Strong Solutions of the Fuzzy Linear Systems
For researchers working on fuzzy linear systems, this work generalizes an existing theorem, but the improvement is incremental as it relaxes a known constraint without introducing a fundamentally new approach.
The paper addresses the restrictive condition for existence of strong solutions in fuzzy linear systems, proving a generalized theorem that applies to more general systems beyond the previously known special case. The new theorem provides necessary and sufficient conditions depending on both the coefficient matrix and the right-hand side.
We consider a fuzzy linear system with crisp coefficient matrix and with an arbitrary fuzzy number in parametric form on the right-hand side. It is known that the well-known existence and uniqueness theorem of a strong fuzzy solution is equivalent to the following: The coefficient matrix is the product of a permutation matrix and a diagonal matrix. This means that this theorem can be applicable only for a special form of linear systems, namely, only when the system consists of equations, each of which has exactly one variable. We prove an existence and uniqueness theorem, which can be use on more general systems. The necessary and sufficient conditions of the theorem are dependent on both the coefficient matrix and the right-hand side. This theorem is a generalization of the well-known existence and uniqueness theorem for the strong solution.