computational method linear programming

7/25/2019 Computational Method linear programming

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CHAPTER 4

Optimization Linear Programming


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LEARNING OUTCOME

This chapter deals with optimizatio pro!lems wherecostraits come ito pla"# The pro!lems where !oth

o!$ecti%e &'ctio ad the costraits are liear will!e disc'ssed# (or s'ch cases) a method called theliear pro*rammi* which co'ld !e 'sed to sol%e

%er" lar*e pro!lems with tho'sads o& %aria!les adcostraits with *reat e+iciec"# The" are 'sed i awide ra*e o& pro!lems i e*ieeri* ad

maa*emet#


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INTRO,UCTIONRoot locatio ad optimizatio are related i the

sese that !oth i%ol%e *'essi* ad searchi* &or a

poit o a &'ctio# The &'dametal di+erece!etwee the two t"pes o& pro!lems are that the root

locatio i%ol%es searchi* &or zeros o& a &'ctio) i

cotrast optimizatio i%ol%es searchi* &or either

the miim'm or the ma-im'm#

The optim'm is the poit where the c'r%e is .at# I

mathematical terms) this correspods to the x%al'e

where the deri%ati%e f (x) is e/'al to zero#

Additioall") i& f(x)


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root

root

root

Ma-im'm

Miim'

m

INTRO,UCTION

The di+erece !etwee roots ad optima is

ill'strated i the dia*ram !elow1

0)0("

0)('

toe :: m>toe @@ m>wee

Prod'ctiotime

:0 hr>toe = hr>toe =0 hr>wee

5tora*e B toes toes

Pro2t D:0>toe D:@>toe


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5ETTING UP TH E L INEARPROGRAMMING PRO7LEM

SOLUTION

The e*ieer operati* this plat m'st decide how

m'ch o& each *as to prod'ce to ma-imize pro2ts# I&the amo'ts o& re*'lar ad premi'm prod'cedweel" are desi*ated as x1adx28!oth are i the'it o& toe9) respecti%el") the total weel" pro2tca !e calc'lated as

Total pro2t 3 :0x1F :@x2

ritte as a liear pro*rammi* o!$ecti%e &'ctio

Ma-imize 3 :0x1F :@x2


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5ETTING UP TH E L INEARPROGRAMMING PRO7LEM

(or the costraits) the total raw *as 'sed ca !ecomp'ted as

Total *as 'sed 3 @x1F ::x2

This total caot e-ceed the a%aila!le s'ppl" o& @@m>wee) so the costrait ca !e represeted as

@x1F ::x2 @@#

The total time tae &or the prod'ctio o& the

amo'ts o& re*'lar ad premi'm m'st ot e-ceed=0 hr>wee) th's) the costraits is represeted as

:0x1F =x2 =0#


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5E TTING UP TH E L INE A RPROGRAMMING PRO7LEM

(or the stora*e costrait) re*'lar ad premi'mamo'ts caot e-ceed B ad toes) respecti%el")

asx1 B ad x2 )

ad the prod'cts m'st ha%e positi%e amo't) as

x1J 0 ad x2J 0#


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These are s'mmarized !elow1

Ma-imize 3 :0x1F :@x2 8ma-imize pro2t9

s'!$ect to@x1F ::x2 @@ 8material costrait9

:0x1F =x2 =0 8time costrait9

x1 B 8Kre*'lar stora*e costrait9

x2 8Kpremi'm stora*e costrait9

x1J 0 ad x2J 0 8positi%it" costrait9


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5E TTING UP TH E L INE A RPROGRAMMING PRO7LEMI this two6dimesioal 8two 'ows) x1adx29 pro!lem)

the sol'tio space is de2ed as a plae with x1adx2#

7eca'se the" are liear) the costraits ca !e plotted othis plae as strai*ht lies# These costrait lies will

delieate a re*io) called thefeasible solution space)ecompassi* all possi!le com!iatios o& x1adx2that

o!e" the costraits ad hece represet &easi!lesol'tios#

x1

x1 Bx2

x2

:0x1F =x2 =0

@x1F ::x2 @@

0

(easi!lesol'tiospace


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The o!$ecti%e &'ctio &or a partic'lar %al'e o& cathe !e plotted as aother strai*ht lie ad

s'perimposed o this space# The %al'e ca the !ead$'sted 'til it is at the ma-im'm %al'e while stillto'chi* the &easi!le space# This %al'e o& represetsthe optimal sol'tio# The correspodi* %al'es o& x1

adx2where to'ches the &easi!le sol'tio space

represet the optimal %al'es &or the acti%ities#


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To do this) A %al'e o& m'st !e chose) &or e-ample) &or30) the o!$ecti%e &'ctio !ecomes 03 :0x1F :@x2

ad sol%i* &or x2we deri%e the lie

x1

x1 Bx2

x2

:0x1F =x2 =0

@x1F ::x2 @@

0

30

300

3:400

Now) sice we are iterested i ma-imizi* ) we caicrease it to sa" 300) ad the o!$ecti%e &'ctio is003 :0x1F :@x2) sol%i* &or x2we deri%e the lie

7eca'se the lie still &alls withi the sol'tio space) o'rres'lt is still &easi!le# There is still room &orimpro%emet# Hece) ca eep icreasi* 'til a&'rther icrease will tae the o!$ecti%e !e"od the

&easi!le re*io#

As show i the 2*'re) the ma-im'm %al'e o& correspods to appro-imatel" :400# At this poit) x1adx2

are e/'al to appro-imatel" 4#B ad #B) respecti%el"# Th'sthe *raphical sol'tio tells 's that i& we prod'ce these

/'atities o& re*'lar ad premi'm) we will reap ama-im'm pro2t o& a!o't D:400#

12

175

150xx = 12

175

150

175

600xx =

9.41 =x

9.32 =x


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e ca %eri&" o'r res'lts !" s'!stit'ti* it !ac tothe costrait e/'atios1

@x1F ::x23 @84#B9F::8#B9 @@:0x1F =x23 :084#B9F=8#B9 =0

x13 4#B B

x23 #B


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CONCLU,ING REMAR5It is clear &rom the plot that prod'ci* at the optimal amo'to& each prod'ct !ri*s 's ri*ht to the poit where we $'stmeet the reso'rce ad time costraits# 5'ch costrait aresaid to !e binding# As ca !e see i the *raphic) either o&

the stora*e costraits 8x1 B ad x2 9 acts as alimitatio# 5'ch costraits are called nonbinding#

x1

x1 Bx2

x2

:0x1F =x2 =0

@x1F ::x2 @@

0



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CONCLU,ING REMAR5

This leads 's to the cocl'sio that we ca icreasepro2ts !" either icreasi* o'r reso'rce s'ppl" 8raw

*as9 or icreasi* o'r prod'ctio time# ('rther) itidicates that icreasi* the stora*e wo'ld ha%e oimpact o pro2t#


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CONCLU,ING REMAR5The res'lt o!taied i this e-ample is oe o& &o'r possi!leo'tcomes1

:9Ui/'e sol'tio < the ma-im'm o!$ecti%e &'ctio itersectsa si*le poit#

;9Alterate sol'tios < i& the o!$ecti%e &'ctio has coe+icietsso that it is precisel" parallel to oe o& the costraits) rathertha a si*le poit) the pro!lem wo'ld ha%e a i2ite'm!er o& optima correspodi* to a lie se*met#

x1

x1 Bx2

x2

:0x1F =x2 =0

@x1F ::x2 @@

0



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CONCLU,ING REMAR59No &easi!le sol'tio < it is possi!le that the pro!lem isset 'p so that there is o &easi!le sol'tio# This ca !ed'e to deali* with a 'sol%a!le pro!lem or d'e to

errors i setti* 'p the pro!lem# The latter ca res'lti& the pro!lem is o%er6costraied to the poit that osol'tio ca satis&" all the costraits#


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CONCLU,ING REMAR549U!o'ded pro!lems < the pro!lem is 'der6costraied ad there&ore ope6eded# As with theo &easi!le case) it ca o&te arise &rom errorscommitted d'ri* pro!lem speci2catio#

x1

x2

0



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CONCLU,ING REMAR5

5'ppose that o'r pro!lem i%ol%es a 'i/'e sol'tio#(rom the pre%io's e-ample it is clear that the

optim'm alwa"s occ'rs at oe o& the corer poitswhere two costraits meet# 5'ch a poit is owas a extreme point# Th's) o't o& the i2ite 'm!ero& possi!ilities i the decisio space) &oc'si* oe-treme poits will arrow dow the possi!leoptios#


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CONCLU,ING REMAR5('rther) we ca reco*ize that ot e%er" e-tremepoit is &easi!le) that is satis&"i* all costraits#Limiti* o'rsel%es tofeasible extreme pointswillarrow the 2eld dow still &'rther#

x1

x1 Bx2

x2

:0x1F =x2 =0

@x1F ::x2 @@

0


Not &easi!le e-tremepoit


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CONCLU,ING REMAR5

Oce all &easi!le e-treme poits are ideti2ed) theoe "ieldi* the !est %al'e o& the o!$ecti%e &'ctio

represets the optim'm sol'tio# (idi* thisoptimal sol'tio co'ld !e doe !" e-ha'sti%el" 8adie+icietl"9 e%al'ati* the %al'e o& the o!$ecti%e&'ctio at e%er" &easi!le e-treme poit# I the e-tlect're we will disc'ss the simple- method whicho+ers a pre&era!le strate*" that charts a selecti%e

co'rse thro'*h a se/'ece o& &easi!le e-tremepoits to arri%e at the optim'm i a e-tremel"e+iciet maer#

computational method linear programming

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