linearprogramming introduction

8/9/2019 LinearProgramming Introduction

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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




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-rotot'pe E;a#p!e


Subect to:

#bectiveFunction

Functionalconstraints

$ariablet%"e

constraints

Decision$ariables



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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

linearprogramming introduction

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