• Aucun résultat trouvé

min f1

N/A
N/A
Protected

Academic year: 2022

Partager "min f1"

Copied!
154
0
0

Texte intégral

(1)

!" #

$ %

&'' %

$ (

&

$

) * + ,

&' !

-

." %

/ (

/ (

/

0+

)

/

)

'"

' 1 2 #

#

(2)

! 3

&' ! 4" 5 (

6

7

!8&

9!)

.:8& #

! 3

&' (

$98& (

)8&

8&

!

)!& #

)!& (

)8&66

&!);!)&

< '

9$ 9*-!

&!$9

= 45 #

$ - %

! "

/ #

! - &!$9 #

$ " - #

-" < <' %

- ," 1 2 %

- '" %

%

#! $ % & '"

6 4Æ5 %%

) %%

$ 3

: >4 5 3

(3)

( # ))

33

: ((

=@ + (

: (

/ 7 7 (

(

" * +

# $

#

!

# < 7

# 7

# : #

# ) - #

# 9 #

# $ %

# %

# ' 3

# &

# / + 7

#

' *

# , "

% 6 4" 5 #

% $ %

% < , / %

% * (

% > (

%

% &4'5 45

% / 4'5

% 45

% 94 5 4'5

%

# )

- ./0 1 -+

(4)

(5)

7 "

" ' @ 7@

- '" *-

> : " >A , +

*-@

*- : +

: " -* * 7@

" :

@@

:&'

! 1&!2 "

" 4

5 @ B @"

' " &'

!

'@ ' + *-

- ' >

+-- @ '

" - 7

+7 @ "

- '

C ' &!@ " +" 7 *-@

" ,' >" 7"

7@ 7 7" @

" 7 7

= - &'!

$ 9*- " ,

7> +

:

9$ ! @ 7

&'

!

6 ' @" -

'' 7 ,

: 71 2 7&'

!

(6)

-D 7

33E" &!$9

' '+- 7"

@ Æ @

> 7" @ " "

' &'

! *-@ "

*-@ " " -

@ " *- @

- '

* " * @

C " &'

!

7 > ) *-@

- @ &!$9@ >

+7 , * " @

> &!$9 6 @ @

- @

@ @ -

: > '

*- " >

' 1 2

: > -? '

Æ - ' @

- " - @ @

@ 1

2 @* " @ ,

" &' !

: 7 1"

7 &!$92 -? 45

@" @"

4 '5 -47 5:'

&!$9 -

: 7

: # %

-

> * :

7 / < 6'@

7@ *-

: B * 68!

!=:" &!$9 - *

' - <

C'

@

& B " > @

* -

(7)

#

@@ -@

> :%

< "

:> ,

' " B -?

(8)

%

(9)

(10)
(11)

!"#

''@

-*- : *- @ '@

7@

12F

12F

12

( (

" ' G

-*- -" 7 "

7

min f1

min f2

0.1 0.3 0.5 0.7 0.9

0 10 20

5 15

Solutions compromis (x2=0) x2=5

H $% &

7@ 7 -D

F 1 I

2 : 7 7

@ F( F : +'

(12)

1

(2 :

F( '

" 7,

" "

7 -

1 2" "

' *- " " @ ,

" > -

" @ " -*-@

7 J%% K@ J% K@ -

7 - *- :

> - ' D

12 F L

( F L

F( FI L

F

1 2

-

: F 1

2 :

-

+

1 : ,,

" ' '

8@

" " '

, 4 '5 , 4- '5 >

- " > @ "

- *- 7 7 )

- ' '@

@ 7 @ " +

1I2 : -

2

#3

" " + - 7@

H : D

J( K ' D

(13)

)+@ ?'' @ +

- 7 @ -

.

+ + "

: > -

+ H7

/" " -12 H7@ " 12(

7

H : 1 2

-*-

7 '- 71

-

7 -

2

*-

"

!"! # $

1

:

1 2

@

+D

1

2

1

2

"

1

2

1

2

)

@ 3

3

min f 1 min f 2

H ! "

"

@ Æ

1 +' 2D

H

L

H

L

H

"

@ " -"

(14)

: @ " + @

H

@ H>L

H

@ *1

2 1

2L

H

@ - 7 " L

H @1

2 ;1

2 " 1

2

: J# K

"

¦

"

1

H ) *-

: @ "

@ - @

" " > 7

" "

-? ' 7 ' 7

*-@ ' >

D

H : *-1 2

" - ) @

' "

H 1 2@

H :'

1 *-2@ '

H @

7 ("

7

1 - 2

+

!"!" $% %

: + -

-? " +$

(15)

@ @

@ @

+ " , *-

+ 4 56

@" * "

" ) @ :

7 1

2 J K

+ 4 !

@ *

" ' Æ H

" >

* & @ ! @"

7 = @

) @ * : 7 4 5

'>" @1

2$@,@

J K

+ 4 Æ

'@ M ' 3# J K@

1 " 2

D

12

H )F @ 12

H )

F 1

2 F 1

2

"

@F @

H

@F

/ 12

: 7 ' (112

2 @ (1122

FJ K

!"!& ' % (

:

1 2 "

-

1 2 1 +' 2 8@

" " - " -@ 7@

(16)

min f 1 min f 2

Domaine de recherche

z *

z nad

H #

$

&B " - " *-

-@ $

-

- 1 +' 2 Æ

" @ % J# K

>

: 7@

- 1@ 7 2

" *-@

-? D

F

!"! # %

Æ

- 1 2 ) " - *-

@

F @ " -

@ F @

@ F I @

> B

# 7/8+

"

1 2@ " 7

( ( 1

@

F (2 "

+D

1

2

1

2F(@

1

2F( F

: > @ 7@ J K ! B@

Æ M

= N *-

# Æ 9+9+-:

) " -

@ F @ 7@ -

@ F

7 -

@ F I 1 @

*-1 2 >7 + 2

(17)

1 2

- J# K

4

5 " @B @

4 " 5 *-

6' P< 3 J#% K

= P 7 *-@

' - " 4* 5 *-

: ' * " " * @ 1 2*-

4+75 * Q *

*-@ , , " @

$

)!@ -*-

1

2) @"

D

!F

!

" !

!

F F

& @ ' 1

2F13

3

2

: 13

3

2 $ F

13

3

2F -

13

3

3

3

3

2

: * " *

" * :

>@ " @ 4 ''5

8@ " Æ

" >>@ -

B: "

1 2 "

" * D

"

F13

3

3

3

3

2 "

13

3

3

3

3

2

13

3

2

6 @ "

" *

B ' " "

>

+

3

"

F( :" @ -@

"

@F

C -? ' 1 " * -2

R J ( K "

@ &'!

(18)

6, "

" > " @

1 '" - 2 "

@

" "

*- =

- *-

? 7 D

'' *-@ Q "

/" - 1 2 +D

F"D

12(F L

12F(FI

!! % %%

@@ ' " @

*

- 7: Æ ' -

*- $ +7 -? @ -

" +-" 1

- J% K2@ -

*- " > '

)

#

12F

$

12

$

( "

$

F : 4 5

1 2 @@ D

#

12 1 2

6 " +

1 2 1 2 1J# K@ J# K2 @

7

-+! : 1 2

*- 1 2 $

-

-+! ) *- 1 2 7

@ 7 - 1$ $

2 "

1 2

(19)

3

: 7

+' 12! B@ 7 *-

45 # F%&' 1 2

# "

1 " ' + $ $

2

> ' -

1 2 :

1 +' 2 1 2@

1 2 1$ $

2

min f 1 min f 2

w w 2

1

A

de recherche Domaine

min f 1 min f 2

B C A

Domaine de recherche

12 12

H ;< &

'! ()'*+';<!

, -

$ $

'

1

@ - 7@

"

B

@ @ > - "

- ' "

@ @ - -

Æ 7 -? "

@ *-

6- @ @+

B 1 2

!+@ - ' 7@

' >

@" " -

+' 12D , ' - " (

" > # -@

(20)

!!" % ¯

! 3# @ . J K '' - 1 2

- D

12

12

F F)L

1 2

)

+ ! @

- " - " *-@"

-

7@ " @ 7

*-@ ,

- 12

:+' B

:

min f 2

C A

Domaine de recherche

ε 1 d ε 1

a ε 1

c min f 1

B

ε 1 b

H &

F 7D- 1 2 -1 2/

1 2 -"

12

+' @ F @ (

! " @

D

-+! F 1

2 " 1 2

@ *- 1 2

1

:+' 1 2 -" B@

'7 - ! 7

"

: 7 Æ @

"

> 7 *-

7@ F

1+' 2 @1 2 -

@

> 1

(21)

!!&

@

1 -?2 *- 1 2 - " @"

=>

12F

$

12

$

J( K"

$

F @

*-

1 29 ' "

1 4 " 5 2 " 1 '

2"

: " @ 1 2

- -? D

12 1 2

Q F@' .% " -

*- D

12F

7

$

12

: ' 7 ' 7

- ' !

=,B B @ "

@ " " @ - *-

B@" 7

' @" *-

+

=+ ?

" @ , +7

-

*- -

'- ! @ D

7

*1(1

122

12

F

1 2

&B" @E

7 $

@ '7

> @ * L @ -

(22)

C -? 7 -

- D

* 1122

1 2

" " >

- > @7@

M, /B J K

!! %

= = @

7 " -

H-B

*- &+ " @

- +

7 G - "

- " @"

-

$ > - B@ > >

7ÆC

'7 -

$@>" @ 7@

=/

!) 7 -

@ 1 " " 2

" " @ - ' - @

- Æ '@

" @

! B@ * ' '

@ R >

!

" @ 7 @ '

7 , 7 +

: " *-

- > @" &' ! @' -

" 7 '

"

@" ' @ -

+ - @ >

> ! @

(23)

7 @ @ @ "

45 7

'

@- 1@

, - 2 " " - 7

@ @ - *- - '

' :

+ @ >

7> @ "

B

! @ , ' 7

@ " 7' =

7 > 7

-

(24)
(25)

' (

)* *

)

&' ! 12 B

: ' 7

&!@ '@ ! @ 7

@ @ " 4'" 5

@ L

&!

"# $

:&!'" '

0 ) 0@

' @

!

@ -

: + 4 5 +

@ @ *- :

- " @

- -

*-

: &! B <' 3 J K@ /'

3# J%# K@ . 3# J K@ 3%3 8'J K :'

' "

&! 4 5 >A -*- -

(26)

"!! '

@

"

H

H 1 # / 2 "

*- 7 +

H -

H +1 2

H - 1 2

H

@@ $@

-@ '+

H "

'- 6'

1'2 7

H + ' "

H

7 -

H ! -

-B -

Æ '

' - &'! +'

000000000 000000000 000000000 111111111 111111111 111111111 000000000

000000000 000000000 111111111 111111111 111111111

00000000 00000000 00000000 11111111 11111111 11111111

00000000 00000000 00000000 11111111 11111111 11111111 00000000000000000

00000000000000000 00000000000000000 11111111111111111 11111111111111111 11111111111111111

0000000000000000 0000000000000000 0000000000000000 1111111111111111 1111111111111111 1111111111111111

000000 000000 111111 111111 0000000 1111111 00000000000 11111111111

Parents Stop ?

Enfants Evaluation

Initialisation

Remplacement

Sélection Meilleur individu

Evaluation

Croisement, Mutation, ...

Initialisation et opérateurs stochastiques Darwinisme (stochastique ou deterministe) Le coût CPU

H+' ,

? +7 " @7@ @

(27)

@ > :

" +7 '

: &!

+ " &

@ " 1 2" -@

:

@

4-5@ " > + "

745-'

" @B @ &! +7

:' '@ @ '> 7

405@ " 1, - - 2

' " 7 " 7

7@ >

>

-

: > > +7 @ "

-

@A +A

: &!@ @ '" @ @

>

B @ " +

-7@ ', > @

," 7 ,

: " - , ', > *D

" , ', : - ',

, : -@ '@ -?

', : 7

' @

'," @ " " -

3 3

& ' @

& 7 @

" Æ &!

: 7 7 B '

: 7Æ -

7

-

= -@7 @

" ' :

'

(28)

: 7 7 ' -?

+:7 '

-@77@@>

'

6 - " 7

7 O ' Æ '

=," @ 7@

" 7 1"

7 ' 2

!B@ @ " '

$ '@

" " @

6- "

-

"!!" )*

:&'! 7@'

" D

* + @> ;@<

:&8'

6 7 . J K : ,

@ ,'S'@

, '> J @ K

:&8" " -P'

3#J K@ " - " @"

"

' " " ' 7@ @

' - 8' J K@ " &8 - >A

+"

* 2 ;2<

: )' !

: B )!

3 E ," J3% K !

3 @/' '1 I 2!)@" -

8 1 2 6 '

J%#K

' 1)I2!) 1)2!) 1 2

)0- 3## J33 @ ((K >@ 7

)

" B ,

(29)

* ; <

: ' ! @ :P <' J K@

+ 1+' 2@

: +'>

- : -

, "

' - @

- @ " ' 1)I)2

' )!

:! 'E<'@+"

J( @ K@

"

1/a 0/b 0/c

0/b

1/c

1/b

A C

B

H 0 / % F ("

+ F( " F*%' 1

" Æ D+ /

1 " / Æ11&

22F1&

2'

* @> ; @<

: ' '"

'' 3% J ( K@ - '"

'''" :' ))8&1-2

: +'8"

' - P MO 33 J K ) *-

- ' '':6)1+' 28> 'MO J K@

8 7 ,

>

@

(30)

and

d0 not

if -

and d1 d0

or d1

d2

d1

H 0 2 !&3 ,(, ,

" *,- !& 4 '

%

"!"! +%

: ' ' &8 J K " 6

1 >A2@" ( D

+ F1* *

2(

1 2

" @ !"

" 1 DÊ

Ê2@

+ " '" Ê

(

#

: 7 &8 ) >

', 7 7 7@'

-' 7 ', 9 ' @

D

H

"

# J KD

-' ' L

H

"

$ J @ KD

7 -'' L

H J ( KD

9 ' " ( 6

>

(31)

Croisement 1 point

Croisement Uniforme Croisement 2 points

H ! 3 ) " * '

=

" ', -

@ ," (@( (@(( 6 + '"

1@ " '" >A2 :

D

H % & 5'J3 @ KD

"

@ %(

,

H # J3 KD

,

Mutation

H : D: ,

"!"!" +% %

&@Ê

12

Ê

12

)' ! J33 K@

7 , &'!

@',+

,D +F

F1

#

@, 7 7 7D

7 1 2 7 1

" 2 7 @

H ( D

F

(32)

. 7

H (

&%

'D

F/

I1 /2

/ - J( K 1/FJ( K2

. 7

C E:T0 146

52@ ! )B J% K@ " '

+ -@ ' / ! B@/

( @1021 I02@ (0 1+' 29

" E:T( "

p1 p2

φΙ I φΙ

H 6!70'

' ' J ((K@ )0- '

, '

-@ ' "

@ ' ' - 1

2@ 7 , 'D'

'

9" 7 J K

H ( D

F

9

" , "

H (

D

F/

I1 /2

/

FJ( K

.

7

(33)

=

H

: " * 8

7B

:Æ *

, 1 ' ! B@ @ 1

>O '- 7

C - - @ '

>

" @ 1 >

@ ' *

@7@'

/'J%# K :' )0- J ((K

H

$0OJ# K@

1 S

2 @"

F1

2@

' -

F 1

2@

N

J K

@

' 2 J122 122K@ +

, @ "

&

"!&! ,%

, ; <

: &8' @ 8

182 8' J K " " 7

@ " "

: '

-1

2D

: @ ' >

1+' #2

:

7D

F

1

2

1

2

: 7 *1

2

- @ *

> @

1

2 @>

(34)

#H 9 1

2F( ('

" " 1 "

'

&+ @EN 3%#

" @ 1)C)2J K& @ -?

8@ " @ "

- $ -@

' " 1+' %2 : B-

-

: -@ '" - "

8

% H : 3 "

" 1"

'

-

: - 7 -

@ - "

7: ' 3

(35)

* " " & '@ ,

" : *

132 C3 -

:' " " 3@

7 # R 7 @

J K

67 @ EO

J (% K@" "

EO

@

''

@.

"!&!" +

,

: ," )' ! )

>- " '

O 6 ) 1 ) 2

D

H ) -D 1)I2!)@

H -D 1)2!)

:1)I2@' -

@ ' @

1)2@ - >A@ '

G7 ' @ '

- '

7@ 7 1)2!)

)0- J ((K

,

C " ' 12@

&8 & @ -

:

' - " @ P' J K

+41222+ 7" '@

+ ' 4 F

' C " - 8- 3% J K "

7 4 " 7 -

"!&!& ,% %

C , ,

= = -@

(36)

7 @ "

" @

@

H * + > 1 2 ; " 2;2

' & " '@

"

- '"

1 2 !+@ -

'

H *

7 , )' ! D 4!)I5

1)I2@ 4!)@5

1)2 & " '@ - '

' -1 " ,

02

9 "

H * + 1 2 ; " 2; 2

@" '@ 7-

7 - -

@ 1

2@ @ >'1 72

" )'! 11)I 252

) @ /' 3# J%# K@

-@ "

H *

: ' !)I@ " "

@ !@-

" D1 I25

$ $ #

:" ,

+ &!D

H ' @

H + " " -*-

@ @ 7@ 4 5

1<2 J K@J% K " ' 1 2

" " "

(37)

"!! $ - .

: 8'/ 3%#J K

: ' - -

' : "

&@ - 1 / 2 #

D

# 1

2F

#1

2

&1,1

22

& ' - ' 1 2" , 7

! 7 " @ (

11 2 '

: - ' D

&1,2F

1

! 2

"

, 1

(

1 2

/ > - ':

7 -'," 1'.'2

," 1' ! 2@ @

7 - 8' 3%3 J 3 K > "

' ," '

: ' " 7 11

22@ "

7 7 &+ 7@

U8,@ 33@ +

' J K " " @" 7

1122

& J K@ "

' " '@

- 7B

@ ?7 1 2

F (&!@

, ' D

1 2@ ' -

@" - ,1 2:

" ' " @

" "

&! $ 9*-

"!/! % % %%

' '1 2

(38)

4'5 7 @ ' '

- - - 7 ' -

" @ 7 ' - >

' " D

Æ

12F

*1

12(2 F

12 FI

12

! - " @

-' D

#$

12F%1'2

%

122 12

,> " "

Æ

12 Æ " + @

#$

12 F (

: 1 2 F -

D

#12F12I

#$

12

F

: " " ' B -? +

- " " 7D

H

J KD

#$

12F

-* -

" -" ,O -

H

J KD

#$

12F

%

Æ

12

Æ+ " :

- +

H

D

H : ' J% KD

#$

12F1('2

"

%

Æ

&

12

(39)

H : ," J# KD

#$

172F

7

%

Æ

12

7 ' - / ) @ "

" '7 " 7

+7 :

'> " 7 ' 7

H

D

: ' -

"

> ' - "

1JK@J ( K2

"!/!"

,++

9 7 'D

- ' -

: '' 33 $0O O, J#K

- 7 D -

1 7 > '" +"

" 2 -

- *- L @ " 7@

45 1 - 2 >

: 7@ 33) TNJ3 K@4E

,5 :' -

" @ -

* " " ' -4 5

> - 7

= ' +"

1 -2 " ' - "

-@ 7 @ ,$0OPN0J#K1

, 82@ $0O ) J3 K@ J3K J3 K 1

- ' -2@ MO $0OJ K 1= 2

=+ +A

:" ,"

-*- 9 7

:

(40)

1J K@ J## K2 & 7 @ &! > "

" *- 1J%K@ J ( K2@

1J% K2

' "# $ $ (

) *

4" 5@ '"

* " " '

1J#@%% @ K2' 12

! @ ' +7@

-@ '' " &

" @

= " " B

-? @ " @ "

: " > 7 'D

':@'@' @ '

-*-:' ' O

:

@ 7' ' Æ - *- :

7 Æ ' @

- 7 7, "

:&' ! @

''> "

6"

" 9='" <1J# @%3 @ ( K2@=6'1J @% @ % K2@

+1J3@ @#( K2

(41)

(42)
(43)

! > '

2;"6:" ?@ A0:'

# B

Æ '

! > " $ '?@

+

@

@ ' @

B 7&'! ' " '

*- @ 7@

@ " + > &!

*- + 1

2:>+

'@+

- *-@

: " " -

& @ " 1212 0 '

*-V

! B@

7 1:

> - * 2 ! &'! @

- '

" @

> " 1+

2@ 7

(44)

- " ' 1<' 2 "

1 2 " 4" 5 &!$9

- B

$

VC - @@* -?

'" "

@ " +

&!$9

! B@ B

' 7D '

- @" " "

' - "

' 1 +' 2

min f1

min f2

1. Dominance de Pareto

2. Maintenance de la diversité

H ;0A@

7 + &!$9 * "

* J @ @ (3K B " '

'77@< <'J K' '

4 *-5

@ WO

J K ' -&!$9

! B@

9$ *- ! :

' '@

" " &!$9 *- !B@

* @ *-@

Æ -

$ ,

&!"! '0

! 3%@)B&!$9

(45)

'

B*-

: D" '

1'2" ,*-@ "

- *- :

" @ -

*- ' B

)

*+

) *- F 3 F 1 *-2

F I1 23 F3@ -

#1

2F

1

2

!B 4 51 2 3

)F@ )@

/ D F

&"

6 " ' "

*- &+

@ >)B,

" 7 *-

7 *-

B@

:' )B - R

) " " ' 7 > &! *- @

" >

!8& 7@

B*- C !8&

J% K

1 B@

: !8&- " *-! >

4

5@ " O '

@-? - 7 ,!8&

1

'2 " '

7" !8& '

- @ " @ -

(46)

&!"!" ',

! 33(@ <N M 0 9!) #@ B 0

J # K @ )' ! +

+ > *-

C 7 " 45

45 &@ 7 D

: + B -?

: - 1 *-2

& " @

+ *-

>

+7 ')

-*-@

- , 1

2

1

2

S D

#

F

1

2

I

1

2

)'! @ )

@ <" '45

45 " S

&" @ *-1 2

" - "

- " : )

D

)F1

2

= ) 7@"

N ! @

* )

'@ '

" > @

'" L '

1 B2

9!) '@

7" $9!&$ @M 0

" " , 7

' 6 Æ -

7

'

6 " 7

" " "

(47)

&!"!& )10

! 33@.*:.:8&J K@

- *- & B

@@ B

"

:

> - 7 -? ) -

" '" @

B+

+

@ 7:"

- " &+ "

' @

D

$

F

$

$

: 7

:

"

' + " * " :

" 7

C

"

#1

2F

$

1

2

1 2

6@

7 *-

&+ @ " ' 1 2

: '

@

'+"

,

D

,

F

:- ' 1,

2 1 / "

' &@ 1 F '+

" >

' 1 F

7 "

B@

: 4 5 %

" -

(48)

" %

" -'

,(+

-*- @

F ,

@2 F !

-' D

1,

2F

!

,

1

(

!+@ 4 5 %

F

1,

2

F @ -#

"

' 'D#

F#

%

Q # @

B &8 :

" 7 @

.:8& ' &!

*- @7 " $9!&

: *- '

' > - D '

- @> -

- -

7 Æ

+

!8& @ B

$

@ 2 F @ ! @

- :

7 ' 1" $

2

'" @@

7 : " $

@ " " + *

& ' @ " $

6, @ "

> ! @

7

'@ '>7

B 6 '

,(+

-

(49)

- #

" "

$

#

!B @ " 7

)2 @ 2

)@ F

)

@*

" @ -@ " >

@ 7

' '> '

&!"!

! - &!

- @ " " '@ - B

7

451)B2 6 '

' -

=," @ Æ 1 2

7 ' - @

!8& .:8&1<' 2

f 1 f 2

VEGA, HLGA

H C ;0A@

!" 9!)@" " " !8&@

< <'J K "

7 -1 *-" 4 5

2

: '!8& .:8& 7

- B &!$9 1@ 7@ J K2 7

1 2@ ' 1 !8&2

>- 7 @

(50)

-> R @!8& > -

'" '

"#

:7 --?*-

C - - -

7 7@ +'

" B- >

1 (x) ] a [ f

2

a = 5

a = 9

a = 1

f (x) ] a [

H $

-

" - @ ' 1

>2 ' 7 -

'+ " - N'

L * ' -

- " $ N' B

7 -

&!&! 0

< <' J K '"

-

'@ " @

@ '-

F1 I

2 1<' ' 2

: ' " ' ' 7

>

'" : -#

"

' : - " - ' -

" > ' > - -

7

E " 4- 5 "

*-@ >

<<' ' >'@

(51)

D

,

F

7 -*- 2

:- '1,

2 " /F

4 5 - D

%

F

1,

2

)1-

2 '-

!+@ -#

' ' '.:8&D

#

F#

%

: -4'5# > >

'@ ' , -

= ! 1

7 *-@ ' 1 >

-? - -

< <'J K '' 1 "

'

= @ 7

" *-

:'

, 8 F

1

2

+@ - 7

) " -

: --

* ' 7

!+@ " 7 " * -

,'+1

21

2

1+' 2@ -- -@

,'D

F

1

2

) " *- @ " > 1

*- &@ - -

7@ > , 1

@

1

(52)

(f min 1 ,f min 2 ,f min 3 )

(f 1 max ,f 2 max ,f max 3 )

f 1

f f

2 3

H " F

& @ @ 1

" D

F

1

I1 2

1

2

1

1 (@ " ,> '

.+

1

6 )1-

2F( ' F ) F

@

@ @ '-

F I

6 ' -

D)1-

2)1-

2I

)@I )@

6+ ' 7 -

-

7 " )1-

2 (

F @ - D

#

F

)122(1)1-

2 2

) -F '

' -@ 4 5 %

> ' @ @ - 4'5 #

&+ > -

,@ #

-? D

#

#

)1-2

#

#

(53)

#3 =@

: - " - 1 2

7' 1

2 ,

7 '@@-4'5

> " 1

2&@7$98&1

2

1 =@

: -? - $98& $@ @ "

' @ B

> - 1 2 >

- " " "

! @ ' > -

-

: ' " ' 4 5

- 4'5 # " '

, ' '

@ , " Æ "

'

&!&!" (,0

:8' > &8J K

R ) 33 )8&$

2 ; J ( K ! -@ *-

@ @ - " -

@ @ " '

>-

)8&

-? D

L @

: "

D

F

' +

" 7 > @ >

*- : @ -

" @ : +' 12 7

+1 "

" 2 N' $98&

Q @

7 -

: - ' L

- ' - =

? -# F

(54)

f 2 1

1 1

1 1 3

2 5

f 1

f 2 1

1 1

1 1 2

f 1

2 3

12$98& 12)8&

H >3 A@2; $2;

&+ > " 1@ "

> 2@ - #

'

- 45

' )8&@ '

@ 7 >

-

"

,

F

- '1,

2"

/F:4 5 @+@-#

'-? '' D#

F#

%

+

1 - #

FI )

- F

= D

1

2F-'12

3

H : -# '

F#

H %

'

-'

H -4'5 # '

F#

'

%

'

#

F1#

'

D3

2 1I 2

) -@ )@

> &8 *- &@ )8&

(55)

#3 C2@

: >

N' ' : N'

1

2 : ' "

>-"

"

B

1" 2@ 7 1

2 &@

> '*11

2 1

22

1 C2@

:' )8& " - 7

" '

- ' -

@ ' '," :

" ' >' " *-

" '+71 "

G - 'J ( K

&!&!& ($0

! > 33@ . J K @

8&$ 2; B'

8&@

7 " J( K 7

*-

C ' * ," J%( K 8& 7

-? ! @ * @

. '

12

. : D

H ) . "

@

H ) 7 .@

.@ 7 - 9

12 4 59 : @" 4

5 ''

& @ " -@ '

Q 7

@ 7 " !

7 7-@

'

" B * " @8& ''

- " :

"

(56)

)

+

/ ) F 9F

!B @

F4' -*:'1I 92

1I2

F4' -*:'1I 92

-% %

"

7

$ * -99% %

1I 2)@ )@9F @

@ ' )@

#3 C @

" @ 7' 7

'

" '

'

@

4 5 9 "

-12@ 1

2

&@ > '1

2

C @

)'

" @> "

*-@ " 8&

* ' *-

1 C @

8& + 7 1 '

9@G

1 ' 8& " )8& ! B@ %

" - 1

: > 1" > ( 1 2@

: '

' ')@

" ' 7 ' )@ @ '@

> '

: * " >

@ ' -

"

@ ' *-@

(57)

- , - ! B@ &!$9

-" - 7 1"

2

' -

:' B -? 4 5

' @ '

&!! ,$

! 33%@ WO =

" 0 ; J ( K :

7 7

& " @ 7

7

6" )!& -

7 '"

'

7 1 452

& " ' '@

#

7

#

@ 1"

>A - * 7

#

2

Q

#

@

* @

#

@ :'

* 7

& *

#

@' :

- " "

)!& * @ )!&

#

'" C - >

7 7

# :

- '

1) 2 :

1

2 "

7 D

F

I

12

: 1I 2 ' "

- @

7 ? -

1 2

! @ - '

4'5 7 D

#

F I

12

:* ' " -

"

-

(58)

f 1

2 2/7

3/7

7/7

9/7 14/7

10/7 10/7

f 9/7

2/7

H 0;

: B

#

-? - --

)

+

= 1

#

2

#

#

#

1

# 2

= 1

#

2 7 +

#

D

#

1

# 2

)

#

@ " * "

: 7 '

#

-

#

"

#

"

&" @ 7

#

#

#

&+ 7

#

#

@

@

#

> 6@

#

! @ "

,

F

(

(

(

(

,12

: ,12 >

@ - )!& & @

2 " ,

F

,

- '=

,

7 -

: * " " "

@ ,

>

(59)

+ %

= D (

F )( F( (

)( @ )@

@ " =

1

2@ "

< (

(

/

" @ ,

>

:' - ' >

11

# 2

2 ! B@ >

1

2! @" '> 1

21

# 2

- $ -@ > " * "

12 O 7 " N -

* " - : , >

@ R ' '

-? " 7 1

2

#3 2

: " -

#

@7'1

2

) >@ > > " : '

> 1

2 : 7 ' :

> )!& 1

2

2

6" )!& @

7! >

* : ' )!& "

- " Æ

1 2

:" 7 G -

') '@, " '

@ @ B > @

7 - >

: 7

" 7 " C

* + "

" - @

(60)

!+@ ' "

7 @ " >

- +

&!!" ,$"

)!& ' > J %K )!&

B

= @ )!& +7@ , Æ

@ 7 " @

" @ " @

: -)!& 4Æ5 " )!&@ "

@

#

# D

F

#

#

! @-

1</ 2

+ " D

F

E" -? - 7

+@Æ "

> :-

-

)!& " 42

5 J ( K@

" - 2

)!&@ 2

7@ "

#

#

*- N

2

1

. @ 2

@ 2 F I :

" " D

"12F

1

I

: -+ )!& D

#12F 12I"12

*

#

@

@ 7 " # - @

' D

F

#

#12

(61)

)

#

F @ * )@1

# 2

#

#

-@'" )!&"

4' )!&5

&@ ' " > )!&

)!& " @ '@

7 7 - &@

'@ "

#

*

* " "

#

F

C B 7 " )!&

&!!& (,0

! (((@ J % K &!$9 " )8&66

" ' ! -@ "

-" )8&66

)8&66@ -9

#

#

! @

# F

# 9

#

@"

: -

#

?@@- "

#

@ - >

' " -

>

@

6 " >

G $ '@ -

#

" @ " '

+

: )8&66@

@ @

& @ )8&66 "

'@ <

7 > ' 1 2

: 9

7

)

+-

&

@ )8&66

D

9

#

#

"

'" ''

/ -

# F

# 9

#

=

#

-

) F ) -F

(62)

="

# I

@

#

#

1I 2

9

4 5 1< L

2

#

,

'

#

6"

#

-- '

#

> B " - " -

@ +

#

) @

N' >

#

@" ,

@ 4 5@ -?

-

) F ) ,

F( ) *-

F

*-@ -?"

)+

F&-'

)

12 @ +

*-

" " +

1 2@ * ,

,

,

I

*

*

! ' 7 7 @

+

F +

F@,

F-

) F @ @ *-

1I 2

:" ,

7

1+' #2

f

f

2

1 i

i+1 i−1

d

i

i (1)

(2)

d

d

i = d (1) i + d (2) i

#H <

(63)

: 7' -

-@ " , > 12 : > 12: >

12

#3 C2@

: )8&66 -@

" 7' 1

2 :

#

@ > 12 : 7

)8&66 1

2

C2@

& B ' * " @

, )8&66 7' +7 " >

1 C2@

* @ )8&66 "

' -

& @ ' 4 5@ >

"

)8&667'N' *

'

&!! $, 2 $,

2

> (((@ M0 '' &!$9@ &!)

; 0 J K " " 1 I 2!) 1 2 :

)'! 7

7 !

*-@ " '

'@ " @ @ "

" : &!)

, , ' *-

@

) %

-

& " ' '@

#

- %

#

@ &!)'

:>

:

# %

#

7 ) @

' '

)

#

%

#

@ %

#

) @%

(64)

#

-' )@@ %

#

" "

@ %

#

"

' )

%

#

- - 1

2@ "

6 " @

# -

@ " @

'

#

%

#

: &!) B "

* " =

1

2

, $@ , + :

" , : +

" > , 1 +' %2

:

Æ '

f

1 2

f

c p

t

t

%H ;0

! @ * -

= @, 7 +)

" 1 2, @*

" , 4 5@ @

#3 2

! ' &!)@ 12 -

" " 1 2

: , @ , "

+@ " > 1,2 = >

' &!) $ 7

&!$9@ - " " ' &!)"

' )8&66 )!&

2

& &!)@ > !)&

(65)

" > " '

!)&@)!&@

'"

+ % +

8 45 1+2

*

C 47

5 152

$ * 152

) > -@ 152

)@ 1+2 * " "

*

7 'D

H &

(

@ 7 152@

- -&1

(

2@

152

/

: * 152" &!)!

1+2@ @ 7 152

: "

4" O-5" @&!)@

> , " " ) ,@

7 @ 4" O-5

''

1

7@ &!) !)&@7' + 7

D , : , '

7 ' ,@ " >Æ >

- @ ' " >

, @ - >A ' " - *-@ "

Æ

,7&!)@ !)&> 1,

2

&!!/ 3

: 7&!$9 J #K -"

- @

- 7

6 Æ " - ' )!&@ )8&66 !)&

7 WO J % K " 7 '@

!)& - @ - + )!&

)8&66> "

!)&> @ @

'@@+ @Æ 7

(66)

6 - " , &!$9

7 " -1 - *

2 : +' 3 " " &'

! $ 9*-@ - 7

'

MOGA NPGA HLGA VEGA SPEA2

SPEA

NSGA−II

NSGA PESA

PAES

Deb Zitzler & Deb

Zitzler Corne & Knowles

3 H % ;0A@D 5

" @ 7 >

R @ 7

7@ @

)8&66" :' )8&66

1, !)&2@

- .-/$

:

1 2 FF(@@

@

" Æ-" " &!$9

" - '

" 7 7

-

&!/! $%

: *&! *-

- 1 2 ' -

7 &!$9

(67)

#

12 F L

F

#

12F

12I

#$

12

6@ -

#$

+ -' '

#$

12F%

1'2

%

122

Æ

12 '

1 " 2&

B *-@

Æ

F

#$

: Æ %

1'2 B

B*- +

#$

> ' "

1

: * > 1 2@ @

+ %

1'26--" &!$9

- *-

7

Æ *C 7

- *- J K@

122FÆ

12 %

1'2F%

&!/!" 4 56

PO J# K *-

" - @

''

7

@,D 7

-@ 7 -@@+@ 7

@ - '' -

@ "

" -@ 7

1 7 , 8& 2 D

/

C - * >

)

7 @'' ) 7

" @ " '

'' '

/

@ -

(68)

- :

>@ 7@ ' -

" " @ ' *

" 7 @ - '

" /F

: 7 1

2

1

: " 7 * 7

' -

+ " @

> Æ

8&@ 7' + 7

, '

" PO 1 B

2@ - - @

+ 7 "

&!/!& %

: " B 1

'+-2 > , -@ "

> +

B N' "

*- 1@ -@ 2 '

)!12 1 2- :-

7 - 4, 5

@ ' "

" !12F( " " !12( 1'

! @ Æ " ' +

!122

: -

)8& $98&@ " "

6" " @@+ !

,

< <'J K@

!12F1Æ 12Æ

% 122

Æ

12'

+" :

(69)

:' !12 F !12 - "

7

@ - >

D

H )!12!12@ " L

H )!12F!12@ *-

@

F

H ) !12 F !12 "

*-@" ' >-

*- "

D

! 33#@ < <'@ E M

J K@" - D

!12F

%

Æ

12

: " " @

E M ' '' '

- 7 " , @

- " - @

7 @ 7

- ' - ' -

D

!(( @X' : ''

&!$9 " & @ ; J K !

" -

D

!12F

%

Æ

12

: < <' "

- "

N' -

1

! B@ < <' > N'

- @ B

'@ 77' *-

-@" > -*-

(70)

@ ' -@

- -

" E M X' :

' * + 7 " !12 @ -

" @

-1 2!B@ 7

'@" " @-Æ

*- *-

(71)

"

'

C >

> "

9$ *-

! 6"

* " :, "

-

% ( .

6 - Æ @ "

! > *-@

, - @ "

" B ' -

" ,"

:"

> 7 ' *- C

" +

B

% 56 - .

6' "

" @

AA *

" *- = '

' - 1+' 2B >'

'> , @" 1 2' Æ

(72)

#

min f 1

f 2 min

f 3 min

H :

+% 5 6 - .

@ 8B 3# J3 K@ ,

" *- 1 +' 2 :

7 O ' + *- :7

7 -*-" '

4 *-5

' : " ' 7 *-

'

1 2 3

0 1

H: , -

' - .

(73)

#

*- ' +' @ -

*- -? " " !

@ 7 *-

C "

' 7 B 1+' 2

2 3

0 1

f 1 f f

H 6 A

" " @ &!$9@ "

+ Æ

7 - .

! 3%@ $ ''

4,5 J 3 KC 1

2 ' 7 *-

" ' @" @

7

0.1 0.3

1 f 2 f 3

f 1

f 2 f 3

f 1

0.4 0.4

0.7

1 0.7 0 0.1

f f

f 3

1

2

H

&B 7@ - *-

7' "

7 - 7 '

1" *- '@ ' +' 2

(74)

#

' 7 *- " 1

+' 2

%%

'

: @

"

F

H

7@ 79<:

+' '"

- ' J% K:

7 " 45

- " - @ "

>' 1- +' 2 @

*-@ -

> ' 1- +' #2

7 " B ' -

!+@*- 7'@

*1A2 '>

: -

*- > ' ' C

*- Æ

,

+

8

: -? Æ

" *-

$' ?@ " O ! B@

(75)

#

# (( %### (( ( %%% (

% %#

% ( %%# 3(( (

H *)' ! ( E+

'

Maximal displacement

Weight

100 200 300 400 500 600

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1 test1

test2

# H ! D

'

' - " @ '

*-@&!$9 - Æ

-

!" A@" *

-' @" ,

:7 > @ @

@ @ " >

, '" ' ,

(76)

#

$ / ( "-

:- 7D"

" > @>

" Q " -@ '

" @ +7-

)' >@-

>

7 !-'

@ > 7

" @ ' "

' > : 7 " 7

, ' > " -

@>-7

" @ " ' , "

7

C > ' -?

Æ Æ@"

Æ

$ +7 @ - - "

" " @ "

" 71 2

@ " ' " ,

&!$9@ @

-* '> 9$ *-

!

!"! *% $ %

@ " 7 7

&!$9" ' - -

' - 7 > "

" &!$9-@"

> " '@ " @

" " 7

@" @" -

7 " 7' - "

B'

@ 9 '

7 1" 2

=>

) "

@

! @7@ @"

(77)

##

>A

@ &!$9

H 0 D

! 333@ O " " " 1 7

2 475 '

- 7J (K6 "

:

7 10 2 + D

5 192F

+

:

9

:

F(

:

F " 9

"

5 192@

9:" " ' 7

9

B " 5

):

+

:

F

( ,1

1 5 1922 9@ " 1 Æ2

@ " " @ ' 7 Æ

H 2 C D

: 7 O J ( K ' 18

2@ + D

4"F 1

+

,

2

9

,

9

G B ,

@

9

" - *- >

,

" 1J @ # K@

" , ,

@

' @ G > 7G

H AD

" '' WOJ K>

7 !+

7 D

!12F

*D*

(78)

#%

min f1

min f2

A B

%H " ( A+

3 !12F % !12F %"

" @ " -

" !12

' !12= 7 !12 !12 >

6 - " "

>

WO > " 7

A 1

2 E " -

' 1 "

2@ @ S"

' > ! B@ " @

" ' '' 7" 1,

,2 @ * 4'5

" 7 17

+' %2

H A D

" ''WOJ K!

-D

"F

* +

+

" ' ' ,

'- 7> 9

*- "

7*-1+' 32 : -

" -

H D

(79)

#3

min f2

min f1

D

3 HA

D

F

9 +

1,

Z

,2

,

F

+

@ Z

, , ,

F 9 7

9 @ 7

:> 19

2

,

" &+ 7@ '

,

1- 2 "

1992

H = D

+ " @" 7 @

'

J 3 @ ( KD

;8 F< :1 +

!

2

!

, ' - $ "

7 *- 1 7 @

7 '2 : +' (

7

,> 8/:E/ F

! 33%@ . PON0O " " " "

4- 5 1 2 "

" J( K

C - " 4 5 "

*- - & @

7 @" 5, -5 "

Références

Documents relatifs

Montrer que si f est libre la restriction du produit scalaire au sous espace engendré par f est encore un produit scalaire, montrer que det G(f ) &gt; 0, on pourra se placer dans

In accordance with Articles (11,12) of the 2003 Convention for the Safeguarding of the Intangible Cultural Heritage, the Cultural Relations Directorate under the

Wortmann über seine &#34;tabu- lose&#34; WM-Doku. Il

Herzleiden, Krebs, SehlayanlaU: Wie Manner den wichtiysten Krankheiten vorbeuyen, haben namhafte Experten der medizinischen Fachgesellschaften für den graBen

MOCN : La diversité des solutions Robotique - Automatisation: Robocaliser encore et encore Organisation: Le Symop au service de la profession Contrôle qualité: Conformes au

Un exercice classique autour de la loi uniforme qui permet en fait (2 ème question) de répondre à un petit « problème pratique »

[r]

min C alcu ls Ich addiere nur Stunden, um so naham Ende wie möglich zu kommen, und ich behalte die Minuten.Ich rechne wieviel Minuten ichaddieren muss,um das Ende