linear prog
TRANSCRIPT
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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