price sensitivity analysis_customized version

8/18/2019 Price Sensitivity Analysis_Customized Version

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BU.520.601

BU.520.601Decision Models

LP: Sensitivity Analysis 1

Sensitivity Analysis

Basic theoryUnderstanding optimum solution

Sensitivity analysis

Summer 2013


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3/20

BU.520.601LP: Sensitivity Analysis 3

"rimal dual relations#i$ 10x1 + 8x2 Max0.7x1 + x2 ≤ 630

(½) x1 + (5/6) x2 ≤ 600

x1 + (2/3) x2 ≤ 708(1/10) x1 + (1/4) x2 ≤ 135

-x1 - x2 ≤ -150

x1 0! x2 0

630y1 + 600y2 + 708y3 + 135y4 - 150y5 Min

0.7y1 + (½)y2 y3 (1/10)y4 -y5 10

y1 + (5/6)y2 + (2/3)y3 + (1/4)4 - y2 8y1 0! y2 0! y3 0! y4 0! y5 0

"#te t$e%#ll#&in'

#nsie* t$e , *#le s$#&n. e &ill all

t$is as a *ial *#le. #* eve*y *ial

*#le! t$e*e is al&ays a #**es#nin' ,*#le alle t$e al *#le.

• Any #ne #% t$ese an e alle *ialt$e #t$e* #ne is al.

• % #ne is #% t$e sie x n! t$e #t$e* is #%t$e sie n x .

• % &e s#lve #ne! &e iliitly s#lve t$e#t$e*.

• 9tial s#lti#ns %#* #t$ $ave ientialvale %#* t$e #:etive %nti#n (i% an

#tial s#lti#n exists).

#tial

Max

Min


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#nsie* a sile t *#t exale&it$ t$*ee *es#*e #nst*aints. ;$e

%easile *e'i#n is s$#&n.

Maxiie 15x1 + 10x2 < =

2x1 + x2 ≤ 800

x1 + 3x2 ≤ >00

+ x2 ≤ 250

x1 0! x2 0

e n#& a sla? va*ialest# ea$ #nst*aint t# #nve*t

t$ese in e@ati#ns.

Max = - 15x1 + 10x2 < 0

2x1 + x2 + S1 < 800

x1 + 3x2 + S2 < >00

+ x2 + S3 < 250

%#e Sim$le& Met#od

"rimal ' dualMa&imi(e 15 &1 ) 10 &2

Minimi(e *00 y1 ) +00 y2 ) 250 y3


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Sta*t &it$ t$e talea %#* $a!imi%e &' !& (

&) !*= x1 x2 S1 S2 S31 -15 -10 0 0 0 0

0 2 1 1 0 0 8000 1 3 0 1 0 >000 0 1 0 0 1 250

A%te* any ite*ati#ns (#vin' %*# #ne#*ne* t# t$e next) &e 'et t$e %inalans&e*.

nitial s#lti#n= < 0! x1 < 0! x2 < 0!

S1 < 800! S2 < >00

an S3 < 250.

= x1 x2 S1 S2 S3

1 0 0 7 1 0 6500

0 1 0 3/5 -1/5 0 300

0 0 1 -1/5 -2/5 0 200

0 0 0 0 0 1 50

%#e Sim$le& Met#od, -ont

"#tie 7! 1! 0 in t$e #:etive *#&.

;$ese a*e t$e vales #% al va*iales! alle shado+prices.

$inimi%e )) y& ( -)) y* ( *') y gives 800B7 + >00B1 +

250B0 < 6500

/$timal solution,= < 6500! x1 < 300! x2 < 200 an S3 < 50.

= < 15 B 300 + 10 B 200 < 6500


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BU.520.601Linear Optimization 6

Ma&imi(e 10 &1 ) * &2 < =

7/10 x1 + x2 630

1/2 x1 + 5/6 x2 600

x1 + 2/3 x2 708

1/10 x1 + 1/4 x2 135

x1 0! x2 0 x1 + x2 150

9tial s#lti#n &1 50 &2 252. 416

Cinin' #nst*aints #nst*aints inte*setin' at

t$e #tial s#lti#n.

!

"#ninin' #nst*aintD! an

SolverAnsere$ort7

/onsiderthe 0olBag

problem.

8o consider t#e

Solver solution.


7/20BU.520.601LP: Sensitivity Analysis

Set up the problem,clic1 2Solve3 and thebo! appears.I you select only

2453, you can readvalues o decisionvariables and theob6ective unction.

7e!t slides sho+s the report 8re9

ormatted:.

Instead o selectingonly 2453, select2;ns+er3 under


8/20BU.520.601LP: Sensitivity Analysis !

#he ans+er report has three tables=&= 4b6ective /ell > or the ob6ectiveunction

*= ?ariable /ells = orconstraints.

Let@s try tointerpretsomeeatures..

Ansere$ort

Aou may+ant torenamethis;ns+er

9 - t i / a l - * # % i t

9- ti/al va*ia.le

val5es

?


9/20BU.520.601LP: Sensitivity Analysis "

Sensitivity Analysis "#& &e &ill #nsie* $an'es int$e #:etive %nti#n #* t$eEFS #e%%iients G #ne

#e%%iient at a tie.

9:etive %nti#n

Ei'$t Fan Sie (EFS).

9ere are some :uestions e ill try to anser.

Ma&imi(e 10 &1 ) * &2 < = 7/10 x1 + x2 630

1/2 x1 + 5/6 x2 600

x1 + 2/3 x2 708

1/10 x1 + 1/4 x2 135

x1 + x2 150

x1 0! x2 0

9tial s#lti#n&1 50 &2 252.

416

&= o+ much the unit proCt o ;ce can go upor do+n rom D +ithout changing thecurrent optimal production EuantitiesF

*=What i per unit proCt or "elu!e model is&*.*'F

= What i an &) more hours o production time

is available in cutting G dyeingF inspectionF


10/20BU.520.601LP: Sensitivity Analysis 10

Sensitivity Analysis

&= o+ much the unitproCt o ;ce can goup or do+n rom D+ithout changing thecurrent optimalproductionEuantitiesF

As lon; as t#e slo$e o< t#e o=>ective



SolverSensitivitye$ort7

Ka*iale ells tale $els s ans&e* @esti#ns *elate t# $an'es int$e #:etive %nti#n #e%%iients.

#nst*aints tale $els s ans&e* @esti#ns *elate t# $an'es int$e EFS #e%%iients.

% y# li? #n Sensitivity! a ne&*?s$eet! alle SensitivityEe#*t is ae. t #ntains ttales Ka*iale ells an#nst*aints.

e &ill isss t$ese tales sea*ately.



Solver Sensitivity e$ort7

Ma&imi(e 10 &1 ) * &2 < =

416&1 50 &2 252

&= o+ much the unit proCt o ;ce can go up or do+nrom D +ithout changing the current optimalproduction EuantitiesF

. to &) ( * &. to ( J.*J

#ry per unit proCt or H* as &.*, &.*-, J.JK and J.JJ

*=What i per unit proCt or "elu!e modelis &*.*'FSli;#t round o


13/20BU.520.601

LP: Sensitivity Analysis 13

L3 A 10 #*e $#*s #% *#ti#n tie %#*

cutting G dyeingF inseti#nD

/utting G dyeing is a inin' #nst*aintin*easin' t$e *es#*e &ill in*ease t$e s#lti#nsae an #ve t$e #tial #int.

nseti#n is a n#ninin'#nst*aint in*easin' t$e *es#*e&ill in*ease t$e s#lti#n sae ant &ill n#t #ve t$e #tial #int.

W#at i< :uestions are a=out t#e 9S?

A $an'e in EFS an $an'e t$e s$ae #% t$e s#lti#n sae(#:etive %nti#n sl#e is n#t a%%ete).



L3 A 10 #*e $#*s #% *#ti#n tie %#*

cutting G dyeingF inseti#nD

Sensitivity e$ort @3

S$a#& *ie *e*esents $an'e in t$e #:etive %nti#n valee* #ne-nit in*ease in t$e EFS #% t$e #nst*aint. n a sinessaliati#n! a s$a#& *ie is t$e axi *ie t$at &e an ay%#* an ext*a nit #% a 'iven liite *es#*e.

#* ttin' yein' t# 52.36 nits an e in*ease. ,*#%it &illin*ease N O2.50 e* nit.

or ins$ection ?


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#st / nitO

SO4

EO5

O3

,O7

O6 Min.

neee

H*as / l.Kitains 10 20 10 30 20 25.00Mine*als 5 7 4 > 2 8.00

,*#tein 1 4 10 2 1 12.50

al#*ies/l 500 450 160 300 500 500

%rail Mi& ,sensitivityanalysis

Sees! Eaisins! la?es!,eans! alnts Min. 3/16#ns ea$;#tal @antity < 2 ls.

Ans&e* Ee#*t


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%rail Mi& ,-ont

nte**etati#n #% all#&ale in*ease #* e*easeD

$at is *ee #stD Als# alle t$e o$$ortunity cost.

9ne inte**etati#n #% t$e *ee #st (%#* t$e minimi(ation *#le) is t$e a#nt y &$i$ t$e #:etive %nti#n #e%%iient %#*a va*iale nees t# e*ease e%#*e t$at va*iale &ill exee t$el#&e* #n (l#&e* #n an e e*#).


17/20


%rail Mi& ,-ont.

Pxlain all#&ale in*ease #* e*ease an s$a#& *ie


18/20

BU.520.601LP: Sensitivity Analysis 1!

&am$le 5

9tial = < 1670!

I2 < 115! I4 < 100

Eee #st (%#*axiiati#n) t$ea#nt y &$i$ t$e#:etive %nti#n#e%%iient %#* a va*ialenees in*ease e%#*et$at va*iale &ill exee

t$e l#&e* #n.

S$a#& *ie *e*esents $an'e in t$e #:etive %nti#n valee* #ne-nit in*ease in t$e EFS #% t$e #nst*aint. n a sinessaliati#n! a s$a#& *ie is t$e axi *ie t$at &e an ay%#* an ext*a nit #% a 'iven liite *es#*e.

Max 2.0x1 + 8.0x2 + 4.0x3 + 7.5x4 < =

x1 + x2 + x3 + x4 200

2.0x1 + 3.0x3 + x4 ≤ 100

+ 4.0x2 + + 5.0x4 ≤ 1250x1 + 2.0x2 ≤ 230

4.0x3 + 2.5x4 ≤ 300

x1 0! x2 0! x3 0! x4 0


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BU.520.601LP: Sensitivity Analysis 1"

/hange one coecient at a time +ithin

allo+able range

/=>ective unction i;#t 9and Side

• #he easible regiondoes not change.

• Since constraintsare not afected,decision variablevalues remain thesame.

• 4b6ective unction

value +ill change.

• Measible regionchanges.

• I a nonbindingconstraint ischanged, thesolution is notafected.

• I a binding

constraint ischanged, the samecorner pointremains optimal

but the variablevalues +ill chan e.


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BU 520 601LP S iti it A l i 20

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