Elsevier

Fuzzy Sets and Systems

Volume 324, 1 October 2017, Pages 74-90
Fuzzy Sets and Systems

Aggregation operators on type-2 fuzzy sets

https://doi.org/10.1016/j.fss.2017.03.015Get rights and content

Abstract

Cubillo et al. in 2015 established the axioms that an operation must fulfill to be an aggregation operator on a bounded poset (partially ordered set), in particular on M (set of fuzzy membership degrees of T2FSs, which are the functions from [0,1] to [0,1]). Previously, Takáč in 2014 had applied Zadeh's extension principle to obtain a set of operators on M which are, under some conditions, aggregation operators on L*, the set of strongly normal and convex functions of M. In this paper, we introduce a more general set of operators on M than were given by Takáč, and we study, among other properties, the conditions required to satisfy the axioms of the aggregation operator on L (set of normal and convex functions on M), which includes the set L*.

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 [0,1], the membership degree of an element in a T2FS is a fuzzy set in [0,1], that is, a T2FS is determined by a membership function μ:XM, where M=[0,1][0,1] is the set of functions from [0,1] to [0,1]. 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 ϕ:[0,1]n[0,1] be a surjective n-ary operator on [0,1], and let :[0,1]n[0,1] be an n-ary operator on [0,1]. We define the n-ary operator on M,ϕ:MnM, as,ϕ(f1,...,fn)(x)=sup{(f1(y1),...,fn(yn)):ϕ(y1,...,yn)=x}, where x,y1,...,yn[0,1] and f1,...,fnM.

Note that if (f1(y1),...,fn(yn))=f1(y1)f2(y2)...fn

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).

References (36)

  • Z. Takáč

    Aggregation of fuzzy truth values

    Inf. Sci.

    (2014)
  • C. Walker et al.

    The algebra of fuzzy truth values

    Fuzzy Sets Syst.

    (2005)
  • C. Wang

    Notes on aggregation of fuzzy truth values

    Inf. Sci.

    (2015)
  • R. Yager

    Aggregation operators and fuzzy systems modeling

    Fuzzy Sets Syst.

    (1994)
  • L. Zadeh

    The concept of a linguistic variable and its application to approximate reasoning-I

    Inf. Sci.

    (1975)
  • S. Zhou et al.

    Type-1 OWA operators for aggregating uncertain information with uncertain weights induced by type-2 linguistic quantifiers

    Fuzzy Sets Syst.

    (2008)
  • H. Bustince et al.

    Generation of interval-valued fuzzy and Atanassov's intuitionistic fuzzy connectives from fuzzy connectives and from Kα operators. Laws for conjunctions and disjunctions. Amplitude

    Int. J. Intell. Syst.

    (2008)
  • H. Bustince et al.

    Interval type-2 fuzzy sets are generalization of interval-valued fuzzy sets: towards a wider view on their relationship

    IEEE Trans. Fuzzy Syst.

    (2014)
  • Cited by (35)

    • Pre-(quasi-)overlap functions on bounded posets

      2022, Fuzzy Sets and Systems
      Citation Excerpt :

      In recent years, the concept of aggregation functions has been broadened by replacing both the unit interval and monotonicity with other options. For one thing, the unit interval can be extended to more general structures such as partially ordered sets (posets, for short) [7,44,50], bounded lattices [22,24,46] and complete lattices [31,37], including some special classes of aggregation functions such as t-norms, semi-t-operators and overlap functions. With boundary conditions and directional monotonicity, definition of pre-aggregation was proposed by Lucca et al. [27].

    • Automorphisms on normal and convex fuzzy truth values revisited

      2022, Fuzzy Sets and Systems
      Citation Excerpt :

      First, membership degrees represent linguistic labels of the “TRUTH” variable so it is not uncommon to require them to be convex and normal. Furthermore, it has been pointed out that this set L contains a bounded and complete lattice structure, and as consequence t-norms, t-conorms, aggregation operators and negations can be properly constructed ([10], [9], [16], [15]). The set of all normal and convex functions of M will be denoted by L.

    • Trapezoidal type-2 fuzzy inference system with tensor unfolding structure learning method

      2022, Neurocomputing
      Citation Excerpt :

      Results show that the TRS of a continuous interval type-2 fuzzy set is a continuous horizontal straight line, and the TRS of a generalised type-2 fuzzy set is a continuous, convex curve. From the point view of the classical set theory, type-2 fuzzy set is constructed on partially ordered set; thus, the operation or aggregation operator on the poset should obey the fundamental rule of axioms of partially ordered set [8]. A low complexity incremental type-2 meta-cognitive extreme learning structure is used in nonlinear system modeling and identification [9], and type-2 fuzzy-based extreme learning structure is also used for electricity load demand forecasting [10].

    • Distributivity between extended nullnorms and uninorms on fuzzy truth values

      2020, International Journal of Approximate Reasoning
    View all citing articles on Scopus
    View full text