Multi-fuzzy sets: An extension of fuzzy sets

2023

The present work is related to a certain extension of intuitionist fuzzy sets, endowed with an addition operation, a scalar product, and a preorder relation. We show that this structure is a bounded left R 0-semimodule. We also discuss some metric properties and develop a new approach to the concept of mean, the application of which is illustrated by an example.

Notes on Intuitionistic Fuzzy Sets

The basic study of fuzzy sets theory was introduced by Lotfi Zadeh in 1965. Many authors investigated possibilities how two fuzzy sets can be compared and the most common kind of measures used in the mathematical literature are dissimilarity measures. The previous approach to the dissimilarities is too restrictive, because the third axiom in the definition of dissimilarity measure assumes the inclusion relation between fuzzy sets. While there exist many pairs of fuzzy sets, which are incomparable to each other with respect to the inclusion relation. Therefore we need some new concept for measuring a difference between fuzzy sets so that it could be applied for arbitrary fuzzy sets. We focus on the special class of so called local divergences. In the next part we discuss the divergences defined on more general objects, namely intuitionistic fuzzy sets. In this case we define the local property modified to this object. We discuss also the relation of usual divergences between fuzzy se...

International Journal of Approximate Reasoning, 2018

In their original and ordinary formulation, fuzzy sets associate each element in a reference set with one number, the membership value, in the real unit interval [0, 1]. Among the various existing generalisations of the concept, we find fuzzy multisets. In this case, membership values are multisets in [0, 1] rather than single values. Mathematically, they can be also seen as a generalisation of the hesitant fuzzy sets, but in this general environment, the information about repetition is not lost, so that, the opinions given by the experts are better managed. Thus, we focus our study on fuzzy multisets and their basic operations: complement, union and intersection. Moreover, we show how the hesitant operations can be worked out from an extension of the fuzzy multiset operations and investigate the important role that the concepts of order and sorting sequences play in the basic difference between these two related approaches.

Complex & Intelligent Systems, 2022

A single-valued neutrosophic multi-set is characterized by a sequence of truth membership degrees, a sequence of indeterminacy membership degrees and a sequence of falsity membership degrees. Nature of a single-valued neutrosophic multi-set allows us to consider multiple information in the truth, indeterminacy and falsity memberships which is pretty useful in multi-criteria group decision making. In this paper, we consider sequences of intuitionistic fuzzy values instead of numbers to define the concept of intuitionistic fuzzy-valued neutrosophic multi-set. In this manner, such a set gives more powerful information. We also present some set theoretic operations and a partial order for intuitionistic fuzzy-valued neutrosophic sets and provide some algebraic operations between intuitionistic fuzzy-valued neutrosophic values. Then, we develop two types of weighted aggregation operators with the help of intuitionistic fuzzy t-norms and t-conorms. By considering some well-known additive ...

Mathematics Letters, 2021

Overtime, mathematics had been used as a tool in modeling real life phenomenon. In some cases, these problems cannot fit-into the classical deterministic or stochastic modeling techniques, perhaps due system complexity arising from lack of complete knowledge about the phenomenon or some uncertainty. The uncertainty could either be due to lack of clear boundaries in the description of the object or perhaps due to randomness. In this article, we study a mathematical tool discovered in 1965 by Zadeh suitable for modeling real life phenomenon and examined operations on such a tool. Motivated by the work of Zadeh, we studied operators on Type-1 Fuzzy Sets (T1FSs) and Type-2 Fuzzy sets (T2FSs) and provided examples, one of which is a variant of the Yager complement function for which the complement operator was graphically illustrated. The joint and the meet operators were also studied and examples provided. Non-standard operators were defined on T1FSs and T2FSs and also classified into two groups; the triangular-norm (t-norm) and triangular-conorm (t-conorm). Using tnorm and t-conorm, an example was adopted from Castillo and Aguilar to illustrate the computation of the standard operation on T2FSs. Finally, future research direction was provided based on what is yet to be achieved in fuzzy set theory.

International Journal for Research in Applied Science & Engineering Technology (IJRASET), 2023

This article proposes some important theoretic aspects of L-fuzzy power set of a finite set. Here, we take a non-empty finite set E and an ordered subset M of the closed interval   0,1 , then the set of mappings from E to M denoted by F is defined as L-fuzzy power set. Considering the disjunctive union '    ' operations between fuzzy subsets of F , the structure   , F  forms a groupoid, which is defined as special fuzzy groupoid. Later, we try to introduce the product of special fuzzy groupoids and their properties.

Lecture Notes in Electrical Engineering, 2015

The theory of soft set and multiset are vital mathematical tools used in handling uncertainties about vague concepts. In this paper we define a new operation product of two intuitionistic fuzzy soft multi setsand present the concept of relations in intuitionistic fuzzy soft multi set. The notions of null relation and absolute relation are to be defined. Also we study their properties and discuss reflexive, symmetric and transitiveintuitionistic fuzzy soft multi relations. Some new results along with illustrating examples have been put forward in our work.

Information Sciences, 2012

The main aim of this work is to present a generalization of Atanassov's operators to higher dimensions. To do so, we use the concept of fuzzy set, which can be seen as a special kind of fuzzy multiset, to define a generalization of Atanassov's operators for n-dimensional fuzzy values (called n-dimensional intervals). We prove that our generalized Atanassov's operators also generalize OWA operators of any dimension by allowing negative weights. We apply our results to a decision making problem. We also extend the notions of aggregating functions, in particular t-norms, fuzzy negations and automorphism and related notions for n-dimensional framework.

2005

In this paper we introduce aggregation operators on the lattice L * , which is the underlying lattice of both intervalvalued fuzzy sets and intuitionistic fuzzy sets in the sense of Atanassov. We consider some particular classes of binary aggregation operators based on t-norms on the unit interval. We investigate the properties of the implicators generated by these classes. In particular, representations in terms of implicators on [0, 1] are obtained, and the Smets-Magrez axioms are studied. We compare the results with those obtained for implicators generated by already known classes of t-norms on L * .

Knowledge-Based Systems, 2012

Archimedean t-conorm and t-norm are generalizations of a lot of other t-conorms and t-norms, such as Algebraic, Einstein, Hamacher and Frank t-conorms and t-norms or others, and some of them have been applied to intuitionistic fuzzy set, which contains three functions: the membership function, the nonmembership function and the hesitancy function describing uncertainty and fuzziness more objectively. Recently, Beliakov et al. [3] constructed some operations about intuitionistic fuzzy sets based on Archimedean t-conorm and t-norm, from which an aggregation principle is proposed for intuitionistic fuzzy information. In this paper, we propose some other operations on intuitionistic fuzzy sets, study their properties and relationships, and based on which, we study the properties of the aggregation principle proposed by Beliakov et al. , and give some specific intuitionistic fuzzy aggregation operators, which can be considered as the extensions of the known ones. In the end, we develop an approach for multi-criteria decision making under intuitionistic fuzzy environment, and illustrate an example to show the behavior of the proposed operators.