Applied Fuzzy Arithmetic: An Introduction with Engineering Applications

Michael Hanss Applied Fuzzy Arithmetic An Introduction with Engineering Applications Michael Hanss Applied Fuzzy Ari...

Author: Michael Hanss

15 downloads 580 Views 5MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Michael Hanss Applied Fuzzy Arithmetic An Introduction with Engineering Applications

Michael Hanss

Applied Fuzzy Arithmetic An Introduction with Engineering Applications With 96 figures and 24 tables

Priv.-Doz. Dr.-Ing. habil. Michael Hanss Institut A für Mechanik Universität Stuttgart Pfaffenwaldring 9 70550 Stuttgart Germany

E-Mail: [email protected]

Library of Congress Control Number: 2004117177 ISBN 3-540-24201-5 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by the author. Final processing by PTP-Berlin Protago-TEX-Production GmbH, Germany Cover-Design: design & production, Heidelberg Printed on acid-free paper 89/3141/Yu – 5 4 3 2 1 0

! " ! #"# ! " $ ! " " ! %

! "#$%

& & '

& & ( ) * ( )

* ) &

( ) + ' " % (

& & + ' $ # &

*

' "

& &

) , ' - * & .

/ & &

' 0 1 2

&

) (

' 3

( &

* 4 ) ' %

&

&

4 5) + ' $ ( & ' 6 7

( ' 8 ( )

4 ) * ) (

9 &

) ) )

: 5 ) , :

) ) + ; (? : ) , : = + + ? 4 +

)

( +

+ ) / +

; ' 5 5

( "- 4 )

O

"

++ 1 ( )

& M

#

$ # H"$I (

) & 4

)

"

,

, ./)( 0 ( ( ( 0

( ( -, - , , 2 , 3 45 6 76 )-- -- 76 * -

1 # $ 1 1 1 # $ 1 # $ # $ 1 # $ # $ 1 # $ # $ # $ 1 # $ # $ 1 1 1 1 1 1 # $ 1 # $ 1 # $ 1 # $ 1 1 1

(

( Æ K L & K L K L Æ J K L HK L K L I K"%%L

K L K L Æ K L Æ

K L J " K L J " L J "D Æ K K"%$L % J 1 2 K"%6L M J K L 1 2 K"%8L J K L 1 2 K"%#L ( Æ

Æ J K L 1 2 K"$DL

++ ! ,

/ &

K L

'

1 2 J D K"$"L D K L J "D K"$-L D ( K L J 1 2 K"$0L (

K L 1 2 4 "0

'

&

& & 9 M B 1 2 ) )P

1 2 1 2

& ) ( K L J 1 2 K"$3L

&

&

) H"0-I 4

M HD "I K"$%L H"-DI &

K L J

K L K L HD "I K"$$L

J K" D -L K- D %L K% " DL K6 D #L K# D %L K"$6L C K L . & 1 2

K L K"$8L K L J " N &K" "DL 4 "0

#$ !

#$

#$ ! !!

!

%"

Æ

!

#$ # , #$ # $

$

- ,

++ & -.

& )

> * M

& 4

K"$#L

( && J " - K L K L 4 K"$$L &

K L

K L J K K"6DL L K L K K L L HD "I

%

& C K L ' &

& "" J K L K"6"L J $ "% 0D J " - % "D (

K L J 1 2 K"6-L K L J 1 2 K"60L J K L K"63L

K L 1 2 K L

& K L

&

K L

( "0

"!

#

$ -

! ! # $

"!

#$

!

! ! ! !! !' ! ! ! ! !' ! ! # $

&

0

( && K L

/

K L J K"6%L

K L

8 ! ) ( K L J K L K

L K & L K LM K L J K L J KL K"6$L K L J " O K L J K L K L K &

L K LM K L J K L J

K"66L L K

( ) ( K L J K L

M

K L J K L J K L J " K"68L K L J K L K L

K L M L J K L J

K K K K"6#L L L J "

K L

) ( K L J K L K

L M

K L J K L J K L D K"8DL

'

K L J K L K L

K L M L J K L J

K K K K"8"L L L D

2 )

( K L J K L

HD "IM

K L J J K L K"8-L (

K L J J K L K"80L

K L J K L

K L HD "IM

K L J J K K K"83L L L (

K L J J K K K"8%L L L

"M

K L J K L K"8$L

K L J K L K"86L

K L J K L J " K"88L K L K L K"8#L 4 9

9

!

L J K"#DL L K J L K

K L K L

K L J KL K"#"L K L J

K

9 K"#DL K"#"L

(

& 5

K --0L (

& 4 "3

#$ 4 #$ 1

!

#$ #$ #$

,,#$

6 4 , , 4 8 8 ,, - *

( )

(

K"-0L K L

( K L J

K L K L

K L J J K L J K L K"#-L

( K L

KL

K L K"#0L

K L J K L J "

C K L K L

M KL

K L

K L J K L J " K"#3L

K L J J K L J

K L

K L

K L

K L

K L

K L J K L J

K L K L

K L

K L J K L J

K"#%L

K L

K"#$L K"#6L

&

&

K L K L M KL JK L K L J K L K L N K" L K L KL HD "I K""D-L ( & M KL

K L K L K L K L K""D0L * / & K L K L 4 "% "$

&

K""D0L

4 4 "$

K L 9 K L K L

K L K L

4 "$ ' 9 &

#$ #$

()

* ()

#$

!

#$

!

1 9 #$ #$:

0,, () - #!"$ #$ () #Ê$8 - #$ * () #Ê$

9 ! : 3 9 ! :

. & "-" 1 2 1 2 12

9 K L (

) 9 (

H"3"I 4 12 & 1 2 1 2 12 12 1 2 ( 9

+ ) &

9 1 2 9 12 12 1 2 12 12 7 4

.&& 4

;

&& && 9

L K-%L J K KL D "NK L K-$L K L J " K L N D N & D " K L K-6L K L J & D " K L

#

#

( K4 -"L (

K-8L J H I J H N I J K L N

$

$

#

#

B -

%

/ .&& 4

/ && : .&& 4

9 J K L J .

J " . & 9 . / D K L . K-$%L J

" K-$$L J #

(

#

#

#

#

# #

#

#

#

) # #

#

#

#

&

#

#

#

#

#

&

#

+ 0 > * > * >

& ( -% Æ

&

&

++ 1 - ,3 - ,3

>

K"#"L 9

9 K L K L J K-""0L K L

%

+ 0

K L 9 K-""3L K L J K L ( HD "I 9

J

4

HD "I K-""%L X J"

&

(

JD " K-""$L J

JD J" JD " " K-""6L J NX (

K-""8L J K N "L

J" K-""#L J J K L K-"-DL J K L J J 4 -"- &

@ H68I K L 9

H I I H J K L J

K L K K-8L K-"0L K-"$L K--"LL K-""8L Æ K N "L

J D " ( 4 J " &

9

#

.

.

/

.

/

.

/

.

.

"

#

.

#

/

.

#

#

#

#

"

.

/

#

.

"

"

+ 0

J K L J N " K-"-"L

J D " ( J K L J

J K LJ J K-"--L JH IJH I H IJ K-"-0L JD " K-"-0L ) H0 0 I ) H#%I

K H- 6$IL (

(

) # #

(

0

!

2

/

.

"

2 #

#

) " "

"

2 "

"

) # #

/

2 #

#

#

#

!

2

!

!

2

.

3

%

H

+ 0

I H

2

IJ

N "

2

2

K-"-3L

&

H I H I D H I > K-"-0L 9 K-"-3L M

(

H

-

H

H

4

6

H

5

INH

IH

I J H N

I J H

K-"-%L

I

K-"-$L

N I

I"H

I J K L &K L 4

4

K-"-6L

J

I H

I J K L L &K 5

5

J

K-"-8L

D H I , N " K-"-%L K-"-8L L &K L H I H I J K K-"-#L J 9

K-"-%L K-"-#L K L J ( K L ) ( J

2

2

6

6

6

)

2

2

2

2

)

2

!

2

+ 0 K0"3L Æ

)

J D

9 & " " ! J " " )

+

K0"8L K0"#L

" 0,, -- 0

&"

B *

& (

4 0" 9 - N JD K0-DL 9

J K0-"L ( 9 " D $" $ " $ # K0--L N %# % J # D % D *

J J" K0-0L /

J D J K0-3L K0--L 9 N ! ! J D! K0-%L

)

+

4 9 K0-%L ) J K0-$L ) * M " - -@ C & K0-%L ( J " K0-6L ' & " K0-8L J N "

&

K0-6L K0-8L K0-%L

&

" -- 0

- 2 / *

& & 4

K0-$L

*

& & . ( ) ( ) ! J N K0-#L

J

(

N

)

( ) ! (

N

)

K00DL

0 6 & *

&

K0-%L * & & ( &

J K00"L )

J J

J K00-L > M 4 & ( * /

J J

N

N

N

K000L K003L

" 0,, -- 0

&

- A 7

J - D " "D ? K00%L J "DD J %DD K00$L

J $ # " "D ? K006L J 6% J %DD K008L ( & J "DDD? K00#L /

- 0 M J D D-%D J D "-"$ K03DL )

)

+

+

- .&&A 7

( Q 2

%" &

' 9 *

& K0-%L & * *

J" K03"L J ( Q 2

9 J D "

4 * ( K "L K N"L

( J " - & B )

( K4 3"L (

K38L K3#L J 0 / &

J D )

J J J K3"#L (

( ) + J ( ) + J K3-DL ( ) + J - J 8 '

K4 3-L (

K3-DL !

/

.

0

(

.

#

#

#

!

0-(- 0 H 3 -

H" "I ( + + +

J J J H" N"I K3-"L + J K" " " " N" N" N" N"L + J K" " N" N" " " N" N"L K3--L + J K" N" " N" " N" " N"L

( K3--L ) & K H#-IL

>

>

>

>

>

>

>

>

>

&

"

%

2 , - - - , * 1 " ,

Æ K L J D K L J " -

J " & )

4 3- 4 # #

#

#

2 - .-- 3 -

!

K3"6L K3"8L & & 5 J "

* & ( * K3"6L K3"8L O *

( )

& &

5 K3"6L K3"8L

& " " J " -

K3"6L K3"8L

9 & & 9 & & ( *

& 0- 03 ) ' 0 ! / &

K L J - -" "8 K3-0L

.

#

7 #

#

#

#

!

0-(- 0 H 3 -

J K" % " % " %L

K3-3L K 4 03L J " , J "% K L 4 30 ( K L K L K L J HD 0I ! &

K L J - K3-%L J K" % " % " %L K3-$L

& 3"

J " , J "% K L 4 30 & K L ( & K L ) J " &

J - 0

& J D & J H0 "I J H0 "I J "% J H0 D ##86I

J "3 ( J "% & J "3

4 J D 9 K0%#L 4 9 K0$-L 4 J "3

H0 D ##86I H0 "I J D "8-%V K3-6L J H0 "I K L J D D3%"V K3-8L #

.

7 #

7 #

7 #

#

, #

#

#

#

.

, #

#

.

,

!

!

.

.

!

.

.

.

9

:

#

2 - .-- 3 -

#$

#$

' #$

!%

( #$

#$ . ' #$ - #$ ( #$8 - - -

! &

K L J N K3-#L

J K- " "L J K3 % D % D %L K30DL K 4 06L

K L & K L J " - J "% K L K L K4 33L !! 4 * 9

Y K L J K" L K L N K L KDL J " K30"L Y K L J K L N K L KDL J " K L J K L N K L K30-L C J " - 0 3 K L K L D

4

& #

#

#

# #

#

#

# #

#

# #

.

# #

#

#

#

#

(

#

#

#

(

!&

0-(- 0 H 3 -

J" - 0 3 K300L J K D " D " L "DV

J D #% J D J D 0 J D DD% K303L & ,

J "D 4 33

K L J D - "D ( * 9 & . = X J D %

#

.

(

/

#$

) # $

#+@ $

1 ! 1 ! 1 !!

+ $ #$ . ) # $ - 4 -- -@ #$ , #+@ $ - 4 4 -

#

++ 2 , 7 7 J " ) (

( * ) 9 9 7

( &

#

(

2 - .-- 3 -

!'

!% / K L J K L K30%L /

J KD 3% D D% D D%L J KD -% D -% D -%L K30$L 4 3% 4 3% '

9

* $"DDV $""V 7 # #

#

#

3#

#

(

#

#

#

# $

#$

#$ *(- - , - @ #$ , ( 1 '# $

9 9 &

K L

( &

K L J K L J K L /

J K L J K L K306L K4 3$L 4

&

$""V $"DDV 7 7 # #

7 # #

#

#

#

#

(

7 #

(

7 #

(

(

#

#

(

#

#

#

#

!

0-(- 0 H 3 -

#$

#$

#$

&M " ( - ( * & 9 0

++ 2- ( + &

J " -

JD" (

&& K4/L H-%I B &

()

! H8- 80I

+# $ J H K38%L % I# $

& J " - J " K L J - K L J 0 !+ & + J H" %I K38$L

#

/

.

5

5

5

5

+Æ , 3 -

"

( 9 & ! + J " " K386L % % J - J " K & L ! L ( 5( > H8-I ++ ! -

3-- & * ( ( & > ) ! H8- 80I

* & ( 1 2 & K3"-L J " JD " J - 0 K3#0L J J D " J N " JD " 4 K3#0L K3"-L 4

H80I

& 9

( )

* &

H80I

+

/

.

+

.

/

.

+

.

/

/

.

/

+Æ , 3 -

++ 0 2$

*

9 Æ ! H8%I

* K H"- "3DIL ( H83I 4 9

& &

9 !2 H""6I

& (

9

4 !2 & ) &

K H$ """ ""#IL H83I & ' '

4 3"D J - 4 3"D

J 0 4 3"D > 9 J 3 $% "66 4 J KD % D % D %L J " - 4 3"" , J % 4 3"" J - 4 3"" J 0 ( #" 33" ( &

#

.

0-(- 0 H 3 -

6* , 4- 6 ? ( 1 #$ * 1 - @ #$ * 1 " -

*

( & )

7 * )

9 ? & H$8 $#I ( 9 && ( H%#I

3" &

9 J K L J K%""L ( J ( K L (

(

# #

# #

#

#

( 0

""

9 J " - 4

( ( ' K%""L

( 9 B J

S

M ( 9 (

J " K & %-L (

9 * C 9 ! J " - J " - K%""L (

M " ; [ [ [M C K%""L J K L J " - ! J K! L ! J " - 9

J K L J K%"-L J K L #

( (

(

# #

#

# #

( (

#

(

;

;

;

# #

( (

#

#

(

(

4 $" /

4 $"

)

)

& ( H30I

'

&

3 +44

,

F - ( 4 #$ ( 8 ,

( - , J @ #$ ( - , J 6 , 6

&

, 2#'$ #$ ' 1 % !!! G@ #$ ' 1 ! !!! G

7 9 7

J J J K L 4 9

7 & # %6V # 30V D D"V K$-%L ( * K L B 9 K L A

A

A

#

=

#

=

#

7

7

A

A

A

7

D 7

7

8 0 8 - - ? K

'

4

K L K L &

7

7

"++ . / & ;

%- & (

& $"- K H$8 "D0 "D-IL / & K 4 $#L 4

= E *

K4 $"3L / *

(

9 =

=

7

5

&

1# - #$ , # $ #

1$

J

- #$ , # $

$

- - 6 6 *, J -

' ) H"D% "-0I K L K L K L K L K L K L

K L K L / KL J KL KL KL K$-$L KL KL KL KL ;(

;)

(

;)

>(

;(

>)

)

>(

>)

;(

;)

!

3 +44

K L J K L K L ! K$-6L

K L

K L

K L N N N K$-8L K L J K L K L ! N KN L N K L K L K L J K L K L ! K$-#L L N K L K L N K N K L J K L K L K$0DL ! L N K L K L N K N & 1 1 J J K$0"L J / J / K$0-L K L J K L J D K$-$L K L K LJD K$00L J D

K L K L ! K$03L L N K L K L J D N K N 4 &

J N K$0%L J K L J - J .K L J &" K$0$L 9

=

&

&

&

=

&

&

&

=

&

&

&

&

=

A

)

&

+

)

A

A

&

)

;(

>)

+

) A

;)

&

&

=

Æ

:

37

Æ

&

:

5

5

7

5

:

8 0 8 - - ? K

( *

9 K$03L ! K$06L J K" L K L N K L =

7

5

=

&

&

&

&

!

K$08L J . K L N K L .K L 4 9 9

&

4 $"% ( "$ 5 ' % 5 "- ( 4 $"$

K$0#L J -"% J 3D ( )

J %D µ ( 9 J -%? 4 7 & 0W6 3W6 &

=

&

&

&

7

5

+

+

4

,

J

(

+), *, , )

( &

-" &

9 -"

K"-L K"6L K-0L K$6L ( &

3 +44 , 6 3 ! ! %!

2 - J

( 9

& (

9 4 $"6 $"8 9 &

4 $"6 $"8 B *

-"

9 4 & ( Q 2

J - "D83 " "D ? J 68"-) K$3DL ( N 9 J - % " "D K$3"L 4

9 4 7

5

7

5

7

)

5

A

)

A

5

+

+

5

5

5

7

5

7

5

8 0 8 - - ? K

"

$

' G

$

N +4> ' -,- , - N ( - - 8 ? 4 - 4 - 3

N ,4 3 -,- , - N ( - - 8 ? 4 - 4 -

9 %- )

J -

J J K$3-L

J K L J K L K$30L 7

=

5

#

=

#

7

(

7 =

(

5 =

5

3 +44

4 [ [

M " ; M (

. #

4 $"6 $"8 >

9 H63I K L

J - N- " N N K"L

$

I $ I

K

H/ 5L 1 N 5 A: K1 L 5 N - E 1 N 5/ 1 A: E

- K

/

$

K L

L K L K L K"L N NN / " / K L K"L N / K$8-L 4 K$$8L K$$#L

J D J K$80L J D K$83L // J D J D J ! " N K L K$8%L " / / N - N K L J D

N

!

H 5/ 1 5 E 1

5

5 1

5 1

A:

$

4

1

$

$

$

1

H

5

1

1

4

1

A:

$

%

3 +44

! " N / K L N -" N K L

$

5 $

5

J - N K"L 3 / N-N K" /NL / / - N K"L K- / / / L / J N- N /

5

5

H 5

5

H H

E

H E

1

5

E

5 1

1

5

1

1

1

H A:

1

5

K$86L

A:

A:

K$8$L

JD

$

K$88L

A:

K$8#L J - N K"L 3 / N-N K" N/ L / - N K"L K- / L K$#DL J N- N / ) Æ K L K L 9 &

5

" -- - J --#

1

H H

5

E

5

N D D

H 5

1

H E

5

5

E

/

1

" - J#

N D

1

A:

A:

5

H A:

1

A:

1

K$#"L

D D

5 1

J
' 8 # 1 '8 , 6 * > 4 !H!! G # " % & ' ' G "" !" & ""! "'!" &" %'! & &! '%%

( K L 4 $-3 K L 4 $-3 K L &

* J D 9 9 (

9 & 9

9

& 9 )

C * 9 9 9 9 # %DB &

& ( 7

7

7

7

7

" > . , ,-

&"

#'$ B

'G

'G

#'$ B

#$ A , - #'$ > , 6 , 1 %"8 % 1 ! "8 - #$ , -4 , , - #'$

9 & ) 9 (

9 K L J " - 0 4 $-% 4 & # %DB K L K L K L & 4 7 K L J " - 0 4 $-$ @

7

A

7

7

@

7

@ 7

@ 7

&

3 +44

8 #'$ B

8 #'$ B

'G

'G

'G

8 #'$ B

--- 4 8 #'$ #$ 8 #'$ 1 $@ #$ 8 #'$ 1 ,@ #$ 8 #'$ 1 +

" > . , ,-

&

, #'$

'G

'G

'G

, #'$

, #'$

B - -4 / ,#'$ #$ , #'$ 1 $@ #$ , #'$ 1 ,@ #$ , #'$ 1 +

&

3 +44

1' # # / 5

& 5 = 4 $-6 > M )

4 $-6 )

) >

Q 2

+

2

/ Q 2

+

2

J - " "D ? J D 0"3 K$""%L )

J 0 ( 9 9 > 4 $-6 J "D 9

! . K L J " - $

)

H

A

)

H

)

H

A

#

#

#

#

A

A

A

A

(

:

7

A

6 - ( ' 1 %%%" G

' 1 ' G

+4 - 4

- ) 4> 6 - ( ' 1 ' G

& 7 9

& 4 $03 * 4 9

7 )

7

7#

- 0 +4 G

-

8 #' $ )

'

0 0

, #' $

1 1

1

0

' ' ' ' ' ' ' ' ' ' ' ' #$ --- 4 8 #' $ - #$ - -4

/ , #' $ 6 4 4 1 8 -F 4 1 8 - F 1 "

$ % ""

6 $ &

( K H% 8 # -D "D# "-"IL ) *

7

(

; 7

* 4 )

H#I *

* ) )

* ( * ) 7

* 4 9 * ( &

( * * (

*

'

% 2 +44

4 + H""%I & *

7

*

* 7

(

*

7 & ! "8%% H08I ( *

& ) , H"3I ! H"8I 4

9

* 7& K* L K

L J ' K6"L * ( * &

* & " $ % J# K6-L

1"2 1-2 102

(

K6"L * ) 4

* J K60L

12 & * ' &

* & " $ D D J # D // D % K63L D D

5

5

5

5

5 5

5

5 5

5

5

5

% 6 7

'

* & K63L

* 4 + H""0I * ) (

7& " K6%L J F

F

.

' K6"L K63L K6%L J '"' K6$L

.

F

2

' J N // / N K66L ' / (

J K68L J I

5

.

I

5

I

F

2

I

5

I

2

I

I I

F

K6$L K68L K6#L '"' J 2 2 ; . * 9 J N // N J K6"DL '"' * )

7

* * )

? 4 & )

5

I

I

I

I

F

5

I

I

F

5

I

I

I

I

'

% 2 +44

- 7 1 =

* ) 9 ( 9

J D J D & * / & M 9 * J D

9 ( * 9 * 9 N // J ID H ID H K6""L

K D LJD ID H K6"-L KD L J D ID H K6"0L K DL J K6"3L K6"-L K6"0L 9 K L ( B 9

* Æ

K H""0IL ( K6""L K6"3L 4 9

* (

* 9 K6""L

J J // K6"%L

* 9 + J ' ID H ID H K6"$L

5

I

5

I

I

I

I

I

I I

5

5

% 6 7

N

K6"6L

+ J '

I

I

I

'%

K D LJD ID H D K6"8L KD L J D ID H D K6"#L K DL J K6-DL (

&

K L J K L " / K L K6-"L K L / K L

J ID H K6--L D K6-0L KD L J D K6-3L K DL J /J / ID H K6-%L D K6-$L / KD L J D K6-6L / K DL J " ( * 9

+ H""0I Æ K6--L K6-3L K6-%L K6-6L ( & ' K6-8L K L J & -&

I

I

I

I

I

I

I

I

I

&

'

-& J &

3

&

3

J

K6-#L

'&

% 2 +44

& ' K60DL / K L J & . K6-8L K60DL ) K6-DL * 9 *

M 4 ' 3 & K L J -& -/ // K60"L

5

5

- 7 1 2. 9

/ &

/ ( ( D ( * ( * 9 * + J K60-L ID H ID / / // H ' (

K D LJD ID H D K600L / K / // L J D ID H D K603L / D K60%L KD L J D ID / // H (

K DL J ID H ID / / // H K60$L / K60-L K60$L , K6-"L (

O Æ ID H / ID / // H H""0I ( +

I

I

5

+

+

+

5

5

+

5

+

5

% 6 7

''

&

) ' & - H" K LI K L J -& K606L ' & ' & //

/ & /

5

3 5 +

3

3

3

+

- 7 1 ) ! )!

/ * * ( * ) ID H ID / H ( J D ( * 9 + J ' ID / H ID / / // H K608L (

/ D K60#L K D LJD ID H / / K / // L J D ID H D K63DL / KD L J D ID / // H D K63"L K / L J D ID / / // H D K63-L (

ID / / // H K630L K DL J ID / H K608L K630L

& ' & ' K L J - H" K LI &

+

+

I I

+

5

+

5

+

5

+

5

+

5

5

+

5

+

5

+

+

5

+

5

.3

.3

- H" K LI & &

3

3

. 3 5 +

3 5

/

+

.3

//

+

'

&

3

/

+

'

K633L

!!

% 2 +44

( & * * / & *

9 ID H ID / H J / J D " ( D 4

* Æ //

J - 9

9 & - H60I ' &

& & 9 & 4

9 ( % H$$ ##I )

>

* M 7 K L & 9 7 K%

&

& Q - +44

K L & ( K4 8"L , K L 6%

6

?0 &

&

! H#6I #/ ! H"--I

; * & * Æ M K L & K L

) ( & & & (

9 ) 9 ( 7 K L &

K L J K LN ' K8"L K L J K LN ' K8-L 1 ! K LJK L J 0K80L ! NK83L ' J

J 8 3 " "D J D % K8%L J # 0 " "D J " D K8$L J 6 "0 " "D J 0 D K86L

F

4 8% 8$ &

J D "D J "D -D K830L C

7 K L J K L ( 7 4 7 O

7

#

(

#

#

#

#

#

6

##

#$

#'

&

& Q - +44

.

( K80%L K83"L J K833L

5

# #

(

7

KL J + K L 6

#'

6

K L

'

K83%L

K83$L J" K L J D J N" K836L K L D ( K L + K L ( 9 9 H""DI ( ( 8"

/ K L J " C

7 > 4 8% 8$

( 8- *

,

+

,

+

6

+

+

*

6

6

=

;

. -- , ( - ? , 3 - , & %

- ? 4 - -

1 ! !

1 ! !

& +- - ? G 2

3

'

( ( 80 * ( M ( + 4 86

K L + K L ( K L C 6

6

6

- ?- ( - , % (* - - , 6 4( 4 ( 4 - ?- ( - ?- ( ( # ( $ #-$ ! !! !" F4 ! !!&!" !""& ! !"! !" M ! !!!! !!!! M ! &" M ! '!& '%! M ! "' !''' M & ! &" &&%! M % ! &"&% &&%! M

9 & ) ( 9 9

(

"!

& Q - +44

2 - #+$ 4 -

2 - #+$

2 - #+$ 4 -

2 #+ $ 4 - -

2 - #+$

+

+ - *

- 4

2 - #+$ - ,*

- ? # $8 - 2 -#+$ ( , - ? # $@ - *

- 4

2 - #+ $ #$

( ""

& Æ &

' 9 )

*

K H#$IL

H$3I

9 K A.L

& ,

9 9

A. ) 4 9 9 ;*

"

' +44

-3

A. 4 #" (

K

L K L K L

K L J K L K L J K L K#"L 9

9 9

& ' J J " K#-L Y ( * )

9 :N :N J K L K#0L :N :N J D K#3L Æ

J N N K#%L J

K L K#$L Y K L K N L J NK N LY Y K#6L J

K L K#8L J N K##L J Y K L

K Y Y L K#"DL 9 K L K#0L K#"DL &

*

5

.

+

L

D

(

(

D

I*

I*

I(

I(

0

M

M

*

0

0

L

. +

5 )

5D

) M

D

. + M

D

D

.

. + D

M

D

. +

L

M

. +

.

D

4

M

.

. +

,

D

M M

4

D

M

. ,

M

D

M

' +44

""

;#+$ # ?: 6

A. K#0DL K#0-L

(

>

A. ( #" ( A.

K L K L

=

D

M

!

' +44

, , 8 1 &8 ( ( - ,- 6

- ( - ( 0

- ( . ( 1 !&! F4 !&' F4 1 !! F4 ! F4 1

# 1 ! !!% 1 # 1 ! ! ! 1 & # !!&& !!"&! 1 & # !!&& !! % 1 ! B - !!! B - & 1 ! B - !!! B -

- .&&( >?: 6

A.

A. (

J 8 ( 9 9 8 #$%'H%&8 !! 3 G

- 0 R 7D. - 4 6 , / 18 4 >8 8 !!" # * . 8 'H%8 ''% &% 2 K R - 0 4 . 2 . G8 +46

- O 8 BK8 '&& && 2 K R - 3 K U 2 &( * 48 G-48 ''' &' 2 K R - Q S = " %

G8 A,, -- .(8 BK8 '' '! 7

A 4 - , *

$ 8 "#"$ H '8 ''% ' 3 7F A,- 4 - 4 - 4 , )- , > 8 %H 8 ''& ' K 7 - 7 0- $) 3 FF8 B6 S F8 ''" '" 78 0 - - 0 -? , ,

4 8 !"!H" 8 '% ' 3 - -, ? - > *

, 8 ! H"8 !! ' . + 3

*G8 +46

- O 8 BK8 ' ' 3 3 - + Y? * *G8 +46

O 8 BK8 '&' '% + 3

F-8 R K 8 Q-8 48 - Q

3 -* 4 , F J - # 8 %%H&%8 '&' '& G 2 B F - ,- 4 , - , , ( ,* 8 H%8 ''& '' 4* 4 - $8 & 'H&8 ''& !& . - K 6 ? ,, , -?- 8 "" "H "%!8 '' !' . R . 68 . 3 D48 - K . Q

F ( A + - B , 8 7 - 8 '' ! R F 6 F , 44 - U 8 G 6 6 4 - 744 , - #8 "&&"H %8 '& 0 #B

8 2 4*04 *A( W 2W 48 !!! Q 6 - 0 F B *G -8 B6 S F8 '&" " 0 (- ! $> / & . = $> . ! $> . A $> / ,4* 48 Q8 !!! 0 (- < F , , , - 6 ( , - - ( 8 " H'8 !!" 0 (- - 3 G

, / , -O ( $8 %H%8 !!" F , ( (%8 &%H'8 '% % 0 F D- - , , -

# 8 !H "8 '" & B -- %/ 326*G8 B6 S F8 ' ' ,4 - , - 6( , 4- - 8 %%H'8 ''& ! 3 4 - 4 H ( 3 3 2, 8 2 B - 8 - Q . 2 8 - 8 &

8 ,4 &'H! 8 B6 S F8 '%% B *G - B Y ,4* 48 Q8 '' Y J

F8 U8 R F 8 0 < 8 - F F* - , J - 4 ( % #8 "& H "8 ''" " G U - U R4 & (& # # Q*U

(48 YW 8 '% K * U4 - * 7 0 - )

8 "&!H"&%8 ''' Q U # $ ' C) & 3(*F 8 3W 8 '&

.

Q U 0444 - , 44 2 3 8 - 8

8 ,4 'H"'8 Q8 '&& ,4* 4 % K U # C) * # &# *

8 Q - 0 W 3F - A( W 4 !!;8 !! & R 7 U

-8 R B < 8 - + R 0

+44 - 4 6 , 8 H !8 '' ' . . S4 <

- 4 3 , ( 8 #$&'H !!8 '%' "! . . S4 < 4

( 8 #"$ "H 8 '&! " G D S48 G S 8 - K K 4 8 %"H &"8 ''" " 7 0 Y- 8 &""&H""8 ' "" 7 0 Y- , 4 ( - ,, ,* , ) 4 3 - +.7*3 8 QF8 8 '%" " 7 0 Y- 7 0 Y-8 R 8 R F8 - 3 8 - 8 %

8 ,4 H"'8 B6 S F8 '% 0-

" 7 0 Y- , 4 ( - ,, ,* , ) 4 H 8 &''H '8 '% " 7 0 Y- , 4 ( - ,, ,* , ) 4 H 8 &"!H"%8 '% "% 7 0 Y- , 4 ( - ,, ,* , ) 4 H 8 '"H&!8 '% "& 7 0 Y- ,

8 "H &8 '%& "' 7 0 Y- 0 , , - 8 'H '%8 '& ! Y4 , 4- U GF 8 - 8 & ! $> 8 ( " - - 8 ,4 H 8 Q 648 '' 64 G*K Y %

R6 0- 8 - 8 - 8 '' G*K Y - Y 7 ( - F* 4 8 "%H8 '&!

*

, 4 "8 "'

, "8 "' -(- '' 444 , " 4 , - "! 4 " * '8 %" *, - !&8 %8 %8 8 &8 &8 !8 8 "8 "& ? &%

( "8 "&8 ! (44 , " - - 8 , - *

4

- ? - 4

/ 6 ' - / 6 & - ! - '

* , - *

4

- J - ) - 8 ' - 4 - % -- -O "! -- "

-

- %

- , - ),-- - ' "8

"

? &8 ! ( "8 "& , , " ,

- !

- ,

"

"%

- '

- - 4 "" ( "8 "' * "8 ! ()

- &

- & , " - 3 45 6 "8 "' - ,

- %" - ,

% - ,

!8 %"

-)

-?

"'

, "

-*F ( "

, - , , "% -4 / 8 8 &8 %%8 &&8 ! -,- , ' - 4 ), ! -O > ' , > , ' , 4 4 '' , *? '& , !!

, '8 !! -O ( ' - " - - - ( "8 "&8 ! - "! - " - -

"

"

,? " , , >

- !

- ), & ) -- - 8 ! ) ,, 8 8 8 8 8 %" ? - &8 &!8 & > , 8 & - % - 4 !

-(- '' ( " 8 "8

-- %' ( ! ,) - ,

- %" -?- "' - - ), & 2

% , %' 7*. " 8 & 4 (

, &! ,

( >- & > *), ' > *2

%8 '8 8 8 %8 &8 !!8 8 8 !

*

4 8 & , 6 & ),

%' ' 4 ! ( 64 - (4 * , 6 *, - '' !&8 %8 %8 8 &8 &8 !8 8 - ? " 8 '8 !&8

!!8 8 %8 8 &!8 &8 !!8 4

4* -- "8 8 " 2

% 4 , % 4 ,

"' 4 "! 4 "

-) 4 - !!8 %8 %8 & 8 4 & 4

% - ?

- - ( ! - %8 ' -, "8 "& - ? *, " 8 '8 !&8 - "8 "& , %8 ( 8 %

- '

- , %' F - * ,-

, !'

, & - , "&

-

- ' ( %8 &!8 -6F ' , % ( ),

&! ( , 4 ' ( " 8 '8 "8 ( - - ,- " , -O !! 7*. " 6 * - "8 "' 6 )-- -- "8 "' 8 & 4 4 ( 4 ( 7D. - 4 "8 ""

( "8 "' * "8 ! 72 - & 30*3B* ,

"% 30*.- & > *), ' > *2

%8 '8 8 8 %8 &8 !!8 8 8 ! -- - !!8 %8 &8 !!8 /)( "8 "&

-)

" ?* "

,, ) &

" WZ( "

,? &8 '

*, !!8 8 %8 8 &!8 &8 !!8

, *4- ,

-- %'

-- ,

-- ,

"%

-- '

-- "

--- 4 8 &&8

J % *, - 6 *, - '

' 4

, & A*3B* ,

"% A*.

Applied Fuzzy Arithmetic: An Introduction with Engineering Applications

Applied Fuzzy Arithmetic: An Introduction with Engineering Applications

Applied Fuzzy Arithmetic: An Introduction with Engineering Applications