Aggregation operators on type-2 fuzzy sets
Introduction
Type-2 fuzzy sets (T2FSs) were introduced by Zadeh in 1975 [35] as an extension of type-1 fuzzy sets (FSs). Whereas for FSs the membership degree of an element in a set is determined by a value in the interval , the membership degree of an element in a T2FS is a fuzzy set in , that is, a T2FS is determined by a membership function , where is the set of functions from to . In [23], [24], Mizumoto and Tanaka gave some first properties of T2FSs. Later, Mendel and John [20] presented a new representation in order to derive formulas for union, intersection and complement of type-2 fuzzy sets without having to use Zadeh's extension principle. Finally, Walker and Walker [30] carried out an exhaustive work on the algebraic properties of the operations in the type-2 fuzzy sets. Because the membership degrees of T2FSs are fuzzy, they are better able to model uncertainty than FSs [18]. Fortunately, new methods have been introduced for the purpose of achieving a computationally efficient and viable framework for representing T2FSs, as well as the T2FLS (type-2 fuzzy logic system) inferencing processes (see, for example, [4], [5], [6], [19], [21]). Thanks to these computational simplifications, the first applications of generalized T2FSs and not just interval type-2 fuzzy sets (IT2FSs), which is a subset of T2FSs, are now being reported, such as, for example, [3], [17], [25], [28].
As it will be pointed out in Section 2, working on T2FSs is equivalent to working on their membership degrees, that is, on M. So, in this paper, we will get results on the set M, as well as on the subset L of normal and convex functions of M.
The theory of aggregation of real numbers is applied in FSs-based fuzzy logic systems (see, for example, [9], [22], [32]). Aggregation operators for real numbers were extended to aggregation operators for intervals (see, for example, [8]). Then, Takáč [26], [27] introduced the definition of aggregation operator on M. We reviewed these ideas in [7] and presented a more general definition of aggregation operator on bounded poset. Furthermore, Takáč applied Zadeh's extension principle ([35]) to extend type-1 aggregation operators to T2FSs. Previously, however, Zhou et al. [36] gave an approximation using the extension of the ordinary aggregation operators called OWA (ordered weighted averaging, see [33]). One of the most significant results reported by [26], [27] are the aggregation operators obtained on L*, the set of strongly normal and convex functions of M. Note that L* is a subset of L, the set of normal and convex functions of M.
The purpose of this paper is to provide in the T2FSs a wider family of aggregation operators than were presented in [26], [27], so that in each application the expert can choose the aggregation operator that best fits the specifications of the problem. So, we introduce new operators on M and determine, among other properties, the conditions under which they are aggregation operators on L. Although the target is to obtain operators of aggregation on M (the set of membership degrees of T2FSs), a first step is to obtain these operators on L, which is a subset of M having a lattice structure.
The article is organized as follows. Section 2 reviews some definitions and properties of FSs, IVFSs (interval-valued fuzzy sets) and T2FSs, and explains the background of the axioms for aggregation operators on FSs and T2FSs ([26], [27]), showing some examples of aggregation operators. Section 3 introduces a set of more general operators on M than were presented in [26], [27], analyzing whether they fulfill the axioms of aggregation operators on L. Section 4 states some conclusions.
Section snippets
Preliminaries
Throughout the paper, X will denote a non-empty set which will represent the universe of discourse. Additionally, ≤ will denote the usual order relation in the lattice of real numbers.
Some aggregation operators on L
In this section, we propose a more general n-ary operator on M than was given in [27] (see Definition 12), and study, among other properties, whether it is an aggregation operator on L.
Definition 14 Let be a surjective n-ary operator on , and let be an n-ary operator on . We define the n-ary operator on , as where and .
Note that if
Conclusions
In this study we introduced a set of more general operators on M than were given by Takáč in [26], [27]. Firstly, we determined the conditions under which they are well defined. Secondly, we focused on the requirements under which they are aggregation operators on L (or L*), which is the set of normal (strongly normal) and convex functions of M. Also, after each result, we have given examples showing that if some condition fails, the conclusion may not be true.
In future research, we plan to
Acknowledgements
This paper was partially supported by UPM (Spain) and UNET (Venezuela).
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