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) 7 1 0 2 E II A ( g n ir e e n i g n E l a ir t s u d n I d n a e c n e g il l e t n I l a i c if it r A n o e c n e r e f n o C l a n o it a n r e t n I d r 3 7 1 0 2 8 7 9 : N B S

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1Schoo lofMathemaitcsandStaitsitcs ,Qingha iNorma lUniverstiy,Xining810008,China

2Schoo lofComputerScience,Qingha iNorma lUniverstiy,Xining810008 ,China

r o h t u a g n i d n o p s e r r o C * : s d r o w y e

K Mulit-objecitve opitmizaiton problem, Evoluitonary algortihm, Opitma l soluitons , n o it a m r o f n i m e l b o r P . .t c a r t s b

A Mulit-objectiveoptimizationproblemsareakindofproblemsoptimizingsimutlaneously l a r e v e

s conflicting objectivesand keeping a balancebetween thediversity and theconvergence of s n o i t u l o

s . In this paper , some nove l techniques are designed to improve the efficiency of i

t l u

m -objectiveevolutionaryalgorithms .Firstly,aspecifics -ub funcitonisseparatedfromaseriesof s e v i t c e j b

o ,whichisapplied toprovideanapproximatesearchdirectionand speed theconvergence , n e h T . m h t i r o g l a e h t f

o the crowding degree scheme ,as in NSGA- II ,is used to selec tpotentia l f o s s e c o r p e h t n i s n o i t u l o s g n i s i m o r

p tierationssuch tha tPareto soluiton se thasmoreuniformand n

e t x

e sive distribution .Finally ,a nove lmutli-objective evolutionary algorithm is presented by g n i d d e b m

e these schemes intoMOEA/D .The simulaiton results show the proposed algorithm is b

i s a e

f leandefficien.t

n o it c u d o r t n I i t l u

M -objectiveoptimization si amainresearchi ssuei nopitmizaitonfield[1] .Theyaregenerally n u d n a d e n i a r t s n o c o t n i d e d i v i

d constrained problems .Themathematica lmode lcan beformulated s

a :

( ) ( )

( )

(

1 2

)

. , , . , x x x x m t s f f f n i m … Ω

∈ ( 1)

n e h

W Ω=Rn ,(1)i sanunconstrainedproblem ;otherwisei tiscalledaconstrainedoptimization . m e l b o r p u l o s y n a o n s t s i x e e r e h t d n a , s e l b a i r a v n o i s i c e d y b d e d i c e d s i e v i t c e j b o h c a

E tion to

t e s n o i t u l o s e s i m o r p m o c a d n i f o t y r a s s e c e n s ’ t i , e r o f e r e h T . s e v i t c e j b o l l a e z i m i t p o y l s u o e n a t l u m i s it l u m f o n o i t a c i l p p a d a e r p s e d i w d n a y t i x e l p m o c e h t g n i r e d i s n o C . s e v i t c e j b o l l a r o

f -objective

t i r o g l a e v it c e f f e y n a m t n e s e r p s e c n e r e f e r e h t , s m e l b o r

p hmsbasedonvariousoptimizationpurposes .

it l u m e v l o s o t d e s u y l r a e s i m h t i r o g l a t n e i d a r g d e t h g i e w e h

T -objective optimizaiton problems ,for

, e l p m a x

e VEGA[2] ,HLGA[3] ,etc .Thedeficiencyoft hesemethodsi st hatt heycanonlyofferone a m it p o d e r r e f e r

p lsolution whenthealgorithmisexecuted ,and thebes tweigh tvectorisalwaysno t e g r a l h t i w g n il a e d e m i t n o it a t u p m o c e c u d e r o t r e d r o n i n e h T . e c n a v d a n i d e n i m r e t e

d -scale

t n e s e r p e r a s t h g i e w d e x i f g n i s u s m h t i r o g l a n o it i s o p m o c e d e h t , s m e l b o r p n o i t a z i m i t p

o ed , for

, e l p m a x

e MOEA/D[4]andMOEA/DVA[5].

i t l u m g n i v l o s n i y c n e i c i f f e s w o h s n o it i s o p m o c e d n o d e s a b D / A E O

M -objective problems ,bu t

e s u a c e B ) 2 ; w o l s i s n o i t u l o s g n i s i m o r p w e n g n i t t e g f o y c n e i c i f f e e h T ) 1 : s k c a b w a r d e m o s e r a e r e h t h g i e w f o y t i m r o f i n u e h

t tvectors canno tcompletely ensure the uniformity of Pareto front ,the t p i r c s u n a m s i h t , s k c a b w a r d e h t e m o c r e v o o t r e d r o n I . n i a t r e c t o n s i t n o r f o t e r a P f o y t i m r o f i n u s e m e h c s w e n o w t s e s o p o r

p .Firstly ,an sub-function is taken from objectives ,and the descending b u s e h t f o n o it c e r i

d -function is used to generate offspring in evolutionary process .Secondly ,the n o n f o y t i m r o f i n u e h t p e e k o t d e t p o d a y l l a c i d o i r e p s i e m e h c s e e r g e d g n i d w o r

c -dominated soluitons

(2)

e m o

S Ba Nisc o itons

n

I m itu -l objectiveoptimizationproblems ,severa lbasicconceptionaredescribedasfollows[1]:

n o it i n if e

D 2.1 (Pareto dominance) F or two ya n solution x1,x2∈Rn i,fthe following conditions

e r

a satisfied:

2 1

2 1

} , , 2 , 1 { , ) ( ) (

} , , 2 , 1 { , ) ( ) (

x x

x x

j j

j j

m j

f f

m j

f f

∈ ∀ ≤

<

 ( 2)

n e h

T the vector(f1(x1) ,f2(x1) … , fm(x1) ) is said to dominate thevector (f1(x2) , f2(x2) … , fm(x2) ,)

d n

a x1dominates x2.

n o it i n if e

D 2.2Ift herei snosoluitondominatingx* inRn,t henx*iscalledanoptima lsolution.

n o it i n if e

D 2.3Se t E(f,Rn)={xRn :xisaParetoopitma lsolution}i saParetooptima lse.t

n o it i n if e

D 2.4Se t FE(f,Rn)={f1(x),f2(x),,fm(x)|xE(f,Rn)} isaParetofron.t

m h ti r o g l A d e s o p o r P

S bu -funciton Seleciton

a i t n e r e f f i d e l p m i s a t c e l e s e

W b bles -u function fwhich i smonotonoust o some objective function . b

u s e h

T -funciton can beobtained by taking variableoneby one .Themorevariablescan betaken b

u s e h t o t n

i -function ,thebetterthesub-function is .Thegradien tornegativegradien tistaken asa .

n o it a r e p o r e v o s s o r c n i d e s u s i h c i h w n o it c e r i d h c r a e

s Ifthereexis tnodifferentiablesub-funcitons ,

an pa proximategradienti savailablet oserveasapproximatedescendingdireciton.

HybridCrossover

r o

F paren tindividual x0 ,the offspring is denoted by x1 . F or probability threshold s .a random r

e b m u

n r isgenerated .Ifr<s ,then

0

1 x *

x = ±af (3)

e r e h

W a= -1 i/Gen ,Gen is the maximum number of iterations .a is a monotonic decreasing n

o i t a r e n e g h ti w d e t a i c o s s a n o i t c n u

f i so tha ta big step is needed a tthebeginning of theiteration h

t m o r f e p a c s e d n a s l a u d i v i d n i r e t t e b d n i f o t r e d r o n i s s e c o r

p e loca loptima ;A tthe end of the

n e e w t e b n o i t a l e r e h t f I . n o it u l o s l a m i t p o e s i c e r p e h t n i a t b o o t d e t p o d a s i h c r a e s g n i n i f e r , m h t i r o g l a

b u s d n a n o i t c n u f l a n i g i r

o -function is monotonic increasing , x1= x0a*∇f ; otherwise ,

0

1 x *

x = +af .When rs,t hecrossoveroperatori st akenast hesamei n[4].

CrowdingDegree

e c n i

S the uniformtiy of weigh tvectors of MOEA/D canno tensure the uniformity of the Pareto t

e s n o i t u l o

s ,andthecomputationamoun tislargei hft ecrowdingdegreemethodisusedin each of s

a s n o i t a r e n e

g i n NSGA- II . Therefore, in the proposed algorithm ,we adop t periodically the e

e r g e d g n i d w o r

c methodofNSGA-IIt okeeptheuniformityoft heParetosolutionse.t

M itu -l objecitveEvolu itonaryAlgorit hmBasedSpeci ifcProblem

e v o b a e h t g n i d d e b m

E schemes into MOEA/D ,a problem-specific mulit-objective evolutionary ,

m h t i r o g l

a P-MOEA/D, isdescribedasfollows.

p e t

S 1 .Generatingani niita lpopulation . t e s l a u d i v i d n i e t a r e n e g y l m o d n a

R {x1,x2,,xPopsize}as an initia lpopulation ,and denote the n

o

n -dominatedsolutionse tbynonD.

p e t

S 2 .Iniitailzingweigh tvectors .

t s e r a e n e h t d n i f d n a , n o i t u b i r t s i d m r o f i n u h t i w s r o t c e v t h g i e w f o p u o r g a e t a e r

C Tweightst oeach

i l c u E o t g n i d r o c c a r o t c e v t h g i e

(3)

p e t

S 3 .Evolutionpopulaiton. b

u s a t c a r t x

E -function f according to theabove designs ,generate crossoveroffspring by hybrid s

u a G g n i t u c e x e y b g n i r p s f f o n o i t a t u m n i a t b o d n a r o t a r e p o r e v o s s o r

c smutation.

p e t

S 4 .Selection. n o i t a r e n e g r o

F iandani ntegermgiven,i f imodm==0,t hecrowdingdegreei sexecutedt oselec t n

o

n -dominated solutions ,otherwise ,the nex tgeneration population is chosen in terms of the c

n u f e v i t c e j b o e h t f o m u s d e t h g i e

w tion .Updatethenon-dominatedsolutionse tnonD.

p e t

S 5 .Terminaitoncriterion.

t u p t u o , p o t s , t e m s i n o i r e t i r c n o i t a n i m r e t e h t f

I nonD ;otherwise ,returnt oStep3.

P-MOEA/DdistinguishesfromMOEA/Dinatl eastt woaspects: )

1 Hybrid crossoveroperatorbasedon thedescending direction ofthesub-function isproposed , l

u f p l e h s i h c i h

w togeneratepromisingsoluitons. )

2 The crowding technology is only used periodically instead of being executed in each of ,

s n o i t a r e t

i whichcansavecomputaitonalt ime.

m u

N era lSimula itonandAnalyssi

f o t n e m n o r i v n e n a g n i v a h C P n a n o d e m r o f r e p e r a s t n e m i r e p x e e h

T i -5 2450MCPUwtih2.50GHz.

f o e l a c s e h t : ] 8 [ n i s a s r e t e m a r a p e m a s e h t e k a t e w , D / A E O M h ti w n o s i r a p m o c f o e s o p r u p e h t r o F

n o i t a l u p o

p Popszie=101,scalingfactorF=0.5,t hedimensionofvariablesi nZDT1



�ZDT2



�ZDT3i s

n=30 , whlie n=10 in ZDT4 and ZDT6 ; mutation probabliity prob=1/n ,crossover probabiltiy

R

C =0.5 ,thenumberofthecloses tweightsto each weigh tvectorT=20 ,thenumberofindependen t l

a e h t f o n o i t u c e x

e gorithm numrun=20 .The generation in MOEA/D is Gen=200 ,while Gen=150 P

r o

f -MOEA/D.

: ] 7 [ n e v i g e r a s e l p m a x e e v i F . s e v it c e j b o e h t f o n o i t a z i m i n i m e r a s e c n a t s n i t s e t l l A

: 1 T D

Z f1(x)=x1 , 2 1

) (

] 1

[ * ) ( ) (

) (

x x

x

x f g

f

g

= ,

e r e h

w 2

) ( * 9 1 ) (

1

x

n

i i

x g

n= + =

, x=(x1,x2,,xn)T∈[0,1]n,n=30.

: 2 T D

Z f1(x)=x1 , 1 2

2

)

( ) ]

( 1 [ * ) ( ) (

) (

x x

x

x f g

f

g

= ,

( g e r e

h x) i sthesameasi nZDT1.

: 3 T D

Z f1(x)=x1 , 1 1

1 2

) ( ) (

] ) * * 0 1 ( n i s * 1

[ * ) ( ) (

) ( ) (

x x x

x

x x

f

f x

g f

g

g − π

= ,

( g e r e h

w x) i sthesameasi nZDT1

: 4 T D

Z f1(x)=x1 , 1

2

) (

] 1

[ * ) ( ) (

) (

x x

x

x f g

f

g

= ,

e r e h

w 2

2

] ) * * 4 ( s o c * 0 1 [ ) 1 ( * 0 1 1 )

(x n i i

i

x x

n

g π

=

− +

− +

=

1

2

1, , , ) [0,1] [ 5,5] , 10

(

x T n

n n

x x

x × =

=  .

: 6 T D

Z f1(x)=1−exp(−4*x1)*sin(6*π*x1), 2 1 2 ) (

] ) ( 1 [ * ) ( ) (

) (

x x

x

x f g

f

g

= ,

e r e h

w 2 0.25

) (

] [

* 9 1 ) (

1

x

n i i

x g

n= + =

,x ( 1, 2, , )T [0,1]n, 10

n n

x x

x ∈ =

=  .

C-metric and Hyper-volume (HV) are selected to compare the n -on dominated sets .We firslty f

o e u l a v e h t e n i m r e t e

d s by an experimen tmethod onZDT1 and ZDT2 ;secondly ,makeuseofthe d

e n i m r e t e

(4)

. D / A E O M h t i w V

H 

) i

( C- metric[8]

D / A E O M f o t e s n o i t u l o s l a m it p o o t e r a P : A t e

S ;

P f o t e s n o i t u l o s l a m it p o o t e r a P : B t e

S -MOEA/D

| } :

| { | ) , (

| |

v v

u B A dominatesu B

A C

B ∈ ∃ ∈

= ( 4)

C (A ,B) is defined as the percentage of the solutions in B tha tare dominated by a tleas tone n

i n o i t u l o

s A. C(A ,B)i sno tnecessarilyequalt o1-C(B ,A). C(A ,B)=1meanst ha tal lsolutionsi n

B are dominated by some solutions in A ,whlie C (A ,B)=0 implies tha tno solution in B is o

i t u l o s a y b d e t a n i m o

d ni nA.

) i i

( HV[8]

e c a p s e v it c e j b o e h t n i t n i o p e c n e r e f e r a t c e l e

S r=(r1,r2,…,rm)T ,compute

V

H ( ,r) ( [ 1, 1] [ m, m])

A f

r f r

f e m l o v A

× ×

=

 ( 5)

t n i o p e c n e r e f e r a r o

F r,t hel agerHVvaluemeansbetterqualtiy . )

i i i

( Selectedsub-function

Table1 .Sub-functionsusedi nP-MOEA/D. .

b o r

P S -ubfunctionf Relations 1

T D Z

2 x

n i i=

f2 :monotonei ncreasing

2 T D Z

2 x

n i i=

f2 :monotonei ncreasing

3 T D Z

2 x

n i i=

f2 :monotonei ncreasing

4 T D

Z 2

2

] ) * * 4 ( s o c * 0 1 [ n

i i

i

x

x π

=

f2 :monotonei ncreasing

6 T D Z

2

x

n i i=

f2 :monotonei ncreasing

We take s=0 .1, 0 .3, 0 .5, and 0.7 f orZDT1-2 for thesakeof selecting appropriates .Resutlsare :

2 e l b a T s a d e b i r c s e d

. 2 e l b a

T Resultsabou tdifferents no ZDT1-2.

s C(A, B) C(B, A) C(A, B) C (B, A) T

D

Z 1 ZDT2

1 .

0 0.9802 0.9901 0.9901 0.9802 3

.

0 0.9802 0.9901 0.9802 0.9901 5

.

0 0.9802 0.9901 0.9802 0.9901 7

.

0 0.9802 0.9901 0.9901 0.9802

2 e l b a

T clearly indicatestha tforZDT1 ,whens=0.1,0.3,0.5,0.7 ,C (A, B) <C (B ,A).ForZDT2, n

e h

w s=0.3and0.5, C(A, B) <C(B, A) .Therefore,wetakes=0.5 . n

e h

W s=0.5 ,wecarryou tP-MOEA/D tota l20runsand providetheaveragevaluesofC-metric V

H d n

a inTable3.

e l b a

T 3. ResultscomparisonwithMOEA/D.

r

Po b. MOEA/DG enP-MOEA/D MOEA/DCPUP-MOEA/D MOEA/D H VP-MOEA/D C(A,B) C(B,A) 1

T D

Z 2 00 1 50 10.0870 7.6222 0.0944 0.0945 0.9802 0.9901 2

T D

Z 2 00 1 50 10.0059 7.6745 9.6847e- 50 9.7060e- 50 0.9802 0.9901 3

T D

Z 2 00 1 50 9.8444 7.7245 0.1136 0.1113 0.9802 0.980 2 4

T D

Z 2 00 1 50 9.8561 7.2501 0.0819 0.0900 0.9802 0.9802 6

T D

(5)

Table3 showsaverageindexvalueof20runsaswel lasgenerationnumbers .HVofP-MOEA/D D

/ A E O M n a h t r e g r a l s

i on al lproblems excep tfor ZDT3 .C (A, B) < C (B, A) holds on al ltes t s

m e l b o r

p ,i tmeanst ha tP-MOEA/Dfoundt hebetternon-dominatedsett hanMOEA/D. I tshouldbe P

t a h t d e t o

n -MOEAexecutedal essnumberofgenerationst hanMOEA/D.

Conclu ison

k a t y

B i ng a sub-funciton from original objecitve functions w , e propose a new multi-objecitve m

h t i r o g l a y r a n o i t u l o v

e on the basis of the existing MOEA/D .Since the objective information is d

e z i l i t

u sucht hatt heproposedalgorithmcanfindpromisingi ndividualsi niteraitons .Inaddition ,in r

o der to keep theuniform Pareto front ,the crowding degree scheme is also adopted periodically . i

c i f f e e h t e t a r t s u l li s t n e m i r e p x e e h t f o s t l u s e r e h

T encyoft heproposedalgorithm.

Acknowledgements

e i c S l a r u t a N l a n o it a N e h t y b d e t r o p p u s s a w k r o w h c r a e s e r e h

T n ceFoundaitonofChinaunderGran t

1 6 . o

N 463045 and the Key Laboratory of the Interne t of Things of Qingha i e

c n i v o r

P (2017- JZ -Y21) .

s e c n e r e f e R

] 1

[ Wang Y.P .Theory and method ofevolutionary computation [M] .Bejiing :SciencePress ,2011 : 0

4 1 -142.

f a h c S ] 2

[ fer J.D . Mulit-objecitve optimization with vector evaluated genetic algorithms [C] . 3

9 : 5 8 9 1 , s m h t i r o g l A c it e n e G . f n o C . t n I t s 1 . c o r

P -100.

H i h c u b i h s I ] 3

[ . ,MurataT .Amulti-objectivegeneticloca lsearch algorithm and its application to w

o l

f -shopscheduling[M] .Oxford :PergamonPress ,1996. Q

g n a h Z ] 4

[ .F. and L i H . MOEA/D : A multi-objective evolutionary algorithm based on n

o i t i s o p m o c e

d [J] .IEEETransactiononEvolutionaryComputation ,2007 ,11(6) :712-731. X

a M ] 5

[ .L. ,LiuF. ,Q iY.T. andWangX.D. ,etc .Amulti-objectiveevolutionaryalgorithmbasedon i

t l u m r o f s e s y l a n a e l b a i r a v n o i s i c e

d -objecitveoptimization problems with large-scalevariables[J] . 5

7 2 : ) 2 ( 0 2 , 6 1 0 2 , n o it a t u p m o C y r a n o i t u l o v E n o n o it c a s n a r T E E E

I -298.

Y n a T ] 6

[ .Y. ,Jiao Y.C. ,L iH. and Wang X.K .MOEA/D+ uniform design :A new version of s

e v i t c e j b o y n a m h t i w s m e l b o r p n o i t a z i m i t p o r o f D / A E O

M [J] .Computers& OperationsResearch ,

8 4 6 1 : ) 0 4 ( , 3 1 0

2 -1660.

E r e l z t i Z ] 7

[ . ,Deb K. ,and Thiele L .Comparison of multi-objective evolutionary algorithms : u

s e r l a c i r i p m

E lts[J] .EvolutionaryComputaiton ,2000 ,8(2) :173-195. E

r e l z t i Z ] 8

[ . and ThieleL .Multi-objecitveevoluitonaryalgorithms :Acomparaitvecasestudyand h

c a o r p p a o t e r a P h t g n e r t s e h

t [J] .IEEE Transaction on Evolutionary Computation ,1999 ,3(4) : 7

5 2 -271.

9

[ ] L iK. ,Deb K. ,Zhang Q.F. and Kwong S .An Evolutionary many -Objective optimization n

o i t i s o p m o c e d d n a e c n a n i m o d n o d e s a b m h t i r o g l

a [ J]. IEEE Transaciton on Evolutionary

4 9 6 : ) 5 ( 9 1 , 5 1 0 2 , n o i t a t u p m o

C -716.

E r e l z t i Z ] 0 1

[ . andLaumannsM .SPEA2 :ImprovingthestrengthParetoevolutionaryalgorithm[J] . 1

: ) 3 ( 5 , 1 0 0 2 , n o i t a c i l p p A d n a e c n e i c S r e t u p m o

C - .2 1

K b e D ] 1 1

[ . and Jain H . An evolutionary many-objective optimization algorithm using e

c n e r e f e

r -point-based non-dominated soritng approach , par t I : solving problems wtih box s

t n i a r t s n o

References

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It also provides tools for training and knowledge capture and sharing and integrates with a wide range of other tools, including identification and image libraries,

With over 1,000 software engineers Comtrade offers industry-leading expertise in data storage management, embedded systems, network systems management, gaming

คุณสมบัติของผูรับจางงานกอสราง 3.1 เปนนิติบุคคลที่มีวัตถุประสงคเพื่อประกอบธุรกิจงานกอสราง และไมมีรายชื่ออยูในบัญชี การละทิ้งงานราชการของกระทรวงการคลัง 3.2

Video, because of its adaptability to many devices has a prominent role in learning..  What is the best use for video

Consequently, improved result was presented (Bas et al., 2013), temporal information was utilized to partition the universe of discourse into intervals with unequal