1) genetic algorithm

7/24/2019 1) Genetic Algorithm

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Genetic algorithmsare searchtechniques based

on the mechanismof naturalselection.


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GA was developed by John Holland,professor of CS and sychology at!niversity of "ichigan in #$%&.

'o understand the adaptive processes of naturalsystems

'o design arti(cial system software that retainsthe robustness of natural systems


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)mportant steps involved in GA#. *ncoding

+. )nitialiation

-. Selection

. /eproduction Crossover

"utation

&. 'ermination criteria


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/epresentation

#.0inary strings

+./eal number coding

-.)nteger coding


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0inary strings

1i2cult to apply because it is not a natural

coding.

'his is a chromosome.

0 1 1 1 0 0 1 1

Gene


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/eal number coding

"ainly useful for constrained optimiation

problem.

3.445

4.659

6.295

1.346

8.271


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)nteger coding

"ainly applicable for combinatorial

optimiation problem.

4 6 5 9 2 3


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GA wor3s on coding space and solution

space alternatively. Genetic operations wor3 on coding space

4chromosomes5.

*valuation and selection wor3 on solutionspace.


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1ecoding

*ncoding

Coding space

Genetic6perations

Solution space

*valuationandSelection


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'hree critical issues emerge concerning withthe encoding and decoding between

chromosomes and solutions.

#.'he feasibility of a chromosome.

+.'he legality of chromosome.-.'he uniqueness of mapping.


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)llegalone

)nfeasible one

7easibleone

7easibility and 8egality

Solution Space

Codingspace


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Codingspace

Solutionspace

1 to nmapping

n to 1mapping

1 to 1mapping

Mapping from cromosomes tosolutions


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Create initial population ofchromosomes.

#./andomly

+.8ocal search

-.7easible solution

0ut in most of the cases initial population israndomly generated.


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*9ample: 8et us assume , initially thepopulation with a chromosome of ; bit and< parents.

1 1 0 1 1 0 0 1

# = = = # # = =

= # # # = # = #

# # = = = # = #

# = = # # # = #

= = # # = # # #

arent no1

+

-

&

) (>)m(>,t1) m(>,t) 1 8 p 8 p ?(>)# 1l


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?here,

a!

m

m(>,t)


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*9ampleD "a9imie Sin4Y5, Nariable bound =ZYZpi

H#I 4 # = U U U 5 , H+I 4 = = U U U 5

@nitia% popu%ation

trin! A

(deodedAa%ue)

B in(B) # i/#

a!.

(atua% ount)

atin! poo%

01001 9 0.912 0.791 1.39 1 01001

10100 (>1) 20 2.027 0.9 1.5 2 10100

00001 (>2) 1 0.101 0.101 0.1 0 10100

11010 26 2.635 0.45 0.5 1 11010

1, p

m0, #

a!0.569


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m4H#,#5[4#5\4=.;$;=.&


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onsider >1(1 0 - - -) 8888 represents the strin! +ith x a%uear*in! #rom 1.621 to 2.330 and #untion a%ue aries 0. 725to 0.999

>2(0 0 - - -) 88888 represents the strin! +here xE0.0 to 0.709

#untion a%ue %ies 88880.0 to 0.651ine our o"Fetie is maximie the #untion, +e +ou%d %iGe to

hae more opies o# strin!s representin! shema >1and >2.

&rom the ine'ua%it*, +e hae to estimate the !ro+th o# >1and

>2.

&or >1(1 0 - - -)


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m(>1, 0) 1

rossoer operation p 1

utation operation pm 0

#(>1) 0.9

?rder o# strin! o(>1) 2

e#inin! %en!th 281 1

The aera!e #itness # a! 0.569)(H

1

0.9 1( ,1) (1) H1 (1.0) 0 (2)I

0.569 5 1

1.14

m H

=

&or >2(0 0 - - -)


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&or >2(0 0 - - -)

m(>2, 0) 1

rossoer operation p 1utation operation pm 0

#(>2) 0.101

?rder o# strin! o(>2) 2

e#inin! %en!th 281 1The aera!e #itness # a! 0.569

)(H

2

0.101 1

( ,1) (1) H1 (1.0) 0 (2)I0.569 5 1

0.133

m H

=


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&rom the a"oe a%u%ation su!!est that the num"er o#strin!s representin! shema >1must inreases.

&or shema >2, there is no representatie strin! exist in

the ne+ popu%ation "eause the !ro+th o# shema>1is more as ompared to shema >2.


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Schema that are #.Short

+.8ow order

-.Above average, are 3nown as Xbuildingbloc3sT.

)n each generation building bloc3scombined to form better and bigger

building bloc3s.


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opulation sie plays signi(cant role inobtaining optimal solution.

)t depends on the comple9ity of theproblem.

Adequate number of strings should bepresent in each region of solution.


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7ig. # represents the bimodal Gaussian function. Bumber of regionsI+ H# , with above average (tness value, becomes building bloc3,

optimal solution achieved

H= H#

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig$re 1

a"erage "a$e


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opulation sie required to solve theabove problem I+J, where Jis

#. #

+.Bumber of copies of Schema requiredin each region to adequately represent

the

(tness variation.


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7ig. + represents the modi(ed bimodal Gaussianfunction.

)f number of regions I+, H=becomes buildingbloc3 , may achieve local optimal solution.

f495

9 H= 7ig. + H#

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

Fig$re 2.


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)f number of regions I. H= or H#becomes building bloc3s, sub optimal

solution may achieved.

f495

9 H= H# H+ H-

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5


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)f number of regions I;.

H


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GA searches solution space byD #.*9ploration: by Selection operation.

+.*9ploitation: by Crossover and

mutation. 0alance between e9ploration and

e9ploitation is required to avoidpremature convergence on local optima.


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1i@erent types of crossover operatorsareD

#. 8inear Crossover

+. Ba]ve crossover

-. 0lend Crossover

. Simulated 0inary Crossover

&. !nimodal Bormally 1istributed Crossover


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8inear CrossoverD^ 'hree Solutions are generated from two

parent solution.^ 0est two are chosen as o@spring.

arents 9i4#,t5

and 9i4+,t5

. tIgeneration number

SolutionsD #. =.&49i4#,t5 9i4+,t55. +. 4#.& 9i4#,t5O =.& 9i4+,t55.

-. 4:=.& 9i4#,t5

#.& 9i4+,t5

5.


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6@spring

9i4#,t5 9i4+,t5


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'his Crossover operator is similar to the

crossover operators used in 0inary codedGATs. Cross:sites are only allowed to be chosen at

the variable boundariesD

arent #D49##,t, 9+#,t, 9-#,t,FFF9n#,t5 arent +D49#+,t, 9++,t, 9-+,t,FFF9n+,t5

6@spring #D49##,t, 9+#,t, 9-+,t,FFF9n#,t5

6@spring +D49#+,t,

9++,t

, 9-#,t

,FFF9n+,t

5

'his Crossover 6perator does not have anadequate search power and thus search hasto mainly rely on mutation operator.


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For two Parent solutions 9i#,t and9i+,t4assuming 9i#,t


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)t has been investigated that 08Y:.&4+',has

given better results than any other .alue,

Property o% BLX- location o% o*springdepends on position o% parent solution,/ewriting 01uation (

49i#,t#: 9i#,t 5 I Ki49i+,t: 9i#,t 5$% di*erence&etween parents is small then thedi*erence &etween o*spring and parent will &esmall,

2hus this allows us to constitute adopti.esearch, $t will allow searching entire spaceearly on and maintain %ocused search whenwe tend to con.erge,


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Spread !actor

)t is the ratio of the absolute di@erence

in o@spring values to that of the parents.)t is denoted by .i

),1(),2(

)1,1()1,2(

ti

ti

ti

ti

i

xx

xx

=

++


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rocedureD^ Select parent strings 9i4#,t5 and 9i4+,t5

^ Choose a random number uiLL=,#M.

^ 1etermine Mqi , where area under thefollowing probability curve between = toMqi is equal to ui.

p4Mi5I =.&4Nc#5 MiN

c, if MiZ#P I =.&4Nc#5 4# MiNcC+5, otherwise.

Ncis any non:negative real number.


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^ Mqi can be calculated by following equationD MqiI 4+ui5#J4NcC#5, if uiZ=.&P I4#+4#:ui55#J4NcC#5, otherwise.^ Calculate the o@spring by following equationD 9i4#,tC#5 I =.&L4#Mqi5 9i4#,t5 4#: Mqi5 9i4+,t5M, 9i4+,tC#5 I =.&L4#: Mqi5 9i4#,t5 4#Mqi5 9i4+,t5M

^ Special case of S0Y, with NcI# and triangularprobability distribution with ape9 at parentsolution and base at d L9i4+,t5 O 9i4#,t5M, is calledfuy recombination crossover.

XdT is a tunable parameter.


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'hree or more parent solution are used to create two or

more o@spring.

6@spring are created from an ellipsoidal probabilitydistribution with one a9is formed along the line >oining twoof the three parent solution and the e9tent of orthogonal

direction is decided by the perpendicular distance of thethird parent from the a9is.

-

+

9+#

9#


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Select Xn#T parent string to use as a simple9,where n is the number of decision variable.

Calculate the Centroid of the parents.

*nlarge the simple9 by e9tending the ape9 at

points away from the Centroid. *ach ape9 is placed on a line >oining the

Centroid and each parents.

?ith uniform probability distribution, a H

number 4HI+== is suggested5 of solutions arecreated from e9tended simple9.


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Generally two parents are replaced by Hsolution set.

7irst o@spring is the best solution of Hand the second one is chosen by ran3based roulette wheel selection.


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'his crossover create o@spring towardsone of the parent solutions.

for two variables nI+.

'he value of O L=, =.&M

+=+

=+=

+=++

=+=

+

+

njiforxx

jiforxxx

njiforxx

jiforxxx

ti

ti

ti

tit

i

ti

ti

ti

tit

i

,...,1)1(

,...,1)1(

,...,1)1(

,...,1)1(

),2(),1(

),2(),1()1,2(

),2(),1(

),2(),1()1,1(

x (1/ t1)!


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x (2/ t)

x (2/ t1)

x (1/ t)

The parameter F is random%* hosen inte!er "et+een 1 and n,

indiatin! the ross site. &or examp%e t+o aria"%e(n 2) and +ith

a ross site at F 2, the "ias in reatin! the o##sprin! is %ear%*

sho+n in the a"oe #i!ure.The shaded re!ion marGs the possi"%e %oation o# the o##sprin!

i


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!nder a crossover operator followingpostulates should be followedD

#.

'he population mean should not change+. 'he population diversity should increase,

in general.


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1i@erent 'ypes of "utationD#. /andom "utation

+. Bon:uniform "utation

-. Bormally distributed distribution

. olynomial "utation


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/andom mutation isD

#. )ndependent of parent solution.

+. *quivalent to random initialiation.-. Can be given by the equationD

9i4#,tC#5I ri49i4!5 O 9i4855, ?here ri LL=,#M


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6ther way of doing /andom "utation isto create a solution in the vicinity ofparent instead of entire search space.

9i4#,tC#5I 9i4#,t5 4ri O =.&5Pi,

?here Pi is the user de(ned perturbation

in the ith iteration. Care must be ta3en to generate solution

within the lower and upper bound.


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Bon:!niform "utation can be given by theequationD

yi4#,t#5I 9i4#,t#5 Q49i4!5O 9i4855L#: ri4#:ttma95_bM,

?here, QI :# or # with equal probability,

tma9I ma9imum number ofgeneration.

bI user de(ned function.

robability of creating solution closer toparent

#. is more than away from it

+. increases with generation.


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Bormally 1istributed "utation can begiven by equationD

yi4#,tC#5I 9i4#,tC#5 B4=,Ri5

?here Riis a user:de(ned parameter and

can change in every generation withprede(ned rule.

Care must be ta3en to generate solutionwithin lower and upper bound.


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olynomial "utation can be given byD yi4#,tC#5I 9i4#,tC#5 49i4!5O 9i4855 =i

?here, =iis calculated from polynomialdistribution,

=iI 4+ri5#J4nmC#5O #, if ri =.&,

I #: L+4# : ri5M#J4nmC#5, if ri[ =.&.

nm is user:de(ned parameter.


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Constrained Handling "ethods have beenclassi(ed in following categoriesD

#. "ethods based on preserving feasibility ofsolutions.

+. "ethods based on penalty functions.

-. "ethod biasing feasible over infeasible

solutions.. "ethod based on decoders


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)n this process decision variables areeliminated by equality constraint. e.g. h495 S+9#O 9++9-I =, 9# can be e9pressed as 9#I =.&9++9-

Getting 9# in terms of 9+ and 9-willsatisfy the constraint h495 S=.

)n optimiing equation with n variablesand 3 equality constraints 4n35, 3

decision variables can be eliminated. 'hiswill automatically satisfy 3 equalityconstraints.


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enalty function is added to the ob>ective function forinfeasible solutions.

7or any optimiation function can be written as, min f495,

s.t. g>495 [ =, for > I # to J, h3495 I =, for 3I # to 3. after adding penalty function new ob>ective function

can be given asD 7495I f495 />gi495 r3`h3495`,

?here, /> and r3are user de(ned penalty parameters. gi495 I `gi495` if gi495 =, I =, elsewhere


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Static penalty functionD 6ne penalty parameter 4/5 is used for

constraint violation.

7495I f495 / L gi495 `h3495` M 1ynamic penalty functionD

penalty parameter is changed withgeneration 4t5.

7495I f495 4C.t5R

L gi495M

`h3495`M

M,?here C ,Rand Mare user de(nedconstants.


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7itness value of best infeasible solution ismade equivalent to worst feasiblesolution. So that,

#. Any feasible solution is preferred overany infeasible solution.

+. Among two feasible solutions, one with

better ob>ective function is preferred. -. Among two infeasible solution, the onehaving smaller constraint violation ispreferred.


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"ethodsD

#. 0y adding Generation dependentpenalty term in static penalty term.

7495I f495 / Lgi495 `h3495` M J4t,95.

J4t, 95 is the di@erence between the static

penalty function between best infeasiblesolution and worst feasible solution.


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+. 0y modifying ob>ective functionD 7495 I f495 , if 9 is feasibleP

I fma9495 Lgi495 `h3495`M,

otherwise. where fma9495 is the ob>ective function

value of worst feasible solution.


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. 1Ue +ou%d %iGe to use !eneti a%!orithms to so%e the#o%%o+in! o+ man* "its are re'uired #or odin! the aria"%esV(") Urite do+n the #itness #untion +hih *ou +ou%d "e

usin! in the se%etion roedure

2

2

2

1 )4()5.1( + xx

4,0

012

015.4

21

21

221

+

xx

xx

xx

1x 2x


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(a) ine and %ies "et+een 0 and 4 To !et three p%aes o# aura* trin! shou%d hae the

minimum a%ue as 4000( &irst di!it #or inte!er part andother 3 parts #or deima%).

ine, There#ore num"er o# "its re'uired is 12.

To !et t+o p%aes o# aura* trin! shou%d hae theminimum a%ue as 400( &irst di!it #or inte!er part and

other 2 parts #or deima%). ine, There#ore num"er o# "its re'uired is 9.

1x 2x

4096212 =

51229

=


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(") The trin! is enoded in the strin! +hih has minimuma%ue 0 and maximum a%ue 4 so,

annot "e as a onstraint.

&ormin! the %a!ran!ian to !et the #itness #untion

Uhere, and are %a!ran!e mu%tip%ier added i# theso%ution !oes to in#easi"%e re!ion.

max(.,0) is added to heG +hether the so%ution is in#easi"%e re!ion or not. @# so%ution is in #easi"%e re!ionthen onstraints are satis#ied and max(.,0) is 0.

4,0 21 xx

)0,12max()0,15.4max()4()5.1()( 2122

211

2

2

2

1 ++++++= xxxxxxxf

1 2


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. 2Wsin! the simu%ated "inar* rossoer (:B) +ith, #ind t+o o##sprin! #rom parent so%utions

and . Wse the random num"er0.723.

2=c

53.10)1( =x

6.0)2( =x


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ine random num"er is !reater than 0.5 so +e use the#ormu%a #or as8

TaGin! and +e !et&ormu%a #or o##sprin! are8

Ue !et,

and

!i

1

1

)1(2

1 +

= c

i

!i

723.0=i 2=c 21.1)723.01(21 3

1

=

=!i

I)1()1H(5.0

I)1()1H(5.0

),2(),1()1,2(

),2(),1()1,1(

t

i!i

t

i!i

t

i

t

i!i

t

i!i

t

i

xxx

xxx

++=

++=

+

+

752.11)6.0-21.053.10-21.2H5.0)1,1( =+=+tix

902.1)6.0-21.253.10-21.0H5.0)1,2( ==+tix


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. Wsin! rea%8oded Xas +ith simu%ated "inar*rossoer(:B) hain! , #ind the pro"a"i%it* o#reatin! hi%dren so%utions in the ran!e +itht+o parents ; . Yea%% that #or :B

operator, the hi%dren so%utions are reated usin! the#o%%o+in! pro"a"i%it* distri"ution

Uhere and 1,2 are hi%dren so%utions

reated #rom parent so%ution p1,p2

2=c

10 x

5.0)1( =x 0.3)2( =x

>++

=+ 1/)1(5.0

1)1(5.0)(

2

P

)12/()12( ppcc =


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a%u%atin! M (pread) #or hi%dren so%ution "et+een(0,1)

ine M is %ess than 1 so usin! the #ormu%a8

Thus, ro"a"i%it* o# reatin! hi%dren so%utions in (0,1)is 0.24

4.0)5.03/()01( ==

)1(5.0)( +=P

24.04.0-)12(-5.0)( 2 =+= P


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. $ T+o parents are sho+n "e%o+

a%u%ate the pro"a"i%it* o# #indin! the hi%d in theran!e #or i1,2 usin!

(a) imu%ated "inar* rossoer operator +ith

(") :%end rossoer(:B) operator +ith

TTxx )0.5,0.5(,)0.3,0.10(

)2()1( ==

I0.3,0.0Hix

2=c

67.0=


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(a) onsiderin! Yan!e o# #or hi%dren so%ution is (0,3) and

parent so%ution aries #rom (5,10) so the spread#ator +i%% "e,

a%u%atin! pro"a"i%it* #or "*

1x1x

6.0)510/()03( ==

c

cP )1(5.0)(1 +=

54.06.0-)12(-5.0)( 21 =+= P


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onsiderin! Yan!e o# #or hi%dren so%ution is (0,3) and

parent so%ution aries #rom (3,5) so the spread #ator+i%% "e,

!reater than 1

a%u%atin! pro"a"i%it* #or "*

There#ore net ro"a"i%it*

2x2x

5.1)35/()03( ==

2

2 /)1(5.0)( +

+= ccP

324.06.0/)12(-5.0)( 32 =+= P

175.0324.0-54.0)(-)( 21 == PP


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(") onsiderin! #or parent so%ution is (10,5), a%u%atin!

hi%dren so%utions +ith

a%u%atin! pro"a"i%it* #or "* %inear #ormu%a,

1x1x

65.1)I105(-67.05H

35.13)I105(-67.010H)1,2(

1

)1,1(1

=+=

==+

+

t

t

x

x

115.0)65.135.13/()65.13()(1 == P

67.0=


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onsiderin! #or parent so%ution is (3,5), a%u%atin! hi%dren

so%utions +ith

a%u%atin! pro"a"i%it* #or "* %inear #ormu%a,

There#ore ro"a"i%it*

2x2x

34.4)I35(-67.03H

66.1)I35(-67.03H

)1,2(1

)1,1(

1

=+=

==

+

+

t

t

x

x

5.0)66.134.4/()66.13()(2 ==P

67.0=

0575.05.0-115.0)(-)( 21 == PP


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. " pp%* the po%*nomia% mutation operator to reatea mutated hi%d o# the so%ution usin! therandom num"er 0.675. TaGe and

0.5)( =tx

10)( =ux 0)( ="x2=m


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&or o%*nomia% mutation +ith ,

Uhere,

a%u%atin! mutated hi%d +ith , ,

i"utt xxx# +=+ IH )()()()1(

5.0)I1(2H1 )1/(1

= +

iii rifr m

675.0=ir

134.0)I675.01(2H1 )12/(1 == +i

10)( =ux 0)( ="x0.5)( =tx

34.6134.0-I010H5)1(

=+=+t

#


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"ost real:world problems involve simultaneous

optimiation of several ob>ective functions. Generally, these ob>ective functions are non:

commensurable and often competing and conKictingob>ectives.

"ulti:ob>ective optimiation having such conKictingob>ective functions gives rise to a set of optimalsolutions, instead of one optimal solution becauseno solution can be considered to be better than any

other with respect to all ob>ectives. 'hese optimalsolutions are 3nown as areto:optimal solutions


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Generally, multi:ob>ective optimiationproblem consisting of a number ofob>ectives and several equality andinequality constraints can be formulated

as followsD

inimie/aximie ( ) 1,.........,

5u"Fet to,

( ) 0, 1,.........,

( ) 0, 1,..........,

i oj

k

l

f x i N

g x k $

h x l "

=

= =

=


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?here, is the ob>ective function, 9 is a

decision vector that represents a solutionand is the number of ob>ectives.

and 8 are the number of equality andinequality constraints respectively.

if

ojN


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"ost optimiation problems naturally haveseveral ob>ectives to be achieved4normally conKicting with each other5, butin order to simplify their solution, the

remaining ob>ectives are normally handledas constraints.


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*volutionary algorithms seem particularlysuitable to solve multi:ob>ectiveoptimiation problems, because they dealsimultaneously with a set of possible

solutions 4the so:called population5.

Capable to (nd several members of the

areto optimal set in a single run of thealgorithm.


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Additionally, evolutionary algorithms areless susceptible to the shape orcontinuity of the areto front

*volutionary algorithm can easily deal

with discontinuous or concave aretofronts


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Classi(cation of *volutionary "ulti:6b>ective optimiation 4*"665 D

Bon:areto 'echniques

areto 'echniques


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Bon:areto 'echniques include the followingD:

Aggregating approaches

Nector evaluated genetic algorithm 4N*GA5 8e9icographic ordering

'he : constraint "ethod

'arget:vector approaches


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areto:based 'echniques include thefollowingD:

"ulti:ob>ective genetic algorithm 4"6GA5

Bon:dominated sort genetic algorithm

4BSGA:))5 "ulti:ob>ective particle swarm optimiation

4"6S65

areto evolution archive strategy 4A*S5 Strength areto evolutionary algorithm

4S*A:))5


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Suggested by Goldberg 4#$;$5 to solvethe problems with Scha@erTs N*GA.

!se of non:dominated ran3ing andselection to move the population towards

the areto front. /equires a ran3ing procedure and a

technique to maintain diversity in the

population.


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)n the absence of weights for ob>ectives,one of the pareto optimal 4non:dominated5solutions cannot be said to be better thanthe otherP therefore it is desirable to (nd all.

Classical optimiation methods including"C1" methods can (nd one such solutionat a time

?ith a population of solutions, GAs seem tobe well:suited for appro9imating the paretooptimal frontier in a single run


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3ecision4pace

)&5ecti.e4pace

Pareto setPareto setappro6imation

Pareto %rontPareto %rontappro6imation

X

X7y7

y

(X!X7!8! Xn F (y!y7!88!yn

4earch

0.aluation


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Convergence topareto optimalfrontier

1iversity

4representation ofthe entire paretooptimal frontier5

Diversity

Convergence

%

%7


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7itness assignmentD Solutions in the (rst

non:dominatedfront have thehighest (tness4they are all ran3ed#5

Solutions in thesame front have the

same (tness 4theyall have the sameran35

%7

%

2hird %ront

4econd %ront

First %ront

Minimization of f1 andf2


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S($A


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!ast nonAdominated sortingD:#. 7or each solution p in population, (nd

D number of solutions that dominate p

D set of solutions that p dominates

+. lace all p with in set , the (rst front-. 7or each , visit each and reduce by

one. )n doing this, if becomes = then place q inset 4 q belong to the second front, 5

. /epeat step - with each member of to (nd thethird front, and so on

pn

ps

0pn = 1& ( 1)p% =

1p F p! & !n

!n 2F

2!% =

2F


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cu&oid

i"i

i-

%

%7

f1 and f2 are to be minimized

QSharingR in BSGA isreplaced withQcrowdedcomparisonR

QCrowded distanceR ofsolution i in a front isthe average sidelength of the cuboids.


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#. Sort all l solution in a front in ascending orderof and compute

#. /epeat step # for each ob>ective and (nd thecrowding distance of solution i as

mf

1 1

max min

( ) ( ), 2,...... 1

( ) ( )

m i m iim

m m

f x f x'D i l

f x f x

+ = =

1

M

i im

m

'D 'D=

=


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Given two solutions i and >, solution i ispreferred to solution > if

0etween two solution with di@erent non:

domination ran3s, the one with lower 4better5ran3 is preferred

?hen two solutions have the same non:domination ran3 4belong to the same front5, the

one located in a less crowded region of thefront is preferred

or ( and )i j i j i j% % % % 'D 'D< = >

StepsD:

# )nitialiation of population


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#. )nitialiation of population

+. 7ast non:dominated sorting-. Calculate crowding distance

. 'ournament selection using crowding comparisonoperator

&. Crossover


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opulation sieD:depends on the nature ofthe problem

robability of crossover 4c5D:=.


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"a9imum number of generation

Bo improvement in (tness value for (9number of generation

*9ample consist of total #; activity


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p y

*ach activity has alternative to completethe pro>ect namely, time, cost and quality.

*ach alternative inKuences three ob>ectivefunctions vi. pro>ect time, cost and quality

ConKicts among three ob>ective functions

)n e9ample there is an average -.#+ optionsto complete each activity, this create nearly

about %& million 4 5combination forscheduling the entire pro>ect.1/

3.12


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ro>ect networ3 has ## path to completethe pro>ect.

ath duration varies with alternative.

)t is very di2cult to decide, which path is

critical


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Goal D: 'o (nd best alternative which balances

three (tness functions vi. pro>ect time,cost and quality.

"inimiation of pro>ect duration "inimiation of pro>ect cost

"a9imiation of pro>ect quality


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:::::::

1Bn

2Bn

3Bn

4Bn

5Bn

6Bn

7Bn

Bn

9Bn

10Bn

B represents !ene

@ndex 1, 2, 3Z..1 indiates atiities

n represents a%ternatie o# atiit*.

[ah a%ternatie represents di##erent time, ost and 'ua%it*.

1Bn

opulation)nitialiation

S'A/'


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*valuate 7itness

7ast non:dominatedSorting

Calculate crowdingdistance

Create child populationusing Genetic operatorsvi. selection, crossoverand "utation

Combine parentpopulation of sie Bwith child population ofsie B7ast non:dominatedsorting on combinedpopulation

Create new populationof sie B using crowdingcomparison operator

Childpopulati

on

8astgeneratio

n

Be9t generation

S'6

9es

o

o

9es

Create initial population of solutions


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Create initial population of solutions /andomly 8ocal search

7easible Solutions

7or optimiation problem "inimie pro>ect time and cost

"a9imie pro>ect quality

opulation sie I -==

D i i


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& # - - # # # # # + - + # - # #

+ # + # + # # # # # # # - # # - # #

+ # - - # # - # # # # # # & # #

# # # - - # # + # # - + # & # +

+ + # - - # + + # # # - + # # #

Solution No

Fitnessvalue

# +

-

-==

'ime

Cost uality

#+-

##ective

space *ach solution is assigned a crowding

distance

:Crowding distance I front density in theneighborhood

:1istance of each solution from its nearestneighbors;

B

%

%7

Solution is morecro!ded than "

erform tournament selection using


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crowding comparison operator Crowding comparison operator guide the

selection process to obtain uniformly spreadout pareto optimal front

*very solution has two attributes

#. Bon:domination ran3

+. Crowding distance

Crossover operation 40asedb bilit 5


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on crossover probability5 Select parents from

population based oncrossover probability

/andomly select two

points between strings toperform crossoveroperation

erform crossoveroperations on selected

strings nown for 8ocal search

operation

Parent

Parent 7

)*spring

)*spring 7

rosso.erPoint

"utation operation 4based on mutation


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probability pm5

*ach bit of every individual is modi(edwith probability pm

main operator for global search 4loo3ingat new areas of the search space5

pm usually small V=.==#,F,=.=#W

rule of thumb pmI #no. of bits inchromosome


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7or optimiationproblem

8etpmI ##; I

=.=&&

Select bits havingprobability less than

pm

)nterchange the bitswith each other

ithsolution string %rom the population

+ - # # F.. +

=.+

-

=.

%

=.=

+

=.&

sing crowdingcomparisonoperator

Combined population is sorted according tonon:domination

tY


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Solution belonging to the non:dominated setconsidered as a best solution in combinedpopulation

)f sie of is smaller than population sie B then

choose all member of for new population 'he remaining member of the population are

chosen from subsequent front in order of theirran3ing

)n general, the count of solution from towould be larger than population sie 4 B 5

!se crowding comparison operator to choosee9actly B population member from last front

1&

1&

1& t1

t1

1& &

&


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0est parameter for BSGA:)) is selected throughseveral test simulation run

BSGA:)) arameter arameter value

opulation sie -==

Bumber of generations #===

Crossover probability4c5

=.;

"utation probability4m5 =.#


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%ime&cost&'uality tradeo(

f


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1) genetic algorithm

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