On Triangular Multisets and Triangular Fuzzy Multisets

Multisets alias bags are similar to fuzzy sets but essentially different in basic concepts and operations. We overview multisets together with basics of fuzzy sets in order to observe differences between the two. We then introduce fuzzy multisets and the combination of the both concepts. There is another concept of real-valued multisets as a generalization of multisets. Rough approximations of multisets and fuzzy multisets are discussed which uses a natural projection of the universal set onto the set of equivalence classes.

Type 2 fuzzy multiset is a new concept formed by integrating type 2 fuzzy set and multiset which are very useful for modelling uncertainties. The purpose of this paper is to introduce the concept of type 2 fuzzy α multicut set and to show how α- cut in secondary level helps to relate type 2 fuzzy α- multicuts and type 2 fuzzy multisets. The unique representation of type 2 fuzzy multisets using either the family of all type 2 fuzzy α multicuts or the family of all type 2 fuzzy α strong multicuts are also showed. The construction of membership grades in different levels are explained. Furthermore, we investigate the multicriteria group decision making based on type 2 fuzzy multisets. Finally, we give an example to show the validity of type 2 fuzzy multiset in group decision making problem.

Several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, Atanassov's intuitionistic fuzzy sets, type 2 fuzzy sets, and fuzzy multisets. In this paper, we propose hesitant fuzzy sets. Although from a formal point of view, they can be seen as fuzzy multisets, we will show that their interpretation differs from the two existing approaches for fuzzy multisets. Because of this, together with their definition, we also introduce some basic operations. In addition, we also study their relationship with intuitionistic fuzzy sets. We prove that the envelope of the hesitant fuzzy sets is an intuitionistic fuzzy set. We prove also that the operations we propose are consistent with the ones of intuitionistic fuzzy sets when applied to the envelope of the hesitant fuzzy sets. © 2010 Wiley Periodicals, Inc.

We introduce a new type of fuzzy set called Fuzzy Distribution Set (FDS). Fuzzy distribution sets are fuzzy sets defined on a finite domain subject to a sum of membership values equal to 1. Such fuzzy sets can serve as models of subjective probability distributions and subjective weight distributions. Considering these distributions as fuzzy sets gives a possibility to extend on such distributions the operations of fuzzy sets and, more generally, the calculus of fuzzy restrictions developed during the last decades. Recently Yager introduced the concept of negation of probability distributions. In our works, we studied several classes of such negations. Here we consider an involutive negation of probability distributions as a complement of FDS. We introduce the operations of union and intersection of fuzzy distribution sets. These basic operations on FDS can serve as a basis for the application of fuzzy logic methods to subjective probability and weight distributions. These operations can be used for the development of reasoning models with subjective probability distributions and subjective weighting functions. Weight distributions can be used in multi-criteria, multi-person, and multi-attribute decision-making models.

Ever since fuzzy set has been introduced, several extensions have been established, such as interval-valued fuzzy sets, Atanassov's intuitionistic fuzzy sets, interval-valued Atanassov's intuitionistic fuzzy sets, fuzzy multisets, hesitant fuzzy sets, interval-valued hesitant fuzzy sets, and dual hesitant fuzzy sets. In this contribution, we propose dual interval-valued hesitant fuzzy sets, which encompass fuzzy sets and its aforementioned extensions as special cases. Because of the importance of correlation measure in data analysis, we propose an approach for deriving the correlation coefficient of dual hesitant fuzzy sets, and then extend the approach to the dual interval-valued hesitant fuzzy set theory. We also put forward some formulas to create new correlation coefficients for fuzzy sets and its extensions in a general way. In addition, we give a practical example to illustrate the application of correlation coefficient for dual hesitant fuzzy sets in medical diagnosis.

Emerging trends in soft set theory and related topics.

Dual hesitant fuzzy set, first proposed by Zhu et al. (Dual hesitant fuzzy sets, J Appl Math, 1–13, 2012) as an extension of hesitant fuzzy sets, which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as a special case. Dual hesitant fuzzy sets consist of two parts, that is, the membership and nonmembership degrees, which are represented by two sets of possible values. Therefore, in accordance with the practical demand, these sets are more flexible than the existing fuzzy sets, and provide much more information about the situation. In this paper, the axiom definition of distance and similarity measures between dual hesitant fuzzy sets is introduced. Some new distance and similarity measures based on the geometric distance model, the set-theoretic approach, and the matching functions are proposed. The proposed distance measures are then applied to the multiple-attribute decision making under dual hesitant fuzzy environments. Through the distance measure between each alternative and the ideal alternative, the ranking order of all alternatives can be determined and the best alternative can be easily identified as well. Finally, a practical example of investment alternatives is given to demonstrate the effectiveness of the developed measures. The advantages of the proposed distance measure over existing measures have been presented.

Several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, Atanassov's intuitionistic fuzzy sets, type 2 fuzzy sets, and fuzzy multisets. In this pape...

Fuzzy sets play a central role in decision-making theory, modeling uncertainty by means of membership grade (MG) and non-membership grade (NMG). Non-linear Diophantine fuzzy sets (N-LDFSs), which are the natural extensions of linear Diophantine fuzzy sets and q-linear Diophantine fuzzy sets, are quite successful in modeling data thanks to their larger domains. However, in an N-LDFS, the MG, NMG and reference parameters (RPs) of an element to a set are given just by a pair of certain numbers from the closed interval [0, 1] that causes a strict modelling. Various types of interval-valued fuzzy sets, multi-fuzzy sets, or circular fuzzy sets change this strict modeling with a sensitive one. In this paper, we propose a theoretical framework for Continuous Non-Linear Diophantine Fuzzy Sets, which introduces continuous functions (CFs) over closed intervals to reduce uncertainty, which represent the MG and NMG functions supported by RPs, which enhance the sensitivity and applicability of decision-making tools. We develop continuous non-linear Diophantine fuzzy algebraic aggregation operators and apply them to a multi-attribute decision-making problem in renewable energy source selection. A case study demonstrates the effectiveness of the proposed CN-LDFS framework using a weighted geometric operator. Comparative analysis with existing methods highlights the superiority and success of our approach in improving decision-making accuracy and reliability.

Typical hesitant fuzzy set is a kind of generalization of classical fuzzy set by possessing a membership degree, called as typical hesitant fuzzy element, of finite non-empty subset of the unitary interval. This paper studies the constructive approach to rough set approximation operators in typical hesitant fuzzy background. Firstly, two novel partial orders are introduced to compare two arbitrary typical hesitant fuzzy elements and typical hesitant fuzzy sets, meanwhile, the basic operations, including intersection, union and α−level sets of typical hesitant fuzzy sets are then proposed and their properties are studied in detail. Secondly, typical hesitant fuzzy rough sets are introduced and studied via the new operations of intersection and union. Furthermore, typical hesitant fuzzy rough approximation operators are represented by the rough approximation operators of the α−level set of typical hesitant fuzzy set. Finally, the connections between typical hesitant fuzzy (and crisp, respectively) relations and typical hesitant fuzzy rough (and rough typical hesitant fuzzy, respectively) approximation operators are further established.

Fuzzy multiset is applicable as a model of information retrieval because it has the mathematical structure that expresses the number and the degree of attribution of an element simultaneously. Ther...

Multisets and Multirelations are extensions of ordinary sets and relations while fuzzy multisets and fuzzy multirelations are fuzzy extensions of these concept. Although many generalizations tend to be meaningless, these are quite useful as one can model things we see every day. In addition, these structures have found uses in the theory of computation. In particular, fuzzy multisets are used in fuzzy models of computation, that is, models of computation where vagueness is manifested by fuzzy sets play central role. These fuzzy models of computation include fuzzy P systems, the fuzzy chemical abstract machine, and fuzzy Petri nets.

The dual hesitant fuzzy sets (DHFSs) were proposed by Zhu et al. (2012), which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as special cases. Correlation measures analysis is an important research topic. In this paper, we define the correlation measures for dual hesitant fuzzy information and then discuss their properties in detail. One numerical example is provided to illustrate these correlation measures. Then we present a direct transfer algorithm with respect to the problem of complex operation of matrix synthesis when reconstructing an equivalent correlation matrix for clustering DHFSs. Furthermore, we prove that the direct transfer algorithm is equivalent to transfer closure algorithm, but its asymptotic time complexity and space complexity are superior to the latter. Another real world example, that is, diamond evaluation and classification, is employed to show the effectiveness of the association coefficient and the algorithm for clustering DHFSs.

A new method for solving dual hesitant fuzzy assignment problems with restrictions based on similarity measure

This paper first presents mathematical models for fuzzy two-term (PI/PD) controllers which employ N1 (³3) number of symmetric fuzzy sets for the input variable displacement, N2 (³3) number of symmetric fuzzy sets for the input variable velocity and N1 + N2 – 1 number of symmetric fuzzy sets for the output variable. These models are derived via triangular membership functions for fuzzification of the inputs and output variables, linear control rules, minimum/algebraic product triangular norm, different triangular conorms, different inference methods and centre of sums (COS) defuzzification method. Properties of such models are investigated. Using the well-known small-gain theorem bounded-input bounded-output (BIBO) stability analysis of feedback systems involving fuzzy PD controller as a subsystem is presented. Next, mathematical models with N1 and N2 number of asymmetric input fuzzy sets and N1 + N2 – 1 number of asymmetric output fuzzy sets (both triangular and trapezoidal) are also presented. Finally, some numerical examples along with their simulation results are included to demonstrate the effectiveness of the fuzzy two-term controllers.