Fuzzification of real-valued functions using the extension principle is frequently used to study engineering and economic problems when fuzzy data are taken into account. In this article, we generate a fuzzy function by fuzzifying a vector-valued function using the extension principle. On the other hand, it is also well known that the membership function of each fuzzy set can be expressed in terms of its α-level sets. Roughly speaking, each fuzzy set can be decomposed into its α-level sets. Therefore, we generate another fuzzy function by fuzzifying a vector-valued function on the basis of the decomposition form. The main focus is to establish their equivalence under some suitable conditions. Without extra conditions, we have difficulty to establish their equivalence. However, their relationship can be realized on the basis of the degree of fuzzy uncertainty. In these settings, we suggest the use of the decomposition form rather than the extension principle to fuzzify vector-valued functions.

当考虑到模糊数据时，使用扩展原理对实值函数进行模糊化经常用于研究工程和经济问题。在本文中，我们使用扩展原理通过对向量值函数进行模糊化来生成模糊函数。另一方面，众所周知，每个模糊集的隶属函数可以用它的α-水平集来表示。粗略地说，每个模糊集都可以分解成它的α级集。因此，我们在分解形式的基础上通过对向量值函数进行模糊化来生成另一个模糊函数。主要关注点是在一些合适的条件下建立它们的等价性。如果没有额外的条件，我们很难确定它们的等价性。但是，它们的关系可以根据模糊不确定程度来实现。在这些设置中，我们建议使用分解形式而不是扩展原理来模糊向量值函数。