curs fuzzy 1

8/16/2019 Curs Fuzzy 1

1/15

Mul!imi fuzzy

Mul!imi

Elementele unei mul!imi

! #$%&'#( $)'*(+,-%. ! (,/( 0(1')'/. 2) $)'*(+,$% 0( 0',3$&'( 4' (- ')3%$0( /5-/((%(#()/(%( 65,'7'%( 3-+( ,( +(8(+. %- 5 6+57%(#. 0-/.9 :-3. 0(1')'# 5 #$%&'#( " 2) #$%&'#(- ! ; *5# -*(- $+#./5-+(- +(%-&'(< A! X 9

=) -3(,/ 3->; ,6$)(# 3. #$%&'#(- " (,/( ')3%$,. 2) #$%&'#(- $)'*(+, ! 9 :-3. " )$(,/( ')3%$,. 2) ! ; -3(-,/. +(%-&'( (,/( +(6+(>()/-/. 3-< A ! X 9

:-3. $) (%(#()/ # (,/( ')3%$, 2) #$%&'#(- "; -3(,/ (%(#()/ (,/( 0()$#'/#(#7+$ -% #$%&'#'' 4' *5# 85%5,' $+#./5-+(- )5/-&'(< x ! A9 :-3. (%(#()/$% # )$

(,/( ')3%$, 2) #$%&'#(- "; *5# 85%5,' $+#./5-+(- )5/-&'(< x ! A 9

=) ?()(+-%; *5# +(6+(>()/- 5 #$%&'#( 6+') ()$#(+-+(- (%(#()/(%5+ ,-%(9 @6+((A(#6%$; (%(#()/(%( $% & $ ( & )))& $* ,$)/ (%(#(/( -%( #$%&'#'' B; 4' *5+ 1'+(6+(>()/-/( -,/8(%< A = a

1, a

2, ... a

n{ } 9

! -%/. #(/50. 0( +(6+(>()/-+( -%( $)(' #$%&'#' (,/( 0-/. 0( ,6(3'1'3-+(-35)0'&''%5+ (%(#()/(%5+9 @6+( (A(#6%$; 0-3. (%(#()/(%( #$%&'#'' + ,-/',8-335)0'&''%( P

1, P

2, ..., P

n; -/$)3' #$%&'#(- + (,/( 0(1')'/. 3-<

B = b | b satisface p1 , p2 , ..., pn{ }

=) -3(,/ 3->; ,'#75%$% CDE -+( 3- 4' 2)&(%(, C-,/8(% 2)3F/E9

G()/+$ - +(6+(>()/- 5 #$%&'#( H$3%'0'-). 0( 0'#(),'$)( I; )$#.+$% 0((%(#()/( (,/( $/'%'>-/ '-+ -3(,/ )$#.+ (,/( 0()$#'/ ,$-./*$0/1$129 J-+0')-%'/-/(-$)(' #$%&'#' " (,/( )5/-/. DBD9 :-3. 3-+0')-%'/-/(- DBD (,/( $) )$#.+ 1')'/;#$%&'#(- " (,/( 5 #$%&'#( 1')'/.9 :-3. DBD (,/( ')1')'/.; " (,/( 5 #$%&'#( ')1')'/.9

=) ?()(+-%; /5-/( 6$)3/(%( 2)/+K$) ,6-&'$ *(3/5+'-% H$3%'0'-); ,$)/ (%(#()/(%(#$%&'#'' $)'*(+,-%( ! 9

1.1.2 Rela!ii între mul!imi

! #$%&'#( 3-+( 35)&')( #$%&'#' (,/( 0()$#'/. 3$4/0/2 .2 450 ! /4/9 @6+( (A(#6%$;5 8-#'%'( 0( #$%&'#' 3-+( 35)&')( #$%&'#'%( "% & " ( & )))& (,/( +(6+(>()/-/. -,/8(%< A

i | i ! I { } ; $)0( / (,/( $) '0()/'1'3-/5+ -% #$%&'#'' '-+ 6 (,/( $) '0()/'1'3-/5+ 0(

#$%&'#(9

:-3. /5-/( (%(#()/(%( 0') #$%&'#(- " ,$)/ 0(-,(#()(- (%(#()/( -%( #$%&'#'' +-/$)3' " (,/( 5 ,$7#$%&'#( - #$%&'#'' +9


2/15

A ! B iff (daca si numai daca) x " A# x " B

@'#75%$% C! E 2),(-#). C'#6%'3.E9

:-3. $+#./5-+(- +(%-&'( (,/( ,-/',8.3$/.< A! B si B ! A

-/$)3' " 4' + -$ -3(%(-4'(%(#()/( 4' 0(3' ,$)/ #$%&'#' (?-%(9 B3(-,/. +(%-&'( (,/( )5/-/. -,/8(%< A = B 9

:-3. $+#./5-+(- +(%-&'( (,/( ,-/',8.3$/. 2)/+( 05$. #$%&'#' " 4' +; A ! B si A " B ; -/$)3' + -+( (%(#()/( 3-+( )$ ,( +(?.,(,3 2) "9 =) -3(,/ 3->; "

(,/( 0()$#'/. !"#$"%!&$' )*+)*&' -% %$' + 4' -3(-,/. +(%-&'( (,/( )5/-/. A ! B 9

! #$%&'#( 3-+( )$ -+( )'3' $) (%(#()/ (,/( )$#'/. $"%!&$', -&./ 4' ,( )5/(->.! 9 ! #$%&'#( *'0. 65-/( 1' 5 ,$7#$%&'#( - 5+'3.+(' #$%&'#'9

1.1.3 Apartenen!a

:-3. $/'%'>.# 8$)3&'- 0( -6-+/()()&. L8$)3&'- 3-+-3/(+',/'3. ,-$ 8$)3&'-0',3+'#')-+(M; 6$/(# +(6+(>()/- 5+'3-+( (%(#()/ # 3-+( -6-+&')( #$%&'#'' " ,-$)$9

0'1&2&!&'< G()/+$ 5 #$%&'#( "; 6$/(# 0(1')' 8$)3&'- 0( -6-+/()()&. µ A

-,/8(%<

µ A =1 iff x ! A

0 iff x " A

#$%

G$/(# ,6$)( 3. 8$)3&'- µ A

#-6(->. (%(#()/(%( 0') #$%&'#(- $)'*(+,-%. ! 3./+(

#$%&'#(- 0,1{ } 9 µ A : X ! 0,1{ }

! #$%&'#( 6$/(+( GLBM (,/( 5 #$%&'#( 8-#'%'( 35)&')F)0 /5-/( ,$7#$%&'#'%(#$%&'#'' "9 :( -3((-; )$#.+$% (%(#()/(%5+ 2) #$%&'#(- 6$/(+( GLBM (,/(+(6+(>()/-/ 0( DGLBMD N O!"!9

34'$)%"% 59 :-3. B N P-; 7; 3Q -/$)3' DBD N R9

GLBM N P! ; P-Q; P7Q; P-; 7Q; P-; 3Q; P7; 3Q; P-; 7; 3QQ

DGLBMD N O# N S9


3/15

1.2. Opera!ii pe mul!imi

1.2.1 Complement

6"%!&$', 7+$)%'$'28 *'%,8&- - #$%&'#'' " 3./+( #$%&'#(- + (,/( 85+#-/. 0')

(%(#()/(%( 3-+( ,$)/ 2) + 4' )$ ,$)/ 2) "9 T$%&'#(- 35#6%(#()/ (,/( 0(1')'/. 0($+#./5-+(- 85+#$%.< B ! A = x | x " B, x # A{ } 9

:-3. #$%&'#(- + (,/( #$%&'#(- $)'*(+, ! ; -/$)3' -3(,/ /'6 0( 35#6%(#()/ (,/(

$) 35#6%(#()/ -7,5%$/ -% #$%&'#'' A 9 B3(,/- (,/(; A = X ! A 9 =) ?()(+-%; 5#$%&'#( 35#6%(#()/ 2),(-#). #$%&'#(- 35#6%(#()/ -7,5%$/9

T$%&'#(- 35#6%(#()/ (,/( 2)/5/0(-$)- ')*5%$/'*.< A = A 9

J5#6%(#()/$% $)(' #$%&'#' *'0( (,/( #$%&'#(- $)'*(+,-%. 4' *'3(*(+,-<! = X si X =!

1.2.2 Reuniunea

U($)'$)(- #$%&'#'%5+ " 4' + (,/( 0(1')'/. 0( 35%(3&'- /$/$+5+ (%(#()/(%5+ " 4' +9

A! B = x | x " A sau x " B{ }

U($)'$)(- 65-/( 1' 0(1')'/. 0(-%$)?$% #$%&'#'%5+ #$%/'6%(9 @6+( (A(#6%$;

+($)'$)(- #$%&'#'%5+ 0') $+#./5-+(- 8-#'%'( 65-/( 1' 0(1')'/. -,/8(%<

A

i! I

! = x | x ! Ai pentruanumiti i ! I { } $)0( 8-#'%'- #$%/'#'%5+ (,/( Ai | i ! I { }

U($)'$)(- 0')/+( 5 -)$#'/. #$%&'#( " 4' #$%&'#(- $)'*(+, ! (,/( +(0$,. %-#$%&'#(- $)'*(+, A! X = X 9

U($)'$)(- $)(' -)$#'/( #$%&'#' " 4' #$%&'#(- *'0. ! (,/( #$%&'#(- B<

A!" = A

U($)'$)(- 0')/+( 5 #$%&'#( " 4' #$%&'#(- 35#6%(#()/ (,/( #$%&'#(-$)'*(+,-%.< A! A = X 9

1.2.3 Intersec!ia

V)/(+,(3&'- A! B (,/( 85+#-/. 0') -3(%( (%(#()/( 3-+( ,$)/ 35#$)(; ')3%$,( 2)-#7(%( #$%&'#' " 4' +9

A! B = x | x " A si x " B

{ }


4/15

V)/(+,(3&'- 65-/( 1' ?()(+-%'>/. 2)/+( #$%&'#' 2)/+K5 8-#'%'( 0( #$%&'#'9

Ai = x | x ! A

i , "i ! I { }

i! I

! $)0( Ai | i ! I { } (,/( 5 8-#'%'( 0( #$%&'#'9

V)/(+,(3&'- 2)/+( #$%&'#(- " 4' #$%&'#(- $)'*(+, ! (,/( "< A! X = A

V)/(+,(3&'- 0')/+( #$%&'#(- B 4' #$%&'#(- *'0. (,/( #$%&'#(- *'0.< A!" ="

V)/(+,(3&'- 0')/+( #$%&'#(- " 4' #$%&'#(- 35#6%(#()/ - ,- (,/( #$%&'#(- *'0.<

A! A ="

JF)0 05$. #$%&'#' " 4' + )$ -$ )'#'3 2) 35#$); +(%-&'- ,( )$#(4/( +(%-&'(0',W$)3/.9 V)/(+,(3&'- 0')/+( 3(%( 05$. #$%&'#' 6+50$3( #$%&'#(- *'0.< A! B ="

1.2.4 Parti!ia unei mul!imi

:(,35#6$)(+(- $)(' #$%&'#' " 2) ,$7#$%&'#' 0',W$)3/( -%( 3.+5+ +($)'$)(+(35),/+$'(4/( #$%&'#(- " (,/( +(8(+'/. 3- 6-+/'&'5)-+(9 G+(,$6$)(# 5 6-+/'&'( -%$' " )5/-/. 3$ ! < ! ( A) = A

i | i " I , A

i # A{ } ; -/$)3' "/ ,-/',8-3( $+#./5-+(%( /+('

35)0'&''<

X9 Ai ! "

O9 Ai ! A j = ",

i # j ,

i,

j $ I

R9

Ai = A

i! I

!

:-3. 35)0'&'- LOM )$ (,/( 2)0(6%')'/.; -/$)3' ! ( A) 0(*')( -356(+'+(- #$%&'#'' "9

1.3. Caracteristicile unei mul!imi crisp

V)*5%$&'- A = A

J5#$/-/'*'/-/( A! B = B! A

A! B = B! A

B,53'-/'*'/-/(- ( A! B)!C = A! ( B!C )

( A! B)!C = A! ( B!C )

:',/+'7$/'*'/-/(- A! ( B"C ) = ( A! B)" ( A!C )

A! ( B"C ) = ( A! B)" ( A!C )


5/15

V0(#65/()&- A! A = A

A! A = A

B7,5+&'- A! ( A" B) = A

A! ( A" B)=

A

B7,5+&'- LY 4' !M A! X = X

A!" ="

V0()/'/-/(- A!" = A

A! X = A

Z(?'%( :( T5+?-) A! B = A" B

A! B = A" B

B7,5+&'- 35#6%(#()/$%$' A! A" B( ) = A! B

A! A" B( ) = A! B

Z(?(- 35)/+-0'3&'(' A! A ="

Z(?(- (A3%$0(+'' #'W%53$%$' A! A = X

1.3.2 Multimea convex"

0'1&2&!&'< [(+#()$% 0( 35)*(A (,/( -6%'3-7'% $)(' #$%&'#' " 2) U$ 0-3.$+#./5-+(%( 35)0'&'' ,$)/ ,-/',8.3$/(<

-9 :5$. 6$)3/( -+7'/+-+( 7 4' - ,$)/ 0(1')'/( 2) "9 r = r

i | i ! N

n( ), s = si | i ! N n( ) K I (,/( 5 #$%&'#( 0( 2)/+(?' 65>'/'*'

79 G()/+$ 5+'3( )$#.+ +(-% ! 2)/+( \ 4' X; 6$)3/$% 1 (,/( '#6%'3-/ 2) " $)0( 1 (,/(<

t = ! ri + 1" ! ( )s

i | i # N

n( )

J$ -%/( 3$*')/(; 5+'3( 6$)3/ -% %')'(' 3-+( 35)(3/(->. 3(%( 05$. 6$)3/( 7 4' - 0') " ,$)/ 0(-,(#()(- 35)&')$/( 2) "99&:"*, 5; B%; B& 4' B# ,$)/ #$%&'#' 35)*(A( 2) U&9


6/15

1.4. Definirea mul!imilor fuzzy

1.4.1 Expresii pentru mul!imea fuzzy

]$)3&'- 0( -6-+/()()&. µ A

2)/+K5 #$%&'#( 3+',6 #-6(->. /5&' #(#7+'' 0')

#$%/'#(- $)'*(+, ! 3./+( #$%&'#(- P\; XQ9

µ A : X ! 0,1{ }

0'1&2&!&'


7/15

A = x,µ A( x)( ){ } ,-$ A = µ A xi( ) / xi

i=1

n

!

G+(,$6$)(# 3. (%(#()/(%( ,$)/ 35)/')$(; 0(3' #$%&'#(- 65-/( 1' +(6+(>()/-/.

-,/8(%< A = µ A x( ) / x!

G()/+$ - 0'8(+()&'- #$%&'#'%( 8$>>^ 0( #$%&'#'%( 3+',6; 8+(3*()/ (,/( 85%5,'/.)5/-&'- A! 9

1.4.2 Exemple de mul!imi fuzzy

34'$)%"% C9 J5),'0(+.# -1'+#-&'-Cg(5+?( (,/( /F).+E9 %- -3(,/ #5#()/;/(+#()$% C/F).+E (,/( *-?9 G()/+$ -+(6+(>()/- 2)&(%(,$% %$' C*-?E 3F/ #-'(A-3/; (,/( )(3(,-+ ,. 0(1')'# 8$)3&'-

0( -6-+/()()&. 3- 2) 1'?$+- 0( #-'W5,9

BA- *(+/'3-%. +(6+(>')/. *-%5-+(-)$#(+'3. - 8$)3&'(' 0( -6-+/()()&.9Z')'- -+-/. 65,'7'%'/-/(- L*-%5-+(-8$)3&'(' 0( -6-+/()()&.M 0( - 1'35)&')$/ 2) #$%&'#(- 8$>>^ C/F).+E9

:-3. $+#.# 0(1')'&'- 6()/+$ C/F).+E;

$) 356'% 0( X\ -)' 65-/( 1' 85-+/( 7')(/F).+9 G5,'7'%'/-/(- 6()/+$ CX\ -)'E 0( - 1' 2) #$%&'#(- 8$>^^ C/F).+E (,/( X9 :(-,(#()(-; 65,'7'%'/-/(- 6()/+$ COh -)'E 3- ,. 1'( 2) #$%&'#(- 8$>>^ C/F).+E (,/(0( \9i9 @( 65-/( 57,(+*- 3. 6()/+$ Cc\ -)'E 65,'7'%'/-/(- 0( - 1' 2) #$%&'#(-C/F).+E (,/( \9

B1'+#-&'- Cg(5+?( (,/( 85-+/( /F).+E 35)0$3( %- 5 )5$. 3$+7. 2) 1'?$+- 0( #-' ,$,;3$+7. ,'/$-/. %- ,/F)?- 6+'#(' 3$+7(9 :(1')'# $+#./5-+(- #$%&'#( 8$>>^< .8$- 92-78$*202 ,$-2 $5 :;-71 75< => .2 $*/ 981 ?/ /*,0572 @* 450 ! /42$ A38$-12 1;*B-C)

B3$#-; 65,'7'%'/-/(- 3- 5 6(+,5-). 3-+( -+( Oh 0( -)' ,. 1'( ')3%$,. 2) #$%&'#(-

C85-+/( /F).+E (,/( 0( \9b

:-3. )5/.# 3$ B #$%&'#(- C/F).+E 4' 3$ j #$%&'#(- C85-+/( /F#.+E -/$)3' 6$/(#,3+'(< µ

A(27) = 0.9 si µ

B(27) = 0.5 9

34'$)%"% D9 ]'( $+#./5-+(- 0(1')'&'( 6()/+$ 5 #$%&'#( 8$>>^< B N P)$#(+(%(+(-%( 2) -6+56'(+(- %$' \Q9 G5,'7'%'/-/(- 3- $) )$#.+ +(-% # ,. 1'( #(#7+$ -%#$%&'#'' 8$>>^ (,/( 0(1')'/. 0( $+#./5-+(- 8$)3&'( 0( -6-+/()()&.<

µ A( x) =

1

1+ x2


8/15

J5)85+# 1'?$+'' #$%&'#(- 8$>>^ 65-/(

1' +(,3+',. -,/8(%< A = µ A

( x) / x! $)0(

µ A( x) =

1

1+ x2 9

g+-0$% 0( -6-+/()()&. - *-%5+'' X (,/(<1

1+12 = 0.5 ; 65,'7'%'/-/(- %$' O (,/( \9O

'-+ 65,'7'%'/-/(- %$' R (,/( \9X

34'$)%"% E9 :(1')'# #$%&'#(- 8$>>^ B -,/8(%< B N P)$#(+(%( +(-%( 85-+/( -6+5-6(0( \Q9 ]$)3&'- 0( -6-+/()()&. 6()/+$- 3 ( - , / . # $ % & ' # ( ( , / ( <

µ B ( x) =1

1+ x2

! " #

$ % &

2

9 G5,'7'%'/-/(- 3-

*-%5-+(- X ,. -6-+&'). #$%&'#'' 8$>>^(,/( \9Ob; - *-%5+'' O (,/( \9\\a '-+ -*-%5+'' R (,/( \9\X9 ]$)3&'- 0(-6-+/()(&. (,/( 3(- 35)85+# 1'?$+'' 0(#-' W5,9

1.4.3. Expansiunea mul!imilor fuzzy

0'1&2&!&' LT$%&'#( 8$>>^ 0( /'6$% )M K k-%5-+(- ?+-0$%$' 0( -6-+/()()&. 65-/(')3%$0( ')3(+/'/$0')'9

:-3. *-%5-+(- 8$)3&'(' 0( -6-+/()&. (,/( 0-/. 0( 5 #$%&'#( 8$>> ;̂ #$%&'#(- (,/( 5#$%&'#( 8$>>^ 0( 8&)"% A9 B3(,/ 35)3(6/ 65-/( 1' (A/'), 3./+( #$%&'#' 8$>>^ 0(8&)"% 29

34'$)%"% F9 J5),'0(+.# #$%&'#(- B N C-0$%/E9 ]$)3&'- 0( -6-+/()()&. - -3(,/('

#$%&'#'; #-6(->. 2)/+(-?- *F+,/. %- C/F).+E; C#-/$+'/-/(E 4' C,()'5+l9 J5)85+#-3(,/5+ 6+(,$6$)(+'; 6()/+$ 5+'3( 6(+,5-). # ; D 4' E *5# -*(-<

µ A( x) = "tanar"

µ A( y) = "matur"

µ A( z) =!

k-%5+'%( 8$)3&''%5+ 0( -6-+/()()&. 6()/+$ C/F).+E 4' C#-/$+'/-/(E ,$)/ 0(-,(#()(- #$%&'#' 8$>>^; 0(3' #$%&'#(- C-0$%/E (,/( 5 $"%!&$' =">>? .' 8&)"% A9


9/15

T$%&'#'%( C/F).+E 4' C#-/$+'/-/(E ,$)/ #$%&'#' 8$>>^ 0( /'6$% X9 =) -3(%-4' #50;0-3. *-%5+'%( 8$)3&'(' 0( -6-+/()()&. 6()/+$ C/F).+E 4' C#-/$+'/-/(E ,$)/ 0( /'6$%O; -/$)3' #$%&'#(- C-0$%/E (,/( 0( /'6$% R9

0'1&2&!&' L#$%&'#( 8$>>^ 0( )'*(% mM K [(+#()$% C#$%&'#( 0( )'*(% OE ')0'3.

#$%&'#'%( 8$>>^ -%( 3.+5+ (%(#()/( ,$)/ #$%&'#' 8$>>^9

[(+#()$% C#$%&'#( 0( )'*(% XE(,/( -6%'3-7'% #$%&'#'%5+ 8$>>^-%( 3.+5+ (%(#()/( )$ ,$)/#$%&'#' 8$>>^9 =)/+K5 #-)'(+., ' # ' % - + . ; 6 $ / ( # 5 7 & ' ) (#$%&'#(- 8$>>^ 0( )'*(% m9

34'$)%"% G9 J5)85+# 1'?$+'' 0(#-' W5,; (A',/. R (%(#()/(#$%/'#' 8$>>^µ

A( A

1) = 0.5 n µ

A( A2 ) = 1.0 n

µ A( A

3) = 0.5

1.4.4. Real!ia dintre mul!imea univers #i mul!imea fuzzy

:-3. (A',/. 5 #$%&'#( $)'*(+, 4' 5 #$%&'#( 3+',6; *5# 35),'0(+- #$%&'#(- 3- $)

,$7,(/ -% #$%&'#'' $)'*(+,9 =)/+K5 #-)'(+. -,(#.)./5-+(; )( *5# +(8(+' %- 5#$%&'#( 8$>>^ " 3- %- 5 ,$7#$%&'#( - #$%&'#'' $)'*(+, ! 9

34'$)%"% H9 ]'( Y N P-; 7; 3Q 5 #$%&'#( $)'*(+,9 B% N PL-; \9bM; L7; X9\M; L3; \9bMQ 4'B& N PL-; X9\M; L7; X9\M; L3; \9bMQ ,$)/ ,$7#$%&'#'# -%( %$' ! < A1 ! X , A2 ! X 9

J5%(3&'- -3(,/5+ ,$7#$%&'#' -%( %$' Y L')3%$,'* #$%&'#'%( 8$>>^M ,( )$#(4/($"%!&$', )"8'*' I>^ 6+(3$# C356'%E; C/F).+E; C-0$%/E 4' C,()'5+E 2) ! 9G5,'7'%'/.&'%( 3- $) (%(#()/ # ,. -6-+&'). $)('- 0')/+( #$%&'#' (,/( 2)35)85+#'/-/( 3$ /-7(%$% 0( #-' W5,9


10/15

vârsta copil tân"r adult senior

5 0 0 0 0

15 0 0.2 0.1 0

25 0 1 0.9 0

35 0 0.8 1 0

45 0 0.4 1 0.1

55 0 0.1 1 0.2

65 0 0 1 0.6

75 0 0 1 1

85 0 0 1 1

G$/(# ?F)0' 5 #$%&'#( 3-+( (,/( 85+#-/. 0') (%(#()/(%( 35)&')$/( 2) B9 B3(-,/.#$%&'#( ,( )$#(4/( $"%!&$', !")+*8 - %$' B9

Suport ( A) = x ! X | µ A( x) > 0{ }

@$65+/$% 6()/+$ #$%&'#(- 8$>>^ C/F).+E (,/(< F598-1G1;*B-H I J%K& (K& LK& =K& KKM 4' (,/( 5 #$%&'#( 3+',69 =) #5, -)-%5?; ,$65+/$% #$%&'#'' C356'%E (,/( #$%&'#(-*'0.9

k-%5-+(- #-A'#. - $)(' 8$)3&'' 0( -6-+/()()&. (,/( 0()$#'/. C2).%&'#(E9G+(,$6$)(# 3. C2).%&'#(-E $)(' #$%&'#' 8$>>^ (,/( X; -3(-,/- 2),(-#). 3.#$%&'#(- 8$>>^ (,/( C)5+#-%'>-/.E9 T$%&'#'%( C/F).+E; C-0$%/E 4' C,()'5+E ,$)/#$%&'#' 8$>>^ )5+#-%'>-/(9

G+(,$6$)(# #$%&'#(- 3+',6 C-05%(,3()/E9 B3(-,/. #$%&'#( 3+',6 (,/( 3%-+0(1')'/. -*F)0 (%(#()/( 05-+ X\KXi 2) $)'*(+,$% #$%&'#'' ! 9 B3(-,/. #$%&'#((,/( 5 $"%!&$' *'!8*&78&-/ 2) 35#6-+-&'( 3$ ! 9

1.5.2. Mul!imi ! -cut

0'1&2&!&'< T$%&'#(- ! K

3$/; A!

(,/( 85+#-/. 0')

#(#7+'' -%( 3.+5+ *-%5+'0( -6-+/()()&. ,$)/ #-'#-+' ,-$ (?-%( 3$ ! 9

A! = x " X | µ

A( x) #! { }

$)0( ! (,/( -%(, -+7'/+-+9B3(-,/. #$%&'#( ! K3$/


11/15

(,/( 5 #$%&'#( 3+',6 4' 65-/( 1'57,(+*-/. 2) 1'?$+- 0( #-' W5,9

HA(#6%$% XX9 T$%&'#(- ! K3$/ (,/(

0(+'*-/. 0') #$%&'#(- 8$>>^ C/F).+E

0-3. 35),'0(+.# ! N\9O9Tânar

0.2 = 12,25,35,45{ }

:-3. ! N\9a -/$)3' [F).+'() N POb;

Rb; abQ9 :-3. ! N\9S -/$)3' [F).+'(*

N POb; RbQ9

J5),'0(+.# 05$. #$%&'' ! K3$/; A!

4' A! '

4' 0-3. ! "! ' -/$)3'

A! " A

! ' 9 @6+( (A(#6%$; [F).+'(& !

[F).+'(*9

0'1&2& ! &'


12/15

! A = 0,0.2,0.4,0.8,1{ } µ

A(t ) ! µ

A(r)

T$%&'#( 8$>>^ )(35)*(A.

µ A (t ) < µ A (r)

1.5.4. Numere fuzzy

CI$#.+$% +(-%E '#6%'3. 5 #$%&'#( 3-+( 35)&')( /5-/( )$#(+(%( +(-%(9

CI$#.+ 65>'/'*E '#6%'3. 5 #$%&'#( 3-+( 35)&')( )$#(+( (A3%$>F)0 )$#(+(%()(?-/'*(9

CI$#.+ 65>'/'* #-' #'3 ,-$ (?-% 3$ X\ L')3%$,'* \ME ,$?(+(->. 5 #$%&'#( 0()$#(+( 0( %- \ %- X\; -0'3. A = x | 0 ! x !10, x "#{ } ,-$

µ A( x) =

1 daca 0 ! x ! 10; x "#

0 daca x 10

$%&

0'1&2&!&'< :-3. 5 #$%&'#( 8$>>^ (,/(-9 35)*(A. 4' )5+#-%'>-/.n79 8$)3&'- ,- 0( -6-+/()()&. (,/( 0(1')'/. 2) ! n39 6-+&'-% 35)/')$.

-/$)3' (- ,( )$#(4/( C2"$/* =">>?E9

d) )$#.+ 8$>>^ +(6+(>')/. $) ')/(+*-% 0( )$#(+( +(-%( -%( 3.+$' #-+?')' ,$)/*-?'9


13/15

1.5.5. Magnitudinea unei mul!imi fuzzy

G()/+$ - -+./- #-?)'/$0')(- $)(' #$%&'#' 8$>>^; (A',/. /+(' #(/50( 0( - #.,$+-3-+0')-%'/-/(- $)(' #$%&'#' 8$>>^9

-9 J-+0')-%'/-/( ,3-%-+. K 2),$#-+(- ?+-0(%5+ 8$)3&''%5+ 0( -6-+/()()&.<

| A |= µ A( x) x! X " 9 =) 35)85+#'/-/( 3$ -3(,/. #(/50.; #-?)'/$0')(- #$%&'#'' 8$>>^

C,()5'+E (,/(< D,()'5+D N \9X f \9O f \9c f X f X N O9i

79 J-+0')-%'/-/( +(%-/'*. K ,( 35#6-+. #-?)'/$0')(- #$%&'#'' 8$>>^ " 3$ 3(- -

#$%&'#'' $)'*(+, ! < || A ||=| A |

| X |9 =) 35)85+#'/-/( 3$ -3(,/. #(/50.;

#-?)'/$0')(- #$%&'#'' 8$>>^ C,()5'+E (,/(< D,()'5+D N O9i 4' DYD N in 0(3'DD,()'5+DD N O9i e i N \9RO9

39 J(- 0( - /+('- #(/50. (A6+'#. 3-+0')-%'/-/(- 3- 4' #$%&'#( 8$>>^

0'1&2&!&'


14/15

-3((-4' #-)'(+.; *5+ 1' a (%(#()/( 2) 3->$% 2) 3-+( ! = 0.2 n R (%(#()/( 2) 3->$% 2)

3-+( ! = 0.6 4' O (%(#()/( 2) 3->$% 2) 3-+( ! = 19

J-+0')-%'/-/(- 8$>>^ 6()/+$ C,()'5+E *- 1'< D,()'5+D N PLb; \9XM; La; \9OM; LR; \9cM; LO;XMQ

1.5.6 Submul!imi ale mul!imilor fuzzy

G+(,$6$)(# (A',/()&- - 05$. #$%&'#' 8$>>^ " 4' +9 JF)0 ?+-0(%( 8$)3&''%5+ 0(-6-+/()()&. ,$)/ (?-%(; ,6$)(# 3. CB 4' j ,$)/ (3o'*-%()/(E -0'3.<

A = B iff µ A( x) = µ B( x);! x " X

:-3. µ A( x) ! µ

B( x) 6()/+$ 5+'3( (%(#()/; -/$)3' A ! B 9

:-3. $+#./5-+(- +(%-&'(< µ A( x) ! µ

B( x);" x # X (,/( 2)0(6%')'/. 6()/+$ 05$.

#$%&'#' 8$>>^ B 4' j -/$)3' B (,/( $) ,$7,(/ -% #$%&'#'' j K A ! B 9

=) 6%$,; 0-3. 35)0'&'-< µ A( x) < µ

B( x);! x " X (,/( 2)0(6%')'/.; -/$)3' B (,/( 5

,$7#$%&'#( 6+56+'( - #$%&'#'' j9 U(%-&'- ,( *- +(,3+'( -,/8(%<

A ! B iff A " B si A # B

1.6. Opera!iile standard pe mul!imile fuzzy

1.6.1 Complementul

J5#6%(#()/$% #$%&'#'' B (,/( )5/-/ A 9 g+-0$% 0( -6-+/()()&. 65-/( 1' 3-%3$%-/-,/8(%< µ

A( x) = 1! µ

A( x); " x # X

:-3. *5# 3-%3$%- 35#6%(#()/$% #$%&'#'' C-0$%/E *5# 57&')(< A N PLb; XM; LXb;\9iM; LOb; \9XMQ9

1.6.2. Reuniunea

k-%5-+(- 0( -6-+/()()&. - #(#7+$%$' # 2) 3->$% +($)'$)'' '- *-%5-+(- 3(- #-'#-+( - 8$)3&'(' 0( -6-+/()()&. 0')/+( B 4' j<

µ A! B

( x) = MAX µ A( x), µ

B( x)[ ]; " x # X

B 4' j ,$)/ ,$7#$%&'#' -%( #$%&'#'' A! B '-+ +($)'$)(- 0')/+( C/F).+E 4' C-0$%/E

(,/(< C/F).+E ! E-0$%/E N PLXb; \9OM; LOb; XM; LRb; XM; Lab; XM; Lbb; XM; Lcb; XM; Lhb; XM;

LSb; XMQ


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1.6.3. Intersec!ia

V)/(+,(3&'- 0')/+( 05$. #$%&'#' 8$>>^ " 4' + 58(+. *-%5-+(- 3(- #-' #'3. 0')/+(*-%5+'%( 0( -6-+/()()&. -%( %$' " 4' +9

µ A! B ( x) = MIN µ A ( x), µ B( x)[ ]; " x # X

V)/(+,(3&'- A! B (,/( 5 ,$7#$%&'#( - %$' B ,-$ j9 @6+( (A(#6%$; ')/(+,(3&'-0')/+( C/F).+E 4' C-0$%/E (,/(< C/F).+E ! E-0$%/E N PLXb; \9XM; LOb; \9iM; LRb; \9SM;Lab; \9aM; Lbb; \9XMQ9

LMN3OPQ"R3< !6(+-&''%( 0(,3+',( 6F). -3$#- ,$)/ *-%-7'%( 4' 2) 3->$% #$%&'#'%5+3+',69

curs fuzzy 1

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