Dual Fuzzy Linear Systems Research Articles

The general algebraic solution of dual fuzzy linear systems and fuzzy Stein matrix equations

In this paper, we present a new method for solving a dual fuzzy linear system (DFLS), AX˜+C˜=BX˜+D˜, where the coefficient matrices A and B are arbitrary real m×n matrices and C˜ and D˜ are given fuzzy number vectors. A necessary and sufficient condition for the R-consistency of the associated system of linear equations is obtained. The straightforward method for solving m×n DFLS based on an arbitrary {1}-inverse of A−B is introduced. Also, as an application, we present the first algorithm for solving the fuzzy Stein matrix equations, based on {1}-inverses. Finally, these results are illustrated by examples.

A straightforward approach for solving dual fuzzy linear systems

The main purpose of this paper is to solve a dual fuzzy linear system algebraically. In considered systems, the coefficient matrices are crisp-valued matrices and the left and right hand sides vectors are fuzzy number-valued vectors. Two types of solutions are defined and the relationship between them is investigated. Finally, based on the obtained results, a simple method is presented to obtain a unique algebraic solution of a dual fuzzy linear system. The main advantage of the proposed method over existing methods is that it does not need to convert a dual fuzzy linear system to two crisp linear systems. Also, a new necessary and sufficient condition for the existence of unique algebraic solution of a dual fuzzy linear system is presented. To illustrate our method, two numerical examples and an applied example in economy are given.

Solving two-dimensional fuzzy Fredholm integral equations via sinc collocation method

In this paper, we present a numerical method to solve two-dimensional fuzzy Fredholm integral equations (2D-FFIE) by combing the sinc method with double exponential (DE) transformation. Using this method, the fuzzy Fredholm integral equations are converted into dual fuzzy linear systems. Convergence analysis is performed in terms of the modulus of continuity. Numerical experiments demonstrate the efficiency of the proposed method.

On the solution of fuzzy dual linear systems of equation

In this paper the exact, multiple and approximation solutions of Dual fuzzy linear systems of equations(DFLSE) with triangular variable are investigated based on a 1-level expansion. To this end, 1-level of DFLSE are solved for calculating the cores of fuzzy solution and then its spreads are obtained by solving an optimization problem with a special objective function. Finally, the existence of solution of DFLSE is proved in details and some numerical examples are solved to illustrate the accuracy and capability of the method

Solving fuzzy Fredholm linear integral equations using Sinc method and double exponential transformation

In this paper, numerical solution of fuzzy Fredholm linear integral equations is considered by applying Sinc method based on double exponential transformation with dual fuzzy linear systems. For this purpose, we convert the given fuzzy integral equation to a fuzzy linear system of equation. In this case, the Sinc collocation method with double exponential transformation is used. Numerical examples are provided to verify the validity of the proposed algorithm.

Approximate solution of dual fuzzy matrix equations

In this paper, we propose a simple and practical method to solve the dual fuzzy matrix equation Ax̃+B∼=Cx̃+D∼, in which A,C are m×n matrices and B∼,D∼ are m×p LR fuzzy numbers matrices. By means of the arithmetic operations on LR fuzzy numbers space, the dual fuzzy matrix equation could be converted into two classical matrix equations, and the LR minimal fuzzy solution and the strong (weak) LR minimal fuzzy solutions of the dual fuzzy matrix equation are derived by solving two classical matrix equations based on the generalized inverses of matrices. Meanwhile, as a special case of the dual fuzzy matrix equation, the fuzzy solutions of the dual fuzzy linear systems are investigated. Finally, some numerical examples are given to illustrate the effectiveness of the proposed method.

Exact and approximate solutions of fuzzy LR linear systems: New algorithms using a least squares model and the ABS approach

We present a methodology for characterization and an approach for computing the solutions of fuzzy linear systems with LR fuzzy variables. As solutions, notions of exact and approximate solutions are considered. We transform the fuzzy linear system into a corresponding linear crisp system and a constrained least squares problem. If the corresponding crisp system is incompatible, then the fuzzy LR system lacks exact solutions. We show that the fuzzy LR system has an exact solution if and only if the corresponding crisp system is compatible (has a solution) and the solution of the corresponding least squares problem is equal to zero. In this case, the exact solution is determined by the solutions of the two corresponding problems. On the other hand, if the corresponding crisp system is compatible and the optimal value of the corresponding constrained least squares problem is nonzero, then we characterize approximate solutions of the fuzzy system by solution of the least squares problem. Also, we characterize solutions by defining an appropriate membership function so that an exact solution is a fuzzy LR vector having the membership function value equal to one and, when an exact solution does not exist, an approximate solution is a fuzzy LR vector with a maximal membership function value. We propose a class of algorithms based on ABS algorithm for solving the LR fuzzy systems. The proposed algorithms can also be used to solve the extended dual fuzzy linear systems. Finally, we show that, when the system has more than one solution, the proposed algorithms are flexible enough to compute special solutions of interest. Several examples are worked out to demonstrate the various possible scenarios for the solutions of fuzzy LR linear systems.

Solution to General Fuzzy Linear System and Its Necessary and Sufficient Condition

This paper investigates general fuzzy linear systems of the form Ax = y and general dual fuzzy linear systems of the form Ax + y = Bx + z with A, B matrices of crisp coefficients and x, y fuzzy number vectors. The aim of this paper is twofold. First, by the unique least Euclidean norm solution we solve the systems with non-full rank matrices A, B. Second, we give the new necessary and sufficient condition for a strong fuzzy solution existence. Moreover, some numerical examples are designed.

Minimal solution of general dual fuzzy linear systems

Fuzzy linear systems of equations, play a major role in several applications in various area such as engineering, physics and economics. In this paper, we investigate the existence of a minimal solution of general dual fuzzy linear equation systems. Two necessary and sufficient conditions for the minimal solution existence are given. Also, some examples in engineering and economic are considered.

Duality in fuzzy linear systems

According to fuzzy arithmetic, dual fuzzy linear system can not be replaced by a fuzzy linear system. In this paper we investigate the existence of a solution of duality fuzzy linear equation systems. Two necessary and sufficient conditions for the solution existence are given.