Linear Prog

7/26/2019 Linear Prog

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Chapter 4:Linear ProgrammingSensitivity Analysis

2007 Pearson Education


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hat i! there is uncertainly a"out one ormore values in the LP model#

Sensitivity analysis allo$s us to determineho$ %sensitive& the optimal solution is to

changes in data values'

(his includes analy)ing changes in:

*' An +",ective -unction Coe!!icient .+-C/2' A ight 1and Side .1S/ value o! aconstraint


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raphical Sensitivity Analysis

e can use the graph o! an LP to see $hathappens $hen:

*' An +-C changes3 or

2' A 1S changes

ecall the -lair -urniture pro"lem


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-lair -urniture Pro"lem

a5 7( 6 C .pro!it/

Su",ect to the constraints:

8( 6 4C 9 2400 .carpentry hrs/2( 6 *C 9 *000 .painting hrs/

C 9 40 .ma5 chairs/

( ; *00 .min ta"les/

(3 C ; 0 .nonnegativity/


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+",ective -unctionCoe!!icient .+-C/ Changes

hat i! the pro!it contri"ution !or ta"leschanged !rom e more ta"les and less chairs#.i'e' ?oes the optimal solution change#/

@


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Characteristics o! +-C Changes

(here is no e!!ect on the !easi"le region

(he slope o! the level pro!it line changes

B! the slope changes enough3 a di!!erentcorner point $ill "ecome optimal


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0 100 200 300 400 500 T

C

500

400

300

200

100

0

Optimal Corner(T=320, C=360)

Still optimal

Feasible

Region

+riginal

+",ective -unction7( 6 C


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C

1000

600

450

0

-easi"le

egion

0 100 500 00 T

hat i! the +-C

"ecame higher#+r lo$er#

**( 6 C


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(here is a range !or each +-C $here the

current optimal corner point remainsoptimal'

B! the +-C changes "eyond that range ane$ corner point "ecomes optimal'

E5cels Solver $ill calculate the +-Crange'


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ight 1and Side .1S/ Changes

hat i! painting hours availa"le changed!rom *000 to *800#

*8002( 6 *C 9 *000 .painting hrs/

(his increase in resources could allo$ us toincrease production and pro!it'

@


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Characteristics o! 1S Changes

(he constraint line shi!ts3 $hich couldchange the !easi"le region

Slope o! constraint line does not change

Corner point locations can change

(he optimal solution can change


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0 100 200 300 400 500 600 T

C

500

400

300

200

100

0

+riginal

-easi"le

egion

2T&1

C=1000

2T&

1C=1300

Feasible regionbe%omes larger

'e$ optimal%orner point

(T=560,C=10)roit=*420

Ol+ optimal%orner point

(T=320,C=360)

roit=*4040


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E!!ect on +",ective -unction Falue

Ge$ pro!it


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Shado$ Price

(he change is the o",ective !unctionvalue per oneHunit increase in the1S o! the constraint'

ill painting hours "e $orth


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ange o! Shado$ Price Falidity

Ieyond some 1S range the value o! eachpainting hour $ill change'

hile the 1S stays $ithin this range3 theshado$ price does not change'

E5cel $ill calculate this range as $ell as theshado$ price'


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Solvers Sensitivity eport

hen E5cel Solver is used to !ind anoptimal solution3 the option o! generatingthe %Sensitivity eport& is availa"le'

o to !ile 4H*'5ls


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Constraint 1S Changes

B! the change in the 1S value is $ithin theallo$a"le range3 then:

(he shado$ price does not change

(he change in o",ective !unction value .shado$ price/ 5 .1S change/

B! the 1S change goes "eyond theallo$a"le range3 then the shado$ price$ill change'


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+",ective -unctionCoe!!icient .+-C/ Changes

B! the change in +-C is $ithin the allo$a"lerange3 then:

(he optimal solution does not change (he ne$ o",ective !unction value can "e

calculated


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Anderson Electronics E5ample

e%ision- 1o$ many o! each o! 4 productsto ma>e#

Ob.e%ti#e- a5imi)e pro!it

e%ision /ariables:F num"er o! FCs

S num"er o! stereos( num"er o! (Fs? num"er o! ?F? players


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a5 2JF 6 82S 6 72( 6 4? .in < o! pro!it/

Su",ect to the constraints:

8F 6 4S 6 4( 6 8? 9 4700 .elec' components/

2F 6 2S 6 4( 6 8? 9 400 .nonelec' components/ F 6 S 6 8( 6 2? 9 200 .assem"ly hours/

F3 S3 (3 ? ; 0 .nonnegativity/

o to !ile 4H2'5ls


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1S Change Kuestions

hat i! the supply o! nonelectricalcomponents changes#

hat happens i! the supply o! electricalcomponents

increased "y 400 .to *00/#

increased "y 4000 .to =700/#


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hat i! $e could "uy an additional 400

elec' components !or


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?ecision Faria"les (hat EMual 0

e are not currently ma>ing any FCs.F0/ "ecause they are not pro!ita"leenough'

1o$ much $ould pro!it need to increase"e!ore $e $ould $ant to "egin ma>ingFCs#


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educed Costo! a ?ecision Faria"le

.marginal contri"ution to the o",' !unc' value/

H .marginal value o! resources used/

Re+%e+ Cost

marginal pro!it o! a FC


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educed Cost is:

(he minimum amount "y $hich the +-Co! a varia"le should change to cause thatvaria"le to "ecome nonH)ero'

(he amount "y $hich the o",ective!unction value $ould change i! the varia"le$ere !orced to change !rom 0 to *'


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+-C Change Kuestions

-or $hat range o! pro!it contri"utions !or?F? players $ill the current solutionremain optimal#

hat happens to pro!it i! this value dropsto


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Alternate +ptimal Solutions

ay "e present $hen there are 0s in

the Allo$a"le Bncrease or Allo$a"le?ecrease values !or +-C values'


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Simultaneous Changes

All changes discussed up to this point haveinvolved only * change at a time'

hat i! several +-Cs change#+r

hat i! several 1Ss change#

Gote: they cannot "e mi5ed


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(he *00N ule

.change O allo$a"le change/ 9 *

1S E5ample

Electrical components decrease 00 00 O J0 0'2D8

Assem"ly hours increase 200

200 O 4DD'D7 0'42=

0'J4=

(he sensitivity report can still "e used


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Pricing Ge$ Faria"les

Suppose they are considering selling a ne$product3 1ome (heater Systems .1(S/

Geed to determine $hether ma>ing 1(Ss$ould "e su!!iciently pro!ita"le

Producing 1(Ss $ould ta>e limitedresources a$ay !rom other products


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(o produce one 1(S reMuires:

electrical components4 nonelectrical components

4 hours o! assem"ly time

Can shado$ prices "e used to calculatereduction in pro!it !rom other products#.chec> *00N rule/

OJ0 6 4OD0 6 4O*82 0'0* 9 *


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eMuired Pro!it Contri"ution per 1(S

elec cpnts 5 < 2


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Geed .1(S pro!it contri"ution/ ;


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Bs 1(S Su!!iciently Pro!ita"le#

ar>eting estimates that selling priceshould not e5ceed


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Sensitivity Analysis !ora inimi)ation Pro"lem

IurnH+!! ma>es a %miracle& diet drin>

e%ision- 1o$ much o! each o! 4ingredients to use#

Ob.e%ti#e- inimi)e cost o! ingredients


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?ata

nits o! Chemical per +unce o! Bngredient

Chemical

Bngredient

eMuirementA I C ?

@ 8 4 = *0 ; 2=0 units

Q 8 D D ; 200 units

R *0 2 20 40 9 *00 units

< per ounce o! ingredient


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in 0'40A 6 0'20I 6 0'D0C 6 0'80? .< o! cost/

Su",ect to the constraints

A 6 I 6 C 6 ? ; 8D .min daily ounces/

8A 6 4I 6 =C 6 *0? ; 2=0 .chem 5 min/

A 6 8I 6 DC 6 D? ; 200 .chem y min/

*0A 6 2I 6 20C 6 40? 9 2=0 .chem ) ma5/

A3 I3 C3 ; 0o to !ile 4H'5ls

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