notes algorithm for mo pso

7/26/2019 Notes Algorithm for MO PSO

1/6

Algorithm for SO-PSO (Single Objective Particle Swarm Optimization)

Step 0)INITIAI!"

Bounds on mdecision variables (positionsof particles):xiL,xiU; i= 1, 2, . . ., m

Number of particles (solutions): Nomputational parameters: !ma", !min, ma"iter, c1, c2

#nitial velocitiesof t$e N particles%solutions:

(& )

1,jV

,

(& )

2, jV

, . . . ,

(&)

,m jV

; for eac$j= 1, 2, . . . , N

(all 'ero; or an ot$er set of values); see able 1 (for m= 2,N= *)

Step #)$"N"%AT" T&" INITIA POSITIONS (

(& )

,i jx

; i= 1, 2, . . . m;j= 1, 2, . . . , N) O'

EACH O' T&" N PA%TI" AN* T&" O+,"TI" '.NTIONS +Ij(&)(

(& )

1,jx

,

(& )

2, jx

, . . . ,

(& )

,m jx

);j= 1, 2,..., N,O' "A& PA%TI"

-enerate t$e initial positions of t$e N points !it$in t$e bounds of xi, usinseveralrandom

numbers (in se/uence),R:

(& ),i jx

=xi,L0R(xiUxiL); & R 1; i= 1, 2, . . . , m;j= 1, 2, . . . ,N

(t$is !ill satisf all t$e bounds). 3nter in able 1 (for m= 2,N= *)

#f constraints are present, use penalt functions

3valuate t$e ob4ective function for all t$e particles:Ij(&)(

(& )

1,jx

,

(& )

2, jx

, . . . ,

(& )

,m jx

);j= 1, 2,...,N

3nter in able 1 (for m= 2,N= *)

Step /)ANA!" T&" %"S.TS TILL NOW

567 eachof t$ej= 1, 2, . . . ,Nparticles, identif t$epersonalbest,

(&)

,i jPbest

, value ofI

(lo!est, obviousl, it isIj(&)) and put t$e correspondinxivalues as

(&)

,i jPbest

=

(& )

,i jx

; i= 1, 2, . . ., m;j= 1, 2, . . . ,N. 3nter in able 1 (for m= 2,N= *)

1


2/6

839Ij(&)for eachof t$ej= 1, 2, . . . , Nparticles, and identif t$e best in t$e s!arm (i.e.,

identif t$egroupbest,

(&)

,i jGbest

; lo!est amon t$eNvalues ofIj(&);j= 1, 2, . . . , N. a, it is

particle, j= k. ut t$e correspondinxivalues as

(&)

,i jGbest

=

(& ),i kx

; i= 1, 2, . . . , m. learl,

(&)

,i jGbestis t$e same for allj. 3nter in able 1 (for m= 2,N= *)

Step 1).P*AT" T&" "OITI"S

(1)

1,jV

(1)

2, jV

2 2 2

(1)

,m jV

'O% "A&j3 # / 2 2 2 N

! = !ma" , iteration No. = &

(1)

,i jV= w

(1)

,i jV0 c1R+

(&)

,i jPbest

(& )

,i jx 0 c2R+

(&)

,i jGbest

(& )

,i jx;

i= 1, 2, . . . , m;j= 1, 2, . . . ,N


Note t$at t$e Vs involve t$e old Vs, t$e particle best and t$e s!arm best (a


3/6

#f t$e

(1)

,i jx

violate t$e bounds, c$ane t$em to satisf t$e bounds

alculate

(1)

jI

. 3nter in able 2 (for m= 2,N= *)

ain, calculate

(1),i jPbest

and

(1),i jGbest

Update iteration number b unit and wusin

w= wma"> +(wma"> wmin)(iteration number)%ma"iter

alculate

(2 )

,i jV

, usin t$e e/uation before


Table /4 (some entries belo! are copies of t$ose in able1)

Particle

No2j

(& )1,jx

(&)

2, jx (1)

1,jV (1)

2, jV (1)

1,jx (1)

2, jx (1)

jI

#

/

1

5

Particl

e No2j

(1)

1,jV (1)

2, jV (1)

1,jx (1)

2, jx (1)

1,jPbest (1)

2,jPbest (1)

1,jGbest (1)

1,jGbest (2 )

1,jV (2 )

2, jV

#

/

1

5

ontinue t$e iterations till converence or !$en iteration number e"ceeds ma"iter

??????????????????????????????????????

@


4/6

*aterial belo! on *O+,-O is not correcte$ et

Step 5)A%&IIN$ O' NON-*O6INAT"* POINTS

ompare 7(< 0 1)!it$ everynon?dominated member present in 5ile 2, one b one

#f 7(< 0 1)is a non?dominated point, cop it in 5ile 2 (as !ell as in 5ile 1)

#f 7(< 0 1)is dominated over b a member alread present in 5ile 2, do not include it in 5ile 2

but c$ec< if ou can cop it in 5ile 1 (tep above)

#f an earlier member is dominated over b t$is ne! point, delete t$e former from 5ile 2 (it is

alread in 5ile 1)

Step 8)%"T.%N TO +AS"

7eturn to an uncro!ded base point ($avin a lare cro!din distance from 5ile 2)

periodically (so as to enerate a more continuous areto set, b locall e"plorin around

uncro!ded points in t$e non?dominated set)

5irst return to base is done after NB,1(= 2N,1) iterations (after t$e first N,1)

Number of iterations before 4t$return to base = NB,4= rBNB,4 ? 1; 4 = 2, @, . . .

& rB 1 (enerall, &.A); s.t.: NB,4 1&

-enerate 5ile @ (of uncro!ded members in 5ile 2)

Normali'e fi(usin ma" and min values) for all points in 5ile 2, so t$at & fi 1

5ind t$e cro!din distances (#C#) of allpoints in 5ile 2 (as in N-?##)

op 4of t$e most uncro!ded points (includin t$e boundar points) into 5ile @, !$ere

4= D4; D401= r D4; D1= 1; r = &.A; 4 E#N= F

Use a random number (diiti'ed appropriatel) to select t$e uncro!ded base point to be

returned to, from 5ile @

IST O' %AN*O6 N.6+"%S $I"N +"O9 (.S" IN T&" S":."N" ;n $I"N) "S" T&" 9ONT +" %AN*O6)

*


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Notes lorit$m E6 6

F

notes algorithm for mo pso

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