7(1).1

Chapter 7Chapter 7

Duality TheoryDuality TheoryChapter 7Chapter 7

Duality TheoryDuality TheoryThe theory of duality is a very elegant and

important concept within the field of operations research. This theory was first developed in relation to linear programming, but it has many applications, and perhaps even a more natural and intuitive interpretation, in several related areas such as nonlinear programming, networks and game theory.

7(1).2

The notion of duality within linear The notion of duality within linear programming asserts that every linear program programming asserts that every linear program has associated with it a related linear program has associated with it a related linear program called its called its dual. The original problem in relation . The original problem in relation to its dual is termed the to its dual is termed the primal..

it is theit is the relationship between the primal and its between the primal and its dual, both on a mathematical and economic dual, both on a mathematical and economic level, that is truly the essence of duality theory. level, that is truly the essence of duality theory.

7(1).3

7.1 Examples7.1 Examples7.1 Examples7.1 Examples There is a small company in Melbourne which has There is a small company in Melbourne which has

recently become engaged in the production of office recently become engaged in the production of office furniture. The company manufactures tables, desks furniture. The company manufactures tables, desks and chairs. The production of a table requires 8 kgs and chairs. The production of a table requires 8 kgs of wood and 5 kgs of metal and is sold for $80; a desk of 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 for uses 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 determine wood and is sold for $50. We would like to determine the revenue maximizing strategy for this company, the revenue maximizing strategy for this company, given that their resources are limited to 100 kgs of given that their resources are limited to 100 kgs of wood and 60 kgs of metalwood and 60 kgs of metal..

7(1).4

Problem P1Problem P1Problem 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

+ + ≤

+ + ≤

≥, ,

7(1).5

Now consider that there is a much bigger Now consider that there is a much bigger company in Melbourne which has been the company in Melbourne which has been the lone producer of this type of furniture for lone producer of this type of furniture for many years. They don't appreciate the many years. They don't appreciate the competition from this new company; so competition from this new company; so they have decided to tender an offer to buy they have decided to tender an offer to buy all of their competitor's resources and all of their competitor's resources and therefore put them out of business. therefore put them out of business.

7(1).6

The challenge for this large company then The challenge for this large company then is to develop a linear program which will is to develop a linear program which will determine the appropriate amount of determine the appropriate amount of money that should be offered for a unit of money that should be offered for a unit of each type of resource, such that the offer each type of resource, such that the offer will be acceptable to the smaller company will be acceptable to the smaller company while minimizing the expenditures of the while minimizing the expenditures of the larger company.larger company.

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

+ ≥

+ ≥

+ ≥

≥,

min

y

w y y= +100 601 2

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 of meat and potatoes, each offering varying degrees of nutritional benefit. He has been warned by his doctor nutritional benefit. He has been warned by his doctor that he must receive at least 400 units of protein, 200 that he must receive at least 400 units of protein, 200 units of carbohydrates and 100 units of fat from his units of carbohydrates and 100 units of fat from his daily diet. Given that a kg of steak costs $10 and daily diet. Given that a kg of steak costs $10 and provides 80 units of protein, 20 units of carbohydrates provides 80 units of protein, 20 units of carbohydrates and 30 units of fat, and that a kg of potatoes costs $2 and 30 units of fat, and that a kg of potatoes costs $2 and provides 40 units of protein, 50 units of and provides 40 units of protein, 50 units of carbohydrates and 20 units of fat, he would like to find carbohydrates and 20 units of fat, he would like to find the minimum cost diet which satisfies his nutritional the minimum cost diet which satisfies his nutritional requirements requirements

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

+ ≥

+ ≥

+ ≥

≥,

min

x

Z x x= +10 21 2

7(1).10

Now consider a chemical company which Now consider a chemical company which hopes to attract this individual away from his hopes to attract this individual away from his present diet by offering him synthetic present diet by offering him synthetic nutrients in the form of pills. This company nutrients in the form of pills. This company would like determine prices per unit for their would like determine prices per unit for their synthetic nutrients which will bring them the synthetic nutrients which will bring them the highest possible revenue while still providing highest possible revenue while still providing an acceptable dietary alternative to the an acceptable dietary alternative to the individual. individual.

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+ + ≤

40 50 20 21 2 3

y y y+ + ≤

y y y1 2 3

0, , ≥

7(1).12

CommentsCommentsCommentsComments

Each of the two examples describes some kind Each of the two examples describes some kind of of competition between two decision makers. between two decision makers.

We shall investigate the notion of We shall investigate the notion of “competition” more formally in 618-261 under “competition” more formally in 618-261 under the heading “the heading “Game Theory”.”.

We shall investigate the economic We shall investigate the economic interpretation of the primal/dual relationship interpretation of the primal/dual relationship later in this chapter.later in this chapter.

7(1).13

7.2 FINDING THE DUAL 7.2 FINDING THE DUAL OF A STANDARD LINEAR OF A STANDARD LINEAR

PROGRAMPROGRAM

7.2 FINDING THE DUAL 7.2 FINDING THE DUAL OF A STANDARD LINEAR OF A STANDARD LINEAR

PROGRAMPROGRAM In this section we formalise the intuitive In this section we formalise the intuitive

feelings we have with regard to the the feelings we have with regard to the the relationship between the primal and dual relationship between the primal and dual versions of the two illustrative examples we versions of the two illustrative examples we examined in Section 7.1examined in Section 7.1

The important thing to observe is that the The important thing to observe is that the relationship - for the standard form - is relationship - for the standard form - is given as a given as a definition..

7(1).14

Standard form of the Primal Standard form of the Primal ProblemProblem

Standard form of the Primal Standard form of the Primal ProblemProblem

a x a x a x 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 20

+ + + ≤

+ + + ≤

+ + + ≤

≥

. . .

. . .

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

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

. . .

, , . . . ,

max

x

j j

j

n

Z c x= ∑

= 1

7(1).15

Standard form of the Dual Standard form of the Dual ProblemProblem

Standard form of the Dual Standard 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 20

+ + + ≥

+ + + ≥

+ + + ≥

≥

. . .

. . .

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

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

. . .

, , . . . ,

min

y

i

i

m

iw b y= ∑

= 1

7(1).16

7.2.1 Definition7.2.1 Definition7.2.1 Definition7.2.1 Definition

z Z cx

s t

Ax b

x

x

*: max

. .

= =

≤

≥ 0

€

w* :=minxw=yb

s.t.

yA≥c

y≥0

Primal Problem Dual Problem

b is not assumed to be non-negative

7(1).17

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

− + − ≤

+ − + + ≤

≥, , , ,

max

x

Z x x x x x= + − + +5 3 8 0 121 2 3 4 5

Primal

7(1).18

min

y

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

+ ≥

− + ≥

− ≥ −

+ ≥

− + ≥

≥,

Dual

7(1).19

Table 7.1: Primal-Dual Table 7.1: Primal-Dual relationshiprelationship

Table 7.1: Primal-Dual Table 7.1: Primal-Dual relationshiprelationship

x 1 0≥ x 2 0≥ x n ≥ 0 w =

y 1 0≥ a 11 a 12 a n1 ≤ b1

D u a l y 2 0≥ a 21 a 22 a n2 ≤ b 2

( m i n w ) . . . . . . . . . . . . . . .

y m ≥ 0 a m 1 a m 2 a mn ≤ b n

≥ ≥ ≥Z = c1 c 2 c n

7(1).20

7.2.3 Example7.2.3 Example7.2.3 Example7.2.3 Example

5 18 5 15

8 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

− + ≤

− + − ≤

− + ≤

− ≤

≥, ,

max

x

Z x x x= + −4 10 91 2 3

7(1).21

x1

0≥ x 2 0≥ x 3 0≥ w =

y 1 0≥ 5 − 1 8 5 ≤ 1 5

D u a l y2

0≥ − 8 1 2 0 ≤ 8

( m i n w ) y3

0≥ 1 2 − 4 8 ≤ 1 0

y4

0≥ 2 0 − 5 ≤ 5

≥ ≥ ≥

Z = 4 1 0 − 9

7(1).22

DualDualDualDual

5 8 12 2 4

18 12 4 10

5 8 5 9

0

1 2 3 4

1 2 3

1 3 4

1 2 3 4

y y y y

y y y

y y y

y y y y

− + + ≥

− + − ≥

+ − ≥ −

≥, , ,

min

y

w y y y y= + + +15 8 10 51 2 3 4

7(1).23

7.3 FINDING THE DUAL 7.3 FINDING THE DUAL OF NONSTANDARD OF NONSTANDARD LINEAR PROGRAMSLINEAR PROGRAMS

7.3 FINDING THE DUAL 7.3 FINDING THE DUAL OF NONSTANDARD OF NONSTANDARD LINEAR PROGRAMSLINEAR PROGRAMS

The approach here is similar to the one we The approach here is similar to the one we used in Section 5.6 when we dealt with non-used in Section 5.6 when we dealt with non-standard formulations in the context of the standard 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 “<=“ constraints by writing them as “<=“ constraints.constraints.

7(1).24

This is possible here because we doThis is possible here because we do not not require here that the RHS is require here that the RHS is non-negative.non-negative.

7(1).25

d x ei

i

k

i

=

∑ =

1

d x e

d x e

i

i

k

i

i

i

k

i

=

=

∑ ≤

∑ ≥

1

1

d x e

d x e

i

i

k

i

i

i

k

i

=

=

∑

∑

≤

− ≤ −

1

1

Standard form!

7(1).26

7.3.1 Example7.3.1 Example7.3.1 Example7.3.1 Example

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

− ≥

− + =

− ≤

≥, ; urs : = unrestricted sign

max

x

Z x x x= + +1 2 3

7(1).27

ConversionConversionConversionConversion

Multiply through the greater-than-or-equal-Multiply through the greater-than-or-equal-to inequality constraint by -1to inequality constraint by -1

Use the approach described above to Use 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 the by the difference of two nonnegative of two nonnegative variables.variables.

7(1).29

DualDualDualDual

y 2 − y 3 + y 4 ≥ 1

− 2 y1 − 3 y 2 + 3 y 3 − 2 y 4 ≥ 1

y1 + 4 y 2 − 4 y 3 ≥ 1

− y1

− 4 y2

+ 4 y3

≥ − 1

y1, y

2, y

3, y

4≥ 0

min

y

w y y y y= − + − +4 5 5 31 2 3 4

7(1).30

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

An equality constraint in the primal generates a constraint in the primal generates a dual variable that is dual variable that is unrestricted in sign.

An unrestricted in sign variable in the primal in sign variable in the primal generates an generates an equality constraint in the dual. in the dual.

Read the discussion in the lecture notesRead the discussion in the lecture notes Good material for a question in the final

exam!

7(1).31

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

min

y

w y y y y= − + − +4 5 5 31 2 3 4

y 2 − y 3 + y 4 ≥ 1

− 2 y1 − 3 y 2 + 3 y 3 − 2 y 4 ≥ 1

y1 + 4 y 2 − 4 y 3 ≥ 1

− y1

− 4 y2

+ 4 y3

≥ − 1

y1, y

2, y

3, y

4≥ 0

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

, ,

, , ,

, ,

, , ,

, ;

+ ≥

− − − ≥

− + =

≥ urs

min

,

, , ,

y

w y y y= − + +4 5 31 2 3

++

+correction

7(1).33

Table 7.2: Primal-Dual Table 7.2: Primal-Dual relationship relationship

Table 7.2: Primal-Dual Table 7.2: Primal-Dual relationship relationship

Primal Problem

opt=max

Constraint i : <= form = form

Variable j: xj >= 0 xj urs

opt=min

Dual Problem

Variable i : yi >= 0 yi urs

Constraint j: >= form = form

7(1).34

7.3.3 Example7.3.3 Example7.3.3 Example7.3.3 Example

3 8 6

6 5

8 10

0

1 2

1 2

1

2 1

x x

x x

x

x x

− ≥ −

+ = −

=

≥ ; urs

max

x

Z x x= +5 41 2

7(1).35

equivalent non-standard formequivalent non-standard formequivalent non-standard formequivalent non-standard form

− + ≤

+ = −

=

≥

3 8 6

6 5

8 10

0

1 2

1 2

1

2 1

x x

x x

x

x x; urs

max

x

Z x x= +5 41 2

7(1).36

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

− + + =

+ ≥

≥

3 8 5

8 6 4

0

1 2 3

1 2

1 2 3

y y y

y y

y y y; , urs

min

y

w y y y= − +6 5 101 2 3

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 the Can use the usual trick of multiplying the objective function by -1 (remembering to objective function by -1 (remembering to undo this when the dual is constructed.)undo this when the dual is constructed.)

It is instructive to use this method to It is instructive to use this method to construct the dual of the dual of the construct the dual of the dual of the standard form.standard form.

i.e, what is the dual of the dual of the i.e, what is the dual of the dual of the standard primal problem?standard primal problem?

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≥c

y≥0

7(1).39

€

maxy

−w=−yb

s.t.

yA≥c

y≥0

€

maxy

−w=−yb

s.t.

−yA≤−c

y≥0

7(1).40

min

. .

x

Z cx

s t

Ax b

x

− = −

− ≥ −

≥ 0

max

. .

x

Z cx

s t

Ax b

x

=

≤

≥ 0

7(1).41

Important ObservationImportant ObservationImportant ObservationImportant Observation

FOR ANY PRIMAL LINEAR FOR ANY PRIMAL LINEAR PROGRAM, THE DUAL OF THE PROGRAM, THE DUAL OF THE DUAL IS THE PRIMALDUAL IS THE PRIMAL

7(1).42

Table 7.3: Primal-Dual Table 7.3: Primal-Dual RelationshipRelationship

Table 7.3: Primal-Dual Table 7.3: Primal-Dual RelationshipRelationship

Primal or Dual

opt=max opt=min

Dual or Primal

Variable i : yi >= 0 yi urs

Constraint j: >= form = form

Constraint i : <= form = form

Variable j: xj >= 0 xj urs

7(1).43

Example 7.3.4Example 7.3.4Example 7.3.4Example 7.3.4

3 5 12

2 8

5 10

0

1 2

1 2

1 2

1 1

x x

x x

x x

x x

− ≥

+ = −

− ≤

≥,

min

x

Z x x= +6 41 2

7(1).44

equivalent formequivalent formequivalent formequivalent form

min

x

Z x x= +6 41 2

3 5 12

2 8

5 10

0

1 2

1 2

1 2

1 2

x x

x x

x x

x x

− ≥

+ = −

− + ≥ −

≥,

7(1).45

DualDualDualDual

3 5 6

5 2 4

0

1 2 3

1 2 3

1 3 2

y y y

y y y

y y y

+ − ≤

− + + ≤

≥, ; urs

max

y

w y y y= − −12 8 101 2 3