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Fuzzification of vector-valued functions
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DOI：10.1016/j.fss.2021.05.011.png)
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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 alpha-level sets. Roughly speaking, each fuzzy set can be decomposed into its alpha-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.(c) 2021 Elsevier B.V. All rights reserved.
Keyword:
Decomposition theorem
Extension principle
Fuzzy functions
Fuzzy intervals
Fuzzification
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