SOLVING FUZZY LINEAR SYSTEMS BY USING THE SCHUR COMPLEMENT WHEN COEFFICIENT MATRIX IS AN M-MATRIX

Iranian Journal of Fuzzy Systems Vol 5, No 3, 2008 pp 15-29 15 SOLVING FUZZY LINEAR SYSTEMS BY USING THE SCHUR COMPLEMENT WHEN COEFFICIENT MATRIX IS AN M-MATRIX M S HASHEMI, M K MIRNIA AND S SHAHMORAD Abstract This paper analyzes a linear system of equations when the righthand side is a fuzzy vector and the coefficient matrix is a crisp M-matrix The fuzzy linear system FLS is converted to the equivalent crisp system with coefficient matrix of dimension 2n 2n However, solving this crisp system is difficult for large n because of dimensionality problems It is shown that this difficulty may be avoided by computing the inverse of an n n matrix instead of Z 1 1 Introduction n n fuzzy linear systems have been studied by many authors [1, 3, 5, 6, 7, 9, 10, 11, 12] In [7] Friedman et al proposed a general model for solving such systems by the embedding approach and stated conditions for the existence of a unique fuzzy solution to n n linear systems A minimum norm solution of the fuzzy under determined systems has been obtained in [10] In this paper we investigate the case where the coefficient matrix in a fuzzy linear system is an M-matrix We first study the properties of this system and then propose a solution using the Schur complement Section 2 provides preliminaries for fuzzy numbers and fuzzy linear systems Several M-matrix properties and the Schur complement formula are also stated The relationship between an M-matrix and its crisp matrix and the existence and expression of the solution to the fuzzy linear system using the Schur complement are discussed in section 3 Numerical examples to illustrate previous sections are given in Section 4 2 Preliminaries 21 Fuzzy Numbers and Fuzzy Linear Systems In this section we recall the basic notion of fuzzy numbers arithmetic and fuzzy linear system Definition 21 [7] A fuzzy number in parametric form is an ordered pair of functions ur, ur, 0 r 1 which satisfy the following conditions: 1 ur is a bounded left continuous nondecreasing function on [0,1] Received: May 2007; Revised: September 2007 and January 2008; Accepted: February 2008 Key words and phrases: Fuzzy linear system, Schur complement, M-matrix, H-matrix

16 M S Hashemi, M K Mirnia and S Shahmorad 2 ur is a bounded left continuous nonincreasing function on [0,1] 3 ur ur, 0 r 1 If ur ur ψ, 0 r 1 then ψ is a crisp number A popular representation for fuzzy number is the trapezoidal representation u x 0, y 0, α, β with defuzzifier interval [x 0, y 0 ], left fuzziness α and right fuzziness β The membership function of this trapezoidal number is as follows: 1 α x x 0 + α, x 0 α x x 0 1, x [x 0, y 0 ] ux 1 β y 0 x + β, y 0 x y 0 + β 0, otherwise The parametric form of the number is ur x 0 α + αr, ur y 0 + β βr The operations of addition and scalar multiplication for arbitrary fuzzy numbers, u u, u and v v, v and λ R 1 are defined by u + vr ur + vr, u + vr ur + vr, λur λur, λur λur, λ 0, λur λur, λur λur, λ 0 A general fuzzy linear system is as follows: 1 a 11 x 1 + a 12 x 2 + + a 1n x n y 1 a 21 x 1 + a 22 x 2 + + a 2n x n y 2 2 a n1 x 1 + a n2 x 2 + + a nn x n y n, where the coefficient matrix A a ij, 1 i, j n is a crisp n n matrix and y y i, 1 i n is a fuzzy vector This system is called a fuzzy linear system FLS By 21, the system 21 is equivalent to the following parametric system: n j1 a ijx j n j1 a ijx j y i n j1 a ijx j 3 n j1 a ijx j y i If for a particular i : a ij > 0, 1 j n, then we obviously get n n a ij x j y i, a ij x j y i j1 j1

Solving Fuzzy Linear Systems by Using the Schur Complement 17 In general, an arbitrary equation for either y i or y i may include a linear combination of x j s and x j s Consequently, in order to solve the system given by Eq?? one must solve a 2n 2n crisp linear system, where the column on the right is the vector y 1, y 2,, y n, y 1, y 2,, y n T Let us now rearrange the linear system of Eqs?? so that the unknowns are x i, x i, 1 i n and the column on the right is Y y 1, y 2,, y n, y 1, y 2,, y n T Define a 2n 2n matrix Z z ij as follows: a ij 0 z ij a ij, z i+n,j+n a ij, a ij < 0 z i,j+n a ij, z i+n,j a ij 4 Any z ij which is not determined by 21 is zero Then the system?? may be written in the following crisp block form: B 0 C 0 X Y ZX Y 5 C 0 B 0 X Y where B, C R n,n and x 1 r X, X x n r x 1 r x n r, Y y 1 r y n r, Y y 1 r y n r The definition of Z z ij implies that B contains the positive entries of A, C contains the absolute values of the negative entries of A and A B C 22 M-matrices and Some of their Properties Definition 22 [2] A real square matrix A a ij is called an M-matrix if a ij 0; i j and A 1 0 For any real or complex n n matrix A a ij, the comparison matrix, denoted by ΞA ξ ij A, and defined by { aij, i j ξ ij A 6 a ij, i j is called an H matrix if ΞA is an M-matrix Definition 23 [2] A square matrix A is said to be generalized diagonally dominant if a ii x i a ij x j, i 1, 2,, n 7 j i

18 M S Hashemi, M K Mirnia and S Shahmorad for some positive vector x x 1, x 2,, x n T, generalized strictly diagonally dominant if 23 is valid with strict inequality and strictly diagonally dominant if 23 is valid for x1, 1,, 1 T Lemma 24 [2] Let A a ij be an n n matrix, with a ij 0, i j and a ii > 0 If A is strictly diagonally dominant, then A is an M-matrix Lemma 25 [2] a A matrix A a ij, with a ij 0, i j, is an M-matrix if and only if there exists a positive vector x, such that Ax is positive b A matrix A a ij is an H-matrix if and only if A is generalized strictly diagonally dominant Theorem 26 [2] Let A a ij be a matrix of order n, with a ij 0, the following are equivalent: a A is generalized strictly diagonally dominant and a ii 0 b A is an M-matrix i j Then 23 Schur Complements Let the matrix A be partitioned into the two-by-two block form A11 A A 12 8, A 21 A 22 where A ii, i 1, 2 are square matrices Definition 27 [2] If A 11 is nonsingular, we define S A/A 11 A 22 A 21 A 1 11 A 12, 9 S is called the Schur complement of A with respect to A 11 Note that S is in the position A 22 of A after we use Gaussian-elimination to convert A 21 to zero in A In fact, the block matrix triangular factorization of A is readily found to be A I 0 A 21 A 1 11 I A11 0 0 S I A 1 11 A 12 0 I 10 Hence we may write the inverse of A in one of the two forms 11 and 12 as follows: A 1 I A 1 11 A 12 A 1 11 0 I 0 11 0 I 0 S 1 A 21 A 1 11 I A 1 A 1 11 + A 1 11 A 12S 1 A 21 A 1 11 A 1 11 A 12S 1 S 1 A 21 A 1 11 S 1 12

If both A 1 11 where Solving Fuzzy Linear Systems by Using the Schur Complement 19 and A 1 22 exist, an alternative expression for A 1 is A 1 S 1 1 A 1 11 A 12S2 1 S2 1 A 21A 1 11 S2 1 S i A/A jj, i j, i, j 1, 2 Theorem 28 [2] Let A be an M-matrix that is partitioned in a two-by-two block form Then the Schur complement exists and is also an M-matrix S A/A 11 A 22 A 21 A 1 11 A 12 3 Solving a Fuzzy Linear System when A Is an M-matrix 31 Solution to a Fuzzy Linear System To study the solution to a fuzzy linear system, it is first necessary to discuss the generalized inverses of the matrix Z in a special structure Definition 31 [5] Let X { x i r, x i r; 1 i n } denote a solution of ZX Y The fuzzy number vector U { u i r, u i r; 1 i n } defined by u i r min { x i r, x i r, x i 1, x i 1 } 13 u i r max { x i r, x i r, x i 1, x i 1 } 14 is called a fuzzy solution of ZX Y If x i r, x i r; 1 i n, are all fuzzy numbers and u i r x i r, u i r x i r, 1 i n, then U is called a strong fuzzy solution Otherwise, U is a weak fuzzy solution From 21 we have the following assertion: Theorem 32 [12] Let matrix Z be in the form 21 Then the matrix Z 1 B + C + B C B + C B C 2 B + C B C B + C + B C 15 is a g-inverse of the matrix Z, where B + C and B C are g-inverses of the matrices B + C and B C, respectively In particular, the Moore-Penrose inverse of the matrix Z is Z 1 B + C + B C B + C B C 2 B + C B C B + C + B C 16 In this paper, we consider only the case when m n; thus Z is nonsingular if and only if the matrices A B C and B + C are both nonsingular [7, Theorem 1] Hence, as a direct consequence of Theorem 32, we have the following corollary

20 M S Hashemi, M K Mirnia and S Shahmorad Corollary 33 [7] If Z 1 exists, it must have the same structure as Z, ie Z 1 T1 T 2 T 2 T 1 where T 1 1 2 [B + C 1 + B C 1 ], T 2 1 2 [B + C 1 B C 1 ] The next result provides a sufficient condition for the solution vector to be a fuzzy vector Theorem 34 [12] For the consistent system 21 and any g-inverse Z of the coefficient matrix Z, X Z Y is a solution to the system and therefore it admits a weak or strong fuzzy solution In particular, if Z is nonnegative with the special structure 32, then X Z Y admits a strong fuzzy solution for arbitrary fuzzy vector Y Corollary 35 [12] For the n n fuzzy linear system, if Z 1 exists, then for an arbitrary fuzzy vector Y, the unique solution X Z 1 Y is a fuzzy vector if Z 1 is nonnegative 32 Properties of the Equivalent Crisp Matrix Z We consider the solution of FLS, Ax ỹ when A is an M-matrix and ỹ is a fuzzy vector Then 21, B C implies that z ij 0, 1 i, j n and Z where B is the diagonal C B matrix containing the diagonal entries of A and C contains the absolute value of the off-diagonal entries of A In other words, B diaga 11, a 22,, a nn, and so B 1 diag 1 1 1 a 11, a 22,, a nn, and C c ij n n where { cii 0, 1 i n c ij a ij, 1 i, j n, i j Theorem 36 If A is an M-matrix then Z, as defined by 21, is an H-matrix Proof From theorem 26, it is clear that A is generalized strictly diagonally dominant and a ii 0 Thus from lemma 25 for some x > 0, x R n we have a ii x i > j i a ij x j, i 1, 2,, n 17 To complete the proof, it is sufficient to show that Z is generalized strictly diagonally dominant Let y [x T, x T ], where x is the vector corresponding to the matrix A Then it may easily be checked that the relation z ii y i > 2n j1 holds for the matrix Z defined in 21 z ij y j, j i, 1 i 2n 18

Solving Fuzzy Linear Systems by Using the Schur Complement 21 33 Solution Using the Schur Complement In this section we employ the Schur complement for computing Z 1 where Z is defined by 21 with block form B C Z where B is a diagonal matrix and C is a nondiagonal matrix whose C B entries are absolute values of the off-diagonal entries of A if A is an M-matrix It is clear from 27, 23 and 21 that S S 1 S 2 B CB 1 C and so Z 1 S 1 B 1 CS 1 19 S 1 CB 1 S 1 Theorem 37 Let A be an M-matrix, in FLS, Z be the extended crisp matrix of A, and S be its Schur complement Then we have S 1 CB 1 B 1 CS 1 Proof Let P S 1, Q 1 B 1 CS 1, Q 2 S 1 CB 1 We must prove Q 1 Q 2 From 33 and 21 we have ZZ 1 B C P Q1 I 0 I, or, C B P 0 I thus { BQ1 + CP 0 CP + BQ 2 0 which completes the proof From theorem 37 we have where P S 1, or Q 2 { Q1 B 1 CP Q 2 B 1 CP Z 1 P Q Q P }, 20 Q P CB 1 Hence 21 and 33 imply that X P Q Y X Q P Y X P Y QY X P Y QY 34 Some Properties of Z 1 To investigate properties of Z 1, we shall need the following definitions and lemmas: Definition 38 [4] Let A a ij be an m n matrix We define a matrix norm A as follows: A max 1 i m n j1 a ij Lemma 39 [4] For any matrices, A and B, we have and consequently A 2 A 2 AB A B, 21

22 M S Hashemi, M K Mirnia and S Shahmorad Lemma 310 [4] The matrix series converges to I A 1 if A < 1 I + A + A 2 + + A k + Theorem 311 Let A be a strictly diagonally dominant matrix with the extended crisp matrix defined by 21 and the inverse defined by by 33 Then P 0, Q 0 Proof We know that P S 1 B CB 1 C 1, thus B 1 S I B 1 CB 1 C Let E B 1 CB 1 C We prove E < 1 From 21 we have 1 a 11 0 0 0 a 12 a 1n 1 B 1 0 a 22 0 a 21 0 a 2n C 1 0 0 a nn a n1 a n2 0 a 0 12 a a 11 1n 11 a 21 a a 22 0 2n a 22 a n1 a n2 a nn a nn 0 Thus by the Lemma 39 and inequalities 32 we obtain Therefore and from lemma 310 B 1 C max 1 i n n j1 a ij a ii < 1 E B 1 C 2 B 1 C 2 < 1, I E 1 I + E + E 2 + 0 Hence S 1 B I E 1 0 and P S 1 S 1 BB 1 0 From Q P CB 1 and P 0, C 0, B 1 0, it is clear that Q 0 and the proof is complete Theorem 312 Let A a ij be an n n matrix with the following properties: a a ij 0 for i j and there exists at least one negative off-diagonal entry b a ii > 0, i 1,, n c A is strictly diagonally dominant X Then there exists a fuzzy vector Y such that X Z 1 Y is not a fuzzy X vector ie x 1 r x 1 r, x 1 r x n r x n r, x n r

Solving Fuzzy Linear Systems by Using the Schur Complement 23 is not a strong fuzzy solution for system 21 Note: If the assumption a fails then C 0 and there is a fuzzy solution by corollary 35 Proof Suppose p 11 p 1n P p n1 p nn, Q q 11 q 1n q n1 q nn From the assumptions, Theorem 311 holds and from second part of assumption a,q 0 Since P is non-singular and non-negative, hence there exist i 0 and j 1 such that q i0,j 1 < 0 and p i0,j 0 > 0, 1 i 0, j 0, j 1 n Now we construct a fuzzy vector Y such that X Z 1 Y is a non-fuzzy vector Let us write p 11 p 1n q 11 q 1n y 1 r X Z 1 p n1 p nn q n1 q nn y n r Y q 11 q 1n p 11 p 1n y 1 r q n1 q nn p n1 p nn y n r x 1 r x n r x 1 r 22 x n r It is sufficient to construct Y such that x i0 r is a non-fuzzy number From 34 we have x i0 r p i01y 1 r + + p i0j 0 y j0 r + +p i0ny n r q i01y 1 r q i0j 1 y j1 r q i0ny n r 23 We have the following two cases: Case1 j 0 j 1 In this case, for the real numbers a i, 1 i n, c j0 and d j0 we define { yi r y i r a i, 1 i n, i j 0 y j0 r a j0 r, y j0 r c j0 r + d j0 24 in which a j0 > 0, c j0 < 0 and c j0 a j0 Then from 34 we will have n x i0 r P i0j q i0ja j + p i0j 0 a j0 r q i0j 0 c j0 r q i0j 0 d j0 j1 j j0

24 M S Hashemi, M K Mirnia and S Shahmorad x i0 r p i0j 0 a j0 q i0j 0 c j0 r + n j1 j j0 P i0j q i0ja j q i0j 0 d j0 Now it is sufficient to define the coefficient of the parameter r so that the function x i0 r is a decreasing function ie dxi 0 r dr < 0 Hence the coefficients c j0 and a j0 are determined so that p i0j 0 a j0 q i0j 0 c j0 < 0 Also, to preserve the fuzziness of Y, it is sufficient to have y j0 y j0, or, equivalently, a j0 r c j0 r + d j0 ; ie a j0 c j0 r d j0 or r dj 0 a j0 c j0 If we set d j0 a j0 c j0, then 0 r 1 Case2 j 0 j 1 In this case, for the real numbers a i, 1 i n, c j0 and d j0 we define y i r y i r a i, 1 i n, i j 0 y j0 r a j0 r, y j0 r c j0 r + d j0 y j1 r a j1 r, y j1 r c j1 r + d j1 25 where a j0 > 0, c j0 < 0, c j0 a j0, a j1 > 0, c j1 < 0 and c j1 a j1 From 34, we have n x i0 r P i0j q i0ja j + p i0j 0 a j0 r q i0j 0 c j0 r q i0j 0 d j0 j1 j j0,j 1 +p i0j 1 a j1 r q i0j 1 c j1 r q i0j 1 d j1 or x i0 r p i0j 0 a j0 + p i0j 1 a j1 q i0j 0 c j0 q i0j 1 c j1 r n + P i0j q i0ja j q i0j 0 d j0 q i0j 1 d j1 j1 j j0,j 1 Choosing a j0, a j1, c j0 and c j1 to satisfy p i0j 0 a j0 + p i0j 1 a j1 < q i0j 0 c j0 + q i0j 1 c j1 guarantees dxi 0 r dr < 0 Thus if d j0 a j0 c j0, d j1 a j1 c j1, the vector Y will be fuzzy, but the number x i0 r will not be fuzzy 4 Numerical Results Example 1: Consider the 2 2 fuzzy linear system { 2x 1 x 2 r, 3 r 3x 1 + 4x 2 r 1, 5 r

where A where Solving Fuzzy Linear Systems by Using the Schur Complement 25 2 1 3 4 From 22 we have is obviously an M-matrix The extended 4 4 matrix is 2 0 0 1 0 4 3 0 Z 0 1 2 0 3 0 0 4 B 2 0 0 4 ΞZ, C 0 1 3 0 2 0 0 1 0 4 3 0 0 1 2 0 3 0 0 4 Clearly ΞZ, is an M-matrix, thus Z is an H-matrix By 33 1 P S 1 B CB 1 C 1 125 0 08 0 0 25 0 04 Q P CB 1 0 02 06 0 and thus 08 0 0 02 0 04 06 0 Let so we have Z 1 P Q Q P X Y x1 r x 2 r 0 02 06 0 r r 1 x1 r, X x 2 r 3 r, Y 5 r 08 0 0 04 and from 33, 08 0 r 0 02 X P Y QY 0 04 r 1 06 0 06r + 1 02r + 14 08 0 3 r 0 02 X P Y QY 0 04 5 r 06 0 06r + 22 02r + 2, 3 r 5 r r r 1

26 M S Hashemi, M K Mirnia and S Shahmorad and thus x 1 r x 1 r, x 1 r 06r + 1, 06r + 22, x 2 r x 2 r, x 2 r 02r + 14, 02r + 2 Obviously, x 1, x 2 are not fuzzy numbers Therefore the corresponding fuzzy solution is a weak fuzzy solution given by { u1 r 06r + 1, 06r + 22 u 2 r 12, 22 Example 2: Now consider a system where the coefficient matrix is the same as in the previous example, but where the fuzzy vector on the right hand side leads to a strongly fuzzy solution { 2x 1 x 2 2r, 3r + 5 where A 2 1 3 4 3x 1 + 4x 2 5r + 1, 4r + 10 and 2 0 0 1 0 4 3 0 Z 0 1 2 0 3 0 0 4 From the previous example we have 08 0 0 02 P, Q, 0 04 06 0 08 0 0 02 Z 1 0 04 06 0 0 02 08 0 06 0 0 04 Also 2r 3r + 5 Y, Y 5r + 1 4r + 10 and thus from 39 it follows that 08 0 2r 0 02 X P Y QY 0 04 5r + 1 06 0 08r + 2 02r + 34 08 0 3r + 5 0 02 X P Y QY 0 04 4r + 10 06 0 14r + 42 04r + 4 so that x 1 r x 1 r, x 1 r 08r + 2, 14r + 42, 3r + 5 4r + 10 2r 5r + 1

Solving Fuzzy Linear Systems by Using the Schur Complement 27 x 2 r x 2 r, x 2 r 02r + 34, 04r + 4 Obviously x 1 r, x 2 r are fuzzy numbers for 0 r 1, and 2 x 1 r 28, 28 x 1 r 42, 34 x 2 r 36, 36 x 2 r 4 Therefore x 1 1 28, x 1 1 28, x 2 1 36, x 2 1 36 x 1 r min{x 1 r, x 1 r, x 1 1, x 1 1}, x 1 r max{x 1 r, x 1 r, x 1 1, x 1 1}, x 2 r min{x 2 r, x 2 r, x 2 1, x 2 1}, x 2 r max{x 2 r, x 2 r, x 2 1, x 2 1}, so X is a strong fuzzy solution Example 3: Consider the 3 3 fuzzy linear system 6x 1 4x 2 x 3 r, 1 r x 1 + 5x 2 2x 3 r + 1, 3 r x 1 + +4x 3 r 2, r The coefficient matrix is which is an M-matrix Also, Y A r r + 1 r 2 6 4 1 1 5 2 1 0 4, Y The extended 6 6 crisp matrix is as follows: 6 0 0 0 5 0 0 0 4 Z 0 4 1 1 0 2 1 0 0 where and B 6 0 0 0 5 0 0 0 4 ΞZ, C 1 r 3 r r 0 4 1 1 0 2 1 0 0 6 0 0 0 5 0 0 0 4 0 4 1 1 0 2 1 0 0 6 0 0 0 4 1 0 5 0 1 0 2 0 0 4 1 0 0 0 4 1 6 0 0 1 0 2 0 5 0 1 0 0 0 0 4

28 M S Hashemi, M K Mirnia and S Shahmorad Since ΞZ is an M-matrix, Z is an H-matrix Moreover, thus Z 1 P B CB 1 C 1 Q P CB 1 413 2031 230 40 and from 33 we have: ie 128 181 774 396 64 15 198 413 2031 230 40 128 181 774 396 64 15 198 160 809 4973 43 221 5223 184 92 495 32 16 160 809 4973 43 221 5223 184 92 495 32 16 X P Y QY X P Y QY,, 160 809 4973 43 221 5223 184 92 495 32 16 413 2031 230 40 7 107 r + 3376 10 107 r + 2851 25 107 r 4161 7 107 r + 2830 10 107 r + 4853 25 107 r + 844 128 181 774 396 ; 64 15 198 x 1 r x 1 r, x 1 r 7 107 r + 3376, 7 107 r + 2830 x 2 r x 2 r, x 2 r 10 107 r + 2851, 10 107 r + 4853 x 3 r x 3 r, x 3 r 25 107 r 4161, 25 107 r + 844 which x 1, x 2, x 3 are fuzzy numbers, but u 1 r x 1 r, u 1 r x 1 r, thus the corresponding fuzzy solution is a weak fuzzy solution given by u 1 r u 1 r, u 1 r x 1 1, x 1 1 2193, 4013 u 2 r u 2 r, u 2 r x 2 r, x 2 r 10 107 r + 2851, 10 107 r + 4853 u 3 r u 3 r, u 3 r x 3 r, x 3 r 25 107 r 4161, 25 107 r + 844,

Solving Fuzzy Linear Systems by Using the Schur Complement 29 The authors thank the referees for their valuable sugges- Acknowledgement tions References 1 T Allahviranloo, The adomain decomposition method for fuzzy system of linear equation, Applied Mathematics and Computation, 163 2005, 553-563 2 O Axelson, Iterative solution methods, Cambridge University Press, 1994 3 J J Buckley and Y Qu, Solving systems of linear fuzzy equations, Fuzzy Sets and Systems 43 1991, 33-43 4 B N Datta, Numerical linear algebra and applications, Brooks/Cole Publishing Company, 1995 5 M Dehgan and B Hashemi, Iterative solution of fuzzy linear systems, Applied Mathematics and Computation, 175 2006, 645-674 6 D Dubois and H Prade, Fuzzy Sets and systems: Theory and Applications, Academic Press, New York, 1980 7 M Friedman, M Ming and A Kandel, Fuzzy linear systems, Fuzzy Sets and Systems, 96 1998, 201-209 8 C R Johnson, Inverse M-matrices, Linear Algebra and its Application, 47 1982, 195-216 9 M Ma, M Friedman and A Kandel, Duality in fuzzy linear systems, Fuzzy Sets and Systems, 109 2000, 55-58 10 M Otadi, SAbbasbandy and A Jafarian, Minimal solution of a fuzzy linear system, 6th Iranian Conference on Fuzzy Systems and 1st Islamic World Conference on Fuzzy Systemms, 2006, pp125 11 X Wang, Z Zhong and M Ha, Iteration algorithms for solving a system of fuzzy linear equations, Fuzzy Sets and Systems, 119 2001, 121-128 12 B Zheng and K Wang, General fuzzy linear systems, Applied Mathematics and Computation, in press M S Hashemi, M K Mirnia and S Shahmorad, Department of Applied Mathematics, Faculty of Mathematical Science, University of Tabriz, Tabriz-Iran E-mail address: hashemi math396@yahoocom E-mail address: mirnia-kam@tabrizuacir E-mail address: shahmorad@tabrizuacir Corresponding author