Class Of Fuzzy Numbers Research Articles

A Few Similarity Measures on the Class of Trapezoidal-Valued Intuitionistic Fuzzy Numbers and Their Applications in Decision Analysis

Similarity measures on trapezoidal-valued intuitionistic fuzzy numbers (TrVIFNs) are functions that measure the closeness between two TrVIFNs, which has a lot of applications in the area of pattern recognition, clustering, decision-making, etc. Researchers around the world are proposing various similarity measures on the generalizations of fuzzy sets. However, many such measures do not satisfy the condition that “the similarity between two fuzzy numbers is equal to 1 implies that both the fuzzy numbers are equal” and this gives a pathway for the researchers to introduce different similarity measures on various classes of fuzzy sets. Also, all of them try to find out the similarity by using a single function, and in the present study, we try to propose a combined similarity measure principle by using four functions (four similarity measures). Thus, the main aim of this work is to introduce a few sets of similarity measures on the class of TrVIFNs and propose a combined similarity measure principle on TrVIFNs based on the proposed similarity measures. To do this, in this paper, firstly, we propose four distance-based similarity measures on TrVIFNs using score functions on TrVIFNs and study their mathematical properties by establishing various propositions, theorems, and illustrations, which is achieved by using numerical examples. Secondly, we propose the idea of a combined similarity measure principle by using the four proposed similarity measures sequentially, which is a first in the literature. Thirdly, we compare our combined similarity measure principle with a few important similarity measures introduced on various classes of fuzzy numbers, which shows the need for and efficacy of the proposed similarity measures over the existing methods. Fourthly, we discuss the trapezoidal-valued intuitionistic fuzzy TOPSIS (TrVIF-TOPSIS) method, which uses the proposed combined similarity measure principle for solving a multi-criteria decision-making (MCDM) problem. Then, we discuss the applicability of the proposed modified TrVIF-TOPSIS method by solving a model problem. Finally, we discuss the sensitivity analysis of the proposed approaches by using various cases.

Total Ordering on Generalized ‘n’ Gonal Linear Fuzzy Numbers

Zadeh introduced fuzzy sets to study imprecision in real life after which many generalizations have been developed in literature. Fuzzy numbers is the major research area of study because of its needfulness for modeling qualitative and imprecise continuous transitions. Most of the time, data involved in multi-criteria decision making (MCDM) will be in the form of fuzzy numbers due to qualitative and continuous deforming criteria. Different methods of defining total ordering on the class of fuzzy numbers have important role in MCDM to find the preference order of alternatives. Many total ordering techniques for various types of piecewise linear fuzzy numbers such as triangular (3-sided), trapezoidal (4-sided), pentagonal (5-sided), hexagonal (6-sided) and so on are available in the literature. In this paper, a generalized ‘n’gonal linear fuzzy number (n-sided) as a generalization of triangular (3-sided), trapezoidal (4-sided), pentagonal (5-sided), hexagonal (6-sided) and so on is defined and a method of defining total ordering on the class of generalized ‘n’gonal linear fuzzy numbers (n-sided) which generalizes total ordering methods defined for triangular (3-sided), trapezoidal (4-sided), pentagonal (5-sided), hexagonal (6-sided) and so on in the literature has been proposed and analyzed. Further, a similarity measure on ‘n’ gonal linear fuzzy numbers using the proposed midpoint score function is also defined and the applicability of the proposed operations, total ordering method and similarity measure on ‘n’ gonal linear fuzzy numbers in MCDM is shown by comparing with some other methods in the literature.

Utility with fuzzy numbers

In this paper, we introduce crisp utility functions with range in the class of fuzzy numbers. These utility functions can be employed to prove a generalization of the classical result of Cantor using partially ordered sets as domain of the utility functions. In particular, for every partially ordered countable set, we prove that it can be represented by an order preserving function (utility).

Fuzzy Random Variable Generation Using $\alpha$-Cuts

A method for fuzzy random variable generation based on $\boldsymbol {\alpha }$ -cuts and uniform random numbers is proposed in this article. A form of $\boldsymbol {\alpha }$ -invariance is proven for a class of fuzzy numbers (FNs) and exact equations for random variable generators of some well-known singleton-core FNs are provided. Applications to univariate/bivariate FNs, two optimization inventory models, and a discussion of the obtained results are provided.

Execution of arithmetic operations involving the second-order fuzzy numbers

The need to improve the adequacy of conventional models of the source data uncertainty in order to undertake research using fuzzy mathematics methods has led to the development of natural improvement in the analytical description of the fuzzy numbers' membership functions. Given this, in particular, in order to describe the membership functions of the three-parametric fuzzy numbers of the (L-R)-type, the modification implies the following. It is accepted that these functions' parameters (a modal value, the left and right fuzzy factors) are not set clearly by their membership functions. The numbers obtained in this way are termed the second-order fuzzy numbers (bi-fuzzy). The issue, in this case, is that there are no rules for operating on such fuzzy numbers. This paper has proposed and substantiated a system of operating rules for a widely used and effective class of fuzzy numbers of the (L-R)-type whose membership functions' parameters are not clearly defined. These rules have been built as a result of the generalization of known rules for operating on regular fuzzy numbers. We have derived analytical ratios to compute the numerical values of the membership functions of the fuzzy results from executing arithmetic operations (addition, subtraction, multiplication, division) over the second-order fuzzy numbers. It is noted that the resulting system of rules is generalized for the case when the numbers-operands' fuzziness order exceeds the second order. The examples of operations execution over the second-order fuzzy numbers of the (L-R)-type have been given.

A complete ranking of trapezoidal fuzzy numbers and its applications to multi-criteria decision making

The problem (or scenario) involving qualitative or imprecise information is not solvable by classical set theory. To overcome the shortcoming of classical set theory, Zadeh (Inf Control 8(3):338–356, 26) introduced the concept of fuzzy sets that generalizes the concept of classical sets. Fuzzy set theory allows modelling and handling of imprecise information in an effective way. As a special class of fuzzy sets, fuzzy numbers (FN) which are very much important in decision making was introduced by Dubois and Prade (Int J Syst Sci 9:631–626, 12). The available methods for solving multi-criteria decision making problems (MCDM) are problem dependent in nature due to the partial ordering on the class of FN. Total ordering on the class of FN by countable number of real-valued parameters was achieved by Wang and Wang (Fuzzy Sets Syst 243:131–141, 21). A complete ranking on the class of trapezoidal fuzzy numbers (TrFNs) using finite number of score functions is achieved in this paper. In this paper, a new ranking procedure (complete) on the class of TrFNs using the concepts of mid-point, radius, left and right fuzziness of TrFN is proposed and further we introduce a method for solving fuzzy multi-criteria decision making (Fuzzy MCDM) problem. Finally, comparisons of our proposed method with familiar existing methods are listed.

Total ordering defined on the set of all intuitionistic fuzzy numbers

L.A.Zadeh introduced the concept of fuzzy set theory as the generalization of classical set theory in 1965 and further it has been generalized to intuitionistic fuzzy sets (IFSs) by Atanassov in 1983 to model information by the membership, nonmembership and hesitancy degree more accurately than the theory of fuzzy logic. The notions of intuitionistic fuzzy numbers in different contexts were studied in literature and applied in real life applications. Problems in different fields involving qualitative, quantitative and uncertain information can be modeled better using intutionistic fuzzy numbers introduced in [17] which generalizes intuitionistic fuzzy values [1,7,17], interval valued intuitionistic fuzzy number (IVIFN) [10] than with usual IFNs [5,11,21]. Ranking of fuzzy numbers have started in early seventies in the last century and a complete ranking on the class of fuzzy numbers have achieved by W.Wang and Z.Wang only on 2014. A complete ranking on the class of IVIFNs, using axiomatic set of membership, non membership, vague and precise score functions has been introduced and studied by Geetha et al.[10]. In this paper, a total ordering on the class of IFNs [17] using double upper dense sequence in the interval [0; 1] which generalizes the total ordering on fuzzy numbers (FNs) is proposed and illustrated with examples. Examples are given to show the proposed method on this type of IFN is better than existing methods and this paper will give the better understanding over this new type of IFNs.

Note on New Solutions of LR Fuzzy Linear Systems Using Ranking Functions and ABS Algorithms

Recently, Ghanbari and Mahdavi-Amiri focused on solving LR fuzzy linear systems by use of ranking functions. They applied a ranking function introduced by Cheng, which is based on the centroid point, to illustrate their method. Also, they presented an important lemma using the centroid formulae provided by Cheng, to determine the centroid point for a class of fuzzy numbers. Unfortunately, they didn't consider that the formulae are incorrect and have led to some misapplications as pointed out by Wang and his colleagues. Therefore, in this paper, we first show that Lemma 19 of Ghanbari and Mahdavi-Amiri's paper is not true and then correct it using the centroid formulae suggested by Wang. Finally, we correct the results obtained in Ghanbari and Mahdavi-Amiri's paper for a special example.

On the Normed Space of Equivalence Classes of Fuzzy Numbers

We study the norm induced by the supremum metric on the space of fuzzy numbers. And then we propose a method for constructing a norm on the quotient space of fuzzy numbers. This norm is very natural and works well with the induced metric on the quotient space.

On The Optimal Index for Comparison fuzzy Quantities Based On Weighted Expected Interval

In the present paper, the researchers discuss the problem of weighted expected interval approximation of fuzzy numbers. This interval can be used as a crisp approximation set with respect to a fuzzy quantity. Then, by using this, the researchers propose a novel approach to ranking fuzzy numbers. The proposed model is studied for a broad class of fuzzy numbers. The calculation of this method is far simpler than the other approaches.

On an Exponential-Type Fuzzy Difference Equation

Our goal is to investigate the existence of the positive solutions, the existence of a nonnegative equilibrium, and the convergence of a positive solution to a nonnegative equilibrium of the fuzzy difference equation , , , where and the initial values belong in a class of fuzzy numbers.

A SOLUTION FOR SYSTEMS OF LINEAR FUZZY EQUATIONS IN SPITE OF THE NON-EXISTENCE OF A FIELD OF FUZZY NUMBERS

In this paper we investigate whether the fuzzy arithmetic based on Zadeh's extension principle could be improved by redefining the fuzzy addition and multiplication so that the class of fuzzy numbers combined with these operations would constitute a field. We will prove that such a fuzzy arithmetic does not exist. This has important consequences for solving systems of linear fuzzy equations. Despite the lack of inverses, we propose a method to solve approximately such systems.

Convexity and local Lipschitz continuity of fuzzy-valued mappings

Two concepts of a convexity and a local Lipschitz continuity are introduced into the class of fuzzy-valued mappings. The convexity is defined through the “fuzzy-max” order on the class of fuzzy numbers. The local Lipschitz continuity of the fuzzy-valued mapping cannot be defined in the same manner as in the case of the real-valued function, since the space of values of the fuzzy-valued mapping is not a linear space. Owing to this reason, the local Lipschitz continuity is defined in a new formula through the fuzzy-max order. A fundamental theorem concerning the local Lipschitz continuity of the fuzzy-valued convex mapping is given.

Topological properties of fuzzy numbers

Fuzzy numbers are defined and a metric is assigned to this class of fuzzy sets. Topological properties of the resulting metric space are studied. The notion of @q-crisp fuzzy numbers is introduced, and, in particular, it is shown that this class of fuzzy numbers is homeomorphic to the Hilbert cube.