Fully fuzzy linear systems of triangular fuzzy numbers (a,b,c)

The purpose of this paper is to study a nascent theory and an emerging concept of solving a fully fuzzy linear system (FFLS) with no non negative restrictions on the triangular fuzzy numbers chosen as parameters. Two new simplified computational methods are proposed to solve a FFLS without any sign restrictions. The first method eliminates the non‐negativity constraint from the coefficient matrix while the second method eliminates the constraint of non‐negativity on the solution vector. The methods are introduced with an objective to broaden the domain of fuzzy linear systems to encompass a wide range of problems occurring in reality.

The design of numerical methods is motivated by decomposing the fuzzy based linear system into its equivalent crisp linear form which can be further solved by variety of classical methods to solve a crisp linear system. Further the paper investigates Schur complement technique to solve the crisp equivalent of the FFLS.

The results that are obtained reveal interesting properties of a FFLS. By using the proposed methods, the authors are able to check the consistency of the fuzzy linear system as well as obtain the nature of obtained solutions, i.e. trivial, unique or infinite. Further it is also seen that an n×n FFLS may yield finitely many solutions which may not be entirely feasible (strong). Also the methods successfully remove the non‐negativity restriction on the coefficient matrix and the solution vector, respectively.

Evolving methods with better computational complexity and that which remove the non‐negativity restriction jointly on all the parameters are left as an open problem.

The proposed methods are new and conceptually simple to understand and apply in several scientific areas where fuzziness persists. The methods successfully remove several constraints that have been employed exhaustively by researchers and thus eventually tend to widen the breadth of applicability and usability of fuzzy linear models in real life situations. Heretofore, the usability of FFLS is largely dormant.