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.

本文提出了一种求解模糊方程组 (SoE) 的新方法，其中它们的参数由模糊区间 (FI) 表示。FI 是一个正态凸模糊集 (FS)，其中所有的α割都是清晰区间 (CI)，即常规区间。由于这些模糊 SoE 的左侧和右侧都存在不确定性，因此既不是作为 FS，也不是作为 FI 或模糊框（FB）——即n 个FI 的笛卡尔积，而是作为不确定的FS。在这个框架中，一个不确定的FS被认为是一个厚模糊集（TFS）。TFS 是一个新概念，它基于联合使用厚集 (TS) 和α-削减原则。因此，一个TS是一个不确定的集合，由一对清晰的集合（CS）表示，描述了它的上下界，即一个TS是一个CS的区间。此外，由于 FS 可以由嵌套的 CS 族表征，因此 TFS 可以由嵌套的 TS 族表示。此外，一个 TFS 可以看作是一个具有 FS 边界的区间。然而，在模糊 SoE 左侧没有不确定性的情况下，TFS 解决方案变成了 FS 解决方案。所提出的方法基于根据铺路和集合投影技术的集合成员方法。所提出方法的独创性在于它适用于任何形式的模糊方程组（线性或非线性），并允许克服 FI（或 FB）对 FS 解的近似假设，在解决模糊的 SoE 时经常被认为是这样。所提出的方法已经使用从文献中发布的应用示例进行了验证。