Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

A New Computational Method for Solving Fully Fuzzy Linear Systems of Triangular Fuzzy Numbers

A New Computational Method for Solving Fully Fuzzy Linear Systems of Triangular Fuzzy Numbers Fuzzy Inf. Eng. (2012) 1: 63-73 DOI 10.1007/s12543-012-0101-5 ORIGINAL ARTICLE A New Computational Method for Solving Fully Fuzzy Linear Systems of Triangular Fuzzy Numbers Amit Kumar· Neetu · Abhinav Bansal Received: 25 May 2010/ Revised: 10 January 2012/ Accepted: 2 February 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, a new computational method is proposed to solve fully fuzzy linear systems (FFLS) of triangular fuzzy numbers based on the computation of row reduced echelon form for solving the crisp linear system of equations. The method is illustrated by solving three numerical examples. As compared to the existing meth- ods, the proposed method is easy to understand and to apply for solving FFLS occur- ring in real life situations and scientific applications. The primary advantage of the proposed method is that, by using it, the consistency of the FFLS can be checked eas- ily and nature of the solutions of an FFLS can also be obtained which may be unique or infinite. Keywords Fully fuzzy linear systems · Row reduced echelon form · Triangular fuzzy numbers 1. Introduction One field of applied mathematics that has many applications in various areas of sci- ence is solving a system of linear equations. Systems of simultaneous linear equations play a major role in various areas such as operational research, physics, statistics, engineering, random variational inclusions [15] and social sciences. When the esti- mation of the system coefficients is imprecise and only some vague knowledge about the actual values of the parameters is available, it may be convenient to represent some or all of them with fuzzy numbers [24]. Fuzzy number arithmetic is widely ap- plied and useful in computation of linear system whose parameters are all or partially represented by fuzzy numbers. Amit Kumar()· Neetu () School of Mathematics and Computer Applications, Thapar University, Patiala-147004, India email: amit rs iitr@yahoo.com neetu babbar83@rediffmail.com Abhinav Bansal () Computer Science and Engineering Department, Thapar University, Patiala-147004, India email: abhinav.bansal8@gmail.com 64 Amit Kumar · Neetu · Abhinav Bansal (2012) Dubois and Prade [11,12] investigated two definitions of a system of fuzzy linear equations, consisting of system of tolerance constraints and system of approximate equalities. The simplest method for finding a solution for this system is creating scenarios for the fuzzy system, which is a realization of fuzzy systems. Based on these actual scenarios, Buckley and Qu [7] extended several methods for this category and proved their approaches are not practicable, because infinite number of scenarios can be driven for a fully fuzzy linear system (FFLS). Zhao and Govind [25] studied the algebraic equations involving generalized fuzzy numbers (which includes fuzzy numbers, fuzzy intervals, crisp numbers and interval numbers) with continuous membership functions. A general model for solving a fuzzy linear system whose coefficient matrix is crisp and the right-hand side column is an arbitrary fuzzy vector was first proposed by Friedman et al [13]. Friedman et al [14] investigated a dual fuzzy linear system by mean of nonnegative matrix theory. Allahviranloo [4] proposed solution of a fuzzy linear system by using iterative method (Jacobi and Gauss Seidel methods), later on the same author proposed the solution of such system using successive over relaxation iterative method [5] and Adomian decomposition method [6]. Abbasbandy et al [3] proposed the Conjugate gradient method, for solving fuzzy symmetric positive definite system of linear equa- tion. Dehghan and Hashemi [9] extended the Adomian decomposition method [6], to find the positive fuzzy vector solution of fully fuzzy linear system. Dehghan et al [8] proposed classic methods such as Cramer’s rule, Gaussian elimination method, LU decomposition method from linear algebra and linear programming for finding the approximated solution of a fully fuzzy linear systems. Abbasbandy and Jafarian [2] applied Steepest descent method for approximation of the unique solution of fuzzy system of linear equation. Abbasbandy et al [1] used LU decomposition method for solving fuzzy system of linear equation when the co- efficient matrix is symmetric positive definite. Muzzioli and Reynaerts [21] pointed out that although several investigations are reported in the literature of the solution of fuzzy systems, very few methods are avail- able for the practical solution of a fuzzy linear system. They introduced an algorithm to find vector solution by transforming the system A x+ b = A x+ b into the FFLS 1 1 2 2 Ax = b, where A = A − A and b = b − b . 1 2 2 1 Dehghan and Hashemi [10] modified the existing methods employed by Allahvi- ranloo [4] for solving fuzzy linear systems. Mosleh et al [18] proposed a method to find the solution of fully fuzzy linear system of the form Ax + b = Cx+ d with A, C square matrices of fuzzy coefficients and b, d fuzzy number vectors and the unknown vector x is vector consisting of n fuzzy numbers. Nasseri et al [19] used a certain decomposition methods of the coefficient matrix for solving fully fuzzy linear system of equations. Yin and Wang [23] considered the general case of Splitting iterative methods for solving fuzzy system of linear equation. Sun and Guo [22] proposed a general model for solving fuzzy linear systems of the form Ax = y and general dual fuzzy linear systems of the form Ax + y = Bx + z with A, B matices of crisp coefficients and x, y are fuzzy number vectors. Nasseri and Zahmatkesh [20] proposed a new method for computing the non-negative solution of fully fuzzy linear system of equations. Fuzzy Inf. Eng. (2012) 1: 63-73 65 ˜ ˜ ˜ ˜ In this paper, we intend to solve A⊗x ˜ = b, where A is a fuzzy matrix and x ˜ and b are fuzzy vectors with appropriate sizes. The proposed method is very easy to understand and to apply for solving fully fuzzy linear systems occurring in real life situations. The method is illustrated by solving numerical examples. The main advantage of the proposed method is that by using it, the consistency of the fully fuzzy linear system can be checked easily and the nature of the infinite solutions of an FFLS can also be obtained. The rest of paper is organized as follows: In Section 2, shortcomings are described in the existing methods to solve fully fuzzy linear system of equation. In Section 3, some basic definitions on fuzzy numbers are reviewed. In Section 4, a new method for solving an FFLS with triangular fuzzy numbers is introduced, explained and verified with numerical examples. Section 5 ends the paper with conclusion. 2. Shortcomings of Existing Methods In this section, the shortcomings are pointed out in the existing methods [1-10, 20, 21]: 1) In almost all the existing methods, it is assumed that the system of equations is consistent and then the methods are developed i.e., consistency of the FFLS can not be checked using the existing methods. 2) In case of consistency, the nature of solution can not be checked, i.e., whether the solution is unique or infinite. To overcome the above shortcomings, in Section 4 a new computational method is proposed to solve FFLS. 3. Preliminary Concepts In this section, some basic definitions of fuzzy set theory are reviewed [8,11]. Definition 3.1 A fuzzy subset A of R is defined by its membership function μ : R → [0, 1], which assigns a real numberμ in the interval [0,1] to each element x ˜ ∈ R, where the value of μ at x shows the grade of membership of x in A. Definition 3.2 A fuzzy number is a convex normalized fuzzy set of the real line R whose membership function is piecewise continuous. ˜ ˜ Definition 3.3 A fuzzy number M is called positive (negative), denoted by M > 0(M < 0) if its membership function μ (x) satisfies μ (x) = 0,∀x ≤ 0(∀x ≥ 0).A ˜ ˜ M M ˜ ˜ ˜ fuzzy number M is called non positive (non negative), denoted by M ≤ 0(M ≥ 0) if its membership functionμ (x) satisfies μ (x) = 0,∀x < 0(∀x > 0). ˜ ˜ M M Definition 3.4 A fuzzy number M is said to be an LR fuzzy number if its membership function is defined as follow: m− x L( ), for x ≤ m,α> 0, x− m μ (x) = R( ), for x ≥ m,β> 0, M ⎪ ⎪ β 0, otherwise, 66 Amit Kumar · Neetu · Abhinav Bansal (2012) where m is the mean value of M and α,β are left and right spreads, respectively and a function L(·) the left shape function satisfying: 1) L(x) = L(−x); 2) L(0) = 1 and L(1) = 0; 3) L(x) is non-increasing on [0,∞). Naturally, a right shape function R(·) is similarly defined as L(·). Using its mean value and left and right spreads and shape functions, such an LR fuzzy number M is ˜ ˜ symbolically written as M = (m,α,β) . Clearly, M is non negative if and only if LR m−α ≥ 0. ˜ ˜ Definition 3.5 Two LR fuzzy number M = (m,α,β) and N = (m,γ,δ) are said to LR LR be equal if and only if m = n,α = γ, β = δ. ˜ ˜ Definition 3.6 For two LR fuzzy number M = (m,α,β) and N = (n,γ,δ) the LR LR formula for extended addition, extended opposite, extended subtraction and extended multiplication are summarized as follows: 1) (m,α,β) ⊕ (n,γ,δ) = (m+ n,α+γ,β+δ), LR LR 2) −M = −(m,α,β) = (−m,β,α) , LR LR ˜ ˜ 3) M N = ((m,α,β) (n,γ,δ) = (m− n,α+δ,β+γ) , LR RL LR ˜ ˜ ˜ ˜ 4) If M > 0 and N > 0, then M⊗ N =(m,α,β) ⊗ (n,γ,δ) = (mn, mγ+ nα, mδ+ LR LR nβ) , LR (λm,λα,λβ) , if λ ≥ 0; ⎨ LR 5) If λ is any scaler, thenλ⊗ (m,α,β) = LR ⎪ (λm,−λβ,−λα) , if λ< 0. RL ˜ ˜ Definition 3.7 A matrix A = (˜ a ) is called a fuzzy matrix if each element of Aisa ij ˜ ˜ ˜ fuzzy number. A will be positive (negative) and denoted by A > 0(A < 0) if each ˜ ˜ element of A is positive (negative). A will be non-positive (non-negative) and denoted ˜ ˜ ˜ by A ≤ 0(A ≥ 0) if each element of A is non-positive (non-negative). We may represent n × m fuzzy matrix A = (˜ a ) , where a ˜ = (a ,α ,β ) with a new ij (n×m) ij ij ij ij LR notation A = (A, M, N), where A = (a ), M = (α ) and N = (β ) are three n × m ij ij ij crisp matrices. ˜ ˜ ˜ Definition 3.8 Let A = (˜ a ) and B = (b ) be two m× n and n× p fuzzy matrices. We ij ij ˜ ˜ ˜ define A⊗ B = C = (˜ c ) which is the m× p matrix, where ij c ˜ = a ˜ ⊗ b . ij ik kj k=1,···,n Definition 3.9 Consider the n× n linear system of equations (˜ a ⊗ x ˜ )⊕ (˜ a ⊗ x ˜ )⊕···⊕ (˜ a ⊗ x ˜ ) = b , 11 1 12 2 1n n 1 (˜ a ⊗ x ˜ )⊕ (˜ a ⊗ x ˜ )⊕···⊕ (˜ a ⊗ x ˜ ) = b , 21 1 22 2 2n n 2 Fuzzy Inf. Eng. (2012) 1: 63-73 67 . . . . . . . . . (˜ a ⊗ x ˜ )⊕ (˜ a ⊗ x ˜ )⊕...⊕ (˜ a ⊗ x ˜ ) = b . n1 1 n2 2 nn n n The matrix form of above equation is ˜ ˜ A⊗ x ˜ = b, where the coefficient matrix A = (˜ a ), 1 ≤ i, j ≤ nisann ⊗ n fuzzy matrix. This ij system is called a fully fuzzy linear system (FFLS). Up to the rest of this paper we want to find the positive solutions of FFLS A⊗ x ˜ = b, ˜ ˜ where A = (A, M, N) ≥ 0, x ˜ = (x, y, z) ≥ 0 and b = (b, g, h) ≥ 0. So we have: (A, M, N)⊗ (x, y, z) = (b, g, h). 4. Proposed Method One of the method to solve the linear system of equations Ax = b is row reduced echelon form of computation [17]. In this sectionˈ the same method is extended to ˜ ˜ solve fully fuzzy linear systems A⊗ x ˜ = b. ˜ ˜ Assuming A = (A, M, N) ≥ 0, x ˜ = (x, y, z) ≥ 0 and b = (b, g, h) ≥ 0. We can write ˜ ˜ A⊗ x = b as: (A, M, N)⊗ (x, y, z) = (b, g, h), (Ax, Ay+ Mx, Az+ Nx) = (b, g, h) (Using Definition 3.6) or Ax = b, Ay+ Mx = g, (Using Definition 3.5) Az+ Nx = h. Now use the following steps to find the solution of FFLS: Step 1: Compute the row reduced echelon form of augmented matrices (A, b), (A, g− Mx), (A, h − Nx) by applying suitable row operations [17]. There may be following three cases, Case 1: If rank(A)  rank(A, b), then the FFLS is inconsistent and no non-negative solution exists. Case 2: If rank(A) = rank(A, b), but there exists at least one-negative element th column of the row reduced echelon form of augmented matrices in the (n + 1) (A, b)or(A, g − Mx)or(A, h − Nx), then the system of equations are inconsistent and no non-negative solution exists. th Case 3: If rank(A) = rank(A, b) and all the elements of (n+ 1) column of the row reduced echelon form of augmented matrices (A, b), (A, g− Mx), (A, h− Nx) are non- negative, then the system of equations are consistent, i.e., non-negative solution exist. There may be two following sub-cases. 68 Amit Kumar · Neetu · Abhinav Bansal (2012) Case 3.a: If rank(A) = rank(A, b)  n, then the FFLS has infinite number of non- negative solutions. Case 3.b: If rank(A) = rank(A, b) = n, then the FFLS has a unique and non-negative solution. Step 2: Compute the values of x, y and z using the row reduced echelon form of i i i augmented matrices (A, b), (A, g − Mx), (A, h − Nx), respectively. The solution of FFLS will be represented by x ˜ = (x, y, z ), ∀i = 1, 2,··· , n. i i i i Example 4.1 Consider the following FFLS and solve it by the proposed method ˜ ˜ ˜ 5⊗ x ˜ ⊕ 6⊗ x ˜ = 50, 1 2 ˜ ˜ ˜ 7⊗ x ˜ ⊕ 4⊗ x ˜ = 48, 1 2 i.e., (5, 1, 1)⊗ (x , y , z )⊕ (6, 1, 2)⊗ (x , y , z ) = (50, 10, 17), 1 1 1 2 2 2 (7, 1, 0)⊗ (x , y , z )⊕ (4, 0, 1)⊗ (x , y , z ) = (48, 5, 7). 1 1 1 2 2 2 Solution: The given FFLS may be written as: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ (5, 1, 1) (6, 1, 2)⎟ ⎜ (x , y , z )⎟ ⎜ (50, 10, 17)⎟ 1 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = , ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (x , y , z ) (7, 1, 0) (4, 0, 1) (48, 5, 7) 2 2 2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 56⎟ ⎜ 11⎟ ⎜ 12⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A= ⎜ ⎟ , M= ⎜ ⎟ , N= ⎜ ⎟ , ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 74 14 10 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 50⎟ ⎜ 10⎟ ⎜ 17⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ b= ⎜ ⎟ , g= ⎜ ⎟ , h= ⎜ ⎟ . ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 48 5 7 The augmented matrix ⎛ ⎞ ⎜ ⎟ ⎜ 5650⎟ ⎜ ⎟ ⎜ ⎟ (A, b)= ⎜ ⎟ . ⎝ ⎠ The row reduced echelon form of this matrix is obtained as follows: Apply the row operation R → R to get: 1 1 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1 10 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 5 . ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ −5 Again we apply row operations in sequence: R → R − 7R , R → R , R → 2 2 1 2 2 1 R − R and we get: 1 2 ⎛ ⎞ ⎜ ⎟ ⎜ 104⎟ ⎜ ⎟ ⎜ ⎟ (A, b)= ⎜ ⎟ . ⎝ ⎠ Using the row reduced echelon form of the augmented matrix (A,b), we obtain a solution x = 4, x = 5. 1 2 Fuzzy Inf. Eng. (2012) 1: 63-73 69 Similarly, the row reduced echelon form of the augmented matrix ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (A, g− Mx)= ⎜ ⎟ ⎝ ⎠ is ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 10 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 11 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 1 1 which gives y = , y = . 1 2 11 11 Similarly, the row reduced echelon form of the augmented matrix ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (A, h− Nx)= ⎜ ⎟ ⎝ ⎠ is ⎛ ⎞ ⎜ 10 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎝ ⎠ which gives z = 0, z = . 1 2 th rd Since Rank(A) = Rank(A, b) = 2 and all the elements of of (n + 1) , i.e., 3 columns of the row reduced echelon form of augmented matrices (A, b), (A, g− Mx), (A, h− Nx) are non-negative, therefore, the given FFLS has the following unique and non-negative solution. Substituting appropriate values in x ˜ = (x, y, z )(∀i = 1, 2), we get i i i i x ˜ = (4, , 0), 1 1 x ˜ = (5, , ). 11 2 As we see, we obtain the same solution as the one with the direct method given in [8]. Example 4.2 Consider the following FFLS and solve it by the proposed method ˜ ˜ ˜ 2⊗ x ˜ ⊕ 4⊗ x ˜ = 10, 1 2 ˜ ˜ ˜ 4⊗ x ˜ ⊕ 2⊗ x ˜ = 20, 1 2 i.e., (2, 1, 1)⊗ (x , y , z )⊕ (4, 1, 1)⊗ (x , y , z ) = (10, 5, 5), 1 1 1 2 2 2 (4, 2, 2)⊗ (x , y , z )⊕ (8, 2, 2)⊗ (x , y , z ) = (20, 10, 10). 1 1 1 2 2 2 Solution: The given FFLS may be written as: 70 Amit Kumar · Neetu · Abhinav Bansal (2012) ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ (2, 1, 1) (4, 1, 1)⎟ ⎜ (x , y , z )⎟ ⎜ (10, 5, 5 ⎟ 1 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ , ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (4, 2, 2) (8, 2, 2) (x , y , z ) (20, 10, 10) 2 2 2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 24 11 11 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A= ⎜ ⎟ , M= ⎜ ⎟ , N= ⎜ ⎟ , ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 48 22 12 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 10⎟ ⎜ 5 ⎟ ⎜ 5 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ b= , g= , h= . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 20 10 10 The augmented matrix ⎛ ⎞ ⎜ ⎟ ⎜ 2410⎟ ⎜ ⎟ ⎜ ⎟ (A, b)= ⎜ ⎟ . ⎝ ⎠ The row reduced echelon form of this matrix is obtained as follows: Apply the row operation R → R to get: 1 1 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ . ⎝ ⎠ Hence the solution is x = u, x = 5− 2u. 2 1 Now compute ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 5 ⎟ ⎜ 11⎟ ⎜ 5− 2u⎟ ⎜ u ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ g− Mx = − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ . ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 10 22 u 2u ⎛ ⎞ ⎜ ⎟ ⎜ 24 u ⎟ ⎜ ⎟ ⎜ ⎟ Hence the augmented matrix (A, g− Mx)= ⎜ ⎟ ⎝ ⎠ 482u ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 12 ⎟ ⎜ ⎟ ⎜ ⎟ is = ⎜ ⎟ . ⎜ ⎟ 00 0 Using the above row reduced echelon form, the solution is y = − 2v, y = v. 1 2 Now compute ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 5 ⎟ ⎜ 11⎟ ⎜ 5− 2u⎟ ⎜ u ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ h− Nx = − = . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 10 22 u 2u ⎛ ⎞ ⎜ ⎟ ⎜ 24 u ⎟ ⎜ ⎟ ⎜ ⎟ Hence the augmented matrix (A, h− Nx)= . ⎜ ⎟ ⎝ ⎠ 482u ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ The row reduced echelon form of augmented matrix = 2 . ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 00 0 Using the above row reduced echelon form, we get the solution y = − 2v, y = v. 1 2 2 Fuzzy Inf. Eng. (2012) 1: 63-73 71 Since rank(A) = rank(A, b) = 1  n(= 2), so the non-negative solution of the FFLS th rd will exist if all the elements of (n+ 1) , i.e., 3 columns of the row reduced echelon form of augmented matrices (A, b), (A, g− Mx), (A, h− Nx) are non-negative. For this particular problem, we chose u ∈ [0, 2], v = 0 and w = 0. Thus the FFLS will have infinite number of non-negative solutions if u ∈ [0, 2], v = 0 and w = 0, else the FFLS will be inconsistent, i.e., no non-negative solution will exist. Thus the solution of this FFLS can be written as follows: u u x ˜ = (5− 2u, − 2v, − 2w), 2 2 x ˜ = (u, v, w); u ∈ [0, 2], v = 0, w = 0. Some particular solutions of this FFLS are: Solution 1: x ˜ = (1, 1, 1), x ˜ = (2, 0, 0). 1 2 Solution 2: x ˜ = (5, 0, 0), x ˜ = (0, 0, 0). 1 2 Solution 3: x ˜ = (3, 1/2, 1/2), x ˜ = (1, 0, 0). 1 2 u u General solution: x ˜ = (5− 2u, , ), x ˜ = (u, 0, 0); u ∈ [0, 2]. 1 2 2 2 Example 4.3 Consider the following FFLS and solve it by the proposed method ˜ ˜ ˜ 2⊗ x ˜ ⊕ 4⊗ x ˜ = 10, 1 2 ˜ ˜ ˜ 4⊗ x ˜ ⊕ 8⊗ x ˜ = 24, 1 2 i.e., (2, 1, 1)⊗ (x , y , z )⊕ (4, 1, 1)⊗ (x , y , z ) = (10, 5, 5), 1 1 1 2 2 2 (4, 2, 2)⊗ (x , y , z )⊕ (8, 2, 2)⊗ (x , y , z ) = (24, 12, 12). 1 1 1 2 2 2 Solution: The given FFLS may be written as: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ (2, 1, 1) (4, 1, 1)⎟ ⎜ (x , y , z )⎟ ⎜ (10, 5, 5 ⎟ 1 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ , ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (4, 2, 2) (8, 2, 2) (x , y , z ) (24, 12, 12) 2 2 2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 24 11 11 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A= ⎜ ⎟ , M= ⎜ ⎟ , N= ⎜ ⎟ , ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 48 22 12 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 10 5 5 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ b= ⎜ ⎟ , g= ⎜ ⎟ , h= ⎜ ⎟ . ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 24 12 12 The augmented matrix ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (A, b)= ⎜ ⎟ . ⎝ ⎠ The row reduced echelon form of this matrix is obtained as follows: ⎛ ⎞ ⎜ ⎟ 1 ⎜ 120⎟ ⎜ ⎟ ⎜ ⎟ Apply the row operation R → R to get ⎜ ⎟ , 1 1 ⎝ ⎠ rank(A) = 1, rank(A, b) = 2. Hence rank(A)  rank(A, b). Therefore the given FFLS is inconsistent i.e., it has no solution. 72 Amit Kumar · Neetu · Abhinav Bansal (2012) 5. Conclusion In this paper, the fully fuzzy linear systems, i.e., fuzzy linear systems with fuzzy coefficients involving fuzzy variables, are investigated and a new method is applied for solving these systems. Row reduced echelon form of matrices is used to construct a new method for solving fully fuzzy linear systems and the validity of the proposed method is examined with numerical examples. The result is obtained easily and is same as compared to the solution described in [8]. The constructed method is efficient in determination of consistency of fully fuzzy linear systems occurring in real life situations and computation of FFLS that may lead to infinite and non negative fuzzy solutions. Acknowledgments The authors would like to thank the Editor-in-Chief and anonymous referees for the various suggestions which have led to an improvement in both the quality and clarity of the paper. References 1. Abbasbandy S, Ezzati R, Jafarian A (2006) LU decomposition method for solving fuzzy system of linear equations. Applied Mathematics and Computation 172: 633-643 2. Abbasbandy S, Jafarian A (2006) Steepest descent method for system of fuzzy linear equations. Applied Mathematics and Computation 175: 823-833 3. Abbasbandy S, Jafarian A, Ezzati R (2005) Conjugate gradient method for fuzzy symmetric positive- definite system of linear equations. Applied Mathematics and Computation 171: 1184-1191 4. Allahviranloo T (2004) Numerical methods for fuzzy system of linear equations. Applied Mathemat- ics and Computation 155: 493-502 5. Allahviranloo T (2004) Successive over relaxation iterative method for fuzzy system of linear equa- tions. Applied Mathematics and Computation 162: 189-196 6. Allahviranloo T (2005) The adomian decomposition method for fuzzy system of linear equations. Applied Mathematics and Computation 163: 553-563 7. Buckley J J, Qu Y (1991) Solving systems of linear fuzzy equations. Fuzzy Sets and Systems 43: 33-43 8. Dehghan M, Hashemi B, Ghatee M (2006) Computational methods for solving fully fuzzy linear systems. Applied Mathematics and Computation 179: 328-343 9. Dehghan M, Hashemi B (2006) Solution of the fully fuzzy linear system using the decomposition procedure. Applied Mathematics and Computation 182: 1568-1580 10. Dehghan M, Hashemi B, Ghatee M (2007) Solution of the fully fuzzy linear system using iterative techniques. Chaos, Solitons and Fractals, 34 : 316-336 11. Dubois D, Prade H (1980) Fuzzy sets and systems: Theory and applications. Academic Press, New York 12. Dubois D, Prade H (1980) Systems of linear fuzzy constraints. Fuzzy Sets and Systems 3: 37-48 13. Friedman M, Ming M, Kandel A (1998) Fuzzy linear systems. Fuzzy Sets and Systems 96: 201-209 14. Friedman M, Ming M, Kandel A (2000) Duality in fuzzy linear systems. Fuzzy Sets and Systems 109: 55-58 15. Gang L H (2010) Generalized fuzzy random set-valued mixed variational inclusions involving ran- dom nonlinear (A ,η )-accretive mappings in Banach spaces. J. Nonlinear Sci. Appl. 3: 63-77 ω ω 16. Kauffmann A, Gupta M M (1991) Introduction to fuzzy arithmetic: Theory and applications. Van Nostrand Reinhold, New York 17. Lipschutz S (2005) Shaum’s outline of theory and problems of linear algebra. 3rd edition, McGraw Hill Book Company, New York Fuzzy Inf. Eng. (2012) 1: 63-73 73 18. Mosleh M, Abbasbandy S, Otadi (2007) Full fuzzy linear systems of the form Ax + b = Cx + d. Proceedings First Joint Congress on Fuzzy and Intelligent Systems 19. Muzzioli S, Reynaerts H (2006) Fuzzy linear systems of the form A x + b = A x + b . Fuzzy Sets 1 1 2 2 and Systems 157: 939-951 20. Nasseri S H, Sohrabi M, Ardil E (2008) Solving fully fuzzy linear systems by use of a certain decom- position of the coefficient matrix. International Journal of Computational and Mathematical Sciences 2: 140-142 21. Nasseri S H, Zahmatkesh F (2010) Huang method for solving fully fuzzy linear system of equations. The Journal of Mathematics and Computer Science 1: 1-5 22. Sun X, Guo S (2009) Solution to general fuzzy linear system and its necessary and sufficient condi- tion. Fuzzy Information and Engineering 3: 317-327 23. Yin J F, Wang K (2009) Splitting iterative methods for fuzzy system of linear equations. Computa- tional Mathematics and Modeling 20: 326-335 24. Zadeh L A (1965) Fuzzy sets. Information and Control 8: 338-353 25. Zhao R, Govind R (1991) Solution of algebraic equations involving generalized fuzzy numbers. In- formation Sciences 56: 199-243 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

A New Computational Method for Solving Fully Fuzzy Linear Systems of Triangular Fuzzy Numbers

Download PDF
11 pages

Loading next page...
 
/lp/taylor-francis/a-new-computational-method-for-solving-fully-fuzzy-linear-systems-of-ZcfiMvm22B

References (25)

Publisher
Taylor & Francis
Copyright
© 2012 Taylor and Francis Group, LLC
ISSN
1616-8666
eISSN
1616-8658
DOI
10.1007/s12543-012-0101-5
Publisher site
See Article on Publisher Site

Abstract

Fuzzy Inf. Eng. (2012) 1: 63-73 DOI 10.1007/s12543-012-0101-5 ORIGINAL ARTICLE A New Computational Method for Solving Fully Fuzzy Linear Systems of Triangular Fuzzy Numbers Amit Kumar· Neetu · Abhinav Bansal Received: 25 May 2010/ Revised: 10 January 2012/ Accepted: 2 February 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, a new computational method is proposed to solve fully fuzzy linear systems (FFLS) of triangular fuzzy numbers based on the computation of row reduced echelon form for solving the crisp linear system of equations. The method is illustrated by solving three numerical examples. As compared to the existing meth- ods, the proposed method is easy to understand and to apply for solving FFLS occur- ring in real life situations and scientific applications. The primary advantage of the proposed method is that, by using it, the consistency of the FFLS can be checked eas- ily and nature of the solutions of an FFLS can also be obtained which may be unique or infinite. Keywords Fully fuzzy linear systems · Row reduced echelon form · Triangular fuzzy numbers 1. Introduction One field of applied mathematics that has many applications in various areas of sci- ence is solving a system of linear equations. Systems of simultaneous linear equations play a major role in various areas such as operational research, physics, statistics, engineering, random variational inclusions [15] and social sciences. When the esti- mation of the system coefficients is imprecise and only some vague knowledge about the actual values of the parameters is available, it may be convenient to represent some or all of them with fuzzy numbers [24]. Fuzzy number arithmetic is widely ap- plied and useful in computation of linear system whose parameters are all or partially represented by fuzzy numbers. Amit Kumar()· Neetu () School of Mathematics and Computer Applications, Thapar University, Patiala-147004, India email: amit rs iitr@yahoo.com neetu babbar83@rediffmail.com Abhinav Bansal () Computer Science and Engineering Department, Thapar University, Patiala-147004, India email: abhinav.bansal8@gmail.com 64 Amit Kumar · Neetu · Abhinav Bansal (2012) Dubois and Prade [11,12] investigated two definitions of a system of fuzzy linear equations, consisting of system of tolerance constraints and system of approximate equalities. The simplest method for finding a solution for this system is creating scenarios for the fuzzy system, which is a realization of fuzzy systems. Based on these actual scenarios, Buckley and Qu [7] extended several methods for this category and proved their approaches are not practicable, because infinite number of scenarios can be driven for a fully fuzzy linear system (FFLS). Zhao and Govind [25] studied the algebraic equations involving generalized fuzzy numbers (which includes fuzzy numbers, fuzzy intervals, crisp numbers and interval numbers) with continuous membership functions. A general model for solving a fuzzy linear system whose coefficient matrix is crisp and the right-hand side column is an arbitrary fuzzy vector was first proposed by Friedman et al [13]. Friedman et al [14] investigated a dual fuzzy linear system by mean of nonnegative matrix theory. Allahviranloo [4] proposed solution of a fuzzy linear system by using iterative method (Jacobi and Gauss Seidel methods), later on the same author proposed the solution of such system using successive over relaxation iterative method [5] and Adomian decomposition method [6]. Abbasbandy et al [3] proposed the Conjugate gradient method, for solving fuzzy symmetric positive definite system of linear equa- tion. Dehghan and Hashemi [9] extended the Adomian decomposition method [6], to find the positive fuzzy vector solution of fully fuzzy linear system. Dehghan et al [8] proposed classic methods such as Cramer’s rule, Gaussian elimination method, LU decomposition method from linear algebra and linear programming for finding the approximated solution of a fully fuzzy linear systems. Abbasbandy and Jafarian [2] applied Steepest descent method for approximation of the unique solution of fuzzy system of linear equation. Abbasbandy et al [1] used LU decomposition method for solving fuzzy system of linear equation when the co- efficient matrix is symmetric positive definite. Muzzioli and Reynaerts [21] pointed out that although several investigations are reported in the literature of the solution of fuzzy systems, very few methods are avail- able for the practical solution of a fuzzy linear system. They introduced an algorithm to find vector solution by transforming the system A x+ b = A x+ b into the FFLS 1 1 2 2 Ax = b, where A = A − A and b = b − b . 1 2 2 1 Dehghan and Hashemi [10] modified the existing methods employed by Allahvi- ranloo [4] for solving fuzzy linear systems. Mosleh et al [18] proposed a method to find the solution of fully fuzzy linear system of the form Ax + b = Cx+ d with A, C square matrices of fuzzy coefficients and b, d fuzzy number vectors and the unknown vector x is vector consisting of n fuzzy numbers. Nasseri et al [19] used a certain decomposition methods of the coefficient matrix for solving fully fuzzy linear system of equations. Yin and Wang [23] considered the general case of Splitting iterative methods for solving fuzzy system of linear equation. Sun and Guo [22] proposed a general model for solving fuzzy linear systems of the form Ax = y and general dual fuzzy linear systems of the form Ax + y = Bx + z with A, B matices of crisp coefficients and x, y are fuzzy number vectors. Nasseri and Zahmatkesh [20] proposed a new method for computing the non-negative solution of fully fuzzy linear system of equations. Fuzzy Inf. Eng. (2012) 1: 63-73 65 ˜ ˜ ˜ ˜ In this paper, we intend to solve A⊗x ˜ = b, where A is a fuzzy matrix and x ˜ and b are fuzzy vectors with appropriate sizes. The proposed method is very easy to understand and to apply for solving fully fuzzy linear systems occurring in real life situations. The method is illustrated by solving numerical examples. The main advantage of the proposed method is that by using it, the consistency of the fully fuzzy linear system can be checked easily and the nature of the infinite solutions of an FFLS can also be obtained. The rest of paper is organized as follows: In Section 2, shortcomings are described in the existing methods to solve fully fuzzy linear system of equation. In Section 3, some basic definitions on fuzzy numbers are reviewed. In Section 4, a new method for solving an FFLS with triangular fuzzy numbers is introduced, explained and verified with numerical examples. Section 5 ends the paper with conclusion. 2. Shortcomings of Existing Methods In this section, the shortcomings are pointed out in the existing methods [1-10, 20, 21]: 1) In almost all the existing methods, it is assumed that the system of equations is consistent and then the methods are developed i.e., consistency of the FFLS can not be checked using the existing methods. 2) In case of consistency, the nature of solution can not be checked, i.e., whether the solution is unique or infinite. To overcome the above shortcomings, in Section 4 a new computational method is proposed to solve FFLS. 3. Preliminary Concepts In this section, some basic definitions of fuzzy set theory are reviewed [8,11]. Definition 3.1 A fuzzy subset A of R is defined by its membership function μ : R → [0, 1], which assigns a real numberμ in the interval [0,1] to each element x ˜ ∈ R, where the value of μ at x shows the grade of membership of x in A. Definition 3.2 A fuzzy number is a convex normalized fuzzy set of the real line R whose membership function is piecewise continuous. ˜ ˜ Definition 3.3 A fuzzy number M is called positive (negative), denoted by M > 0(M < 0) if its membership function μ (x) satisfies μ (x) = 0,∀x ≤ 0(∀x ≥ 0).A ˜ ˜ M M ˜ ˜ ˜ fuzzy number M is called non positive (non negative), denoted by M ≤ 0(M ≥ 0) if its membership functionμ (x) satisfies μ (x) = 0,∀x < 0(∀x > 0). ˜ ˜ M M Definition 3.4 A fuzzy number M is said to be an LR fuzzy number if its membership function is defined as follow: m− x L( ), for x ≤ m,α> 0, x− m μ (x) = R( ), for x ≥ m,β> 0, M ⎪ ⎪ β 0, otherwise, 66 Amit Kumar · Neetu · Abhinav Bansal (2012) where m is the mean value of M and α,β are left and right spreads, respectively and a function L(·) the left shape function satisfying: 1) L(x) = L(−x); 2) L(0) = 1 and L(1) = 0; 3) L(x) is non-increasing on [0,∞). Naturally, a right shape function R(·) is similarly defined as L(·). Using its mean value and left and right spreads and shape functions, such an LR fuzzy number M is ˜ ˜ symbolically written as M = (m,α,β) . Clearly, M is non negative if and only if LR m−α ≥ 0. ˜ ˜ Definition 3.5 Two LR fuzzy number M = (m,α,β) and N = (m,γ,δ) are said to LR LR be equal if and only if m = n,α = γ, β = δ. ˜ ˜ Definition 3.6 For two LR fuzzy number M = (m,α,β) and N = (n,γ,δ) the LR LR formula for extended addition, extended opposite, extended subtraction and extended multiplication are summarized as follows: 1) (m,α,β) ⊕ (n,γ,δ) = (m+ n,α+γ,β+δ), LR LR 2) −M = −(m,α,β) = (−m,β,α) , LR LR ˜ ˜ 3) M N = ((m,α,β) (n,γ,δ) = (m− n,α+δ,β+γ) , LR RL LR ˜ ˜ ˜ ˜ 4) If M > 0 and N > 0, then M⊗ N =(m,α,β) ⊗ (n,γ,δ) = (mn, mγ+ nα, mδ+ LR LR nβ) , LR (λm,λα,λβ) , if λ ≥ 0; ⎨ LR 5) If λ is any scaler, thenλ⊗ (m,α,β) = LR ⎪ (λm,−λβ,−λα) , if λ< 0. RL ˜ ˜ Definition 3.7 A matrix A = (˜ a ) is called a fuzzy matrix if each element of Aisa ij ˜ ˜ ˜ fuzzy number. A will be positive (negative) and denoted by A > 0(A < 0) if each ˜ ˜ element of A is positive (negative). A will be non-positive (non-negative) and denoted ˜ ˜ ˜ by A ≤ 0(A ≥ 0) if each element of A is non-positive (non-negative). We may represent n × m fuzzy matrix A = (˜ a ) , where a ˜ = (a ,α ,β ) with a new ij (n×m) ij ij ij ij LR notation A = (A, M, N), where A = (a ), M = (α ) and N = (β ) are three n × m ij ij ij crisp matrices. ˜ ˜ ˜ Definition 3.8 Let A = (˜ a ) and B = (b ) be two m× n and n× p fuzzy matrices. We ij ij ˜ ˜ ˜ define A⊗ B = C = (˜ c ) which is the m× p matrix, where ij c ˜ = a ˜ ⊗ b . ij ik kj k=1,···,n Definition 3.9 Consider the n× n linear system of equations (˜ a ⊗ x ˜ )⊕ (˜ a ⊗ x ˜ )⊕···⊕ (˜ a ⊗ x ˜ ) = b , 11 1 12 2 1n n 1 (˜ a ⊗ x ˜ )⊕ (˜ a ⊗ x ˜ )⊕···⊕ (˜ a ⊗ x ˜ ) = b , 21 1 22 2 2n n 2 Fuzzy Inf. Eng. (2012) 1: 63-73 67 . . . . . . . . . (˜ a ⊗ x ˜ )⊕ (˜ a ⊗ x ˜ )⊕...⊕ (˜ a ⊗ x ˜ ) = b . n1 1 n2 2 nn n n The matrix form of above equation is ˜ ˜ A⊗ x ˜ = b, where the coefficient matrix A = (˜ a ), 1 ≤ i, j ≤ nisann ⊗ n fuzzy matrix. This ij system is called a fully fuzzy linear system (FFLS). Up to the rest of this paper we want to find the positive solutions of FFLS A⊗ x ˜ = b, ˜ ˜ where A = (A, M, N) ≥ 0, x ˜ = (x, y, z) ≥ 0 and b = (b, g, h) ≥ 0. So we have: (A, M, N)⊗ (x, y, z) = (b, g, h). 4. Proposed Method One of the method to solve the linear system of equations Ax = b is row reduced echelon form of computation [17]. In this sectionˈ the same method is extended to ˜ ˜ solve fully fuzzy linear systems A⊗ x ˜ = b. ˜ ˜ Assuming A = (A, M, N) ≥ 0, x ˜ = (x, y, z) ≥ 0 and b = (b, g, h) ≥ 0. We can write ˜ ˜ A⊗ x = b as: (A, M, N)⊗ (x, y, z) = (b, g, h), (Ax, Ay+ Mx, Az+ Nx) = (b, g, h) (Using Definition 3.6) or Ax = b, Ay+ Mx = g, (Using Definition 3.5) Az+ Nx = h. Now use the following steps to find the solution of FFLS: Step 1: Compute the row reduced echelon form of augmented matrices (A, b), (A, g− Mx), (A, h − Nx) by applying suitable row operations [17]. There may be following three cases, Case 1: If rank(A)  rank(A, b), then the FFLS is inconsistent and no non-negative solution exists. Case 2: If rank(A) = rank(A, b), but there exists at least one-negative element th column of the row reduced echelon form of augmented matrices in the (n + 1) (A, b)or(A, g − Mx)or(A, h − Nx), then the system of equations are inconsistent and no non-negative solution exists. th Case 3: If rank(A) = rank(A, b) and all the elements of (n+ 1) column of the row reduced echelon form of augmented matrices (A, b), (A, g− Mx), (A, h− Nx) are non- negative, then the system of equations are consistent, i.e., non-negative solution exist. There may be two following sub-cases. 68 Amit Kumar · Neetu · Abhinav Bansal (2012) Case 3.a: If rank(A) = rank(A, b)  n, then the FFLS has infinite number of non- negative solutions. Case 3.b: If rank(A) = rank(A, b) = n, then the FFLS has a unique and non-negative solution. Step 2: Compute the values of x, y and z using the row reduced echelon form of i i i augmented matrices (A, b), (A, g − Mx), (A, h − Nx), respectively. The solution of FFLS will be represented by x ˜ = (x, y, z ), ∀i = 1, 2,··· , n. i i i i Example 4.1 Consider the following FFLS and solve it by the proposed method ˜ ˜ ˜ 5⊗ x ˜ ⊕ 6⊗ x ˜ = 50, 1 2 ˜ ˜ ˜ 7⊗ x ˜ ⊕ 4⊗ x ˜ = 48, 1 2 i.e., (5, 1, 1)⊗ (x , y , z )⊕ (6, 1, 2)⊗ (x , y , z ) = (50, 10, 17), 1 1 1 2 2 2 (7, 1, 0)⊗ (x , y , z )⊕ (4, 0, 1)⊗ (x , y , z ) = (48, 5, 7). 1 1 1 2 2 2 Solution: The given FFLS may be written as: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ (5, 1, 1) (6, 1, 2)⎟ ⎜ (x , y , z )⎟ ⎜ (50, 10, 17)⎟ 1 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = , ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (x , y , z ) (7, 1, 0) (4, 0, 1) (48, 5, 7) 2 2 2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 56⎟ ⎜ 11⎟ ⎜ 12⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A= ⎜ ⎟ , M= ⎜ ⎟ , N= ⎜ ⎟ , ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 74 14 10 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 50⎟ ⎜ 10⎟ ⎜ 17⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ b= ⎜ ⎟ , g= ⎜ ⎟ , h= ⎜ ⎟ . ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 48 5 7 The augmented matrix ⎛ ⎞ ⎜ ⎟ ⎜ 5650⎟ ⎜ ⎟ ⎜ ⎟ (A, b)= ⎜ ⎟ . ⎝ ⎠ The row reduced echelon form of this matrix is obtained as follows: Apply the row operation R → R to get: 1 1 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1 10 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 5 . ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ −5 Again we apply row operations in sequence: R → R − 7R , R → R , R → 2 2 1 2 2 1 R − R and we get: 1 2 ⎛ ⎞ ⎜ ⎟ ⎜ 104⎟ ⎜ ⎟ ⎜ ⎟ (A, b)= ⎜ ⎟ . ⎝ ⎠ Using the row reduced echelon form of the augmented matrix (A,b), we obtain a solution x = 4, x = 5. 1 2 Fuzzy Inf. Eng. (2012) 1: 63-73 69 Similarly, the row reduced echelon form of the augmented matrix ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (A, g− Mx)= ⎜ ⎟ ⎝ ⎠ is ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 10 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 11 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 1 1 which gives y = , y = . 1 2 11 11 Similarly, the row reduced echelon form of the augmented matrix ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (A, h− Nx)= ⎜ ⎟ ⎝ ⎠ is ⎛ ⎞ ⎜ 10 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎝ ⎠ which gives z = 0, z = . 1 2 th rd Since Rank(A) = Rank(A, b) = 2 and all the elements of of (n + 1) , i.e., 3 columns of the row reduced echelon form of augmented matrices (A, b), (A, g− Mx), (A, h− Nx) are non-negative, therefore, the given FFLS has the following unique and non-negative solution. Substituting appropriate values in x ˜ = (x, y, z )(∀i = 1, 2), we get i i i i x ˜ = (4, , 0), 1 1 x ˜ = (5, , ). 11 2 As we see, we obtain the same solution as the one with the direct method given in [8]. Example 4.2 Consider the following FFLS and solve it by the proposed method ˜ ˜ ˜ 2⊗ x ˜ ⊕ 4⊗ x ˜ = 10, 1 2 ˜ ˜ ˜ 4⊗ x ˜ ⊕ 2⊗ x ˜ = 20, 1 2 i.e., (2, 1, 1)⊗ (x , y , z )⊕ (4, 1, 1)⊗ (x , y , z ) = (10, 5, 5), 1 1 1 2 2 2 (4, 2, 2)⊗ (x , y , z )⊕ (8, 2, 2)⊗ (x , y , z ) = (20, 10, 10). 1 1 1 2 2 2 Solution: The given FFLS may be written as: 70 Amit Kumar · Neetu · Abhinav Bansal (2012) ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ (2, 1, 1) (4, 1, 1)⎟ ⎜ (x , y , z )⎟ ⎜ (10, 5, 5 ⎟ 1 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ , ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (4, 2, 2) (8, 2, 2) (x , y , z ) (20, 10, 10) 2 2 2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 24 11 11 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A= ⎜ ⎟ , M= ⎜ ⎟ , N= ⎜ ⎟ , ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 48 22 12 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 10⎟ ⎜ 5 ⎟ ⎜ 5 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ b= , g= , h= . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 20 10 10 The augmented matrix ⎛ ⎞ ⎜ ⎟ ⎜ 2410⎟ ⎜ ⎟ ⎜ ⎟ (A, b)= ⎜ ⎟ . ⎝ ⎠ The row reduced echelon form of this matrix is obtained as follows: Apply the row operation R → R to get: 1 1 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ . ⎝ ⎠ Hence the solution is x = u, x = 5− 2u. 2 1 Now compute ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 5 ⎟ ⎜ 11⎟ ⎜ 5− 2u⎟ ⎜ u ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ g− Mx = − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ . ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 10 22 u 2u ⎛ ⎞ ⎜ ⎟ ⎜ 24 u ⎟ ⎜ ⎟ ⎜ ⎟ Hence the augmented matrix (A, g− Mx)= ⎜ ⎟ ⎝ ⎠ 482u ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 12 ⎟ ⎜ ⎟ ⎜ ⎟ is = ⎜ ⎟ . ⎜ ⎟ 00 0 Using the above row reduced echelon form, the solution is y = − 2v, y = v. 1 2 Now compute ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 5 ⎟ ⎜ 11⎟ ⎜ 5− 2u⎟ ⎜ u ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ h− Nx = − = . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 10 22 u 2u ⎛ ⎞ ⎜ ⎟ ⎜ 24 u ⎟ ⎜ ⎟ ⎜ ⎟ Hence the augmented matrix (A, h− Nx)= . ⎜ ⎟ ⎝ ⎠ 482u ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ The row reduced echelon form of augmented matrix = 2 . ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 00 0 Using the above row reduced echelon form, we get the solution y = − 2v, y = v. 1 2 2 Fuzzy Inf. Eng. (2012) 1: 63-73 71 Since rank(A) = rank(A, b) = 1  n(= 2), so the non-negative solution of the FFLS th rd will exist if all the elements of (n+ 1) , i.e., 3 columns of the row reduced echelon form of augmented matrices (A, b), (A, g− Mx), (A, h− Nx) are non-negative. For this particular problem, we chose u ∈ [0, 2], v = 0 and w = 0. Thus the FFLS will have infinite number of non-negative solutions if u ∈ [0, 2], v = 0 and w = 0, else the FFLS will be inconsistent, i.e., no non-negative solution will exist. Thus the solution of this FFLS can be written as follows: u u x ˜ = (5− 2u, − 2v, − 2w), 2 2 x ˜ = (u, v, w); u ∈ [0, 2], v = 0, w = 0. Some particular solutions of this FFLS are: Solution 1: x ˜ = (1, 1, 1), x ˜ = (2, 0, 0). 1 2 Solution 2: x ˜ = (5, 0, 0), x ˜ = (0, 0, 0). 1 2 Solution 3: x ˜ = (3, 1/2, 1/2), x ˜ = (1, 0, 0). 1 2 u u General solution: x ˜ = (5− 2u, , ), x ˜ = (u, 0, 0); u ∈ [0, 2]. 1 2 2 2 Example 4.3 Consider the following FFLS and solve it by the proposed method ˜ ˜ ˜ 2⊗ x ˜ ⊕ 4⊗ x ˜ = 10, 1 2 ˜ ˜ ˜ 4⊗ x ˜ ⊕ 8⊗ x ˜ = 24, 1 2 i.e., (2, 1, 1)⊗ (x , y , z )⊕ (4, 1, 1)⊗ (x , y , z ) = (10, 5, 5), 1 1 1 2 2 2 (4, 2, 2)⊗ (x , y , z )⊕ (8, 2, 2)⊗ (x , y , z ) = (24, 12, 12). 1 1 1 2 2 2 Solution: The given FFLS may be written as: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ (2, 1, 1) (4, 1, 1)⎟ ⎜ (x , y , z )⎟ ⎜ (10, 5, 5 ⎟ 1 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ , ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (4, 2, 2) (8, 2, 2) (x , y , z ) (24, 12, 12) 2 2 2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 24 11 11 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A= ⎜ ⎟ , M= ⎜ ⎟ , N= ⎜ ⎟ , ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 48 22 12 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 10 5 5 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ b= ⎜ ⎟ , g= ⎜ ⎟ , h= ⎜ ⎟ . ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 24 12 12 The augmented matrix ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (A, b)= ⎜ ⎟ . ⎝ ⎠ The row reduced echelon form of this matrix is obtained as follows: ⎛ ⎞ ⎜ ⎟ 1 ⎜ 120⎟ ⎜ ⎟ ⎜ ⎟ Apply the row operation R → R to get ⎜ ⎟ , 1 1 ⎝ ⎠ rank(A) = 1, rank(A, b) = 2. Hence rank(A)  rank(A, b). Therefore the given FFLS is inconsistent i.e., it has no solution. 72 Amit Kumar · Neetu · Abhinav Bansal (2012) 5. Conclusion In this paper, the fully fuzzy linear systems, i.e., fuzzy linear systems with fuzzy coefficients involving fuzzy variables, are investigated and a new method is applied for solving these systems. Row reduced echelon form of matrices is used to construct a new method for solving fully fuzzy linear systems and the validity of the proposed method is examined with numerical examples. The result is obtained easily and is same as compared to the solution described in [8]. The constructed method is efficient in determination of consistency of fully fuzzy linear systems occurring in real life situations and computation of FFLS that may lead to infinite and non negative fuzzy solutions. Acknowledgments The authors would like to thank the Editor-in-Chief and anonymous referees for the various suggestions which have led to an improvement in both the quality and clarity of the paper. References 1. Abbasbandy S, Ezzati R, Jafarian A (2006) LU decomposition method for solving fuzzy system of linear equations. Applied Mathematics and Computation 172: 633-643 2. Abbasbandy S, Jafarian A (2006) Steepest descent method for system of fuzzy linear equations. Applied Mathematics and Computation 175: 823-833 3. Abbasbandy S, Jafarian A, Ezzati R (2005) Conjugate gradient method for fuzzy symmetric positive- definite system of linear equations. Applied Mathematics and Computation 171: 1184-1191 4. Allahviranloo T (2004) Numerical methods for fuzzy system of linear equations. Applied Mathemat- ics and Computation 155: 493-502 5. Allahviranloo T (2004) Successive over relaxation iterative method for fuzzy system of linear equa- tions. Applied Mathematics and Computation 162: 189-196 6. Allahviranloo T (2005) The adomian decomposition method for fuzzy system of linear equations. Applied Mathematics and Computation 163: 553-563 7. Buckley J J, Qu Y (1991) Solving systems of linear fuzzy equations. Fuzzy Sets and Systems 43: 33-43 8. Dehghan M, Hashemi B, Ghatee M (2006) Computational methods for solving fully fuzzy linear systems. Applied Mathematics and Computation 179: 328-343 9. Dehghan M, Hashemi B (2006) Solution of the fully fuzzy linear system using the decomposition procedure. Applied Mathematics and Computation 182: 1568-1580 10. Dehghan M, Hashemi B, Ghatee M (2007) Solution of the fully fuzzy linear system using iterative techniques. Chaos, Solitons and Fractals, 34 : 316-336 11. Dubois D, Prade H (1980) Fuzzy sets and systems: Theory and applications. Academic Press, New York 12. Dubois D, Prade H (1980) Systems of linear fuzzy constraints. Fuzzy Sets and Systems 3: 37-48 13. Friedman M, Ming M, Kandel A (1998) Fuzzy linear systems. Fuzzy Sets and Systems 96: 201-209 14. Friedman M, Ming M, Kandel A (2000) Duality in fuzzy linear systems. Fuzzy Sets and Systems 109: 55-58 15. Gang L H (2010) Generalized fuzzy random set-valued mixed variational inclusions involving ran- dom nonlinear (A ,η )-accretive mappings in Banach spaces. J. Nonlinear Sci. Appl. 3: 63-77 ω ω 16. Kauffmann A, Gupta M M (1991) Introduction to fuzzy arithmetic: Theory and applications. Van Nostrand Reinhold, New York 17. Lipschutz S (2005) Shaum’s outline of theory and problems of linear algebra. 3rd edition, McGraw Hill Book Company, New York Fuzzy Inf. Eng. (2012) 1: 63-73 73 18. Mosleh M, Abbasbandy S, Otadi (2007) Full fuzzy linear systems of the form Ax + b = Cx + d. Proceedings First Joint Congress on Fuzzy and Intelligent Systems 19. Muzzioli S, Reynaerts H (2006) Fuzzy linear systems of the form A x + b = A x + b . Fuzzy Sets 1 1 2 2 and Systems 157: 939-951 20. Nasseri S H, Sohrabi M, Ardil E (2008) Solving fully fuzzy linear systems by use of a certain decom- position of the coefficient matrix. International Journal of Computational and Mathematical Sciences 2: 140-142 21. Nasseri S H, Zahmatkesh F (2010) Huang method for solving fully fuzzy linear system of equations. The Journal of Mathematics and Computer Science 1: 1-5 22. Sun X, Guo S (2009) Solution to general fuzzy linear system and its necessary and sufficient condi- tion. Fuzzy Information and Engineering 3: 317-327 23. Yin J F, Wang K (2009) Splitting iterative methods for fuzzy system of linear equations. Computa- tional Mathematics and Modeling 20: 326-335 24. Zadeh L A (1965) Fuzzy sets. Information and Control 8: 338-353 25. Zhao R, Govind R (1991) Solution of algebraic equations involving generalized fuzzy numbers. In- formation Sciences 56: 199-243

Journal

Fuzzy Information and Engineering – Taylor & Francis

Published: Mar 1, 2012

Keywords: Fully fuzzy linear systems; Row reduced echelon form; Triangular fuzzy numbers

There are no references for this article.