Post on 01-Jun-2018
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Linear Programming
REFERENCES:
FREDERICK HILLIER & GERALD LIEBERMAN.Introduction to Operations Research.Ninth Edition
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WARNING!ides in"or#ation $as ta%en "ro# re"erencedoo%. As the' #a' contain t'pin( #ista%e) it
is reco##ended to consu!t "ro# oo%s!ocated at the !irar'. *he s!ides are a +uic%and (enera! (uide "or the topics co,ered inc!ass.
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Linear -ro(ra##in( L-/
0hat%ind o"pro!e
#sdoes itaddress1
*he (enera! pro!e# o" a!!ocatin( !i#itedresources a#on( co#petin( acti,ities inthe est possi!e $a'.
*he one co##on in(redient in each o"these situations is the necessit' "ora!!ocatin( resources to acti,ities 'choosin( the !e,e!s o" those acti,ities.
An' pro!e# $hose #athe#atica! #ode!2ts the ,er' (enera! "or#at "or the !inearpro(ra##in( pro!e#.
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De2nition
• L- uses a#athe#atica!#ode! to descrie
the pro!e# o"concern.
• *he ad3ecti,elinear #eans that
a!! the#athe#atica!"unctions in this#ode! are re+uiredto e linear
Fuente4http455$$$6.ha$aii.edu57suthers5courses5ics899"995Notes5*opic:665!inear:pro(ra##in(:e;a#p!e:6a:no!ines.3p(
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De2nition
• *he $ord programming doesnot re"er here to
co#puterpro(ra##in(<rather) it=sessentia!!' a
s'non'# "or planning.
• L- in,o!,es the planning o"acti,ities to otain
Fuente4http455t8.(static.co#5i#a(es1+>tn4ANd?Gc@(?DrB0?t;@9es0r,AHR(epd,9e"D!+enIR'
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o#e situations in $hich app!iesL-
• ta=s pro(ra##in(
• A(ricu!tura! p!annin(
• -orta"o!io se!ection
• e!ection o" shippin(patterns
• A!!ocations o"productions "aci!ities
• *he desi(n o"radiation therap'
• -roduct #i; t'pe
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L- tep ' tep
DecisionJaria!es
0hat shou!d'ou decide1
-ara#eters
0hatin"or#ation isa,ai!a!e to#a%e thedecision1
Constraints
0hich are theconditions
that !i#it thedecision1
O3ecti
,e
Ho$ to+uanti"' thei#pact o" the
decision1
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*he L- Mode!
Jaria!es-ara#eter
s
-er"or#ance
#easure
-ro!e#sie
,aria!es
"unctiona!constraints
Jaria!es-ara#eter
s
-er"or#ance
#easure
-ro!e#sie
,aria!es
"unctiona!constraints
*he input constants "or the
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*he L- Mode!
• > Ja!ue o" o,era!! #easure o" per"or#ance
• >Le,e! o" acti,it' "or >9)6)) /. Decisionvariables.
• > Increase in that $ou!d resu!t "ro# each unitincrease in !e,e! o" acti,it' . Contribution toobective !unction.
• > A#ount o" resource that is a,ai!a!e "or
a!!ocation to acti,ities "or >9)6)) /.Resources.
• > A#ount o" resource consu#ed ' each unit o"acti,it' . Resource consum"tion.
•
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nderstandin( su# notation
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Subectto:
.
.
.
A tandard For# o" the Mode!
Subect to:
# b e c
t i v e
F u n c t
i
F u n c t i o n a l
c o n s t r a i n t s
$
a r i a b l e
t % " e
c o
n s t r a i n t
s
nn xc xc xc Z +++= ...max 2211
11212111
... b xa xa xann
≤+++
22222121 ... b xa xa xa
nn ≤+++
mnmnmm b xa xa xa ≤+++ ...
2211
0,...,,21
≥n
x x x
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Subectto:
.
.
.
A tandard For# o" the Mode!
Subect to:
# b e c
t i v e
F u n c t
i
F u n c t i o n a l
c o n s t r a i n t s
$
a r i a b l e
t % " e
c o
n s t r a i n t
s
nn xc xc xc Z +++= ...max 2211
11212111
... b xa xa xann
≤+++
22222121 ... b xa xa xa
nn ≤+++
mnmnmm b xa xa xa ≤+++ ...
2211
0,...,,21
≥n
x x x
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A tandard For# o" theMode!
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Other For#s
Mini#iin( rather than #a;i#iin( the o3ecti,e"unction4
o#e "unctiona! constraints $ith a (reater:than:or:e+ua!:to
ine+ua!it'4• "or so#e ,a!ues o"
o#e "unctiona! constraints in e+uation "or#4
• "or so#e ,a!ues o"
De!etin( the nonne(ati,it' constraints "or so#e decisions,aria!es4
• unrestricted in si(n/ "or so#e ,a!ues o"
Mini#iin( rather than #a;i#iin( the o3ecti,e"unction4
o#e "unctiona! constraints $ith a (reater:than:or:e+ua!:to
ine+ua!it'4• "or so#e ,a!ues o"
o#e "unctiona! constraints in e+uation "or#4
• "or so#e ,a!ues o"
De!etin( the nonne(ati,it' constraints "or so#e decisions,aria!es4
• unrestricted in si(n/ "or so#e ,a!ues o"
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Assu#ptions o" L-
-roportiona!it'
*he contriution o" each acti,it' tothe ,a!ue o" the o3ecti,e "unction isproportiona! to the !e,e! o" theacti,it'
*he contriution o" each acti,it' tothe !e"t:hand side o" each "unctiona!constraint / is proportiona! to the!e,e! o" the acti,it'
Conse+uent!') this assu#ption ru!esout an' e;ponent other than 9 "oran' ,aria!e in an' ter# o" an'"unction.
-roportiona!it'
*he contriution o" each acti,it' tothe ,a!ue o" the o3ecti,e "unction isproportiona! to the !e,e! o" theacti,it'
*he contriution o" each acti,it' tothe !e"t:hand side o" each "unctiona!constraint / is proportiona! to the!e,e! o" the acti,it'
Conse+uent!') this assu#ption ru!esout an' e;ponent other than 9 "oran' ,aria!e in an' ter# o" an'"unction.
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Assu#ptions o" L-
• -roportiona!it'
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Assu#ptions o" L-
0 1 2 3 40
2
4
6
8
10
12
14
16
18
20
Proportionality
X1
Contribution from X1 to Z
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Assu#ptions o" L-
Additi,it'
E,er' "unction in a L- #ode! is the su#o" the indi,idua! contriutions o" therespecti,e acti,ities.
*he 3oint pro2t or costs/ is dierentthan the su# o" the indi,idua! pro2tsor costs/ $hen each one is produced' itse!".
Conse+uent!') this assu#ption ru!es outan' cross:product ter#s) na#e!') ter#sin,o!,in( the product o" t$o or #ore,aria!es /.
Additi,it'
E,er' "unction in a L- #ode! is the su#o" the indi,idua! contriutions o" therespecti,e acti,ities.
*he 3oint pro2t or costs/ is dierentthan the su# o" the indi,idua! pro2tsor costs/ $hen each one is produced' itse!".
Conse+uent!') this assu#ption ru!es outan' cross:product ter#s) na#e!') ter#sin,o!,in( the product o" t$o or #ore,aria!es /.
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Case8
Case
Assu#ptions o" L-
• Additi,it'
2121 5.023 x x x x ++
2
2
121 1.023 x x x x −+
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Assu#ptions o" L-
(2,0) (0,3) (2,3)
0
2
4
6
8
10
12
14
16
Additivity
(X1,X2)
Value of Z
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Assu#ptions o" L-
Di,isii!it'
It concerns the ,a!ues a!!o$ed "or thedecision ,aria!es.
Decision ,aria!es are a!!o$ed to ha,ean' ,a!ues) inc!udin( noninte(er ,a!ues)that satis"' the "unctiona! andnonne(ati,it' constraints.
*hese ,aria!es are not restricted to 3ust inte(er ,a!ues. It=s ein( assu#edthat the acti,ities can e run atfractional levels.
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Assu#ptions o" L-
Certain
t'
It concerns the para#eters o" the#ode!) na#e!') the coePcients in theo3ecti,e "unction) the coePcients inthe "unctiona! constraints and the ri(ht:hand sides o" the "unctiona! constraints
/. *he ,a!ue assi(ned to each para#etero" a L- #ode! is assu#ed to e a&no'n constant.
In rea! app!ications) the certaint'assu#ption is se!do# satis2edprecise!'. For this reason it=s usua!!'i#portant to conduct sensitivit%anal%sis a"ter a so!ution is "ound that isopti#a! under the assu#ed para#eter
,a!ues.
Certain
t'
It concerns the para#eters o" the#ode!) na#e!') the coePcients in theo3ecti,e "unction) the coePcients inthe "unctiona! constraints and the ri(ht:hand sides o" the "unctiona! constraints
/. *he ,a!ue assi(ned to each para#etero" a L- #ode! is assu#ed to e a&no'n constant.
In rea! app!ications) the certaint'assu#ption is se!do# satis2edprecise!'. For this reason it=s usua!!'i#portant to conduct sensitivit%anal%sis a"ter a so!ution is "ound that isopti#a! under the assu#ed para#eter
,a!ues.
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-rotot'pe E;a#p!e
*he 0'ndor G!ass CO9
• Deter#ine $hat the production rates shou!d e "or thet$o products in order to #a;i#ie their tota! pro2t)su3ect to the restrictions i#posed ' the !i#itedproduction capacities a,ai!a!e in the three p!ants.
• Each product $i!! e produced in atches o" 6Q) so theproduction rate is de2ned as the nu#er o" atchesproduced per $ee%.
• An' co#ination o" production rates that satis2esthese restrictions is per#itted) inc!udin( producin(none o" one product and as #uch as possi!e o" theother.
ee printed #ateria!
LIER & LIEBERMAN. Introduction to Operations Research.
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-rotot'pe E;a#p!e
*he 0'ndor G!ass CO9
LIER & LIEBERMAN. Introduction to Operations Research.
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-rotot'pe E;a#p!e
*he 0'ndor G!ass CO9
Subect to:
#bectiveFunction
Functionalconstraints
$ariablet%"e
constraints
Decision$ariables
LIER & LIEBERMAN. Introduction to Operations Research.
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-rotot'pe E;a#p!e *he 0'ndor G!ass CO9
Graphica!
o!ution
IER & LIEBERMAN. Introduction to Operations Research.
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*er#ino!o(' "or o!utions o" theMode!
Feasi!e Re(ion
• *he resu!tin( re(ion o" per#issi!e ,a!ues o" decision ,aria!es that satis"' all o! t(econstraints oth "unctiona! and ,aria!es t'pe/. It=s the co!!ection o" a!! "easi!eso!utions.
In"easi!eso!ution
• It=s a so!ution "or $hich at least one constraint is violate).
Opti#a! so!ution
• It=s a "easi!e so!ution that has t(e most !avorable value o" the o3ecti,e "unction
C-F/Corner:-ointFeasi!e• It=s a so!ution that !ies at a corner o" the "easi!e re(ion. C-F so!utions a!so are
co##on!' re"erred to as extreme points or vertices.
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Re!ationship et$eenopti#a! so!utions and C-F so!utions
*he est
C-Fso!ution
must ean opti#a!
so!ution
One
opti#a!so!ution
It must e a C-F
so!ution
! i hi
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Re!ationship et$eenopti#a! so!utions and C-F so!utions
9. Identi"' a!!C-F and itsrespecti,e ,a!ue
6. Find the estC-F
8. Ca!cu!ateso#e points
inside "easi!ere(ion thatsurround theest C-F
IER & LIEBERMAN. Introduction to Operations Research. .
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Find Opti#a! o!ution '
Graphica! Method
Gradient Jector4
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Other cases
Mu!tip!e opti#a!so!utions
No opti#a! so!utions
nounded o3ecti,e
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Mu!tip!e opti#a! so!utions
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Mu!tip!e opti#a! so!utionsC#N$E*
C#+,INA-I#N
E,er' point on the !ine se(#ent et$een 6)/ and )8/ is opti#a!.A!! opti#a! so!utions are a weighted average o" these t$o opti#a!
C-F so!utions.
For e;a#p!e4
182321
=+= x x Z
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No opti#a! so!utions
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nounded o3ecti,e