SOLVING FUZZY LINEAR SYSTEMS OF EQUATIONS

Transcription

1 ROMAI J, 4, 1(2008), SOLVING FUZZY LINEAR SYSTEMS OF EQUATIONS A Panahi, T Allahviranloo, H Rouhparvar Depart of Math, Science and Research Branch, Islamic Azad University, Tehran, Iran panahi53@gmailcom Abstract Systems of linear equations, with uncertainty on the parameters, play a major role in various problems in economics and finance Fuzzy system of linear equations has been discussed in [1] using LU decomposition when the matrix A in Ax = b is a crisp matrix Also the Adomian decomposition method and iterative methods have been studied in [2, 6] for fuzzy system of linear equations In this paper we study such a system with fuzzy coefficients, ie the matrix A is a fuzzy matrix We find two fuzzy matrices, the lower triangular L and the upper triangular U such that A = LU and give a procedure to solve the fuzzy system of linear equations Keywords: fuzzy matrix, fuzzy linear system, decomposition 2000 MSC: 15A33 1 INTRODUCTION In this paper we intend to find a new solution for x in the matrix equation Ax = b, (1) where A is a fuzzy square matrix and x and b are fuzzy vectors Buckly and Qu [4] argued that the classical solution, based on extension principle and regular fuzzy arithmetic, should rejected since it too often fails to exist They defined six other solutions and showed that five of them are identical Basically, in their work the solutions of all systems of linear crisp equations formed by the α-levels are calculated In [8] another method for solving system of linear fuzzy equations based on using parametric functions in which the variables are given by the fuzzy coefficients of the system, was proposed 207

2 208 A Panahi, T Allahviranloo, H Rouhparvar 2 NOTATIONS AND BASIC DEFINITIONS First we recall some definitions concerning fuzzy numbers We denote by E 1 the set of all fuzzy numbers Definition 21 A fuzzy subset u of the real line R with membership function u(t) : R [0, 1] is called a fuzzy number if: (a) u is normal, ie, there exist an element t 0 such that u(t 0 ) = 1; (b) u is fuzzy convex, ie, u(λt 1 + (1 λ)t 2 ) min{u(t 1 ), u(t 2 )}, t 1, t 2 R, λ [0, 1]; (c) u(t) is upper semicontinuous; (d) supp u is bounded, where supp u =cl ({t R : u(t) > 0}), and cl is the closure operator Definition 22 We represent an arbitrary fuzzy number by an ordered pair of functions [x](α) = [x 1 (α), x 2 (α)], 0 α 1, which satisfy the following requirements [6]: (a) x 1 (α) is a bounded left continuous nondecreasing function over [0, 1]; (b) x 2 (α) is a bounded left continuous nonincreasing function over [0, 1]; (c) x 1 (α) x 2 (α), 0 α 1 For arbitrary [x] α = [x 1 (α), x 2 (α)] and [y] α = [y 1 (α), y 2 (α)] and k > 0 we define addition [x y](α) and scalar multiplication by k as [x y] α = [x](α) + [y](α) = [x 1 (α) + y 1 (α), x 2 (α) + y 2 (α)], and [kx 1 (α), kx 2 (α)], k 0 [kx] α = [kx 2 (α), kx 1 (α)], k < 0 respectively, for every α [0, 1] We denote by x = ( 1)x E 1 the symmetric of E 1 The product x y of two fuzzy numbers x and y, based on Zadeh s extension principle, is defined by (x y) 2 (α) = max{x 1 (α)y 1 (α), x 1 (α)y 2 (α), x 2 (α)y 1 (α), x 2 (α)y 2 (α)}, (x y) 1 (α) = min{x 1 (α)y 1 (α), x 1 (α)y 2 (α), x 2 (α)y 1 (α), x 2 (α)y 2 (α)} Definition 23 A fuzzy number x E 1 is said to be positive if x 1 (1) 0, strictly positive if x 1 (1) > 0, negative if x 2 (1) 0 and strictly negative if

3 Solving fuzzy linear systems of equations 209 x 2 (1) < 0 We say that x and y have the same sign if they are either both positive or both negative [3] Definition 24 A matrix A = [a ij ] is called a fuzzy matrix, if each element of A is a fuzzy number A is positive (negative) and denoted by A > 0 (A < 0) if each element of A is positive (negative) Similarly, nonnegative and nonpositive fuzzy matrices can be defined [5] The product of fuzzy numbers defined based on Zadeh s extension principle is not very practical from the computational point of view but the cross product is a computational method Now we study summary from the theoretical properties of the cross product of fuzzy numbers, for more details see [3] Firstly we begin with a theorem which was obtained by using the stacking theorem [7] Definition 25The binary operation on E 1 that will be introduced by Theorem 21 and Corollary 21 is called the cross product of fuzzy numbers Theorem 21 If x and y are positive fuzzy numbers, then w = x y, defined by [w] α = [w 1 (α), w 2 (α)], where w 1 (α) = x 1 (α)y 1 (1) + x 1 (1)y 1 (α) x 1 (1)y 1 (1), w 2 (α) = x 2 (α)y 2 (1) + x 2 (1)y 2 (α) x 2 (1)y 2 (1), for every α [0, 1], is a positive fuzzy number Corollary 21 Let x and y be two fuzzy numbers (a) If x is positive and y is negative then x y = (x ( y)) is a negative fuzzy number (b) If x is negative and y is positive then x y = (( x) y) is a negative fuzzy number (c) If x and y are negative then x y = ( x) ( y) is a positive fuzzy number Remark 21 The below formulas of calculus can be easily proved (α [0, 1]): (x y) 1 (α) = x 2 (α)y 1 (1) + x 2 (1)y 1 (α) x 2 (1)y 1 (1), (3) (x y) 2 (α) = x 1 (α)y 2 (1) + x 1 (1)y 2 (α) x 1 (1)y 2 (1), (2)

4 210 A Panahi, T Allahviranloo, H Rouhparvar if x is positive and y is negative, (x y) 1 (α) = x 1 (α)y 2 (1) + x 1 (1)y 2 (α) x 1 (1)y 2 (1), (x y) 2 (α) = x 2 (α)y 1 (1) + x 2 (1)y 1 (α) x 2 (1)y 1 (1), (4) if x is negative and y is positive In the last possibility, if x and y are negative, then (x y) 1 (α) = x 2 (α)y 2 (1) + x 2 (1)y 2 (α) x 2 (1)y 2 (1), (x y) 2 (α) = x 1 (α)y 1 (1) + x 1 (1)y 1 (α) x 1 (1)y 1 (1) Remark 22 The cross product extends the scalar multiplication of fuzzy numbers Indeed, if one of operands is the real number k identified with its characteristic function, then k 1 (α) = k 2 (α), α [0, 1] and using the above formulas of calculus we get the result 3 NEW METHOD FOR FINDING THE SOLUTION OF A FUZZY SYSTEM OF LINEAR EQUATIONS In the previous section we have analyzed the properties and main features of the cross product for multiplying fuzzy numbers (5) In this section we are going to show some ideas about the use of this operation to solve fuzzy system of linear equations of the form (1) We consider that the fuzzy coefficients and the elements of fuzzy right-hand side vector are all triangular fuzzy numbers To find our new solution, first we find two fuzzy matrices L and U such that A = L U and L is lower triangular with diagonals l ii = 1 and U is an upper triangular fuzzy matrix Consider A = [a ij ], L = [l ij ] and U = [u ij ] be three fuzzy matrices such that [a ij ] α = [a 1 ij(α), a 2 ij(α)], [l ii ] α = [1, 1], 1 i n, 1 j n For α = 1 we have and u 1i (1) = a 1i (1), l i1 (1) = a i1(1), i = 1,, n, (6) u 11 (1)

5 Solving fuzzy linear systems of equations 211 r 1 u ri (1) = a ri (1) l rk (1)u ki (1), k=1 l ir (1) = a ir(1) r 1 k=1 l ik(1)u kr (1), (7) u rr (1) for r = 2, 3,, n and i = r, r + 1,, n In case α [0, 1) we obtain 1 [u 1 1j (α), u2 1j (α)] = [a1 1j (α), a2 1j (α)], j = 1, 2,, n [li1 1 (α), l2 i1 (α)] [u1 11 (α), u2 11 (α)] = [a1 i1 (α), a2 i1 (α)], i = 1, 2,, n and [u 1 ri (α), u2 ri (α)] r 1 k=1 [l1 rk (α), l2 rk (α)] [u1 ki (α), u2 ki (α)] = [a1 ri (α), a2 ri (α)], [l 1 ir (α), l2 ir (α)] [u1 rr(α), u 2 rr(α)] r 1 k=1 [l1 ik (α), l2 ik (α)] [u1 kr (α), u2 kr (α)] = [a1 ir (α), a2 ir (α)], r = 2, 3,, n; i = r, r + 1,, n Now suppose the elements of matrices L and U are positive, therefore according to (2) we get for i = 1,, n and u 1 1j(α) = a 1 1j(α), u 2 1j(α) = a 2 1j(α), j = 1,, n (8) l 1 i1(α) = a1 i1 (α) l1 i1 (1)u1 11 (α) + l1 i1 (1)u1 11 (1) u 1 11 (1), (9) l 2 i1(α) = a2 i1 (α) l2 i1 (1)u2 11 (α) + l2 i1 (1)u2 11 (1) u 2 11 (1), u 1 rj (α) = a1 rj (α) r 1 k=1 (l1 rk (α)u1 kj (1) + l1 rk (1)u1 kj (α) l1 rk (1)u1 kj (1)), u 2 rj (α) = a2 rj (α) r 1 k=1 (l2 rk (α)u2 kj (1) + l2 rk (1)u2 kj (α) l2 rk (1)u2 kj (1)), j = r, r + 1,, n, r = 2, 3,, n, (10) also

6 212 A Panahi, T Allahviranloo, H Rouhparvar lir(α) 1 = a1 ir (α) r 1 k=1 ψ l1 ir (1)u1 rr(α) + lir 1 (1)u1 rr(1) u 1, (11) rr(1) lir(α) 2 = a2 ir (α) r 1 k=1 ω l2 ir (1)u2 rr(α) + lir 2 (1)u2 rr(1) u 2, rr(1) where j = r, r+1,, n, r = 2, 3,, n, ψ = l(1) ik (α)u(1) kr (1)+l(1) ik (1)u(1) kr (α) l(1) ik (1)u(1) kr (1) and ω = lik 2 (α)u2 kr (1) + l2 ik (1)u2 kr (α) l2 ik (1)u2 kr (1) Now we solve the system Ly = b, and after finding y we solve the system Ux = y to find the solution x for the fuzzy system Ax = b 4 AN EXAMPLE Example 41 Consider the 3 3 fuzzy system of linear equations Ax = b and let A be a fuzzy matrix and b be a fuzzy vector in the α-cut representation as follows [1 + α, 3 α] [5 + α, 10 4α] [3 + 2α, 7 2α] [1 + 2α, 4 α] b = [α, 2 α], A = [4 + 8α, 32 20α] [2 + 12α, 29 15α] [8α, 18 10α] [ 3, 2 α] [ α, 58 34α] [2 + 30α, 62 30α] [1 + 19α, 44 24α] For α = 1 we have = l l 31 l 32 1 u 11 u 12 u 13 0 u 22 u u 33, where u ij and l ij are u ij (1) and l ij (1) respectively In view of relations (6) and (7) we have L(1) = 2 1 0, U(1) =

7 Solving fuzzy linear systems of equations 213 Therefore, according to the sign of elements of matrixes L and U, also relations (8-11) for α [0, 1), we obtain L and U as follows L = [1 + α, 4 2α] 1 0, [3 + α, 7 3α] [1 + 2α, 4 α] 1 U = [5 + α, 10 4α] [3 + 2α, 7 2α] [1 + 2α, 4 α] 0 [1 + 3α, 5 α] [1 + α, 4 2α] 0 0 [1 + α, 5 3α] Now we solve the system Ly = b, considering the cross product in each needed multiplication [y 1 1, y2 1 ] [1 + α, 3 α] [1 + α, 4 2α] 1 0 [y 2 1, y2 2 ] = [α, 2 α] [3 + α, 7 3α] [1 + 2α, 4 α] 1 [y3 1, y2 3 ] [ 3, 2 α] After finding y we solve in the same way the system Ux = y to find the solution x for the fuzzy system Ax = b [5 + α, 10 4α] [3 + 2α, 7 2α] [1 + 2α, 4 α] 0 [1 + 3α, 5 α] [1 + α, 4 2α] 0 0 [1 + α, 5 3α] [x 1 1, x2 1 ] [x 1 2, x2 2 ] [x 1 3, x2 3 ] = [y 1 1, y2 1 ] [y 1 2, y2 2 ] [y 1 3, y2 3 ] 5 CONCLUSION In this paper we studied fuzzy linear system of the form Ax = b with A square matrix of fuzzy coefficients and b fuzzy number vector We introduced two fuzzy matrices, the lower triangular L and the upper triangular U such that A = LU and solved the fuzzy system of linear equations Ly = b and Ux = y respectively

8 214 A Panahi, T Allahviranloo, H Rouhparvar References [1] S Abbasbandy, R Ezzati, A Jafarian, LU decomposition method for solving fuzzy system of linear equations, Appl Math and Comput, 172 (2006), [2] T Allahviranloo, Numerical methods for fuzzy system of linear equations, Appl Math and Comput, 155 (2004), [3] B Bede, J Fodor, Product type operations between fuzzy numbers and their applications in geology, Acta Polytechnica Hungarica, 3, 1 (2006), [4] J J Buckly, Y Qu, Solving systems of linear fuzzy equations, Fuzzy Sets and Systems, 43 (1991) [5] M Dehghan, B Hashemi, M Ghatee, Computational methods for solving fully fuzzy linear systems, Appl Math and Comput, 179 (2006), [6] M Ma, M Friedman, A Kandel, A new fuzzy arithmetic, Fuzzy Sets and Systems, 108 (1999), [7] M L Puri, D A Ralescu, Differentials of fuzzy functions, J Math Anal Appl, 91 (1983), [8] A Vroman, G Deschrijver, E E Kerre, Solving systems of equations by parametric functions-an improved algorithm, Fuzzy Sets and Systems