Abstract
Since Lotfi A. Zadeh introduced the concept of fuzzy sets in 1965, many authors have devoted their efforts to the study of these new sets, both from a theoretical and applied point of view. Fuzzy sets were later extended in order to get more adequate and flexible models of inference processes, where uncertainty, imprecision or vagueness is present. Type 2 fuzzy sets comprise one of such extensions. In this paper, we introduce and study an extension of the fuzzy numbers (type 1), the type 2 generalized fuzzy numbers and type 2 fuzzy numbers. Moreover, we also define a partial order on these sets, which extends into these sets the usual order on real numbers, which undoubtedly becomes a new option to be taken into account in the existing total preorders for ranking interval type 2 fuzzy numbers, which are a subset of type 2 generalized fuzzy numbers.
Highlights
In the framework of fuzzy systems, in order to give a response or to make a decision, fuzzy numbers (FNs) and fuzzy quantities (FQs) need to be compared, i.e., to be ranked
Since Jain [1,2] and Dubois and Prade [3] introduced the concept of fuzzy numbers, many authors have given different methods that, they do not produce total orders, allow one to compare the FNs and to order some of their subsets
Definition 3. [5,6] A generalized fuzzy number (GFN) A is a convex fuzzy sets (FSs) on R whose membership function f A
Summary
In the framework of fuzzy systems, in order to give a response or to make a decision, fuzzy numbers (FNs) and fuzzy quantities (FQs) need to be compared, i.e., to be ranked. We define and compare type 2 fuzzy numbers (T2FNs) and type 2 generalized fuzzy numbers (T2GFNs), which are type 2 fuzzy sets (T2FSs) on the real line, with certain conditions. These sets are more general than the ones considered in [19,20,22], so that in each application, the expert can choose the linguistic labels that best fit the specifications of the problem.
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