Bonfring International Journal of Man Machine Interface, Vol. 2, No. 2, June 2012 1 ISSN 2277 - 5064 | © 2012 Bonfring Abstract--- System of simultaneous linear equations plays a vital role in mathematics, Operations Research, Statistics, Physics, Engineering and Social Sciences etc. In many applications at least some of the system’s parameters and measurements are represented by fuzzy numbers rather than crisp numbers. Therefore it is imperative to develop mathematical models and numerical procedures to solve such a fuzzy linear system. The general model of a fuzzy linear system whose coefficient matrix is crisp and the right hand side column is an arbitrary fuzzy vector. In the fully fuzzy linear system all the parameters are considered to be fuzzy numbers. Since triangular fuzzy numbers is a special case of trapezoidal fuzzy numbers, hence in this paper we considered fully fuzzy linear system with trapezoidal fuzzy numbers. LU decomposition method for a crisp matrix is well known in solving linear system of equations. We discuss LU decomposition of the coefficient matrix of the fully fuzzy linear system, in which the coefficients are trapezoidal fuzzy numbers. Keywords--- Fully Fuzzy Linear System, LU Decomposition, Trapezoidal Numbers I. INTRODUCTION HE concept of fuzzy numbers and fuzzy arithmetic operations were first introduced by Zadeh. M. Fridamn, M. Ming , M. Kandel [1] introduced a general model for solving a fuzzy n x n linear system whose coefficient matrix is crisp and the right hand side column is a fuzzy vector of positive fuzzy numbers. M. Dehghan, B. Hashemi, M. Ghatee, [3 ] are solved n x n fully fuzzy linear system using direct method, Cramer’s rule, Gauss Elimination, Doolittle & Crout factorization methods and Linear programming approach. Amit Kumar, Neetu, Abhinav Bansal [8] discussed consistency of the fully fuzzy linear system and the nature of solutions. S.H. Neseri, M. Sohrabi, E. Ardil [5] Proposed a method for solving fully fuzzy linear systems by certain decomposition (LU) of the coefficient matrix with triangular S. Radhakrishnan, Research Scholar, University of madras, Assistant Professor, Department of Mathematics, D.G .Vaishnav College, Chennai, India. E-mail:adithyakrish18@gmail.com R. Sattanathan, Associate Professor & Head, Department of Mathematics, D.G .Vaishnav College, Chennai, India. P. Gajivaradhan, Associate Professor, Department of Mathematics, Pachaiyapp’s College, Chennai, India. fuzzy numbers. M. Mosleh, M. Otadi and A. Khanmirzaie [6] introduced ST decomposition for 2x2 fully fuzzy linear systems with triangular fuzzy numbers. P. SenthilKumar, G Rajendran, [9] have solved n x n fully fuzzy linear system using Cholesky method. Amit kumar, Abhinav Bansal, Neetu [10] are bring in a new method for finding the non negative solution of fully fuzzy linear system without any restriction on the coefficient matrix. kumar, Neetu, Abhinav Bansal [11] are put in a new method for finding the non negative solution of the m x n fully fuzzy linear system without any restriction on the coefficient matrix using Linear programming problem method. Nasseri et al. [7] proposed Greville’s method to find the positive solution of fully fuzzy linear system. T. Allahyiranloo [2] proposed solution of a fuzzy linear system by iterative methods such as Jacobi and Gauss Seidel methods. A.K. Shyamal and M. Pal [4] discussed Triangular Fuzzy matrices. The structure of this paper is organized as follows In Section II, we present some basic concepts of fuzzy set theory and define a fully fuzzy linear system of equations. In Section III, A numerical method for computing the solution of Fully fuzzy linear system is discussed. Section IV deals with a Numerical example to illustrate the above method. Section V ends this paper with conclusion and References. II. PRELIMINARIES Definition 2.1: A fuzzy subset of R is defined by its membership function : R [0,1], which assigns a real number in the interval [0, 1], to each element x R, Where the value of at x shows the grade of membership of x in Definition 2.2: A fuzzy number = (m, n, , ) is said to be a trapezoidal fuzzy number if its membership function is given by (x) = Definition 2.3: A fuzzy number is called positive (negative), denoted by > 0 ( < 0), if its membership function (x) satisfies (x) = 0, x 0 ( x 0 ). LU Decomposition Method for Solving Fully Fuzzy Linear System with Trapezoidal Fuzzy Numbers S. Radhakrishnan, R. Sattanathan and P. Gajivaradhan T