On the weakness of linear programming to interpret the nature of solution of fully fuzzy linear system

One of the applications of linear programing is to get solutions for fully fuzzy linear system (FFLS) when the near-zero fuzzy number is considered.This usage could be applied to interpret the nature of FFLS solution according to the nature of FFLS solution in the work of Babbar et al.(Soft Comput. 17:1-12, 2012) and Kumar et al. (Advances in Fuzzy Systems 2011:1-8, 2011).This paper shows that the nature of FFLS solutions must not depend upon the nature of linear programming (LP) solutions, because LP is not enough to obtain all the exact solutions for FFLS which contradicts the claims of researchers.Counter examples are provided in order to falsify those claims.Numerically, we confirm that the nature of the possible way of solving FFLS is completely different from that of the linear system. For instance, FFLS may have two unique solutions which contradict the uniqueness that can be obtained through only one unique solution

A note on “The nearest symmetric fuzzy solution for a symmetric fuzzy linear system”

This paper provides accurate approximate solutions for the symmetric fuzzy linear systems in (Allahviranloo et al.[1])

Solution of LR-fuzzy linear system with trapezoidal fuzzy number using matrix theory

This study provides solutions to a LR-fuzzy linear system (LR-FLS) with trapezoidal fuzzy number using matrix theory. The components of the LR-FLS are represented in block matrices and vectors to produce an equivalent linear system. Then, the solution can be obtained using any classical linear system, such as an inversion matrix. In this method, fuzzy operations are not required and the solution obtained is either fuzzy or non-fuzzy exact solution. Finally, several examples are given to illustrate the ability of the proposed method

Solving Fully Fuzzy Linear System with the Necessary and Sufficient Condition to have a Positive Solution

This paper proposes new matrix methods for solving positive solutions for a positive Fully Fuzzy Linear System (FFLS). All coefficients on the right hand side are collected in one block matrix, while the entries on the left hand side are collected in one vector. Therefore, the solution can be gained with a non-fuzzy common step. The necessary theorems are derived to obtain a necessary and sufficient condition in order to obtain the solution.The solution for FFLS is obtained where the entries of coefficients are unknown. The methods and results are also capable of solving Left-Right Fuzzy Linear System (LR-FLS). To best illustrate the proposed methods, numerical examples are solved and compared to the existing methods to show the efficiency of the proposed method. New numerical examples are presented to demonstrate the contributions in this paper

Solving Fully Fuzzy Linear System with the Necessary and Sufficient Condition to have a Positive Solution

Abstract: This paper proposes new matrix methods for solving positive solutions for a positive Fully Fuzzy Linear System (FFLS). All coefficients on the right hand side are collected in one block matrix, while the entries on the left hand side are collected in one vector. Therefore, the solution can be gained with a non-fuzzy common step. The necessary theorems are derived to obtain a necessary and sufficient condition in order to obtain the solution.The solution for FFLS is obtained where the entries of coefficients are unknown. The methods and results are also capable of solving Left-Right Fuzzy Linear System (LR-FLS). To best illustrate the proposed methods, numerical examples are solved and compared to the existing methods to show the efficiency of the proposed method. New numerical examples are presented to demonstrate the contributions in this paper

An algorithm for a positive solution of arbitrary fully fuzzy linear system

This paper proposes a new computational method to obtain a positive solution for arbitrary fully fuzzy linear system (FFLS).The new method transforms the coefficients in FFLS to a one-block matrix.As a result, none of the fuzzy operations are needed.This method can provide a solution regardless of the size of a system. Some necessary theorems are proved and new numerical examples are presented to illustrate the proposed method

Row Reduced Echelon Form for Solving Fully Fuzzy System with Unknown Coefficients

This study proposes a new method for finding a feasible fuzzy solution in positive Fully Fuzzy Linear System (FFLS), where the coefficients are unknown. The fully fuzzy system is transferred to linear system in order to obtain the solution using row reduced echelon form, thereafter; the crisp solution is restricted in obtaining the positive fuzzy solution. The fuzzy solution of FFLS is included crisp intervals, to assign alternative values of unknown entries of fuzzy numbers. To illustrate the proposed method, numerical examples are solved, where the entries of coefficients are unknown in right or left hand side, to demonstrate the contributions in this study

A note on “Solving fully fuzzy linear systems by using implicit gauss–cholesky algorithm”

This paper shows that the exact solutions for fully fuzzy linear systems for all examples in [1] are nonfuzzy solutions, and that the proposed solutions in [1] do not correspond to these systems.In addition, approximate fuzzy solutions are provided for all the examples.Finally, this paper shows the efficiency of the provided solutions by using the distance metric function introduced in [13]