A METHOD FOR SOLVING DUAL FUZZY GENERAL LINEAR SYSTEMS*

Transcription

1 Appl Comput Math 7 (2008) no2 pp A METHOD FOR SOLVING DUAL FUZZY GENERAL LINEAR SYSTEMS* REZA EZZATI Abstract In this paper the main aim is to develop a method for solving an arbitrary m n dual fuzzy linear system as A x = B x + ỹ which m n and ỹ is symmetric fuzzy number vector By using the parametric form of fuzzy numbers we show that the least square solution of this system is symmetric Keywords: Dual fuzzy linear system Least square solution AMS Subject Classification: 14C20 1 Introduction System of simultaneous linear equations are important for studying and solving a large proportion of the problems in many topics in applied mathematics Usually in many applications at least some of the system s parameters are represented by fuzzy rather than crisp numbers and hence it is important to develop mathematical models and solving methods that would appropriately treat general fuzzy linear systems and solve them A general model for solving an arbitrary n n FSLE (fuzzy system of linear equation) whose coefficients matrix is crisp and the right hand side column is an arbitrary fuzzy number vector were first proposed by Friedman et al 5 and they studied duality fuzzy linear systems in 6 They have used the parametric form of fuzzy numbers 7 and replace the original fuzzy linear system by 2n 2n crisp system In 2 Asady et al solved m n(m n) original fuzzy linear system by 2m 2n crisp system 2 Recently the authors of 1 obtained the least square symmetric solution of m n fuzzy general linear systems which m n In section 2 some basic definitions and results on fuzzy linear systems are introduced In section 3 model is proposed for solving of dual fuzzy linear systems The proposed model is illustrated by solving some examples in section 4 and conclusions are drawn in section 5 2 Preliminaries Definition 1 A fuzzy number is a fuzzy set like ũ : R I = 0 1 which satisfies 4 1 ũ is upper semi-continuous 2 ũ(x) = 0 outside some interval c d 3 there are real numbers a b such that c a b d and 31 ũ(x) is monotonic increasing on c a 32 ũ(x) is monotonic decreasing on b d 33 ũ(x) = 1 a x b *This work is presented in the 2nd International Conference on Control and Optimization with Industrial Applications 2-4 June 2008 Baku Azerbaijan Department of Mathematics Islamic Azad University Karaj Branch Karaj Iran ezati@kiauacir Manuscript received 30 Juny

2 236 APPL COMPUT MATH VOL 7 NO The set of all these fuzzy numbers is denoted by E 1 In this paper we suppose a = b An equivalent parametric is also given in as follows Definition 2 A fuzzy number ũ in parametric form is a pair (u u) of functions u(r) u(r) 0 r 1 which satisfy the following requirements: 1 u(r) is a bounded monotonic increasing left continuous function 2 u(r) is a bounded monotonic decreasing left continuous function 3 u(r) u(r) 0 r 1 Remark 1 A crisp number α is simply represented by u(r) = u(r) = α 0 r 1 A popular fuzzy number is the triangular fuzzy number ũ = (α c β) with the membership function x α c α α x c ũ(x) = x β c β c x β where c α c β and hence u(r) = α + (c α)r u(r) = β + (c β)r If c = α+β 2 or u(r) + u(r) is independent of r then the triangular fuzzy number is called symmetric Let T F (R) be the set of all triangular fuzzy numbers The addition and scalar multiplication of fuzzy numbers are defined by the extension principle and can be equivalently represented as follows For arbitrary ũ = (u u) ṽ = (v v) and scalar k > 0 we define addition (ũ+ṽ) and multiplication as (u + v)(r) = u(r) + v(r) (u + v)(r) = u(r) + v(r) (ku)(r) = ku(r) (ku)(r) = ku(r) 3 Solving dual fuzzy general linear system Usually there is no inverse element for an arbitrary fuzzy number ã E 1 ie there exists no element b E 1 such that ã + b = 0 Actually for all non-crisp fuzzy number ã E 1 we have ã + ( ã) 0 Therefore the fuzzy system a 11 x 1 + a a 1n x n = b 11 x 1 + b b 1n x n + ỹ 1 a 21 x 1 + a a 2n x n = b 21 x 1 + b b 2n x n + ỹ 2 a m1 x 1 + a m2 + + a mn x n = b m1 x 1 + b m2 + + b mn x n + ỹ m can not be equivalently replaced by the fuzzy system (A B) x = ỹ where the coefficients matrix A B = (a ij b ij ) 1 i m 1 j n is a crisp m n matrix (m n) ỹ = (ỹ 1 ỹ 2 ỹ m ) T and x = ( x 1 x m ) T are fuzzy number vectors The system of (1) is called dual fuzzy general linear system (DFGLS) Definition 3 A fuzzy number vector ( x 1 x n ) T given by x j = (x j (r) x j (r)) j = 1 2 n 0 r 1 is called a solution of (1) if a ij x j = a ij x j = b ij x j + y i i = 1 2 m (2) (1)

3 REZA EZZATI:A METHOD FOR SOLVING DUAL FUZZY GENERAL 237 a ij x j = a ij x j = b ij x j + y i i = 1 2 m (3) Theorem 1 If rank of the coefficients matrix A B in (1) is m (ie A B is row full rank) and ỹ = (ỹ 1 ỹ 2 ỹ m ) T is an arbitrary symmetric fuzzy number vector then the least square solution of this system is symmetric Proof Suppose the parametric form of x j be x j = (x j x j ) j = 1 2 n and let a ij = a ij a ij b ij = b ij b ij such that a ij a ij b ij and b ij are positive and a ij a ij = 0 b ij b ij = 0 If we consider (1) in parametric form then for i = 1 2 m we have: and hence By addition of (5) and (6) we have: (a i1 a i1 )(x 1 x 1 ) + + (a in a in )(x n x n ) = (b i1 b i1 )(x 1 x 1 ) + + (b in b in )(x n x n ) + (y i y i ) a i1 x 1 a i1 x 1 + a i2 x 2 a i2 x a in x n a in x n = b i1 x 1 b i1 x 1 + b i2 x 2 b i2 x b in x n b in x n + y i a i1 x 1 a i1 x 1 + a i2 x 2 a i2 x a in x n a in x n = b i1 x 1 b i1 x 1 + b i2 x 2 b i2 x b in x n b in x n + y i (a i1 a i1 )(x 1 + x 1 ) + (a i2 a i2 )(x 2 + x 2 ) + + (a in a in )(x n + x n ) = (b i1 b i1 )(x 1 + x 1 ) + (b i2 b i2 )(x 2 + x 2 ) + + (b in b in )(x n + x n )+ (y i + y i ) (4) (5) (6) (7) and hence (a i1 b i1 )(x 1 + x 1 ) + (a i2 b i2 )(x 2 + x 2 ) + + (a in b in )(x n + x n ) = (y i + y i ) Since A B the coefficients matrix of above system is row full rank so the least square solution of this system is X = (A B) T ((A B)(A B) T ) 1 Y where X = (x 1 + x 1 x 2 + x 2 x n + x n ) T and Y = (y 1 + y 1 y 2 + y 2 y m + y m ) T Since every ỹ i is symmetric so Y and hence X is independent of r for all r 0 1 So the least Square solution of (1) is symmetric For solving (1) we first solve the following system: n (a 1j b 11 )(x j (0) + x j (0)) = (y 1 (0) + y 1 (0)) n (a 2j b 2j )(x j (0) + x g (0)) = (y 2 (0) + y 2 (0)) n (a mj b mj )(x j (0) + x j (0)) = (y m (0) + y m (0)) and suppose the solution of this system be as d 1 d 2 d = = x(0) + x(0) = d n x 1 (0) + x 1 (0) x 2 (0) + x 2 (0) x n (0) + x n (0) Let matrix C 1 contains the positive entries of A matrix D 1 contains the absolute of the negative entries of A matrix C 2 contains the positive entries of B and matrix D 2 contains the absolute of the negative entries of B hence A = C 1 D 1 and B = C 2 D 2 Now by using matrix (8)

4 238 APPL COMPUT MATH VOL 7 NO notation for (1) we get A x = B x + ỹ or (C 1 D 1 ) x = (C 2 D 2 ) x + ỹ and in parametric form (C 1 D 1 )(x(r) x(r)) = (C 2 D 2 )(x(r) x(r))+(y(r) y(r)) We can write this system as follows: C1 x(r) D 1 x(r) = C 2 x(r) D 2 x(r) + y(r) C 1 x(r) D 1 x(r) = C 2 x(r) D 2 x(r) + y(r) and for r = 0 we have (C1 C 2 )x(0) (D 1 D 2 )x(0) = y(0) (C 1 C 2 )x(0) (D 1 D 2 )x(0) = y(0) By substituting of x(0) = d x(0) and x(0) = d x(0) in the first and second equation of above system respectively we have and (C 1 C 2 + D 1 D 2 )x(0) = y(0) + (D 1 D 2 )d (9) (C 1 C 2 + D 1 D 2 )x(0) = y(0) + (D 1 D 2 )d (10) If the matrix F = C 1 C 2 + D 1 D 2 be row full rank then x(0) = F T (F F T ) 1 (y(0) + (D 1 D 2 )d) x(0) = F T (F F T ) 1 (y(0) + (D 1 D 2 )d) Definition 4 Let x = (x j (0) x j (0)) 1 j n} denotes the least square solution of (8) and (9) (or (8) and (10)) The symmetric fuzzy number vector Ũ = (u j (0) u j (0)) 1 j n} defined by u j (0) = minx j (0) x j (0)} u j (0) = maxx j (0) x j (0)} is called the fuzzy solution of (1) If u j (0) = x j (0) and u j (0) = x j (0) then Ũ is called a strong fuzzy solution Otherwise Ũ is called weak fuzzy solution Theorem 2 If rank of the matrices A B = C 1 D 1 +C 2 D 2 in (1) and F = C 1 C 2 +D 1 D 2 are m (ie A B and F are row full rank) and ỹ = (ỹ 1 ỹ 2 ỹ m ) T is an arbitrary symmetric fuzzy number vector then (1) has strong symmetric fuzzy solution if F T (F F T ) 1 be positive ie (F T (F F T ) 1 ) ij 0 i = n j = 1 2 mm Proof According to (9) and (10) we can write By using the least square method we have F (x(0) x(0)) = y(0) y(0) x(0) x(0) = F T (F F T ) 1 (y(0) y(0)) Since (y i (0) y i (0)) 0 for 1 i m and (F T (F F T ) 1 ) ij 0 therefore (x j (0) x j (0)) 0 for j = 1 2 m

5 REZA EZZATI:A METHOD FOR SOLVING DUAL FUZZY GENERAL Numerical examples Example 1 Consider the 1 2 fuzzy linear system It is clear that x 1 + = (r 2 r) x1 = ( r r) ( r ) is the solution of this system x 1 and are not symmetric Now for finding symmetric solution we use Theorem 1 This example has strong fuzzy solution Therefore we have x1 (0) + x 1 (0) 1 = x 2 (0) + x 2 (0) 1 and x1 (0) x 2 (0) 0 = 0 Hence the strong solution of this example is x1 = ( r 2 1 r 2 ) ( r 2 1 r 2 ) Example 2 Consider the 2 3 fuzzy linear system x1 + + x 3 = x 1 + (1 + r 3 r) 2 x 1 + x 3 = + x 3 + (2 + r 4 r) In this example A = and by notice to (8) we have x 1(0) + x 1 (0) x 2 (0) + x 2 (0) x 3 (0) + x 3 (0) B = = (11) From (9) we have the following system x1 (0) + x 2 (0) + x 3 (0) = x 1 (0) + x 2 (0) x 1 (0) + x 2 (0) + x 3 (0) = x 2 (0) + x 3 (0) By using the least square method we have 8 9 and hence x(0) = ( )T x(0) = ( )T

6 240 APPL COMPUT MATH VOL 7 NO Since x i (0) x i (0) for i=123 therefore the strong solution of (11) is follows: x ( r r) 1 = (0 0) x 3 ( r 5 9 r Example 3 Consider the 2 3 fuzzy linear system x x 3 = x (6 + 4r 14 4r) 2 x x 3 = x 1 x 3 + (5 + 4r 13 4r) In this example A = B = by notice to (8) and (9) we have x 8 1(0) + x 1 (0) x 2 (0) + x 2 (0) = 14 x 3 (0) + x 3 (0) 6 (12) x(0) = ( )T and hence x(0) = (0 7 7) T Here x i (0) x i (0) i=123 Therefore the strong solution of (12) is x ( r 4 3 r) 1 = ( 5 3 x r rr) ( r r) 5 Conclusion In this paper we are proposed a general model for solving a system of m dual fuzzy linear equations with n (m n) variables as A x = B x + ỹ which ỹ is symmetric fuzzy number vector The original system is solved by two m n crisp linear systems where solving of this linear systems is done by least square method We are proved a theorem for the existence of a symmetric solution to the dual fuzzy general linear system References 1 Abbasbandy S Allahviranloo T and Ezzati R A method for solving fuzzy general linear systems The Journal of Fuzzy Mathematics 15 (2007) Asady B Abbasbandy S and Alavi M Fuzzy general linear systems Applied Mathematics and Computation 169 (2005) Barrett R Berry M and Chan T Templates for the solution of linear systems SAIM press Dubois D and Prade H Theory and Application Fuzzy Sets and Systems Academic Press New York Friedman M Ming M and Kandel A Fuzzy linear systems Fuzzy Sets and Systems 96 (1998)

7 REZA EZZATI:A METHOD FOR SOLVING DUAL FUZZY GENERAL Friedman M Ming M and Kandel A Duality in fuzzy linear systems Fuzzy Sets and Systems 109 (2000) Ma M Friedman M and Kandel A A new fuzzy arithmetic Fuzzy Sets and Systems 108 (1999) Minc H Nonnegative Matrices Wily New York 1998 Reza Ezzati - was born in 1974 in Tabriz Iran He received BSc (1997) in pure mathematics and MSc in applied mathematics from Azarbijan Teacher Education University Tarbyat Modarres University of Tehran respectively and PhD (2006) in applied mathematics from Islamic Azad University Science and Research Branch Tehran Iran He is an assistant professor in the department of mathematics at Islamic Azad University Karaj Branch Karaj in Iran His current interest is in fuzzy mathematics especially on numerical solution of fuzzy systems and fuzzy interpolation iterative methods for solving nonlinear equations