Abstract
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).
Highlights
Most of the real-life problems are involved with complex forms of uncertain information which are continuous transitions
In this paper, it is proposed to define a better approximation of non-linear fuzzy numbers by a Hexagonal fuzzy number (HXFN) which preserves the core of the given fuzzy number and reduces the loss of information as much as possible than existing methods
In “Hexagonal approximation of fuzzy number preserving the core”, we present a procedure of new hexagonal approximation of fuzzy number and parabolic fuzzy number preserving the core with algorithms and illustrations
Summary
The interval approximation of fuzzy numbers (FNs) is initially started using Hamming distance on intervals by Stephen chanas [23] and later is developed using Euclidean metric on intervals based on the α-cuts by Grzegorzewski [33]. Abbasbandy and Amirfakhrian [1] have recommended an ideal method to enumerate the proximate approximation of a fuzzy number as a polynomial They [2] suggested a nearest trapezoidal form of a FN using pseudo metric on the set of all FNs by considering generalized LR type fuzzy number. Continuing with that, Grzegorzewski [34] has expressed the algorithms and properties for computing the proper trapezoidal approximation for a fuzzy number by preserving the expected interval. In this paper, it is proposed to define a better approximation of non-linear fuzzy numbers by a Hexagonal fuzzy number (HXFN) which preserves the core of the given fuzzy number and reduces the loss of information as much as possible than existing methods.
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