Algorithm for Fully Fuzzy Linear System with Hexegonal Fuzzy Number Matrix by Singular Value Decompo

Integrated Intelligent Research (IIR)

International Journal of Computing Algorithm Volume: 03 Issue: 03 December 2014 Pages: 218-222 ISSN: 2278-2397

Algorithm for Fully Fuzzy Linear System with Hexegonal Fuzzy Number Matrix by Singular Value Decomposition A.Venkatesan1, A.Virginraj2 1,2

PG & Research Department of Mathematics, St. Joseph’s College of arts And Science (Autonomous), Cuddalore Email :suresh11venkat@gmail.com, avirginraj@gmail.com Abstract - A general fuzzy linear system of equation is investigated using embedding approach. In the literature of fuzzy system: 1) Fuzzy linear system, 2) Fully fuzzy linear system, are more important. In both classes of these system usually the author considered triangular and trapezoidal type of fuzzy numbers. In this paper, we introduce hexagonal fuzzy number and computing algorithm for n  n fully fuzzy linear

non-membership; the value one is used to represent complete membership and the values in between are used to represent intermediate degrees of membership. The mapping A is also called the membership function of fuzzy set A B.

Definition

system A  x  b (where A is a fuzzy matrix x and b are fuzzy vectors) with Hexagonal fuzzy numbers, by Using the singular value Decomposition method.

A Fuzzy number “A” is a convex normalized fuzzy set on the real line R such that  There exists at least one with  is piecewise continous

Key Words - Fully Fuzzy Linear system, Hexagonal fuzzy number, Hexagonal fuzzy number matrices, Singular Value Decomposition.

C.

I.

INTRODUCTION

Linear system of equation has applications in many areas of science, Engineering, Finance and Economics. Fuzzy linear system whose coefficient matrix is crisp and right hand side column is an arbitrary fuzzy number was first proposed by Friedman.

Several methods based on numerical algorithms and generalize inverse were used for solving fuzzy linear systems have been introduced by many authors.A linear system is called a fully fuzzy linear system (FFLS) if all coefficients in the system are all fuzzy numbers. Nasser was investigated linear system of equation with trapezoidal fuzzy numbers using embedding approach. Amitkumar was solved FFLS with Hexagonal fuzzy numbers using row reduce echelon form. In this paper, we introduce hexagonal number and computing algorithm for n  n fully fuzzy linear system A  x  b

A fuzzy number is a triangular fuzzy number denoted by ( where are real number and its membership function is given below.



A

D.

 ( x  a1 )  ( a 2  a1 )   (a3  x ) (x)   (a3  a2 )    0

fo r

a1  x  a 2

fo r

a2  x  a3

o th e r w is e

Definition

A fuzzy set A  ( a 1 , a 2 , a 3 ) is said to be trapezoidal fuzzy number if its membership function is given by where a1  a 2  a 3  a 4

(where A is a fuzzy matrix x and b are fuzzy vectors) with Hexagonal fuzzy numbers, by Using the singular value Decomposition method (SVD). Most direct methods fail when dealing with system of linear equations that are singular because matrix inversion is not possible. SVD is very powerful and efficient in dealing with such systems.



II. PRELIMINARY A.

Definition

Definition

E.

Let X be a nonempty set. A fuzzy set A in X is characterized by its membership function ,and A(x) is interpreted as the degree of membership of element x in fuzzy A for each .The value zero is use to represent complete

A

0  ( x  a1 )   ( a 2  a1 )  ( x )  1  (a4  x)   (a4  a3 )  0

fo r x  a1 fo r a1  x  a 2 fo r a 2  x  a 3 fo r

a3  x  a4

fo r

x  a4

Definition

A Trapezoidal fuzzy number A  ( p , q , r , s ) is said to be zero trapezoidal fuzzy number if and only if

218

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F.

International Journal of Computing Algorithm Volume: 03 Issue: 03 December 2014 Pages: 218-222 ISSN: 2278-2397

Definition

Two fuzzy number A  ( p , q , r , s ) and B  ( x , y , z ,  ) equal if and only if p  x , q  y , r  z , s   G.

Definition



AH

For two fuzzy number A  ( p , q , r , s ) and B  ( x , y , z ,  ) the operation extended addition, extended opposite, extended multiplication are ( p , q , r , s )  ( x, y , z , )  ( p  x, q  y , r  z , s   )

 A   ( p , q , r , s )  (  p , q , r , s ) .

0     1  x  a1       2  a 2  a1     1 1  x  a2      2 2  a3  a2    (x)   1   1  x  a4     1    2  a5  a4     1  a6  x        2  a6  a5     0 

f or

   

x  a1

f or

a1  x  a

2

f or

a

 x  a

3

f or

a

f or

2

3

a

4

 x  a

 x  a

 x  a

f or

a

f or

x  a

5

4

5

6

6

If M  0 and N  0 then

A.

( p , q , r , s )  ( x , y , z ,  )  ( px , qy , rz , s  )

For scalar multiplication  p , q , r , s   0   ( p, q, r, s)    p ,  q ,  r , s   0 H.

An Hexagonal fuzzy number denoted by A H =(P1(u),Q1(v),Q2(v),P2(u))

Definition

A matrix A  ( a ij )

is called a fuzzy matrix if each element

of A is a fuzzy number .A fuzzy matrix A is positive denoted

by A

Definition

if each element of A is positive. Fuzzy matrix

A  ( a i j ) which is n  n matrix can be represented such that

AH

is defined as

for u  [ 0 , 0 .5 ]

and

v  [ 0 .5 , w ] where,  P1(u) is a bounded left continuous non decreasing function over [0,0.5]  Q1(v) is a bounded left continuous non decreasing function over [0.5,w]  Q2(v) is a bounded continuous non decreasing function over [w,0.5]  P2(u) is a bounded left continuous non increasing function over [0.5,0]

( a ij )  ( p ij , q ij , rij , s ij ) where A  ( p , q , r , s ) .

a) Remark If w=1, then the hexagonal fuzzy number is called a normal

I.

hexagonal fuzzy number. Here A w represents a fuzzy number in which “w” is the maximum membership value that a fuzzy number takes on whenever a normal fuzzy number is meant,

Definition

A square matrix A  ( a i j ) is symmetric if a i j  a

ji

the fuzzy number is shown by A H for convenience.

III. HEXAG0NAL FUZZY NUMBERS

b)

A fuzzy number A H is a hexagonal fuzzy number denoted by

Remark

A H is ordered quadruple Hexagonal fuzzy number P1(u),Q1(v),Q2(v),P2(u) for [0,0.5] and v  [ 0 .5 , w ] where,

A H  ( a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ) where a1, a2, a3, a4, a5 ,a6 are real

numbers and its membership function  A ( x ) is given below

P1 ( u ) 

H

Q1 (v ) 

1

(

2 1 2

1 2

a 2  a1



1

(

2

Q 2 (v )  1  P2 ( u ) 

u  a1

(

1 2

(

)

v  a2

)

a3  a2 v  a4

a5  a4

a6  u a6  a5

)

)

c) Remark Membership function  A ( x ) are continuous functions.

H

219

Integrated Intelligent Research (IIR)

B.

International Journal of Computing Algorithm Volume: 03 Issue: 03 December 2014 Pages: 218-222 ISSN: 2278-2397

Definition

A positive hexagonal fuzzy number

AH

is denoted as

A H =(a1,a2,a3,a4,a5,a6) where all ai’s>0 f or all i=1,2,3,4,5,6.

Example: A=(1,2,3,5,6,7)

C.

Definition

A negative hexagonal fuzzy number

A H is denotes as

A H =(a1,a2,a3,a4,a5,a6) where all ai’s<0 for all i=1,2,3,4,5,6.

Example: A H = (-8,-7,-6,-4,-3,-2) Note: A negative hexagonal fuzzy number can be written as the negative multiplication of a positive hexagonal fuzzy number.

D.

Let A H = (1,2,3,5,6,7) and B

H

=(2,4,6,10,12,14)be two fuzzy numbers then

AH -B

H

= (-1,-2,-3,-5,-6,-7)

Definition

Let A H  ( a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ) and B

H

 ( b1 , b 2 , b 3 , b 4 , b 5 , b 6 ) be two hexagonal fuzzy number,

If A H is identically equal of B a4=b4, a5=b5, a6=b6.

H

only if a1=b1, a2=b2, a3=b3,

Let A H =(1,2,3,5,6,7)and

IV. OPERATIONS OF HEXAGONAL FUZZY NUMBERS

B

Following are the three operations that can be performed on hexagonal fuzzy numbers, Suppose

H

=(2,4,6,8,10,12)be two fuzzy numbers then

AH *B

H

= (2,8,18,40,60,84)

A H  ( a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ) and

B H  ( b1 , b 2 , b 3 , b 4 , b 5 , b 6 ) be two hexagonal fuzzy number

then Addition: A H Subtractio A H

 B H  ( a 1  b1 , a 2  b 2 , a 3  b 3 , a 4  b 4 , a 5  b 5 , a 6  b 6 )

B

H

 ( a 1  b1 , a 2  b 2 , a 3  b 3 , a 4  b 4 , a 5  b 5 ,a 6  b 6 )

Multiplication: AH *B

H

 ( a 1 * b 1 , a 2 * b 2 , a 3 * b 3 , a 4 * b 4 , a 5 * b 5 , a 6 *b 6 )

Let A H =(1,2,3,5,6,7)and B

H

=(2,4,6,8,10,12)be two fuzzy numbers then

AH +B

H

= (3,6,9,13,16,19)

A.

Definition

Consider

n n

fuzzy

linear

system

if

equation

( a 11  x 1 )  ( a 12  x 1 )  .......... ...  ( a 1 n  x 1 )  b 1 ( a 21  x 1 )  ( a 22  x 1 )  .......... ...  ( a 2 n  x 2 )  b 2 .......... .......... .......... .......... .......... ...... ( a n 1  x 1 )  ( a n 2  x 1 )  .......... ...  ( a nn  x n )  b n

The matrix of the above equation is A  X  b where the coefficient

matrix

A  ( a ij ) is

an  n

fuzzy

matrix

and X j , b j  F ( R ). This system is called a fully fuzzy linear system (FFLS).

220

Integrated Intelligent Research (IIR)

B.

International Journal of Computing Algorithm Volume: 03 Issue: 03 December 2014 Pages: 218-222 ISSN: 2278-2397

Definition

T

T

T

T

T

T

(U 1 D 1 V 1 , U 2 D 2 V 2 , U 3 D 3 V 3 , U 4 D 4 V 4 , U 5 D 5 V 5 , U 6 D 6 V 6 )

 ( x , y , z , w ,  ,  )  (b , g , h , i, j , k ) U1D1V1Tx=b  x= V1D1+U1Tb

For solving n  n FFLS A  X  b where A  ( P , Q , R , S , T , U ) , X  ( x , y , z , w ,  ,  ) and

Where D1+ is the Moore-Penrose generalized inverse of D1 and is obtained by taking the reciprocal of each non-zero entry on the diagonal of D1, leaving zeros in place.U2D2V2Ty=g  y= V2D2+U2TgWhere D2+ is the Moore-Penrose generalized inverse of D2 and is obtained by taking the reciprocal of each non-zero entry on the diagonal of D2, leaving zeros in place.

b  (b , g , h , i, j, k )

( P , Q , R , S , T ,U )  ( x , y , z , w , ,  )  (b , g , h , i, j , k ) ( P * x , Q * y , R * z , S * w , T *  ,U *  )  (b.g , h , i, j , k ) P*x  b Q* y  g

Similarly z= V3D3+U3Th, w= V4D4+U4 T I, = V5D5+U5T j , = V6D6+U6T k.If A and B are square, symmetric and positive definite then U=V.

R*Z  h S *w  i T *  j U *  k

V. SOLVING FFLS USING HEXAGONAL FUZZY NUMBER MATRICES BY SINGULAR VALUE DECOMPOSITION Any real m  n matrix A can be decomposed as A = UDVT where U is m  n orthogonal matrix and its columns are eigenvectors of AAT, D is a diagonal matrix having the positive square roots of eigenvalues of ATA or AAT as its main diagonal called the singular values of A arranged in descending order and V is n  n orthogonal matrix its columns are eigenvectors of ATA..ATA= (VDUT) (UDVT) = VD2VT. AAT= (UDVT) (VDUT) = UD2UT. The product (UDVT) is called the singular value decomposition (SVD) of A.The SVD construction is based on the following result: A. Singular value decomposition Theorem For every m  n real matrix A there exists an orthogonal mxm matrix U, an orthogonal nxn matrix and a m  n diagonal matrix D such that A= UDVT. B. Moore-Penrose Generalized inverse Moore-Penrose generalized inverse or pseudo inverse can be computed using SVD. When A is a singular taking A= UDVT in Ax= b the solution is x=A+ b where A+=V D+ UT. The solution is not exact but is closed set in the least squares sense. If A is a non singular then A+=A-1.

VI. SOLVING FFLS BY SINGULAR VALUE DECOMPOSITION METHOD Consider the FFLS A  x  b Taking trapezoidal fuzzy

number

matrices

( P , Q , R , S , T , U )  ( x , y , z , w ,  ,  )  ( b , g , h , i , j , k ) Let

P = U 1 D 1 V1

T

T

T

T

, Q = U 2D 2 V2 , R = U 3D 3V3 ,

S = U 4D 4V4 , T = U 5D 5V5

T

and U = U 6 D 6 V 6

Algorithm for solving FFLS by singular value decomposition method  For the crisp linear system p x  b find the eigenvalues of PPT and PTP. The eigenvectors of PPTmake up columns of U1 and eigenvectors of PTP make up columns of V1. Square roots of eigenvalues from P TP orPTP called singular values are the diagonal entries of D 1 in descending order. Compute P=U1D1V1T.  Compute x= V1D1+U1Tb  For the crisp linear system Qy=g find the eigenvalues of QQT and QTQ. The eigenvectors of QQT make up columns of U2 and eigenvectors of QTQ make up columns of V2. Square roots of eigenvalues from BB T orBTB called singular values are the diagonal entries of D2 in descending order. Compute Q=U2D2V2T.  Compute y= V2D2+U2Tb  For the crisp linear system Rz=h find the eigenvalues of RRT andRTR. The eigenvectors of RRT make up columns of U2 and eigenvectors of RTR make up columns of V2.

Square roots of eigenvalues from RRT orRTR called singular values are the diagonal entries of D 2 in descending order. Compute R=U3D3V3T.  Compute z= V3D3+U3Th  For the crisp linear system Sw=i find the eigenvalues of SST and STS The eigenvectors of SST make up columns of U2 and eigenvectors of STS make up columns of V2.

Square roots of eigenvalues from SST orSTS called singular values are the diagonal entries of D 2 in descending order. Compute S=U4D4V4T.  = V5D5+U5T j VII. CONCLUSION In this paper solution of FFLS A  x  b is obtained by SVD by a new methodology in the form of hexagonal fuzzy number matrix. By using orthogonal matrices throught SVD reduces the risk of numerical error. The Moore-Penrose generalized inverse can be easily obtained from SVD. Hence the SVD is very efficient for the system A  x  b when A is singular.

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