1) genetic algorithm
TRANSCRIPT
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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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