A note on fuzzy-transform approximation of fuzzy numbers

Coroianu, Lucian; Stefanini, Luciano

doi:10.1109/nafips-wconsc.2015.7284184

Cited by 4 publications

(1 citation statement)

References 16 publications

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“…Further research involves some theoretical aspects about several topics: arithmetic operations: using an approach similar to probabilistic arithmetic, which is based on convolutions with density functions (as in [32] and [26], [20]), we will try to express fuzzy arithmetic in terms of AC functions, as we have done for scalar multiplication and addition in subsection 2.1. This approach has been extensively addressed by developing a very e¢ cient software like the packages "distr" and "distrEx" in R language ( [23]); membership estimation through observations (see for example [15] and [7]); possible metrics on ACFs that focuse on useful topological structures (see for example [31] and [35]); ACF approximation through F-transform: the ACF-representation based on monotonic functions eases the search of approximation methods and algorithms (as in [3] and [4]);…”

Section: Conclusion and Further Researchmentioning

confidence: 99%

On possibilistic representations of fuzzy intervals

Stefanini

Guerra

2017

Information Sciences

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It is well known that a fuzzy interval has two equivalent representations given in terms of the so called Left and Right sides of the membership function (LR-representation) or in terms of the Lower and Upper branches de…ning the endpoints of the-cuts (LU-representation). In this paper we suggest an additional representation of fuzzy intervals called ACF-representation (using an average cumulative function instead of the membership function), based on possibility theory. We illustrate how to build the new representation and we state its basic properties. The main result is that the Average Cumulative (AC) function can be uniquely de…ned for any fuzzy interval and it is possible to go from one representation to the others through appropriate transformations. An interesting link can be established between ACF-representation and quantile functions, with a possible statistical interpretation useful in real application. We also recommend a parametric form of the AC function.

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Section: Conclusion and Further Researchmentioning

confidence: 99%

On possibilistic representations of fuzzy intervals

Stefanini

Guerra

2017

Information Sciences

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Triangular approximation of intuitionistic fuzzy numbers on multi-criteria decision making problem

GomathiNayagam

Murugan

2020

Soft Comput

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Hexagonal fuzzy approximation of fuzzy numbers and its applications in MCDM

Nayagam

Murugan

2021

Complex Intell. Syst.

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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).

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A note on fuzzy-transform approximation of fuzzy numbers

Cited by 4 publications

References 16 publications

On possibilistic representations of fuzzy intervals

On possibilistic representations of fuzzy intervals

Triangular approximation of intuitionistic fuzzy numbers on multi-criteria decision making problem

Hexagonal fuzzy approximation of fuzzy numbers and its applications in MCDM

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