Lesson 29: Linear Programming I

Lesson 29 (Section 19.1)Linear Programming: The Corner Principle

Math 20

November 30, 2007

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Outline

Motivation: LP Problems

The Corner Principle

Solving 2D or 3D LP Problems

More Rolls and Cookies

The Duality Theorem

Motivational Example

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

This is an optimization problem with inequalities. We have tomaximize

f (x , y) = 8x + 10y

subject to the constraints

2x + y ≤ 50 x ≥ 0

x + 2y ≤ 70 y ≥ 0.

Motivational Example

Suppose a baker makes rolls and cookies. Rolls contain two ouncesof flour and one ounce of sugar per each. Cookies contain twoounces of sugar and one ounce of flour per each. The profit oneach roll sold is 8 cents, while the profit on each cookie sold is 10cents. The baker has 50 ounces of flour and 70 ounces of sugar onhand. How much of each bakery product should he make that day?This is an optimization problem with inequalities. We have tomaximize

f (x , y) = 8x + 10y

2x + y ≤ 50 x ≥ 0

x + 2y ≤ 70 y ≥ 0.

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

The feasibility set

These are the points representing combinations of rolls and cookiesthat can be produced.

x5 10 15 20 25

I x ≥ 0

I y ≥ 0

I 2x + y ≤ 50

I x + 2y ≤ 70

The completefeasiblity set

The feasibility set

x5 10 15 20 25

I x ≥ 0

I y ≥ 0

I 2x + y ≤ 50

I x + 2y ≤ 70

The feasibility set

x5 10 15 20 25

I x ≥ 0

I y ≥ 0

I 2x + y ≤ 50

I x + 2y ≤ 70

The feasibility set

x5 10 15 20 25

I x ≥ 0

I y ≥ 0

I 2x + y ≤ 50

I x + 2y ≤ 70

The feasibility set

x5 10 15 20 25

I x ≥ 0

I y ≥ 0

I 2x + y ≤ 50

I x + 2y ≤ 70

The feasibility set

x5 10 15 20 25

I x ≥ 0

I y ≥ 0

I 2x + y ≤ 50

I x + 2y ≤ 70

Outline

The Duality Theorem

x5 10 15 20 25

(0, 0) (25, 0)

(10, 30)

(0, 35)

Look at the feasibleset as a subset of acontour plot of theobjective functionf (x , y):

I isoquants arelines

I Where thecritical points?

x5 10 15 20 25

(0, 0) (25, 0)

(10, 30)

(0, 35)

x5 10 15 20 25

(0, 0) (25, 0)

(10, 30)

(0, 35)

x5 10 15 20 25

(0, 0) (25, 0)

(10, 30)

(0, 35)

x5 10 15 20 25

(0, 0) (25, 0)

(10, 30)

(0, 35)

x5 10 15 20 25

(0, 0) (25, 0)

(10, 30)

(0, 35)

x5 10 15 20 25

(0, 0) (25, 0)

(10, 30)

(0, 35)

x5 10 15 20 25

(0, 0) (25, 0)

(10, 30)

(0, 35)

x5 10 15 20 25

(0, 0) (25, 0)

(10, 30)

(0, 35)

x5 10 15 20 25

(0, 0) (25, 0)

(10, 30)

(0, 35)

x5 10 15 20 25

(0, 0) (25, 0)

(10, 30)

(0, 35)

x5 10 15 20 25

(0, 0) (25, 0)

(10, 30)

(0, 35)

Theorem of the Day

Theorem (The Corner Principle)

In any linear programming problem, the extreme values of theobjective function, if achieved, will be achieved on a corner of thefeasibility set.

A Diet Problem

Example

A nutritionist is planning a menu that includes foods A and B asits main staples. Suppose that each ounce of food A contains 2units of protein, 1 unit of iron, and 1 unit of thiamine; each ounceof food B contains 1 unit of protein, 1 unit of iron, and 3 units ofthiamine. Suppose that each ounce of A costs 30 cents, while eachounce of B costs 40 cents. The nutritionist wants the meal toprovide at least 12 units of protein, at least 9 units of iron, and atleast 15 units of thiamine. How many ounces of each of the foodsshould be used to minimize the cost of the meal?

We want to minimize the cost

c = 30x + 40y

2x + y ≥ 12 x + y ≥ 9 x + 3y ≥ 15 x ≥ 0 y ≥ 0

A Diet Problem

Example

A nutritionist is planning a menu that includes foods A and B asits main staples. Suppose that each ounce of food A contains 2units of protein, 1 unit of iron, and 1 unit of thiamine; each ounceof food B contains 1 unit of protein, 1 unit of iron, and 3 units ofthiamine. Suppose that each ounce of A costs 30 cents, while eachounce of B costs 40 cents. The nutritionist wants the meal toprovide at least 12 units of protein, at least 9 units of iron, and atleast 15 units of thiamine. How many ounces of each of the foodsshould be used to minimize the cost of the meal?

We want to minimize the cost

c = 30x + 40y

2x + y ≥ 12 x + y ≥ 9 x + 3y ≥ 15 x ≥ 0 y ≥ 0

Feasibility Set for the diet problem

x2 4 6 8 10 12 14

I x ≥ 0

I y ≥ 0

I 2x + y ≥ 12

I x + y ≥ 9

I x + 3y ≥ 15

The completefeasiblity set. Noticeit’s unbounded.

x2 4 6 8 10 12 14

I x ≥ 0

I y ≥ 0

I 2x + y ≥ 12

I x + y ≥ 9

I x + 3y ≥ 15

x2 4 6 8 10 12 14

I x ≥ 0

I y ≥ 0

I 2x + y ≥ 12

I x + y ≥ 9

I x + 3y ≥ 15

x2 4 6 8 10 12 14

I x ≥ 0

I y ≥ 0

I 2x + y ≥ 12

I x + y ≥ 9

I x + 3y ≥ 15

x2 4 6 8 10 12 14

I x ≥ 0

I y ≥ 0

I 2x + y ≥ 12

I x + y ≥ 9

I x + 3y ≥ 15

x2 4 6 8 10 12 14

I x ≥ 0

I y ≥ 0

I 2x + y ≥ 12

I x + y ≥ 9

I x + 3y ≥ 15

Outline

The Duality Theorem

Go back to the baker.

f (x , y) = 8x + 10y

We need only check the corners:

I f (0, 0) = 0

I f (25, 0) = 200

I f (0, 35) = 350

I f (10, 30) = 380

So the baker should make 10 rolls and 30 cookies. Mmmm . . .cookies.

f (x , y) = 8x + 10y

I f (0, 0) =

I f (25, 0) = 200

I f (0, 35) = 350

I f (10, 30) = 380

f (x , y) = 8x + 10y

I f (0, 0) = 0

I f (25, 0) = 200

I f (0, 35) = 350

I f (10, 30) = 380

f (x , y) = 8x + 10y

I f (0, 0) = 0

I f (25, 0) =

I f (0, 35) = 350

I f (10, 30) = 380

f (x , y) = 8x + 10y

I f (0, 0) = 0

I f (25, 0) = 200

I f (0, 35) = 350

I f (10, 30) = 380

f (x , y) = 8x + 10y

I f (0, 0) = 0

I f (25, 0) = 200

I f (0, 35) =

I f (10, 30) = 380

f (x , y) = 8x + 10y

I f (0, 0) = 0

I f (25, 0) = 200

I f (0, 35) = 350

I f (10, 30) = 380

f (x , y) = 8x + 10y

I f (0, 0) = 0

I f (25, 0) = 200

I f (0, 35) = 350

I f (10, 30) =

f (x , y) = 8x + 10y

I f (0, 0) = 0

I f (25, 0) = 200

I f (0, 35) = 350

I f (10, 30) = 380

f (x , y) = 8x + 10y

I f (0, 0) = 0

I f (25, 0) = 200

I f (0, 35) = 350

I f (10, 30) = 380

So the baker should make 10 rolls and 30 cookies.

Mmmm . . .cookies.

f (x , y) = 8x + 10y

I f (0, 0) = 0

I f (25, 0) = 200

I f (0, 35) = 350

I f (10, 30) = 380

For the diet problem, we have to evaluate the cost function at eachof the corner points.

c = 30x + 40y

I c(15, 0) = 450

I c(0, 12) = 480

I c(6, 3) = 300

I c(3, 6) = 330

So we should use 6 units of ingredient A and 3 units of ingredientB.

c = 30x + 40y

I c(15, 0) =

I c(0, 12) = 480

I c(6, 3) = 300

I c(3, 6) = 330

c = 30x + 40y

I c(15, 0) = 450

I c(0, 12) = 480

I c(6, 3) = 300

I c(3, 6) = 330

c = 30x + 40y

I c(15, 0) = 450

I c(0, 12) =

I c(6, 3) = 300

I c(3, 6) = 330

c = 30x + 40y

I c(15, 0) = 450

I c(0, 12) = 480

I c(6, 3) = 300

I c(3, 6) = 330

c = 30x + 40y

I c(15, 0) = 450

I c(0, 12) = 480

I c(6, 3) =

I c(3, 6) = 330

c = 30x + 40y

I c(15, 0) = 450

I c(0, 12) = 480

I c(6, 3) = 300

I c(3, 6) = 330

c = 30x + 40y

I c(15, 0) = 450

I c(0, 12) = 480

I c(6, 3) = 300

I c(3, 6) =

c = 30x + 40y

I c(15, 0) = 450

I c(0, 12) = 480

I c(6, 3) = 300

I c(3, 6) = 330

c = 30x + 40y

I c(15, 0) = 450

I c(0, 12) = 480

I c(6, 3) = 300

I c(3, 6) = 330

Outline

The Duality Theorem

Let’s go back to the baker. Suppose a factory wants to buyingredients from the baker and use them to make their own bakedgoods. What prices should they quote to the baker?

If the baker bakes and sells x rolls and y cookies, he makes8x + 10y dollars. He used 2x + y ounces of flour and x + 2y ouncesof sugar to make these products. If instead the baker sold theseingredients 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, thefactory needs to insure

2p+ q ≥ 8

p+2q ≥10

while minimizing their total payout 50p + 70q.

Let’s go back to the baker. Suppose a factory wants to buyingredients from the baker and use them to make their own bakedgoods. What prices should they quote to the baker?If the baker bakes and sells x rolls and y cookies, he makes8x + 10y dollars. He used 2x + y ounces of flour and x + 2y ouncesof sugar to make these products.

If instead the baker sold theseingredients at prices p for flour and q for sugar, he would make

p(2x + y) + q(x + 2y) = (2p + q)x + (p + 2q)y

2p+ q ≥ 8

p+2q ≥10

Let’s go back to the baker. Suppose a factory wants to buyingredients from the baker and use them to make their own bakedgoods. What prices should they quote to the baker?If the baker bakes and sells x rolls and y cookies, he makes8x + 10y dollars. He used 2x + y ounces of flour and x + 2y ouncesof sugar to make these products. If instead the baker sold theseingredients 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, thefactory needs to insure

2p+ q ≥ 8

p+2q ≥10

Let’s go back to the baker. Suppose a factory wants to buyingredients from the baker and use them to make their own bakedgoods. What prices should they quote to the baker?If the baker bakes and sells x rolls and y cookies, he makes8x + 10y dollars. He used 2x + y ounces of flour and x + 2y ouncesof sugar to make these products. If instead the baker sold theseingredients at prices p for flour and q for sugar, he would make

p(2x + y) + q(x + 2y) = (2p + q)x + (p + 2q)y

2p+ q ≥ 8

p+2q ≥10

The Dual Problem

This concept can be applied in general. A linear programmingproblem is in standard form if it seeks to maximize a functionf (x) = p′x subject to constraints Ax ≤ b (or b− Ax ≥ 0). It turnsout that each such problem has a dual problem: to minimizeg(x) = b′x subject to A′q ≥ p.

Solving the Dual Problem

2 4 6 8 10

Problem: Minimize 50p + 70qsubject to 2p + q ≥ 8,p + 2q ≥ 10

The feasibility set can bedrawn. The corner points arefound and tested. Thechoices are

I z(10, 0) = 500

I z(0, 8) = 560

I z(2, 4) = 380

The factory should quotep = 2 and q = 4 to the bakerand the baker would get 380for it. This is the sameamount of money as if he hadbaked the goods and soldthem!

2 4 6 8 10

Problem: Minimize 50p + 70qsubject to 2p + q ≥ 8,p + 2q ≥ 10The feasibility set can bedrawn.

The corner points arefound and tested. Thechoices are

I z(10, 0) = 500

I z(0, 8) = 560

I z(2, 4) = 380

2 4 6 8 10

Problem: Minimize 50p + 70qsubject to 2p + q ≥ 8,p + 2q ≥ 10The feasibility set can bedrawn. The corner points arefound and tested. Thechoices are

I z(10, 0) =

I z(0, 8) = 560

I z(2, 4) = 380

2 4 6 8 10

I z(10, 0) =

I z(0, 8) = 560

I z(2, 4) = 380

2 4 6 8 10

I z(10, 0) = 500

I z(0, 8) =

I z(2, 4) = 380

2 4 6 8 10

I z(10, 0) = 500

I z(0, 8) =

I z(2, 4) = 380

2 4 6 8 10

I z(10, 0) = 500

I z(0, 8) = 560

I z(2, 4) =

2 4 6 8 10

I z(10, 0) = 500

I z(0, 8) = 560

I z(2, 4) =

2 4 6 8 10

I z(10, 0) = 500

I z(0, 8) = 560

I z(2, 4) = 380

2 4 6 8 10

I z(10, 0) = 500

I z(0, 8) = 560

I z(2, 4) = 380

The factory should quotep = 2 and q = 4 to the bakerand the baker would get 380for it.

This is the sameamount of money as if he hadbaked the goods and soldthem!

2 4 6 8 10

I z(10, 0) = 500

I z(0, 8) = 560

I z(2, 4) = 380

The dual problem determines what are often called the shadowprices of the choice variables. They are somehow the value of thestock on hand.

Discuss: should the baker really be indifferentbetween selling his products or his ingredients? Besides thephysical ingredients, we haven’t counted the use of his labor andhis overhead. Or just his preference for staying as a baker.

The dual problem determines what are often called the shadowprices of the choice variables. They are somehow the value of thestock on hand. Discuss: should the baker really be indifferentbetween selling his products or his ingredients?

Besides thephysical ingredients, we haven’t counted the use of his labor andhis overhead. Or just his preference for staying as a baker.

The dual problem determines what are often called the shadowprices of the choice variables. They are somehow the value of thestock on hand. Discuss: should the baker really be indifferentbetween selling his products or his ingredients? Besides thephysical ingredients, we haven’t counted the use of his labor andhis overhead. Or just his preference for staying as a baker.

Outline

The Duality Theorem

The equality of payoff in the primal and dual problem is not anaccident. It will always be the case! The shadow prices are verymuch related to Lagrange multipliers.

Lesson 29: Linear Programming I

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