ABSTRACT

The extension principle is always used to fuzzify the crisp functions to obtain the fuzzy functions in which the α-level sets of fuzzy sets are involved in the calculation. In this paper, we shall consider the concepts of upper and lower α-level sets of fuzzy sets. On the other hand, we shall establish the so-called dual extension principle using the concept of dual fuzzy set, which can also be used to fuzzify the crisp function into fuzzy functions. This kind of technique can also be called the dual fuzzification. The benefit of this paper is to provide another technique called dual fuzzification to fuzzify the crisp function into fuzzy functions. The relationship between the fuzzified function using the extension principle and the dual-fuzzified function using the dual extension principle will be established in this paper.

摘要

扩展原理总是用于模糊函数的模糊化，以获得模糊函数，其中模糊集的α级集合参与了计算。在本文中，我们将考虑上下α的概念级集的模糊集。另一方面，我们将使用对偶模糊集的概念建立所谓的对偶扩展原理，也可以将模糊函数模糊化为模糊函数。这种技术也可以称为双重模糊化。本文的好处是提供另一种称为双重模糊化的技术，以将清晰函数模糊化为模糊函数。本文将建立使用扩展原理的模糊函数和使用双重扩展原理的双重模糊函数之间的关系。