04.1 Chapter4_Duality.pdf

8/10/2019 04.1 Chapter4_Duality.pdf

1/35

Duality in Linear Programming

based on:Bradley, Hax, and Magnanti; Applied

Mathematical Programming, Addison-Wesley,1977Chapter 4, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6 and 4.7

(FEUP | DEGI) Operations Research March 26, 2014 1 / 35
http://find/http://goback/


2/35


In the preceding chapter on sensitivity analysis, we saw that the shadow-price interpretation of theoptimal simplex multipliers is a very useful concept.First, these shadow prices give us directly the marginal worth of an additional unit of any of theresources.Second, when an activity is priced out using these shadow prices, the opportunity cost ofallocating resources to that activity relative to other activities is determined.Duality in linear programming is essentially a unifying theory that develops the relationshipsbetween a given linear program and another related linear program stated in terms of variableswith this shadow-price interpretation.The importance of duality is twofold:First, fully understanding the shadow-price interpretation of the optimal simplex multipliers canprove very useful in understanding the implications of a particular linear-programming model.Second, it is often possible to solve the related linear program with the shadow prices as the

variables in place of, or in conjunction with, the original linear program, thereby taking advantageof some computational efficiencies.

http://find/


3/35


Learning Objectives

Write the dual problem of a linear programming problem.

Understand the:

weak duality property,optimality property,unboundedness property,strong duality property.

Verify that the complementary slackness conditions hold for a given problem.

Apply the dual simplex method.

Apply the parametric primal-dual algorithm.

http://find/


4/35

A Preview of Duality



5/35

Production Planning at BA

The factory of Barbosa & Almeida (BA) in Avintes produces glass containers through mouldinjection.BA recently won two new clients and intends to assign this production to one of its furnaces. Theorders were of bottles of Sandeman Ruby Port (75cl), one liter bottles of olive oil Oliveira da Serra

and one liter bottles of Favaios Moscatel. The customers buy all the bottles that BA can produce.Due to differences in the production (number of cavities and different cycle times), each batch ofportwine bottles takes 50 hours to produce, whereas each batch of oil bottles requires only 30hours and a batch of favaios takes 50 hours to produce. There are a total of 2000 hours availableat the oven.Moreover we must also consider the capacity available in the decoration sector (e.g. labels,packaging), which is 200 hours. The portwine bottles spend 3 hours per batch, the olive oil

bottles 5 hours per batch and the favaios bottles also 5 hours per batch.Barbosa & Almeida wants to maximize the profit from these two orders, knowing that the profitper batch is 50e 60e and 20e, respectively.



6/35

Production Planning at BA

Decision variables

xVP Number of batches of Vinho do Porto Ruby Sandeman bottles to produce;

xA Number of batches of Azeite Oliveira da Serra bottles to produce;

xF Number of batches of Favaios bottles to produce.

Objective:

max Z= 50xVP+ 60xA+ 20xF

Subject to:

50xVP +30xA +50xF 2000 (time in the oven)

3xVP +5xA +5xF 200 (capacity in the decoration sector)

xVP, xA, xF 0

In this model xVP, xA and xFmay not be integer.

http://find/


7/35


8/35

Economic properties of the shadow prices associated withthe resources

Shadowx1 . . . xn s1 . . . sm price

s1 a11 . . . a1n 1 . . . 0 b1 y1...

.

.

....

.

.

....

.

.

....

sm am1 . . . amn 0 . . . 1 bm ymZ c1 . . . cn 0 . . . 0 0

At the final tableau:Z c

1 . . . c

n c

n+1 . . . c

n+m z

0Reduced costs in terms of shadow prices:

cj =cjm

i=1aijyi (j= 1 . . . n)

aij is the amount of resource iused per unit of activity j, and yi is the imputed value of thatresource.

The termmi=1aijyi is the total value of the resources used per unit of activity jand is thus themarginal resource cost per unit of activity j.If we think of the objective coefficients cjas being marginal revenues, the reduced costs

cj =cjm

i=1aijyi (j= 1 . . . n)

are simply net marginal revenues (i.e., marginal revenue minus marginal cost).

http://find/


9/35

Production Planning at BAReduced cost (net marginal revenue) for each type of

bottle (decision variable)xVP xA xF sO sD

sO 50 30 50 1 0 2000sD 3 5 5 0 1 200Z 50 60 20 0 0 0At the final tableau:

xVP xA xF sO sD

xVP 1 0 5

81

32 316 25

xA 0 1 5

8 3160

516 25

Z 0 0 48 34 716 9

38 2750

For the basic variables the reduced costs are zero.The marginal revenue is equal to the marginal cost for theseactivities.

For all nonbasic variables the reduced costs are negative.The marginal revenue is less than the marginal cost for theseactivities, so they should not be pursued.

Shadow price for the oven capacity: yO = 716 .

Shadow price for the decoration capacity: yD = 93

8 .Net marginal revenue for the production of portwine bottles:

cVP = cVPm

i=1aiVPyi = 50 (50 716 + 3 9

38 ) = 0

Net marginal revenue for the production of azeite bottles:

cA = cA m

i=1aiAyi = 60 (30 716

+ 5 9 38

) = 0

Net marginal revenue for the production of favaios bottles:

cF

= cFmi=1aiFyi = 20 (50 716 + 5 9 38 ) = 48 34

http://find/


10/35


11/35

Production Planning at BACan shadow prices be determined directly?

For a maximization decision problem, the shadow prices must satisfy the requirement thatmarginal revenue be less than or equal to marginal cost for all activities.

The shadow prices must be nonnegative since they are associated with less-than-or-equal-toconstraints in a maximization decision problem.

Imagine that BA does not own its productive capacity but has to rent it. In this case theshadow prices could be seen as rent rates.

Objective:

max Z = 50xVP+ 60xA+ 20xF

Subject to:

50xVP+ 30xA+ 50xF 2000

3xVP+ 5xA+ 5xF 200

xVP, xA, xF 0

Objective:

min V= 2000yO+ 200yD

Subject to:

50yO+ 3yD 50

30yO+ 5yD 60

50yO+ 5yD 20

yO, yD 0

http://find/


12/35

Production Planning at BAPrimal and Dual Decision Problems

PRIMAL Decision Problem:

Objective:

max Z = 50xVP+ 60xA+ 20xF

Subject to:

50xVP+ 30xA+ 50xF 2000

3xVP+ 5xA+ 5xF 200

xVP, xA, xF 0

Optimal solution:xVP = 25

xA = 25xF = 0Z= 2750with shadow prices:yO=

716

yD= 938

DUAL Decision Problem:

Objective:

min V= 2000yO+ 200yD

Subject to:

50yO+ 3yD 5030yO+ 5yD 60

50yO+ 5yD 20

yO, yD 0

Optimal solution:

yO= 716yD = 9

38

V= 2750with shadow prices:xVP= 25xA = 25x

F = 0

http://find/


13/35

Production Planning at BARequirements of the optimality conditions of the Simplex

Method

The reduced costs of the basicvariables are zero.

If a decision variable of the primal is positive, then thecorresponding constraint in the dual must hold with

equality.

IfxVP 0 then cVP = 50(50yO+ 3yD) = 0If

xA 0 then cA = 60(30

yO+ 5

yD) = 0

The variables with negativereduced costs are nonbasicvariables (zero valued)

If a constraint holds as a strict inequality, then the

corresponding decision variable must be zero.

IfcF = 20(50yO+ 5yD) < 0 thenxF = 0

If a shadow price is positive,then the correspondingconstraint must hold withequality.

IfyO>0 then 50xVP+ 30xA + 50xF = 2000IfyD>0 then 3xVP+ 5xA + 5xF= 200

http://find/


14/35

Complementary Slackness Conditions

If a primal variable ispositive,then its corresponding(complementary) dualconstraint holds withequality.

If a dual constraint holdswith strict inequality,then its corresponding(complementary) primalvariable must be zero.

http://find/


15/35

Finding the dual in general

Definition of the dual problem


PRIMAL
http://find/


16/35

PRIMAL is a maximization problem

PRIMAL

Objective:

max Z =n

j=1

cjxj

Subject to:

nj=1

aijxj bi (i = 1,2, . . . , m

)

nj=1

aijxj bi (i = m

,2, . . . , m

)

nj=1

aijxj = bi (i = m

,2, . . . , m)

xj 0 (j = 1,2, . . . , n)

DUAL

Objective:

min V =

mi=1

biyi +m

i=m+1

biy

i +m

i=m+1

biy

i

Subject to:

mi=1

aijyi +m

i=m+1

aijy

i +m

i=m+1

aijy

i cj (j = 1,2, . . . , n)

yi 0 (i = 1, . . . , m)

y

i 0 (i = m + 1 , . . . , m)

y

i R (i = m

+ 1 , . . . , m)


PRIMAL
http://find/


17/35

PRIMAL is a minimization problem

PRIMAL

Objective:

min Z =n

j=1

cjxj

Subject to:

nj=1

aijxj bi (i = 1,2, . . . , m)

nj=1

aijxj bi (i = m

,2, . . . , m

)

nj=1

aijxj = bi (i = m

,2, . . . , m)

xj 0 (j = 1,2, . . . , n)

DUAL

Objective:

max V =

mi=1

biyi +m

i=m+1

biy

i +m

i=m+1

biy

i

Subject to:

mi=1

aijyi +m

i=m+1

aijy

i +m

i=m+1

aijy

i cj (j = 1,2, . . . , n)

yi 0 (i = 1, . . . , m)

y

i 0 (i = m + 1 , . . . , m)

y

i R (i = m

+ 1 , . . . , m)


S a o co s o c s PRIMAL a
http://find/


18/35

Summary of the correspondences between PRIMAL andDUAL decision problems

Primal (Maximize) Dual (Minimize)ith constraint ith variable 0ith constraint ith variable 0

ith constraint = ith variable unrestrictedjth variable 0 jth constraint jth variable 0 jth constraint jth variable unrestricted jth constraint =

http://find/


19/35

Weak duality property


20/35

Weak duality property

PRIMAL

Objective:

max Z =n

j=1cjxj

Subject to:n

j=1

aijxj bi (i= 1, 2, . . . ,m)

xj 0 (j= 1, 2, . . . , n)

DUAL

Objective:min V =

mi=1biyi

Subject to:m

i=1

aijyi cj (j= 1, 2, . . . ,n)

yi 0 (i= 1, . . . ,m)

nj=1cjxj

mi=1biyi

Ifxj, j= 1, 2, . . . , n is a feasible solution tothe primal problemandyi, i= 1, 2, . . . ,m is a feasible solutionto the dual problem.

DUAL feasible V decreasing

PRIMAL feasible Z increasing


Optimality property
http://find/


21/35

Optimality property

PRIMAL

Objective:

max Z =n

j=1cjxj

Subject to:n

j=1

aijxj bi (i= 1, 2, . . . ,m)

xj 0 (j= 1, 2, . . . , n)

DUAL

Objective:

min V =m

i=1biyi

Subject to:m

i=1

aijyi cj (j= 1, 2, . . . ,n)

yi 0 (i= 1, . . . ,m)

Ifxj, j = 1, 2, . . . , n is a feasible solution to the primal problemand

yi, i = 1, 2, . . . , m is a feasible solution to the dual problem

andn

j=1cjxj = m

i=1biyithenxj, j= 1, 2, . . . , n is an optimal solution for the primal problemandyi, i= 1, 2, . . . ,m is an optimal solution for the dual problem

http://find/


22/35

Strong Duality Property


23/35

Strong Duality Property

If the PRIMAL problem has afinite optimal solution,then so does the DUALproblemand these two values areequal

If the DUAL problem has afinite optimal solution,then so does the PRIMALproblemand these two values areequal

http://find/


24/35

The complementary slackness


Complementary Slackness Property
http://find/


25/35


In an optimal solution of a linear program:

For the PRIMAL linear programposed as a maximization problem with

less-than-or-equal-to constraints, thismeans:

If the value of the dual variable(shadow price) associated with aconstraint is nonzero, then that

constraint must be satisfied withequality.

If

yi>0, then

nj=1aij

xj =bi

If a constraint is satisfied withstrict inequality, then its

corresponding dual variable must bezero.

If

n

j=1aijxj


26/35


In an optimal solution of a linear program:

For the DUAL linear program posed asa minimization problem with greater-

than-or-equal-to constraints, thismeans:

If the value of the dual variable(shadow price) associated with aconstraint is nonzero, then that

constraint must be satisfied withequality.

If

xj>0, then

mi=1aij

yi =cj

If a constraint is satisfied withstrict inequality, then its

corresponding dual variable must bezero.

If

mi=1aij

yi


27/35

Ifxj,j = 1, 2, . . . , n is a feasible solution to thePRIMAL problemand

yi, i = 1, 2, . . . , m, is a feasible solution to the

DUAL problem,then they are optimal solutions to these problems if,and only if, the complementary-slackness conditionshold for both the primal and the dual problems.

http://find/


28/35

The dual Simplex Algorithm


The DUAL Simplex Algorithm
http://find/


29/35

Solving the primal problem, moving throughsolutions (simplex tableaux) that are dual feasiblebut primal unfeasible.a

aDifferent from solving the DUAL problem with the (PRIMAL) simplex method

PRIMAL feasible: bi 0

DUAL feasible: ci 0

An optimal solution is a solution that is both primaland dual feasible.


DUAL Simplex Algorithm
http://find/


30/35

Initialization: problem in canonical form with all cj 0.

Optimality test :ifbi 0 for all i= 1, 2, . . . ,m, then stop optimal solution found;else:

Iterate :

Choose the variable to leave the basis by1:

br= mini

bi|bi


31/35

x1 x2 x3 x4

x3 1 1 1 0 1 x4 2 -3 0 1 2

Z 3 1 0 0 0

x1 x2 x3 x4

x3 1

3 0 1 1

3 1

3

x2 2

3 1 0 13

23

Z 73 0 0 13

23

x1 x2 x3 x4

x4 1 0 3 1 1x2 1 1 1 0 1Z 2 0 1 0 1

x4 leaves the basis because:minibi|bi< 0 = min {1,2} = 2

x2 enters the basis because:

minj

cjarj|arj


32/35

Parametric PRIMAL-DUAL Algorithm an example I


33/35

Objective:

max Z = 2x1+ 3x2

Subject to:

x1+ x2 6

x1+ 2x2 1

2x1 3x2 1

x1, x2 0

x1 x2 x3 x4 x5x3 1 1 1 0 0 6x4 1 2 0 1 0

12

x5 1 3 0 0 1 1Z 2 3 0 0 0 0

x1 x2 x3 x4 x5x3 1 1 1 0 0 6x4 1 2 0 1 0

12

+x5 1 3 0 0 1 1 +

Z 2 3 0 0 0 0

This example can easily be put in canonicalform by addition of slack variables.

However, neither primal feasibility nor primaloptimality conditions will be satisfied.

We will arbitrarily consider the example as a

function of the parameter .If we choose large enough, i.e. 3,this system of equations satisfies the primalfeasibility and primal optimality conditions.

http://find/


34/35

Parametric PRIMAL-DUAL Algorithm an example III


35/35

After the pivot, the new canonical form is represented in the next tableau.

x1 x2 x3 x4 x5x3 0 0 1 4 3 1 + 7x2 0 1 0 1 1

32 2

x1 1 0 0 3 2 7

2 5Z 0 0 0 3 1 (3 )( 14 +

12 ) + (

12 +

12 )(

72 5)

As we continue to decrease to zero, the optimality conditions remain satisfied. Thus the optimal final tableaufor this example is given by setting equal to zero.

x1 x2 x3 x4 x5x3 0 0 1 4 3 1x2 0 1 0 1 1

32

x1 1 0 0 3 2 7

2

Z 0 0 0 3 1 52

http://find/

Documents

04.1 Chapter4_Duality.pdf