primal and dual relationship

8/3/2019 primal and dual relationship

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7(1).1

Chapter 7Chapter 7

Duality TheoryDuality Theory

Chapter 7Chapter 7

Duality TheoryDuality TheoryThe theory of duality is a very elegant

and important concept within thefield of operations research. Thistheory was first developed in relationto linear programming, but it hasmany applications, and perhaps even

a more natural and intuitiveinterpretation, in several relatedareas such as nonlinearprogramming, networks and game

theory.


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The notion of duality within linearThe notion of duality within linear

programming asserts that every linearprogramming asserts that every linearprogram has associated with it a relatedprogram has associated with it a related

linear program called itslinear program called its dual. The original. The original

problem in relation to its dual is termed theproblem in relation to its dual is termed theprimal..

it is theit is the relationshipbetween the primalbetween the primal

and its dual, both on a mathematical andand its dual, both on a mathematical and

economic level, that is truly the essence ofeconomic level, that is truly the essence of

duality theory.duality theory.


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7(1).3

7.1 Examples7.1 Examples

7.1 Examples7.1 Examples

There is a small company in Melbourne which hasThere is a small company in Melbourne which has

recently become engaged in the production of officerecently become engaged in the production of office

furniture. The company manufactures tables, desksfurniture. The company manufactures tables, desks

and chairs. The production of a table requires 8 kgsand chairs. The production of a table requires 8 kgsof wood and 5 kgs of metal and is sold for $80; a deskof wood and 5 kgs of metal and is sold for $80; a desk

uses 6 kgs of wood and 4 kgs of metal and is sold foruses 6 kgs of wood and 4 kgs of metal and is sold for

$60; and a chair requires 4 kgs of both metal and$60; and a chair requires 4 kgs of both metal and

wood and is sold for $50. We would like to determinewood and is sold for $50. We would like to determinethe revenue maximizing strategy for this company,the revenue maximizing strategy for this company,

given that their resources are limited to 100 kgs ofgiven that their resources are limited to 100 kgs of

wood and 60 kgs of metalwood and 60 kgs of metal..


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7(1).4

Problem P1Problem P1

Problem P1Problem P1

max

x

Z x x x! 80 60 501 2 3

8 6 4 100

5 4 4 60

0

1 2 3

1 2 3

1 2 3

x x x

x x x

x x x

e

e

u, ,


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7(1).5

Now consider that there is a much biggerNow consider that there is a much bigger

company in Melbourne which has been thecompany in Melbourne which has been thelone producer of this type of furniture forlone producer of this type of furniture for

many years. They don't appreciate themany years. They don't appreciate the

competition from this new company; socompetition from this new company; so

they have decided to tender an offer to buythey have decided to tender an offer to buy

all of their competitor's resources andall of their competitor's resources and

therefore put them out of business.therefore put them out of business.


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7(1).6

The challenge for this large company then isThe challenge for this large company then is

to develop a linear program which willto develop a linear program which willdetermine the appropriate amount of moneydetermine the appropriate amount of money

that should be offered for a unit of each typethat should be offered for a unit of each type

of resource, such that the offer will beof resource, such that the offer will be

acceptable to the smaller company whileacceptable to the smaller company while

minimizing the expenditures of the largerminimizing the expenditures of the larger

company.company.


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7(1).7

Problem D1Problem D1Problem D1Problem D1

8 5 80

6 4 60

4 4 50

0

1 2

1 2

1 2

1 2

y y

y y

y y

y y

u

u

u

u,

min

y

w y y! 100 601 2


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7(1).8

A Diet ProblemA Diet ProblemA Diet ProblemA Diet Problem An individual has a choice of two types of food to eat,An individual has a choice of two types of food to eat,

meat and potatoes, each offering varying degrees ofmeat and potatoes, each offering varying degrees of

nutritional benefit. He has been warned by his doctornutritional benefit. He has been warned by his doctor

that he must receive at least 400 units of protein, 200that he must receive at least 400 units of protein, 200

units of carbohydrates and 100 units of fat from hisunits of carbohydrates and 100 units of fat from hisdaily diet. Given that a kg of steak costs $10 anddaily diet. Given that a kg of steak costs $10 and

provides 80 units of protein, 20 units of carbohydratesprovides 80 units of protein, 20 units of carbohydrates

and 30 units of fat, and that a kg of potatoes costs $2and 30 units of fat, and that a kg of potatoes costs $2

and provides 40 units of protein, 50 units ofand provides 40 units of protein, 50 units ofcarbohydrates and 20 units of fat, he would like to findcarbohydrates and 20 units of fat, he would like to find

the minimum cost diet which satisfies his nutritionalthe minimum cost diet which satisfies his nutritional

requirementsrequirements


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7(1).9

Problem P2Problem P2Problem P2Problem P2

80 40 400

20 50 200

30 20 100

0

1 2

1 2

1 2

1 2

x x

x x

x x

x x

u

u

u

u,

minx

Z x x! 10 21 2


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7(1).10

Now consider a chemical company whichNow consider a chemical company which

hopes to attract this individual away fromhopes to attract this individual away from

his present diet by offering him synthetichis present diet by offering him synthetic

nutrients in the form of pills. This companynutrients in the form of pills. This company

would like determine prices per unit forwould like determine prices per unit for

their synthetic nutrients which will bringtheir synthetic nutrients which will bringthem the highest possible revenue while stillthem the highest possible revenue while still

providing an acceptable dietary alternativeproviding an acceptable dietary alternative

to the individual.to the individual.


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7(1).11

Problem D2Problem D2Problem D2Problem D2

max

y

w y y y! 400 200 1001 2 3

80 20 30 101 2 3 y y y e

40 50 20 21 2 3 y y y e

y y y1 2 3 0, , u


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7(1).12

CommentsCommentsCommentsComments

Each of the two examples describes someEach of the two examples describes some

kind ofkind ofcompetitionbetween two decisionbetween two decision

makers.makers.

We shall investigate the notion ofWe shall investigate the notion of

competition more formally in 618competition more formally in 618--261261

under the heading under the heading Game Theory..

We shall investigate the economicWe shall investigate the economic

interpretation of the primal/dual relationshipinterpretation of the primal/dual relationship

later in this chapter.later in this chapter.


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7(1).13

7.2 FINDING THE DUAL7.2 FINDING THE DUAL

OF A STANDARD LINEAROF A STANDARD LINEARPROGRAMPROGRAM


OF A STANDARD LINEAROF A STANDARD LINEARPROGRAMPROGRAM

In this section we formalise the intuitiveIn this section we formalise the intuitive

feelings we have with regard to the thefeelings we have with regard to the therelationship between the primal and dualrelationship between the primal and dual

versions of the two illustrative examples weversions of the two illustrative examples we

examined in Section 7.1examined in Section 7.1

The important thing to observe is that theThe important thing to observe is that the

relationshiprelationship -- for the standard formfor the standard form -- isis

given as agiven as a definition..


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7(1).14

Standard form of the PrimalStandard form of the Primal

ProblemProblem

Standard form of the PrimalStandard form of the Primal

ProblemProblem

ax

ax

ax b

a x a x a x b

a x a x a x b

x x x

n n

n n

m m mn n m

n

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

1 2 0

e

e

e

u

. . .. ..

. .. .. . . . . .. .

. .. .. . . . . .. .. ..

, , .. .,

maxx

j jj

n Z c x!

!1


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7(1).15

Standard form of the DualStandard form of the Dual

ProblemProblem

Standard form of the DualStandard form of the Dual

ProblemProblem

a y a y a y c

a y a y a y c

a y a y a y c

y y y

m m

m m

n n mn m n

m

11 1 21 2 1 1

12 1 22 2 2 2

1 1 2 2

1 2 0

u

u

u

u

.. .

. . .

. .. .. . . . . .. .

. .. .. . . . . .. .

. ..

, ,.. . ,

miny

ii

m

iw b y! !1


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7(1).16

7.2.1 Definition7.2.1 Definition7.2.1 Definition7.2.1 Definition

z Z cx

s t

Ax bx

x*: max

. .

! !

eu 0

w* :! minx w ! yb

s.t.

yA u c

y u 0

Primal Problem Dual Problem

b is not assumed to be non-negative


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7.2.2 Example7.2.2 Example7.2.2 Example7.2.2 Example

3 8 9 15 20

18 5 8 4 12 30

0

1 2 4 5

1 2 3 4 5

1 2 3 4 5

x x x x

x x x x x

x x x x x

e

e

u, , , ,

maxx Z x x x x x! 5 3 8 0 121 2 3 4 5

Primal


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7(1).18

miny

w y y! 20 301 2

3 18 5

8 5 3

8 8

9 4 0

15 12 12

0

1 2

1 2

2

1 2

1 2

1 2

y y

y y

y

y y

y y

y y

u

u

u

u

u

u,

Dual


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Table 7.1: PrimalTable 7.1: Primal--DualDual

relationshiprelationship



x1 0u x2 0u xn u 0 w=

y1 0u a11 a12 a n1 e b1

Dual y2 0u a21 a22 a n2 e b2(minw) ... ... ... ... ...

ym u 0 am1 am2 amn e bnu u u

Z= c1 c2 cn


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7(1).20


5 18 5 158 12 8 8

12 4 8 10

2 5 5

0

1 2 3

1 2 3

1 2 3

1 3

1 2 3

x x x

x x x

x x x

x x

x x x

e

e

e

e

u, ,

maxx

Z x x x! 4 10 91 2 3


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7(1).21

x1 0u x2 0u x3 0u w=

y1 0u 5 - 18 5 e 15Dual y2 0u 8 12 0 e 8

(min w) y3 0u12

- 48

e10

y4 0u 2 0 - 5 e 5

u u uZ= 4 10 - 9


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7(1).22

DualDualDualDual

5 8 12 2 4

18 12 4 10

5 8 5 90

1 2 3 4

1 2 3

1 3 4

1 2 3 4

y y y y

y y y

y y yy y y y

u

u

u

u, , ,

miny

w y y y y! 15 8 10 51 2 3 4


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7(1).23


OF NONSTANDARDOF NONSTANDARDLINEAR PROGRAMSLINEAR PROGRAMS


OF NONSTANDARDOF NONSTANDARDLINEAR PROGRAMSLINEAR PROGRAMS

The approach here is similar to the one weThe approach here is similar to the one we

used in Section 5.6 when we dealt with nonused in Section 5.6 when we dealt with non--standard formulations in the context of thestandard formulations in the context of the

simplex method.simplex method.

There is one exception:There is one exception: we do not add

artificial variables. We handle =We handle =

constraints by writing them as


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7(1).24

This is possible here because we doThis is possible here because we do notnot

require here that the RHS isrequire here that the RHS is nonnon--negative.negative.


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7(1).25

dx eii

k

i!!

1

d x e

d x e

ii

k

i

ii

k

i

!

!

e

u

1

1

d

x e

dx e

ii

k

i

ii

k

i

!

!

e

e

1

1

Standard form!


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7(1).26


2 4

3 4 5

2 3

0

2 3

1 2 3

1 2

1 2 3

x x

x x x

x x

x x x

u

!

e

u, ; urs:= unrestricted sign

maxx

Z x x x! 1 2 3


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7(1).27

ConversionConversionConversionConversion

Multiply through the greaterMultiply through the greater--thanthan--oror--equalequal--

to inequality constraint byto inequality constraint by --11 Use the approach described above toUse the approach described above to

convert the equality constraint toconvert the equality constraint to a pair of

inequality constraints.

Replace the variable unrestricted in sign, ,Replace the variable unrestricted in sign, ,

by theby the difference of two nonnegativeof two nonnegative

variables.variables.


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7(1).28

e

e

e

e

u

2 4

3 4 4 5

3 4 4 5

2 3

0

2 3 3

1 2 3 3

1 2 3 3

1 2

1 2 3 3

x x x

x x x x

x x x x

x x

x x x x

, ,,

, ,,

, ,,

, ,,, , ,

max , ,,

x Z x x x x!

1 2 3 3


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7(1).29

DualDualDualDual

y2

y3 y4 u 12y13y2 3y3 2y4 u 1

y1 4y2 4y3 u 1

y1 4y2 4y3 u 1

y1,y2,y3,y4 u 0

miny

w y y y y! 4 5 5 31 2 3 4


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7(1).30

Streamlining the conversion ...Streamlining the conversion ...Streamlining the conversion ...Streamlining the conversion ...

An equality constraint in the primalconstraint in the primal

generates a dual variable that isgenerates a dual variable that is

unrestricted in sign.

An unrestricted in sign variable in thein sign variable in the

primal generates anprimal generates an equality constraint inin

the dual.the dual.

Read the discussion in the lecture notesRead the discussion in the lecture notes

Good material for a question in the final

exam!


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7(1).31

Example 7.3.1 (Continued)Example 7.3.1 (Continued)Example 7.3.1 (Continued)Example 7.3.1 (Continued)min

yw y y y y! 4 5 5 31 2 3 4

y2

y3 y4 u 12y13y2 3y3 2y4 u 1

y1 4y2 4y3 u 1

y1 4y2 4y3 u 1

y1,y2,y3,y4 u 0


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7(1).32

y y

y y y

y y

y y y

2 3

1 2 3

1 2

1 3 2

1

2 3 2 1

4 1

0

, ,

, , ,

, ,

, , ,, ;

u

u

!

u urs

min,

, , ,y

w y y y!

4 5 31 2 3

++

+

correction


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7(1).33





Primal Problem

opt=max

Constraint i :

= 0

xj urs

opt=min

Dual Problem

Variable i :

yi >= 0

yi urs

Constraint j:

>= form

= form


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7(1).34


3 8 66 5

8 100

1 2

1 2

1

2 1

x xx x

xx x

u

!

!u ; urs

maxx

Z x x! 5 41 2


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7(1).35

equivalent nonequivalent non--standard formstandard formequivalent nonequivalent non--standard formstandard form

e

!

!

u

3 8 6

6 5

8 10

0

1 2

1 2

1

2 1

x x

x x

x

x x; urs

maxx

Z x x! 5 41 2


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7(1).36

Dual from the recipeDual from the recipeDual from the recipeDual from the recipe

!

u

u

3 8 5

8 6 4

0

1 2 3

1 21 2 3

y y y

y y

y y y; , urs

miny

w y y y! 6 5 101 2 3


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7(1).37

What about opt=min ?What about opt=min ?What about opt=min ?What about opt=min ?

Can use the usual trick of multiplying theCan use the usual trick of multiplying the

objective function byobjective function by --1 (remembering to1 (remembering toundo this when the dual is constructed.)undo this when the dual is constructed.)

It is instructive to use this method toIt is instructive to use this method to

construct the dual of the dual of theconstruct the dual of the dual of the

standard form.standard form.

i.e, what is the dual of the dual of thei.e, what is the dual of the dual of the

standard primal problem?standard primal problem?


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7(1).38

What is the dual ofWhat is the dual ofWhat is the dual ofWhat is the dual of

w* :!minxw ! yb

s.t.

yA u c

y u 0


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7(1).39

maxy

w ! yb

s.t.

yA u c

y u 0

maxy

w ! yb

s.t.

yA e c

y u 0


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7(1).40

min

. .

x Z cx

s t Ax b

x

!

u

u 0

max

. .

x Z cx

s t Ax b

x

!

e

u 0


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7(1).41

Important ObservationImportant ObservationImportant ObservationImportant Observation

FOR ANY PRIMAL LINEARFOR ANY PRIMAL LINEAR

PROGRAM, THE DUAL OF THEPROGRAM, THE DUAL OF THEDUAL IS THE PRIMALDUAL IS THE PRIMAL


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7(1).42


RelationshipRelationship


RelationshipRelationship

Primal or Dual

opt=max opt=min

Dual orPrimal

Variable i :

yi >= 0

yi urs

Constraint j:

>= form

= form

Constraint i :

= 0

xj urs


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7(1).43

Example 7.3.4Example 7.3.4Example 7.3.4Example 7.3.4

3 5 122 8

5 100

1 2

1 2

1 2

1 1

x xx x

x xx x

u

!

e

u,

minx

Z x x! 6 41 2


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7(1).44

equivalent formequivalent formequivalent formequivalent form

minx

Z x x! 6 41 2

3 5 122 8

5 100

1 2

1 2

1 2

1 2

x xx x

x xx x

u

!

u

u,


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7(1).45

DualDualDualDual

3 5 6

5 2 4

0

1 2 3

1 2 31 3 2

y y y

y y y

y y y

e

e

u, ; urs

max

y

w y y y! 12 8 101 2 3

primal and dual relationship

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