Suppose a factory wants to buy
ingredients from the baker and use them to make their own baked
goods. What prices should they quote to the baker?
If the baker bakes and sells x rolls and y cookies, he makes
8x + 10y dollars. He used 2x + y ounces of flour and x + 2y ounces
of sugar to make these products.

59. Let’s go back to the baker. Suppose a factory wants to buy
ingredients from the baker and use them to make their own baked
goods. What prices should they quote to the baker?
If the baker bakes and sells x rolls and y cookies, he makes
8x + 10y dollars. He used 2x + y ounces of flour and x + 2y ounces
of sugar to make these products. If instead the baker sold these
ingredients at prices p for flour and q for sugar, he would make
p(2x + y ) + q(x + 2y ) = (2p + q)x + (p + 2q)y
dollars.

60. Let’s go back to the baker. Suppose a factory wants to buy
ingredients from the baker and use them to make their own baked
goods. What prices should they quote to the baker?
If the baker bakes and sells x rolls and y cookies, he makes
8x + 10y dollars. He used 2x + y ounces of flour and x + 2y ounces
of sugar to make these products. If instead the baker sold these
ingredients at prices p for flour and q for sugar, he would make
p(2x + y ) + q(x + 2y ) = (2p + q)x + (p + 2q)y
dollars. In order to make the deal attractive to the baker, the
factory needs to insure
2p+ q ≥ 8
p+2q ≥10
while minimizing their total payout 50p + 70q.

61. The Dual Problem
This concept can be applied in general. A linear programming
problem is in standard form if it seeks to maximize a function
f (x) = p x subject to constraints Ax ≤ b (or b − Ax ≥ 0). It turns
out that each such problem has a dual problem: to minimize
g (x) = b x subject to A q ≥ p.

62. Solving the Dual Problem
Problem: Minimize 50p + 70q
subject to 2p + q ≥ 8,
p + 2q ≥ 10
q
8
6
4
2
p
2 4 6 8 10

63. Solving the Dual Problem
Problem: Minimize 50p + 70q
subject to 2p + q ≥ 8,
p + 2q ≥ 10
The feasibility set can be
drawn.
q
8
6
4
2
p
2 4 6 8 10

64. Solving the Dual Problem
Problem: Minimize 50p + 70q
subject to 2p + q ≥ 8,
p + 2q ≥ 10
The feasibility set can be
drawn. The corner points are
q
8 found and tested. The
choices are
6
z(10, 0) =
4
2
p
2 4 6 8 10

65. Solving the Dual Problem
Problem: Minimize 50p + 70q
subject to 2p + q ≥ 8,
p + 2q ≥ 10
The feasibility set can be
drawn. The corner points are
q
8 found and tested. The
choices are
6
z(10, 0) =
4
2
p
2 4 6 8 10

66. Solving the Dual Problem
Problem: Minimize 50p + 70q
subject to 2p + q ≥ 8,
p + 2q ≥ 10
The feasibility set can be
drawn. The corner points are
q
8 found and tested. The
choices are
6
z(10, 0) = 500
4 z(0, 8) =
2
p
2 4 6 8 10

67. Solving the Dual Problem
Problem: Minimize 50p + 70q
subject to 2p + q ≥ 8,
p + 2q ≥ 10
The feasibility set can be
drawn. The corner points are
q
8 found and tested. The
choices are
6
z(10, 0) = 500
4 z(0, 8) =
2
p
2 4 6 8 10

68. Solving the Dual Problem
Problem: Minimize 50p + 70q
subject to 2p + q ≥ 8,
p + 2q ≥ 10
The feasibility set can be
drawn. The corner points are
q
8 found and tested. The
choices are
6
z(10, 0) = 500
4 z(0, 8) = 560
z(2, 4) =
2
p
2 4 6 8 10

69. Solving the Dual Problem
Problem: Minimize 50p + 70q
subject to 2p + q ≥ 8,
p + 2q ≥ 10
The feasibility set can be
drawn. The corner points are
q
8 found and tested. The
choices are
6
z(10, 0) = 500
4 z(0, 8) = 560
z(2, 4) =
2
p
2 4 6 8 10

70. Solving the Dual Problem
Problem: Minimize 50p + 70q
subject to 2p + q ≥ 8,
p + 2q ≥ 10
The feasibility set can be
drawn. The corner points are
q
8 found and tested. The
choices are
6
z(10, 0) = 500
4 z(0, 8) = 560
z(2, 4) = 380
2
p
2 4 6 8 10

71. Solving the Dual Problem
Problem: Minimize 50p + 70q
subject to 2p + q ≥ 8,
p + 2q ≥ 10
The feasibility set can be
drawn. The corner points are
q
8 found and tested. The
choices are
6
z(10, 0) = 500
4 z(0, 8) = 560
z(2, 4) = 380
2
p The factory should quote
p = 2 and q = 4 to the baker
2 4 6 8 10
and the baker would get 380
for it.

72. Solving the Dual Problem
Problem: Minimize 50p + 70q
subject to 2p + q ≥ 8,
p + 2q ≥ 10
The feasibility set can be
drawn. The corner points are
q
8 found and tested. The
choices are
6
z(10, 0) = 500
4 z(0, 8) = 560
z(2, 4) = 380
2
p The factory should quote
p = 2 and q = 4 to the baker
2 4 6 8 10
and the baker would get 380
for it. This is the same
amount of money as if he had
baked the goods and sold
them!

73. The dual problem determines what are often called the shadow
prices of the choice variables. They are somehow the value of the
stock on hand.

74. The dual problem determines what are often called the shadow
prices of the choice variables. They are somehow the value of the
stock on hand. Discuss: should the baker really be indifferent
between selling his products or his ingredients?

75. The dual problem determines what are often called the shadow
prices of the choice variables. They are somehow the value of the
stock on hand. Discuss: should the baker really be indifferent
between selling his products or his ingredients? Besides the
physical ingredients, we haven’t counted the use of his labor and
his overhead. Or just his preference for staying as a baker.

76. Outline
Motivation: LP Problems
The Corner Principle
Solving 2D or 3D LP Problems
More Rolls and Cookies
The Duality Theorem

77. The Duality Theorem
The equality of payoff in the primal and dual problem is not an
accident. It will always be the case! The shadow prices are very
much related to Lagrange multipliers.