Hexagonal fuzzy approximation of fuzzy numbers and its applications in MCDM

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.

A Novel Inverse Credibility Distribution Approach for the Membership Functions of LRFuzzy Intervals: A Case Study on a Completion Time Analysis

Fuzzy arithmetic is of great significance in dealing with vague information, especially the basic arithmetic operations (i.e., ⊕, ⊖, ⊗, ⊙). However, the classical and widely accepted accurate and approximate approaches, the interval arithmetic approach and standard approximation method, cannot output accurate or well-approximated expressions of the membership function, which may prevent decision makers from making the right decisions in real applications. To tackle this problem, this paper first discusses the relationships among the membership function, the credibility distribution, and the inverse credibility distribution and summarizes the relationships as several theorems. Then, by means of the theorems and the newly proposed operational law, this paper proposes an inverse credibility distribution approach that can output the accurate expression of the membership function for continuous and strictly monotone functions of regular LR fuzzy intervals. To better demonstrate the effectiveness of the raised approach, the commonly-used LR fuzzy interval, the symmetric trapezoidal fuzzy number, is employed, and several comparisons with the other two methods are made. The results show that the proposed approach can output an exact or well-approximated expression of the membership function, which the others cannot. In addition, some comparisons of the proposed approach with other methods are also made on a completion time analysis of a construction project to show the effectiveness of the proposed approach in real applications.

On the Neutrosophic, Pythagorean and Some Other Novel Fuzzy Sets Theories Used in Decision Making: Invitation to Discuss

In this short paper, a critical analysis of the Neutrosophic, Pythagorean and some other novel fuzzy sets theories foundations is provided, taking into account that they actively used for the solution of the decision-making problems. The shortcomings of these theories are exposed. It is stated that the independence hypothesis, which is a cornerstone of the Neutrosophic sets theory, is not in line with common sense and therefore leads to the paradoxical results in the asymptotic limits of this theory. It is shown that the Pythagorean sets theory possesses questionable foundations, the sense of which cannot be explained reasonably. Moreover, this theory does not completely solve the declared problem. Similarly, important methodological problems of other analyzed theories are revealed. To solve the interior problems of the Atanassov’s intuitionistic fuzzy sets and to improve upon them, this being the reason most of the criticized novel sets theories were developed, an alternative approach based on extension of the intuitionistic fuzzy sets in the framework of the Dempster–Shafer theory is proposed. No propositions concerned with the improvement of the Cubic sets theory and Single-Valued Neutrosophic Offset theory were made, as their applicability was shown to be very dubious. In order to stimulate discussion, many statements are deliberately formulated in a hardline form.