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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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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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BU.520.601LP: Sensitivity Analysis 4
#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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BU.520.601LP: Sensitivity Analysis 5
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.
8/18/2019 Price Sensitivity Analysis_Customized Version
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
8/18/2019 Price Sensitivity Analysis_Customized Version
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
?
8/18/2019 Price Sensitivity Analysis_Customized Version
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
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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$e o< t#e o=>ective
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SolverSensitivitye$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.
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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
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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).
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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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BU.520.601Linear Optimization 15
#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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BU.520.601Linear Optimization 16
%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$el#&e* #n (l#&e* #n an e e*#).
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BU.520.601Linear Optimization 1
%rail Mi& ,-ont.
Pxlain all#&ale in*ease #* e*ease an s$a#& *ie
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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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