A new methodology for solving fuzzy systems of equations: Thick fuzzy sets based approach

This paper presents a new method for solving fuzzy systems of equations (SoEs) where their parameters are represented by fuzzy intervals (FIs). A FI is a normal and convex fuzzy set (FS), where all its α-cuts are crisp intervals (CIs), i.e., conventional intervals. Due to the presence of uncertainty in the left-hand and right-hand sides of these fuzzy SoEs, the solutions are sought neither as FSs, nor as FIs or fuzzy boxes (FBs)—i.e., a Cartesian product of n FIs, but as uncertain FS. In this framework, an uncertain FS is regarded as a thick fuzzy set (TFS). A TFS is a new concept that is based on the joint use of thick sets (TSs) and the α-cuts principle. Therefore, a TS is an uncertain set and is represented by a pair of crisp sets (CSs), which describe its upper and lower bounds, i.e., a TS is an interval of CSs. Moreover, as a FS can be characterized by a family of nested CSs, a TFS can be represented by a family of nested TSs. Furthermore, a TFS can be regarded as an interval with FS boundaries. Nevertheless, in absence of uncertainty in the left-hand side of the fuzzy SoEs, the TFS solution becomes a FS solution. The proposed method is based on a set membership methodology according to paving and set projection techniques. The originality of the proposed approach resides in the fact that it applies whatever the form of the fuzzy system of equations (linear or nonlinear) and allows overcoming the approximation assumption of FS solutions by FIs (or FBs), often supposed in solving fuzzy SoEs. The proposed method has been validated using application examples that are issued from the literature.