Lesson 30 (Section 19.2–3)Duality in Linear Programming

Math 20

December 3, 2007

Announcements

I Problem Set 11 on the WS. Due December 5.

I next OH: Monday 1–2 (SC 323)

I next PS: Sunday 6–7 (SC B-10)

I Midterm II review: Tuesday 12/4, 7:30-9:00pm in Hall E

I Midterm II: Thursday, 12/6, 7-8:30pm in Hall A

Outline

Recap

Example

Shadow Prices

The Dual Problem

DefinitionA linear programming problem is a constrained optimizationproblem with a linear objective function and linear inequalityconstraints.

DefinitionAn LP problem is in standard form if it is expressed as

max z = c1x1 + c2x2 + · · ·+ cnxn

subject to the constraints

a11x1 + a12x2 + · · ·+ a1nxn ≤ b1

a21x1 + a22x2 + · · ·+ a2nxn ≤ b2

......

am1x1 + am2x2 + · · ·+ amnxn ≤ bm

x1, x2, . . . , xn ≥ 0

DefinitionA linear programming problem is a constrained optimizationproblem with a linear objective function and linear inequalityconstraints.

DefinitionAn LP problem is in standard form if it is expressed as

max z = c1x1 + c2x2 + · · ·+ cnxn

subject to the constraints

a11x1 + a12x2 + · · ·+ a1nxn ≤ b1

a21x1 + a22x2 + · · ·+ a2nxn ≤ b2

......

am1x1 + am2x2 + · · ·+ amnxn ≤ bm

x1, x2, . . . , xn ≥ 0

In vector notation, an LP problem in standard form looks like

max z = c · x

subject to constraints

Ax ≤ b x ≥ 0

Theorem of the Day for Friday

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.

Outline

Recap

Example

Shadow Prices

The Dual Problem

Example

Example

We are starting a business selling two Harvard insignia products:sweaters and scarves. The profits on each are $35 and $10,respectively. Each has a pre-bought embroidered crest sewn on it;we have 2000 crests on hand. Sweaters take four skeins of yarnwhile scarves only take one, and there are 2300 skeins of yarnavailable. Finally, we have available storage space for 1250 scarves;we could use any of that space for sweaters, too, but sweaters takeup half again as much space as scarves.What product mix maximizes revenue?

Formulating the problem

Let x be the number of sweaters and y the number of scarvesmade. We want to

max z = 35x + 10y

subject to

x + y ≤ 2000

4x + y ≤ 2300

3x + 2y ≤ 2500

x , y ≥ 0

Finding the corners

x+

y=

2000

4x+

y=

23003x

+2y

=2500

575 83313

2000

1250

2000

2300

•

(420, 620)

Notice one constraint issuperfluous!

z(0, 0) = 0

z(575, 0) = 20, 125

z(0, 1250) = 12, 500

z(420, 620) = 20, 900

Answer

We should make 420 sweaters and 620 scarves.

Outline

Recap

Example

Shadow Prices

The Dual Problem

Suppose our business were suddenly given

I one additional crest patch?

I one additional skein of yarn?

I one additional unit of storage space?

How much would profits change?

One more patch

x+

y=

2001

4x+

y=

23003x

+2y

=2500

575 83313

2000

1250

2000

2300

•

(420, 620)

I Since we weren’t “up against”this constraint in the first place,one extra doesn’t change ouroptimal product mix.

I At this product mix, themarginal profit of patches is 0.

One more skein

x+

y=

2000

4x+

y=

23013x

+2y

=2500

575 83313

2000

1250

2000

2300

•

(420.4, 619.4)

I We’ll make a little more sweaterand less scarf

I The marginal profit is

∆z = 35(0.4) + 10(−0.6) = 8

One more storage unit

x+

y=

2000

3x+

2y=

2501575 8331

32000

1250

2000

2300

•(419.8, 620.8)

I We’ll make a little less sweaterand more scarf

I The marginal profit is

∆z = 35(−0.2) + 10(0.8) = 1

Shadow Prices

DefinitionIn a linear programming problem in standard form, the change inthe objective function obtained by increasing a constraint by one iscalled the shadow price of that constraint.

Example

In our example problem,

I The shadow price of patches is zero

I The shadow price of yarn is 8

I The shadow price of storage is 1

We should look into getting more yarn!

Shadow Prices

DefinitionIn a linear programming problem in standard form, the change inthe objective function obtained by increasing a constraint by one iscalled the shadow price of that constraint.

Example

In our example problem,

I The shadow price of patches is zero

I The shadow price of yarn is 8

I The shadow price of storage is 1

We should look into getting more yarn!

Outline

Recap

Example

Shadow Prices

The Dual Problem

QuestionSuppose an entrepreneur wants to buy our business’s resources.What prices should be quoted for each crest? skein of yarn? unitof storage?

Answer.Suppose the entrepreneur quotes p for each crest patch, q for eachskein of yarn, and r for each storage unit.

I Each sweater takes one patch, 4 skeins, and 3 storage units,so effectively p + 4q + 3r is bid per sweater

I Likewise, p + q + 2r is bid per scarf.

So we must have

p + 4q + 3r ≥ 35

p + q + 2r ≥ 10

for us to sell out. The entrepreneur’s goal is to minimize the totalpayout

w = 2000p + 2300q + 2500r

QuestionSuppose an entrepreneur wants to buy our business’s resources.What prices should be quoted for each crest? skein of yarn? unitof storage?

Answer.Suppose the entrepreneur quotes p for each crest patch, q for eachskein of yarn, and r for each storage unit.

I Each sweater takes one patch, 4 skeins, and 3 storage units,so effectively p + 4q + 3r is bid per sweater

I Likewise, p + q + 2r is bid per scarf.

So we must have

p + 4q + 3r ≥ 35

p + q + 2r ≥ 10

for us to sell out. The entrepreneur’s goal is to minimize the totalpayout

w = 2000p + 2300q + 2500r

DefinitionGiven a linear programming problem in standard form, the duallinear programming problem is

min w = b1y1 + · · ·+ bmym

subject to constraints

a11y1 + a21y2 + · · ·+ am1ym ≥ p1

a12y1 + a22y2 + · · ·+ am2ym ≥ p2

......

a1ny1 + a2ny2 + · · ·+ amnym ≥ pn

y1, . . . , ym ≥ 0

In fancy vector language, the dual of the problem

max z = p · x subject to Ax ≤ b and x ≥ 0

ismin w = b · y subject to A′y ≥ p and y ≥ 0

Solving the Dual Problem

I The feasible set is unbounded (extending away from you)

I w(0, 8, 1) = 20, 900 is minimal

•

(35, 0, 0)

•

(0, 10, 0)

•

(0, 0, 83/4)

•

(0, 8, 1)

•

(12/3, 81/3, 0)

•

w = 70, 000

•

w = 23, 000

•

w = 21, 875

•

w = 20, 900

•

w = 22, 500

p

q

r

Solving the Dual Problem

I The feasible set is unbounded (extending away from you)

I w(0, 8, 1) = 20, 900 is minimal

•

(35, 0, 0)

•

(0, 10, 0)

•

(0, 0, 83/4)

•

(0, 8, 1)

•

(12/3, 81/3, 0)

•

w = 70, 000

•

w = 23, 000

•

w = 21, 875

•

w = 20, 900

•

w = 22, 500

p

q

r

Solving the Dual Problem

I The feasible set is unbounded (extending away from you)

I w(0, 8, 1) = 20, 900 is minimal

•

(35, 0, 0)

•

(0, 10, 0)

•

(0, 0, 83/4)

•

(0, 8, 1)

•

(12/3, 81/3, 0)

•

w = 70, 000

•

w = 23, 000

•

w = 21, 875

•

w = 20, 900

•

w = 22, 500p

q

r

Solving the Dual Problem

I The feasible set is unbounded (extending away from you)

I w(0, 8, 1) = 20, 900 is minimal

•

(35, 0, 0)

•

(0, 10, 0)

•

(0, 0, 83/4)

•

(0, 8, 1)

•

(12/3, 81/3, 0)

•

w = 70, 000

•

w = 23, 000

•

w = 21, 875

•

w = 20, 900

•

w = 22, 500

p

q

r

The Big Idea

I The shadow prices are the solutions to the dual problem

I The payoff is the same in both the primal problem and thedual problem