Exact soultion of full interval linear equation AX=B based on Kaucher arithmetic

(1)

Int. J.IndustrialMathematisVol. 2,No. 4(2010)319-327

Exat Solution of Full Interval (Fuzzy) Linear

Equation A

2nn X

n1 = b

2n1

based on Kauher

Arithmeti

M.AdabitabarFirozja

, N.Ardin

Department ofmathematis,QaemsharBranh, IslamiAzadUniversity,Qaemshahr,Iran.

Reeived10 August 2010;aepted17Otober 2010.

|||||||||||||||||||||||||||||||-Abstrat

Inthispaper,rst,we proposeKauherarithmetiforintervalnumberandthensolvethe

systemA

2nn X

n1 =b

2n1

where a

ij =[a

i;j ;a

i;j

℄ and x

j =[x

j ;x

j

℄ and b

i =[b

i ;b

i

℄ based

on Kauher arithmeti suh that a

i;j : a

i;j

0. Then, we extend this method by use of

levelsetfor fullyfuzzy systemof ~

A

2nn ~

X

n1 =

~

b

2n1 .

Keywords: intervalnumber;Kauherarithmeti;fuzzynumber;linearsystem.

||||||||||||||||||||||||||||||||{

1 Introdution

Reently, intervallinearequationshavebeenusedtostudyavarietyofproblems. Some

of researhes have worked to solve the fuzzy linear equations. Muzzioli et al. [12 , 13 ℄

solved the Fuzzy linear systems of the form A

1 X +b

1 = A

2 X +b

2

with onditions in

whih the system has a vetor solution and showed that linear systems Ax = b with

A=A

1 A

2

and b=b

2 b

1

, havethesamevetor solutionsandproposedamethodology

to nd a unique vetor solution for the system with non-linear optimization problem.

Abbasbandy et al. [2 ℄ introdued a numerial method for nding minimal solution of

a mn AX +F = BX +C based on pseudo-inverse alulation, when the matrix of

oeÆientsisfullrankroworfullrankolumn,andA,B arerealmnmatrix,unknown

vetor X is vetor onsisting of n fuzzy numbers and the onstants F and C are vetors

onsisting of m fuzzy numbers. Sevastjanov et al. [15 ℄ disussed the treatment of the

intervalzero as an intervalentered around zero and proposeda new method forsolving

interval and fuzzy equations. Bukley et al.[5, 6 , 7℄ presented neessary and suÆient

onditions forequations and onstruted solutions forthe fuzzy matrix equation Ax=b

(2)

whentheelementsinAandbaretriangularfuzzynumbersandshowedthatwithextensive

priniple many fuzzy equations do not have solutions. Allahviranloo [3 , 4℄ proposed the

numerialmethodforndingthesolutionoffuzzysystemoflinearequationAx=bwhenA

isrealmatrixandbisfuzzyvetor. Dehghanetal.[8 ,9 ℄introduedomputationalmethod

forsolvingfullyfuzzylinearsystems. Abbasbandyetal. [1 ℄introduednumerialmethods

for fuzzy system of linear equations. Wang et al.[16 ℄ disussed fuzzy linear equations,

X =AX+U with iterationalgorithmsifkAk

1 <1.

The paper is organized as follows: Some basi onepts are presented in Setion 2. In

Setion 3, we proposean approah for solvingthe fullinterval linear equations basedon

Kauher arithmeti for interval number. Subsequently, we extend this approah for full

intervallinear equationsinsetion 4. Finally,onlusion isdrawn inSetion5.

2 Preliminaries

Denition 2.1. By IR we denote the setof all nonempty, losed, bounded intervals in R

identied withordered pairsof numbers[x;x ℄ wherexx. Kauher intervalarithmetiis

an algebrai struture <KR ;+; ;:;= > where the four laws are dened by the formulae

below (see[5,6℄).

Let x=[x;x ℄ and y=[y;y℄ with u +

=maxfu;0g,u =maxf u;0g then

x+y=[x+y;x+y℄ ; x y =[x y;x y℄

x:y=[x:y;x:y℄

x:y=maxfx +

y +

;x y g maxfx +

y ;x y +

g

x:y=maxfx y ;x +

y +

g maxfx y +

;x +

y g

x=y=x:[1=y;1=y ℄ ; yy>0

let, x=[x;x℄and y=[y;y℄ be two intervalsthen

(a) [x;x ℄=[y;y℄ ifand onlyifx=y and x=y.

(b) x0 ifandonlyifx0and sox0 ifand onlyifx0.

RegardingKauherintervalarithmeti

[x;x℄:[y;y℄ = (

[x y +

xy ;x y +

x y ℄; x0

[x y +

xy ;x y +

x y ℄; x0

(2.1)

Denition 2.2. A fuzzy set ~a = (a

1 ;a

2 ;a

3 ;a

4

) is a generalized left right fuzzy numbers

(GLRFN) of Dubois and Prade [7℄, if its membership funtion satises the following:

~ a

(x)= 8

>

<

>

: L(

a2 x

a

2 a

1 ) a

1

xa

2

1 a

2

xa

3

R ( x a3

a

4 a

3 ) a

3

xa

4

(3)

WhereL and Rarestritly dereasingfuntions dened on [0;1℄and satisfyingthe

ondi-tions:

L(t)=R (t)=1 if t0

L(t)=R (t)=0 if t1

(2.3)

For a

2 = a

3

, we have the lassial denition of left right fuzzy numbers (LRFN) of

Dubois and Prade [7 ℄. Trapezoidal fuzzy numbers (TRFN) are speial ases of GLRFN

with L(t) = R (t) = 1 t. Triangular fuzzy numbers (TFN) are also speial ases of

GLRFN withL(t)=R (t)=1 tand a

2 =a

3 .

A GLRFN ~ais denoted asa~=(a

1 ;a

2 ;a

3 ;a

4 )

LR

and an r levelinterval of fuzzy number

~ aas:

[ ~a℄ r

=[a(r);a (r)℄=[a

2 (a

2 a

1 )L

1

a (r);a

3 +(a

4 a

3 )R

1

a

(r)℄ (2.4)

let,~aand ~

b be two fuzzy numbers then

(a) ~a= ~

b ifand onlyif[ ~a℄ r

=[ ~

b ℄ r

ora(r)=b(r);a (r)=b(r)8r.

(b) ~a0 ifand onlyifa(0)0 and ~a0 ifandonlyifa(0) 08r.

We applysomeoperations2f+; ;:;=g toget[a (r);a(r)℄[b (r);b(r)℄byKauher

arith-metifor r2[0;1℄ (inthease ofdivisionb(r)b(r)>0 forall r).

3 Solution of full interval Linear equation

Inthissetion,wewanttosolvethethefollowingfullintervalsystemsformA

2nn X

n1 =

b

2n1 with

8

>

<

>

: [a

1;1 ;a

1;1 ℄:[x

1 ;x

1

℄++[a

1;n ;a

1;n ℄:[x

n ;x

n ℄=[b

1 ;b

1 ℄

.

[a

2n;1 ;a

2n;1 ℄:[x

1 ;x

1

℄++[a

2n;n ;a

2n;n ℄:[x

n ;x

n ℄=[b

2n ;b

2n ℄

(3.5)

Wherea

i;j :a

i;j

0forall i;j.

With Eq. (2.1) and restrition a

i;j :a

i;j

0 and equality of two interval numbers,

system(3.5), 2nnonverts to system 4n4nand isrewrittenas follows:

8

<

: P

j2 a

i;j x

j +a

i;j x

j +

+ P

j2 a

i;j x

j +a

i;j x

j +

=b

i

i=1;:::;2n

P

j2 a

i;j x

j +a

i;j x

j +

+ P

j2 a

i;j x

j +a

i;j x

j +

=b

i

i=1;:::;2n

(3.6)

With =fi;jj a

i;j

0g and =fi;jj a

i;j

0gand T

=.

Matrix formof theaboveequation is asfollows:

SX =b

or

s

1 s

2

s s

x

=

b

(4)

where

X =[x;x℄;

x=[x

1 ;x

1 +

;:::;x

n ;x

n +

℄;

x=[x

1 ;x

1 +

;:::;x

n ;x

n +

℄;

b=[b ;b℄;

b=[b

1 ;b

1 +

;:::;b

n ;b

n +

℄;

b=[b

1 ;b

1 +

;:::;b

n ;b

n +

℄

and s

ij

;i=1;:::;2n; j=1;:::;2n aredetermined as

s

i;j =

8

>

<

>

: s

i+2n;j+2n+1

= a

ij

if a

ij

0;j is odd

s

i+2n;j+2n 1 =a

ij

if a

ij

0;j is even

0 otherwise

(3.7)

s

i+2n;j =

8

>

<

>

: s

i;j+2n+1

= a

ij

if a

ij

0;j is odd

s

i;j+2n 1 =a

ij

if a

ij

0;j is even

0 otherwise

(3.8)

By solvingthesystem(3:6), aording to denition(2.1):

x

j +

: x

j

=0; x

j +

+x

j 0

x

j +

: x

j

=0; x

j +

+x

j 0

(3.9)

Therefore,

8

>

<

>

:

if x

j +

=0 ! x

j

= xj

if x

j

=0 ! xj =x

j +

if x

j +

=0 ! x

j

= xj

if x

j

=0 ! x

j =xj

+

(3.10)

Theorem 3.1. The solution of system Eq.(5) with respet to multipliation in interval

kauher arithmeti exists ifand only if det (S)6=0:

Theorem 3.2. If in E.q. (3.5) a

i;j

0 (a

i;j

0) for all i;j then in system SX = b,

s

2 =s

3 =0 (s

1 =s

4 =0).

Proof: Ifa

i;j

0thenregardingto(3.6) =fi;jj a

i;j

0g=?and ifa

i;j

0then

=fi;jj a

i;j

(5)

exat solution [x

1 ;x

1

℄=[2;3℄ , [x

2 ;x

2

℄=[1;5℄

8 > > > > > > < > > > > > > :

[ 3; 2℄:[x

1 ;x

1

℄+[1 ; 5℄:[x

2 ;x

2

℄=[ 8; 21℄

[ 3; 1℄:[x

1 ;x

1

℄+[0 ; 3℄ :[x

2 ;x

2

℄=[ 9; 13℄

[1 ; 2℄:[x

1 ;x

1

℄+[ 2; 1℄:[x

2 ;x

2

℄=[ 8; 5℄

[2 ; 4℄:[x

1 ;x

1

℄+[ 3; 1℄:[x

2 ;x

2

℄=[ 11;11℄

(3.11)

Aording to Kauher arithmetioperation, Eq. (2.1) onverts tothe following form:

8 > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > : 3 x 1 + + 2x 1 5 x 2 + x 2 + = 8 3 x 1 + + x 1 3 x 2 = 9 2 x 2 + + x 2 2 x 1

+1 x

1 + = 8 3 x 2 + + x 2 4 x 1

+2 x

1 + = 11 5 x 2 + x 2

+3 x

1 2 x 1 + =21 3 x 2 +

+3 x

1 1 x 1 + =13 2 x 1 + x 1

+2 x

2 x 2 + =5 4 x 1 + 2x 1

+ 3 x

2

x

2 +

=11

and matrix form SX =bis

0 B B B B B B B B B B

0 0 5 1 2 3 0 0

0 0 3 0 1 3 0 0

2 1 0 0 0 0 1 2

4 2 0 0 0 0 1 3

3 2 0 0 0 0 1 5

3 1 0 0 0 0 0 3

0 0 2 1 1 2 0 0

0 0 3 1 2 4 0 0

1 C C C C C C C C C C A : 0 B B B B B B B B B B x 1 x 1 + x 2 x 2 + x 1 x 1 + x 2 x 2 + 1 C C C C C C C C C C A = 0 B B B B B B B B B B 8 9 8 11 21 13 5 11 1 C C C C C C C C C C A

then the solutions of the above systemareas:

x

1

=0, x

1 +

=2:0000, x

2

= 0, x

2 +

= 1:0000, x

1 +

=3:0000, x

1

= 0, ,x

2 +

=

5:0000,x

2

=0andwithondition(3:10)solutioninitialsystemasx

1

=[2;3℄ x

2

=[1;5℄.

4 Solution of full fuzzy Linear equation

In thissetion, we want to solve the thefollowing fullfuzzy systems form ~ A 2nn ~ X n1 = ~ b 2n1

suhthat a~

i;j

0ora~

i;j

0 forall i;j.

8 > < > : ~ a 1;1 :x~

1

++a~

1;n :x~

n = ~ b 1 . . . ~

a :x~ ++a~ :x~ = ~

b

(6)

Thisequation is equivalentto the followingfullyparametri intervalequation 8 > < > : [a 1;1 (r);a 1;1 (r)℄:[x 1 (r);x 1

(r)℄++[a

1;n (r);a 1;n (r)℄:[x n (r);x n

(r)℄=[b

1 (r);b 1 (r)℄ . . . [a 2n;1 (r);a 2n;1 (r)℄:[x 1 (r);x 1

(r)℄++[a

2n;n (r);a 2n;n (r)℄:[x n (r);x n

(r)℄=[b

2n (r);b 2n (r)℄ (4.13) wherea i;j (r):a i;j

(r)0 forall i;j and r2[0;1℄.

WithKuherarithmetiandrestrition a

i;j (0):a

i;j

(0)0and equalityof twofuzzy

num-bers,parametriintervalsystem(4.13), 2nnonvertsto parametrisystem4n4nand

isrewrittenasfollows:

8 < : P j2 a i;j (r) x j

(r)+a

i;j (r)x j + (r)+ P j2 a i;j (r) x j

(r)+a

i;j (r)x

j +

(r)=b

i (r) P j2 a i;j (r) x j

(r)+a

i;j (r)x j + (r)+ P j2 a i;j (r)x j

(r)+a

i;j (r) x

j +

(r)=b

i (r)

(4.14)

wherei=1;:::;2n, =fi;jj a

i;j

(r)0g,=fi;jj a

i;j

(r)0g and T

=.

Matrix formof theaboveequation is asfollows:

SX =b

or s 1 s 2 s 3 s 4 x(r) x(r) = b(r) b(r) ; where ~

X =[x (r);x(r)℄;

x(r)=[x

1 (r);x

1 +

(r);:::;x

n (r);x

n +

(r)℄;

x(r)=[x

1

(r);x

1 +

(r);:::;x

n (r);x n + (r)℄; ~

b=[b(r);b (r)℄;

b(r)=[b

1 (r);b

1 +

(r);:::;b

n (r);b

n +

(r)℄;

b=[b

1

(r);b

1 +

(r);:::;b

n (r);b n + (r)℄ and s ij

;i=1;:::;2n; j=1;:::;2n aredetermined as

s i;j = 8 > > > < > > > : s i+2n;j+2n+1 = a ij

(r) if a

ij

(r)0;j is odd

s

i+2n;j+2n 1 =a

ij

(r) if a

ij

(r)0;j is even

0 otherwise (4.15) s i+2n;j = 8 > > > < > > > : s i;j+2n+1 = a ij

(r) if a

ij

(r)0;j is odd

s

i;j+2n 1 =a

ij

(r) if a

ij

(r)0;j is even

0 otherwise

(4.16)

By solvingthesystem(4:14) aording to denition1:

x

j +

(r):x

j

(r)=0; x

j +

(r)+x

j

(r)0

x +

(r):x (r)=0; x +

(r)+x (r)0

(7)

Therefore,

8

>

<

>

:

if xj +

(r)=0 ! xj(r)= xj (r)

if x

j

(r)=0 ! xj(r)=xj +

(r)

if x

j +

(r)=0 ! xj(r)= xj (r)

if x

j

(r)=0 ! xj(r)=xj +

(r)

(4.18)

Theorem 4.1. The solution of system Eq. (4.12) with respet to multipliation in fuzzy

kauher arithmeti exists ifand only if det (S)6=0 8r:

Example4.1. Weonsiderthe followingsystemof fuzzy linear equationwithexat

solu-tion x~

1

=[r+1; r+3℄,x~

2

=[3r+1; 2r+6℄.

8

>

<

>

:

[r 3; r℄:[x

1 ;x

1

℄+[r;7 2r℄:[x

2 ;x

2

℄ =[ r 2

+9r 8;3r 2

27r+42℄

[r 4; r℄:[x

1 ;x

1

℄+[0;4 r℄:[x

2 ;x

2

℄=[ r 2

+7r 12;r 2

15r+24℄

[r;2℄:[x

1 ;x

1

℄+[r 3; r℄:[x

2 ;x

2

℄=[ r 2

+13r 18; 3r 2

3r+6℄

[r+1;5 r℄:[x

1 ;x

1

℄+[r 4; r℄:[x

2 ;x

2

℄=[ r 2

+16r 23; 2r 2

9r+15℄

Aording to the Kauher arithmeti operation above system onverts to the following

forms:

8

>

<

>

:

(r 3)x

1 +

r x

1

+1x

2 +

(7 2r)x

2

= r 2

+9r 8

(r 4)x

1 +

+rx

1

(4 r)x

2

= r 2

+7r 12

(r)x

1 +

2x

1

+(r 3)x

2 +

r x

2

= r 2

+13r 18

(r+1)x

1 +

(5 r)x

1

+(r 4)x

2 +

rx

2

= r 2

+16r 23

( r) x

1 +

(r 3) x

1

+(7 2r) x

2 +

x

2

=3r 2

27r+42

( r)x

1 +

(r 4)x

1

+(4 r)x

2 +

=r 2

15r+24

2x

1 +

rx

1

rx

2 +

(r 3)x

2

= 3r 2

3r+6

(5 r)x

1 +

(r+1)x

1

rx

2 +

(r 4)x

2

= 2r 2

9r+15

where, by solving the above system for r = 0;0:1;:::;1 solution is obtined in Table 1 and

with(4:18) solution of initial equation isshown in Table 2 and Fig. 1.

Table 1

r x

1

(r) x

1 +

(r) x

2

(r) x

2 +

(r) x

1

(r) x

1 +

(r) x

2

(r) x

2 +

(r)

0 0 1.0000 0 1.0000 0 3.0000 0 6.0000

0:1 0 1.1000 0 1.3000 0 2.9000 0 5.8000

0:2 0 1.2000 0 1.6000 0 2.9000 0 5.6000

0:3 0 1.3000 0 1.9000 0 2.9000 0 5.4000

0:4 0 1.4000 0 2.2000 0 2.9000 0 5.2000

0:5 0 1.5000 0 2.5000 0 2.9000 0 5.0000

0:6 0 1.6000 0 2.8000 0 2.9000 0 4.8000

0:7 0 1.7000 0 3.1000 0 2.9000 0 4.6000

0:8 0 1.8000 0 3.4000 0 2.9000 0 4.4000

0:9 0 1.9000 0 3.7000 0 2.9000 0 4.2000

(8)

Table 2

r x

1

(r) x

1

(r) x

2

(r) x

2 (r)

0 1.0000 3.0000 1.0000 6.0000

0:1 1.1000 2.9000 1.3000 5.8000

0:2 1.2000 2.8000 1.6000 5.6000

0:3 1.3000 2.7000 1.9000 5.4000

0:4 1.4000 2.6000 2.2000 5.2000

0:5 1.5000 2.5000 2.5000 5.0000

0:6 1.6000 2.4000 2.8000 4.8000

0:7 1.7000 2.3000 3.1000 4.6000

0:8 1.8000 2.2000 3.4000 4.4000

0:9 1.9000 2.1000 3.7000 4.2000

1:0 2.0000 2.0000 4.0000 4.0000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6 membership function

x

Fig. 1. Theexat solutionx~

1 , ~x

2

andsolutions withthis approah.

5 Conlusions

In this paper, rst, we proposed Kauher arithmeti for interval number and then we

solved the systemA

2nn X

n1 =b

2n1

wherea

i;j : a

i;j

0 and we extended thismethod

by use of r level set for full fuzzy system. We hope that this approah will be use

forother problemsas, linear(nonlinear) system, linear(nonlinear) programmingand the