Australian Journal of Basic and Applied Sciences, 5(9): , 2011 ISSN Fuzzy M -Matrix. S.S. Hashemi

Transcription

1 ustralian Journal of Basic and pplied Sciences, 5(9): , 20 ISSN Fuzzy M -Matrix S.S. Hashemi Young researchers Club, Bonab Branch, Islamic zad University, Bonab, Iran. bstract: The theory of M-matrices plays an important role in the more sciences such as operations research, Markov chains, numerical treatment of ordinary, elliptic partial differential equations and Economical Leontief model. We present an algorithm which gives the membership degree of any fuzzy matrix in M-matrices class, by using the relationship between interval matrices and fuzzy matrices. Key words: Fuzzy matrix, Fuzzy number, Interval matrix, Interval arithmetic, M-matrix, Z- matrix. INTRODUCTION Fuzzy sets provide a widely appreciated tool to introduce uncertain parameters into engineering systems. In many positions, some parameters of the system should be represented by fuzzy numbers rather than crisp numbers. Hence, it is immensely important to develop crisp definitions in fuzzy definitions. On the other hand, theory of M-matrices is the most important part of linear algebra which are used in many sciences. Therefore, in this paper we analyze the relationship between M-matrices and fuzzy theory. We know, definitions in fuzzy environment are uncertainly, hence for define fuzzy M-matrix, it is sufficient to determine its membership degree. Clearly, any fuzzy matrix has a membership degree of M-matrices class that is determined in the given algorithm. In section 2, some preliminaries of the interval arithmetic (Numaier 990) and some topics about M- matrices (xelson 994: Datta 995: Hashemi et al., 2008: Johnson 982: Numaier 990). are mentioned and then we recall fuzzy numbers and arithmetic operators (Dubois et al., 980: Hashemi et al., 2008) n efficient algorithm for computing the membership degree of fuzzy M-matrices is presented in section 3. Finally, one example is given to clarify more this paper. 2 Preliminaries: 2. Interval arithmetic: Interval arithmetic is an elegant tool for practical work with inequalities, approximate numbers, error bounds, and more generally with certain convex and bounded sets. In this section we define real intervals and operations with intervals. (real) interval is a set of the form x [ xx, ] where x is lower bound and x is upper bound. If x is a more complex expression, we also write x = inf ( x ), x = sup( x ). If S is a nonempty bounded subset of we denote by S =[ inf ( S), sup( S)] the hull of S, i.e. the tightest interval enclosing S e.g. [ ab, ], a b {,}= ab. [, ba], a> b Elementary operations o:= {,,, /} are defined on the set of intervals by putting xoy := { xoy : xx, y y} for all xy, such that xoy is defined for all x x, y y. Now we generalize the definitions and rules of interval arithmetic to the matrix case. n m n interval matrix is a rectangular array = ( ) = m n mn of intervals. Clearly an interval matrix =( ) is interpreted as a set of real m n matrices by = m n :, i =,, m, j =,, n. If we suppose = inf ( ):=( ) and the convention Corresponding uthor: Hashemi Young researchers Club, Bonab Branch, Islamic zad University, Bonab, Iran. hashemi_math396@yahoo.com 2096

2 = sup( ):=( ) then the interval matrix can be denoted by =[, ]= m n :. We now extend the definitions of addition, substraction and multiplication to interval matrices. If mn mn B, then we define B by B:= B:, BB mn np and if, B then B B:= B :, BB m p defines by we also define scalar multiplication of a and a:= a : aa, nn by Proposition 2.: mn Error! Reference source not found. Let,, BB,, C, then ) B=[ B, B], B=[ B, B] 2) ( B) = B B= B:, B B. 3) Let is a bounded set of real m n matrices, then inf ( ) and sup( ) exist, and if, b vary over bounded intervals, then (, b) is bounded, and the hull of the solution set which we denote as H b:= (, b) is defined. nn If is a regular square matrix (i.e. every matrix has rank n ), then we define a matrix inverse by := :. Now we recall the definition of crisp M-matrix and interval M-matrix. Definition 2.2: (xelson, 994; Datta, 995) square crisp matrix is an M-matrix if and only if a 0, i, j =,, n, i j and 0. Definition 2.3: (xelson, 994) square matrix is said to be generalized diagonally dominant if a x a x, i =,2,, n () i j ji for some positive vector x =( x, x2,, x ) T n and generalized strictly diagonally dominant if () is valid with strict inequality and it is (strictly) diagonally dominant if () is valid for x = (,,,) T. Lemma 2.4: (xelson, 994) Let =( a ) be an n n matrix, with a 0, j and a >0. If is strictly 2097

3 diagonally dominant, then is an M-matrix. For definition of interval M-matrix, we call an interval matrix inverse positive if is regular and 0. ny interval inverse positive with non positive off-diagonal entries is an interval M-matrix. Theorem 2.5: nn (i)if is an M -matrix and B then B is an M-matrix; in particular, all are M - matrices. nn () is an M -matrix iff, are M -matrices. nn (i) Every M -matrix is regular and =[, ] 0. Proposition 2.6: Let nn. If, are regular and, 0 then is regular and =[, ] 0. Definition 2.7: (xelson, 994; Numaier, 990) crisp Matrix =( a ) is a Z -matrix if a 0,, j =,, n, i j and similarly, an interval matrix is a Z-matrix if for all ; be a Z-matrix. 2.2 Fuzzy numbers and arithmetic operators: Now, we recall the basic notations of fuzzy number arithmetic. Definition 2.8: fuzzy number is an upper semi continuous, normal and convex fuzzy subset of the real line so that : [0,] where ( x ) is the membership function of, i.e. there exists an x so that ( x )=, and ( ) 2 ( ), ( 2) x x min x x, for [0,]. In addition, a fuzzy number is called positive (negative), shown as >0 ( <0), if its membership function ( x ) satisfies ( x )=0, x <0 ( x >0). Definition 2.9: ( -level set). The -level set of is defined as an ordinary set [ ] of which the degree of membership function exceeds the level [ ] = { x ( x), (0,]} we separately define the 0-cut of, written [ ] 0, as the closure of the union of [ ], 0<. Definition 2.0: ( fuzzy number). fuzzy number is said to be an fuzzy number if ( x ) = a x L, x a, > 0, x a R, x a, > 0, where a is the mean value of and and are left and right spreads, respectively. The function L(.), which is called left shape function, satisfies: () L(x)=L(-x), (2) L(0)= and L()=0, (3) L(x) is non-increasing on [0, ). 2098

4 The definition of a right shape function R(.) is usually similar to that of L(.). The mean value, left and right spreads, and the shape function of an fuzzy number are symbolically shown as = a,,. Clearly, = a,, is positive, if and only if, a >0 lso, two fuzzy numbers a and B b a= b, =, =. =,, (since L()=0). =,, are said to be equal, if and only if On the other hand, since each fuzzy number is a set, we can define its subset as follows: n fuzzy number = a,, is said to be a subset of the fuzzy number B = b,,, if and only if a b and a b. Definition 2.: matrix = a is called a fuzzy matrix if each element of is a fuzzy number. The fuzzy matrix will be positive (negative) and is shown by >0 ( <0) if each element of be a positive (negative) number. The nonnegative and non positive fuzzy matrices may be defined similarly. We may represent = a that a =,, a, with new notation = M,, N, where, M and N are three crisp matrices, with the same size of, so that =( a ), M =( ) and N =( ), are called the center matrix and the right and the left spread matrices, respectively. 3 Membership degree of fuzzy M -matrix: We know, the function : X [0,] denotes the degree of specific property that x have in fuzzy number ( x, ( x)). Thus we can define the fuzzy M -matrix (, ( )), where ( ) is the membership degree of to M -matrices class. In this section we introduce an efficient algorithm to obtain the membership degree of any fuzzy M -matrix. Let =( a ) be an fuzzy matrix where a =( a,, ). If 0-cut of diagonal entries be a subset of (,0] or 0-cut of off-diagonal entries be a subset of [0, ), then membership degree of in Z -matrices class is zero, and consequently ( )=0. Otherwise, if for every diagonal entry a we have been a 0 then [ a,0] is a disorder interval. Now we find out h = L (0) and then h cut of a that omit the disorder interval. Obviously a produce the interval [0, ] (Figure ) where h = R L(0)= R ( h ). (2) = L R(0)= L ( h ). (3) 2099

5 Fig. : diagonal disorder interval. lso for off-diagonal entries (Figure 2) a =( a,, ),, j =,, n, i j if a >0 then (0, a ] is disorder interval, and similar to previous case we obtain [ a ] where h = R (0). This h cut, produce the interval [,0] where μ(x) h L R h a α η a a +β Fig. 2: off-diagonal disorder interval. Suppose I ={( i, i): a 0, i =,, n}, J ={( i, j): a >0, i, j =,, n, i j}. If be a membership degree of in Z-matrix with positive diagonal entries then it can be obtain from = where = thus (, i j) J (,) i i I { } { } (,) I (, i j) J (4) = (,) I (, i j) J { } { } (,) I (, i j) J (5) It is obvious [ ] h where h= Max{ h } i, jn is an interval matrix with =. Now, we define the interval [, ],, j =,, n as following: 200

6 0, ( i, i) I,( i, j) J =, = a,( i, i) I i j a,( i, j) J,,(, ) 0,(, ) i i I i j J =, = a,(, ) i j a,( i, j) J i i I There are three cases: I) If there exist an i=,, n such that j=, n ji < j=, n ji II) If > for every i =,, n, then ( )=. III) Otherwise, define ={ i: >, i =,, n}. Suppose there exist j=, n ji then ( )=0. = min x, x [, a ] and = min y, y [, a ] (6) i n i, jn i j such that x > y, i =,, n j=,, n j i then, by definition = n { } { } i= i i, j= i, ji n i, j= n { } (7) membership degree of M -matrix will be ( )= where (8) = R L( ), i =,, n and = R L( ), i, j =,, n, i j. Fig. 3:, positions in diagonal entries. 20

7 Fig. 4:, positions in off-diagonal entries. 4 Numerical example: Let a a 2 (4,5,) ( 2,2,) = = a 2 a 22 ( 3,,4) (4,6,2) where a a L : y = ( x ) L : y = x 2 = 5, a = 2, R : y = 5 x, R : y = x 2 L : y = x 4 L : y = ( x 2) 6 =, a =. R : y = ( x) 4 R : y = x For a we have a = <0 thus [,0] is the disorder interval. By hypothesis h = L(0)= (0 )= and from (2) and R =5 x we will have = R L(0)= R ( )=5 =. The interval [ 2,0] is disorder interval for a 22 because a = 2<0, thus h22 = L(0)= (0 2)= and R =6 2x yields = R L(0)= R ( )=. 3 3 lso for off-diagonal entry a 2 we have a = <0 2 2 thus there is no disorder interval. Finally [0,] is the disorder interval of a 2 because a =>0 2 2, thus substitution h2 = R (0)= 4 and L = x 4 in (3) concludes 5 2 = L R (0) = L ( ) =. 4 4 Clearly I = {(,),(2, 2)}, J ={(2,)} thus 202

8 (,) I (, i j) J = = =. { } { } 9 40 (,) I (, i j) J On the other hand 5 =0, 22 =0, 2 = a2 2 = 22= 4, 2 = 2 =, = =, 22 = 22 =, 2 = a2 2 = 2=, 2 = Clearly, > 2 and 22 > 2, thus ( ) 0 and from =0=, =0 0=, we have = therefore from (6) for every small positive, we can choose 5 =2, 22 =3, 2 = 4, 2 =, 4 consequently (8) yields = R L( )=5 =, 22 = R L( 22)=6 2( )= = R L( 2)= R L( 4)=, 2 = R L( 2)= R L( )=0. 4 Hence from (7) we will have { } { } = { } { } = { 2 } { 3 } { 4 4 } { 0} = thus ( ) = = = REFERENCES xelson, O., 994. Iterative Solution Methods, Cambridge University Press. Datta, B.N., 995. Numerical Linear lgebra and ppl, Brooks/Cole Publishing Company. Dubois, D., H. Prade, 980. Fuzzy Sets and systems: Theory and pplications, cademic Press, New York. Hashemi, M.S., M. K. Mirnia, S. Shahmorad, Solving fuzzy linear systems by using the Schur complement when coefficient matrix is an M -matrix, Iranian Journal of Fuzzy Systems, 5(3):

9 Johnson, C.R., 982. Inverse M-Matrices, Linear lgebra, ppl, 47: Numaier,., 990. Interval methods for systems of equations, Cambridge University Press, Cambridge. 204