Nearest symmetric trapezoidal approximation of fuzzy numbers

Abstract Many authors analyzed triangular and trapezoidal approximation of fuzzy numbers. But, to best of our knowledge, there is no method for symmetric trapezoidal fuzzy number approximation of fuzzy numbers. So, in this paper, we try to convert any fuzzy number into symmetric trapezoidal fuzzy number by using metric distance. This approximation helps us to avoid the computational complexity in the process of decision making problems. Moreover, we investigate some reasonable properties of this approximation. An application of this new method is also provided

A new approach for trapezoidal approximation of fuzzy numbers using WABL distance

In this paper, we present a new approach to obtain trapezoidal approximation of fuzzy numbers with respect to weighted distance proposed by Nasibov [5] which the main property of this metric is flexibility in the decision maker's choice. Also, we prove some properties of the proposed method such as translation invariance, scale invariance and identity. Finally, we illustrate the efficiency of proposed method by solving some numerical examples

On improving trapezoidal and triangular approximations of fuzzy numbers

AbstractRecently, various researchers have proved that the approximations of fuzzy numbers may fail to be fuzzy numbers, such as the trapezoidal approximations of fuzzy numbers. In this paper, we show by an example that the weighted triangular approximation of fuzzy numbers, proposed by Zeng and Li, may lead to the same result. For filling the gap, improvements of trapezoidal and triangular approximations are proposed. The formulas for computing the two improved approximations are provided. Some properties of the two improved approximations are also proved

One-Factor ANOVA Model Using Trapezoidal Fuzzy Numbers Through Alpha Cut Interval Method

Most of our traditional tools in descriptive and inferential statistics is based on crispness (preciseness) of data, measurements, random variable, hypotheses, and so on.  By crisp we mean dichotomous that is, yes-or-no type rather than more-or-less type.  But there are many situations in which the above assumptions are rather non-realistic such that we need some new tools to characterize and analyze the problem.  By introducing fuzzy set theory, different branches of mathematics are recently studied.  But probability and statistics attracted more attention in this regard because of their random nature.  Mathematical statistics does not have methods to analyze the problems in which random variables are vague (fuzzy). In this regard, a simple and new technique for testing the hypotheses under the fuzzy environments is proposed.  Here, the employed data are in terms of trapezoidal fuzzy numbers (TFN) which have been transformed into interval data using  interval method and on the grounds of the transformed fuzzy data, the one-factor ANOVA test is executed and decisions are concluded.  This concept has been illustrated by giving two numerical examples. Keywords: Fuzzy set, , Trapezoidal fuzzy number (TFN), Test of hypotheses, One-factor ANOVA model, Upper level data, Lower level data

Trapezoidal operator preserving the expected interval and the support of fuzzy numbers

The problem to find the trapezoidal fuzzy number which preserves the expected interval and the support of a given fuzzy number is discussed. Properties of this new trapezoidal approximation operator are studied

Generalized Modus Ponens using Fodor’s Implication and T-norm Product with Threshold

Using Generalized Modus Ponens reasoning, we examine the values of the inferred conclusion depending on the correspondence between the premise of the rule and the observed fact. The conclusion is obtained using Fodor’s implication in order to represent a fuzzy if-then rule with a single input single output and the tnorm with threshold generated by t-norm product, as a compositional operator. A comparison study with the case when the standard t-norm product is used is made. Some comments and an example are presented in order to show how the obtained results can be used

Quantified trapezoidal fuzzy numbers

The aim of this work is to construct quantified trapezoidal fuzzy numbers as an extension of trapezoidal fuzzy numbers, by using modal intervals and accepting the possibility that the α-cuts of a trapezoidal fuzzy number may also be improper intervals. In addition, this paper addresses the inclusion relationship which is deduced from the inclusion of modal intervals and is related to the classical set-inclusion relationship between trapezoidal fuzzy numbers. Moreover, in this paper we also study the extensions of real continuous functions over the set of quantified trapezoidal fuzzy numbers. Using the semantic interpretation of the calculations over modal intervals will enable us to interpret the meaning of the calculus accurately over quantified trapezoidal fuzzy numbers. With quantified trapezoidal fuzzy numbers, we will be able to overcome some operational limitations that are usually faced when working with trapezoidal fuzzy numbers from a classical point of view. In order to show the applicability of quantified trapezoidal fuzzy numbers, we propose fuzzy equations which have no solution in the set of proper fuzzy numbers yet do have solutions that are improper fuzzy numbers. We also propose two applications of quantified trapezoidal fuzzy numbers, one of them about financial calculations and the other one in an optical problem

A Generalization of Trapezoidal Fuzzy Numbers Based on Modal Interval Theory

We propose a generalization of trapezoidal fuzzy numbers based on modal interval theory, which we name 'modal interval trapezoidal fuzzy numbers'. In this generalization, we accept that the alpha cuts associated with a trapezoidal fuzzy number can be modal intervals, also allowing that two interval modalities can be associated with a trapezoidal fuzzy number. In this context, it is difficult to maintain the traditional graphic representation of trapezoidal fuzzy numbers and we must use the interval plane in order to represent our modal interval trapezoidal fuzzy numbers graphically. Using this representation, we can correctly reflect the modality of the alpha cuts. We define some concepts from modal interval analysis and we study some of the related properties and structures, proving, among other things, that the inclusion relation provides a lattice structure on this set. We will also provide a semantic interpretation deduced from the modal interval extensions of real continuous functions and the semantic modal interval theorem. The application of modal intervals in the field of fuzzy numbers also provides a new perspective on and new applications of fuzzy numbers