Examining fuzzy number approximation through a topological algebraic approach

In this paper, we discussed the fully fuzzy linear programming (FFLP) problems in which all the parameters and variables are triangular fuzzy numbers. We proposed a new approach of solving FFLP problem using the concept of nearest symmetric triangular fuzzy number approximation with preserve expected interval. Using this technique, FFLP problems model turned into two crisp linear problems: first problem is designed in which the centre objective value will be calculated. Second is designed to obtain the margin from the objective of the principal problem. Finally, the fuzzy approximate solution of the FFLP problem is obtained from the solution of two linear programming problems.

Weighted pseudometric approximation of 2-dimensional fuzzy numbers by fuzzy 2-cell prismoid numbers preserving the centroid

Fuzzy number approximation by trapezoidal fuzzy numbers which preserves the expected interval is discussed. New operators that fulfill additional requirements for the core and support of the fuzzy number are suggested. These supplementary conditions guarantee the proper interpretation of the solution even for very skew original fuzzy numbers.

Natural trapezoidal approximations of fuzzy numbers

Trapezoidal approximations of fuzzy numbers—revisited

Fuzzy number approximation via shadowed sets

Trapezoidal approximations of fuzzy numbers preserving the expected interval — Algorithms and properties

Numerous research papers and several engineering applications have proved that the fuzzy set theory is an intelligent effective tool to represent complex uncertain information. In fuzzy multi-criteria decision-making (fuzzy MCDM) methods, intelligent information system and fuzzy control-theoretic models, complex qualitative information are extracted from expert’s knowledge as linguistic variables and are modeled by linear/non-linear fuzzy numbers. In numerical computations and experiments, the information/data are fitted by nonlinear functions for better accuracy which may be little hard for further processing to apply in real-life problems. Hence, the study of non-linear fuzzy numbers through triangular and trapezoidal fuzzy numbers is very natural and various researchers have attempted to transform non-linear fuzzy numbers into piecewise linear functions of interval/triangular/trapezoidal in nature by different methods in the past years. But it is noted that the triangular/trapezoidal approximation of nonlinear fuzzy numbers has more loss of information. Therefore, there is a natural need for a better piecewise linear approximation of a given nonlinear fuzzy number without losing much information for better intelligent information modeling. On coincidence, a new notion of Generalized Hexagonal Fuzzy Number has been introduced and its applications on Multi-Criteria Decision-Making problem (MCDM) and Generalized Hexagonal Fully Fuzzy Linear System (GHXFFLS) of equations have been studied by Lakshmana et al. in 2020. Therefore, in this paper, approximation of nonlinear fuzzy numbers into the hexagonal fuzzy numbers which includes trapezoidal, triangular and interval fuzzy numbers as special cases of Hexagonal fuzzy numbers with less loss/gain of information than other existing methods is attempted. Since any fuzzy information is satisfied fully by its modal value/core of that concept, any approximation of that concept is expected to be preserved with same modal value/core. Therefore, in this paper, a stepwise procedure for approximating a non-linear fuzzy number into a new Hexagonal Fuzzy Number that preserves the core of the given fuzzy number is proposed using constrained nonlinear programming model and is illustrated numerically by considering a parabolic fuzzy number. Furthermore, the proposed method is compared for its efficiency on accuracy in terms of loss of information. Finally, some properties of the new hexagonal fuzzy approximation are studied and the applicability of the proposed method is illustrated through the Group MCDM problem using an index matrix (IM).

Fuzzy number approximation has been investigated by many researchers. It is still useful to develop new approximations in order to better fit real world problems. This paper proposes a method for symmetric trapezoidal fuzzy number approximation which preserves the expected interval. Some properties of approximation are proved and a fuzzy partition is generated by using the proposed method.

Parameterized approximation of fuzzy number with minimum variance weighting functions

Algebraic approach is widely used in the study of rough sets. In the algebraic approach, one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators. Axiomatic approaches have been used in the study of intuitionistic fuzzy rough sets. Intuitionistic fuzzy rough approximation operators are defined by axioms, and different axiom sets of lower and upper intuitionistic fuzzy rough approximation operators guarantee the existence of different types of intuitionistic fuzzy relations. However, the independence of the axioms in each of those axiom sets has not been proved. If the axioms are not independent, one of the axioms is redundant. In this paper, we prove that no axioms can be derived from other axioms, which means the axioms in the axiom set are independent. For each axiom set, we examine the independence of axioms and present the independent axiom sets characterizing intuitionistic fuzzy rough approximation operators. Thus, we improve the intuitionistic fuzzy rough set theory.

Approximation of fuzzy numbers using the convolution method

In this paper, an approximation method is given using step type fuzzy number to approximate a general fuzzy number. Say concretely, for any given general fuzzy number, a group of calculation formulas are obtained for the coordinates of the left end point of the left step and the right end point of the right step of the 2-2-step type fuzzy number that is the best approximation of the general fuzzy number, i.e., is nearest to the general fuzzy number. At last, we present specific example to show the effectiveness and usability of our calculation formulas.

Using simple fuzzy numbers to approximate general fuzzy numbers is an important research aspect of fuzzy number theory and application. The existing results in this field are basically based on the unweighted metric to establish the best approximation method for solving general fuzzy numbers. In order to obtain more objective and reasonable best approximation, in this paper, we use the weighted distance as the evaluation standard to establish a method to solve the best approximation of general fuzzy numbers. Firstly, the conceptions of I-nearest r-s piecewise linear approximation (in short, PLA) and the II-nearest r-s piecewise linear approximation (in short, PLA) are introduced for a general fuzzy number. Then, most importantly, taking weighted metric as a criterion, we obtain a group of formulas to get the I-nearest r-s PLA and the II-nearest r-s PLA. Finally, we also present specific examples to show the effectiveness and usability of the methods proposed in this paper.

The fuzzy transform setting (F-transform) is proposed as a tool for representation and approximation of type-1 and type-2 fuzzy numbers; the inverse F-transform on appropriate fuzzy partition of the membership interval [0,1] is used to characterize spaces of fuzzy numbers in such a way that arithmetic operations are defined and expressed in terms of the F-transform of the results. A type-2 fuzzy number is represented as a particular fuzzy-valued function and it is expressed in terms of a two-dimensional F-transform where the first dimension represents the universe domain and the second dimension represents the membership domain. Operators on two dimensional F-transform are then proposed to approximate arithmetic operations with type 2 fuzzy numbers.