chapter 04 over

8/13/2019 Chapter 04 Over

1/12

Primal and Dual LP Problems

Economic theory indicates that scarce (limited) resources

have value. In LP models, limited resources are allocated,so they should be, valued.

Whenever we solve an LP problem, we implicitly solvetwo problems: the primal resource allocation problem,and the dual resource valuation problem.

ere we cover the resource valuation, or as it iscommonly called, the Dual LP

Primal

!all"or#$

iall"orb$a..

$c

!

i

!

!i!

!

!!

ts

Max

Dual

iall"or#%

!all"orca%s.t.

b%&in

i

!

i

i!i

i

ii

CH04-OH-1


2/12

Primal Dual Pair and 'heir %nitsPrimal

!all"or#$

iall"orb$a..

$c

!

i

!

!i!

!

!!

ts

Max

where is the variableand euals units sold

ma sum (per unit pro"its)* (units sold)s.t. sum (per unit res. use)*(units sold) + res on hand

Dual

iall"or#%

!all"orca%s.t.

b%&in

i

!

i

i!i

i

ii

% is the variableand eualsper unit resource value

min sum (per unit res value) * (res on hand)s.t. sum (per unit res value) * (per unit res use)

per unit pro"its

-o resource values shadow prices are set up so per unitpro"its are ehausted.

CH04-OH-2


3/12

Primal Dual Pair and Eample

#$,$,$

/0#12$/#$/3$

1/$$$s.t.

1/##$14##$/###$ma(

new"ine"ancy

new"ine"ancy

new"ine"ancy

new"ine"ancy

++

++

++

ity)(none5ativ%,%

)($1/##12%%

)($14##/#%%

)($/###/3%%s.t.

Payments)(6esource/0#%1/%min

(pro"its)(labor)cap)(van

/1

new/1

"ine/1

"ancy/1

/1

+

+

+

+

Primal rows become dual columns

CH04-OH-3


4/12

Primal Dual 7b!ective 8orrespondence

Dual 9ariable

6elationiven ;easible $* and %* to Partial

Derivative

b%*8$*

b%*?8>u

1@

?

*

8omplementary -lacAness =ero Pro"its c

CH04-OH-4


5/12

Primal Dual Interrelations8onstructin5 Dual -olutions

Cote we can construct a dual solution "rom optimal primalsolution without resolvin5 the problem.iven optimal primal $?

*> ?@1b and $C?*> #. 'his

solution must be "easible$?

*> ?@1b # and $ #

and must satis"y nonne5ative reduced cost8??

@1 8??@1as a dual solution.

;irst, is this "easible in the dual constraints.'o be "easible, we must have %* < 8 and %* #.

We Anow %*


6/12


Cow are dual nonne5ativity conditions % # satis"ied.

We can looA at this by looAin5 the implication o" thenonne5ative reduced costs over the slacAs (%8??@1is a "easible

dual solution.

CH04-OH-6


7/12


Cow the uestion becomes, is this choice optimal

In this case the primal ob!ective "unction = euals 8$*>8?$?

*F 8C?$C?*> 8??

@1b F 8C?# > 8??@1b.

-imultaneously, the dual ob!ective euals %b > 8??@1b

which euals the primal ob!ective. 'here"ore, the primal

and dual ob!ectives are eual at optimality.

;urthermore, since "or a "easible primal we have


8/12

Primal Dual InterrelationsInterpretin5 Dual -olutions

In addition 5iven the derivation in the last chapter we canestablish the interpretation o" the dual variables. Inparticular, since the optimal dual variables eual 8??

@1(which are called the primal shadow prices) then the dualvariables are interpretable as the mar5inal value producto" the resources since we showed

.%>?8>b= *1

?


9/12

Primal Dual Interrelations

Primal Dual 7b!ective 8orrespondence

Primal -olution ItemPrimal -olution

In"ormation

Dual -olution Item8orrespondin5 Dual -olution

In"ormation

7b!ective "unction 7b!ective "unction

-hadow prices 9ariable values

-lacAs 6educed costs

9ariable values -hadow prices

6educed costs -lacAs

CH04-OH-9


10/12

De5eneracy and Duality

'he above interpretations "or the dual variables depend

upon whether the basis still eists a"ter the chan5e occurs.

When a basic primal variable euals Gero, dual hasalternative optimal solutions. 'he cause o" this situationis 5enerally primal constraints are redundant at thesolution point and the ran5e o" ri5ht hand sides is Gero.

#$,$

3#$

3#$

1##$$/$H$&a

/1

/

1

/1

/1

++

3#, $/> 3#, constraints areredundant.

I" "irst slacA variable is basic then $1> 3#, $/> 3#, -1> #while -1is basic. -hadow prices are #, H, and /.

I" -Hbasic $1> 3#, $/> 3# -H> # with shadow prices /,1, #. -ame ob!ective value @@ multiple solutions.

De5eneracy and Duality

CH04-OH-10


11/12

#$,$

3#$

3#$

1##$$

/$H$&a

/1

/

1

/1

/1

+

+

u1 u/ uH > # H /J or / 1 # J

'he main di""iculty with de5eneracyis in interpretin5 theshadow prices as they taAe on a direction.

I" one were to increase the "irst ri5ht hand side "rom 1##

to 1#1 this would lead to a Gero chan5e in the ob!ective"unction and $1and $/would remain at 3#.

Decrease "irst constrain rhs "rom 1## to 22 then ob!ective"unction which is two units smaller because $/wouldneed to be reduced "rom 3# to K2.

'his shows that the two alternative shadow prices "or the"irst constraint (i.e., # and /) each hold in a direction.

-imilarly i" bound on $131, ob! increases by 1, whereas,i" moved downward to K2, it would cost H.&eanwhile, reducin5 $/bound costs / and increasin5 by#. this eplains all shadow prices

CH04-OH-11
http://agrinet.tamu.edu/mccarl/gms/ch04/Degen.GMShttp://agrinet.tamu.edu/mccarl/gms/ch04/Degen.GMS


12/12

Primal 8olumns are Dual 8onstraints

8olumns in the primal, "orm constraints on the dualshadow price in"ormation.

'hus, "or eample, when a column is entered into a modelindicatin5 as much o" a resource can be purchased at a"ied price as one wants, then this column "orms an upperbound on the shadow price o" that resource.

Cote that it would not be sensible to have a shadow priceo" that resource above the purchase price since one couldpurchase more o" that resource.

-imilarly, allowin5 5oods to be sold at a particular pricewithout restriction provides a lower bound on the shadow

price.In 5eneral, the structure o" the columns in a primal linearpro5rammin5 model should be eamined to see whatconstraints they place upon the dual in"ormation.

'he linear pro5rammin5 modelin5 chapter etends this

discussion.

CH04-OH-12

chapter 04 over

Documents