Arithmetic Of Fuzzy Numbers Research Articles

Notes on the compatibility between defuzzification and t-norm based fuzzy arithmetic operations

Recently, Oussalah 〔Fuzzy Sets and Systems 128(2002) 247-260〕 investigated some theoretical results about some invariance properties concerning the relationships between the defuzzification outcomes and the arithmetic of fuzzy numbers. But, in this note we introduce some explicit calculations of the resulting fuzzy set or possibility distribution when the matter is the determination of the defuzzified value pertaining to the result of some manipulation of fuzzy quantities under t-norm based fuzzy arithmetic operations.

A Fuzzy TOPSIS Method for Robot Selection

A fuzzy TOPSIS method for robot selection is proposed, where the ratings of various alternatives versus various subjective criteria and the weights of all criteria are assessed in linguistic terms represented by fuzzy numbers. The values of objective criteria are converted into dimensionless indices to ensure compatibility between the values of objective criteria and the linguistic ratings of subjective criteria. The membership function of each weighted rating is developed by interval arithmetic of fuzzy numbers. To avoid complicated aggregation of fuzzy numbers, these weighted ratings are defuzzified into crisp values by the ranking method of mean of removals. A closeness coefficient is defined to determine the ranking order of alternatives by calculating the distances to both the ideal and negative-ideal solutions. A numerical example demonstrates the computational process of the proposed method.

Selecting Plant Location via a Fuzzy TOPSIS Approach

A fuzzy TOPSIS approach for selecting plant location is proposed, where the ratings of various alternative locations under various criteria and the weights of various criteria are assessed in linguistic terms represented by fuzzy numbers. In the proposed method, the ratings and weights assigned by decision makers are averaged and normalised into a comparable scale. The membership function of each normalised weighted rating can be developed by interval arithmetic of fuzzy numbers. To avoid complicated aggregation of fuzzy numbers, these normalised weighted ratings are defuzzified into crisp values. A closeness coefficient is defined to determine the ranking order of alternative locations by calculating the distances to both the ideal and negative-ideal solutions. Using the suggested method, the decision makers' fuzzy assessments with different rating viewpoints and the trade-off among different criteria are considered in the aggregation procedure to assure more convincing decision making. A numerical example demonstrates the feasibility of the proposed method.

On the compatibility between defuzzification and fuzzy arithmetic operations

Since the introduction of the extension principle by Zadeh, the arithmetic of fuzzy numbers has gained importance both from the theoretical and the practical points of view. For the former, many works were accomplished on the topological level as well as on the parametrization level in order to improve the foundation of the theory and to simplify the performance of different combination operations. For the latter, in many practical applications, the need for a permanent switch from a fuzzy representation to a numerical representation is patent. This transformation is usually carried out by the defuzzification process. This paper addresses some theoretical results about some invariance properties concerning the relationships between the defuzzification outcomes and the arithmetic of fuzzy numbers. One of the benefits of such analysis is the fact that when the matter is the determination of the defuzzified value pertaining to the result of some manipulation of fuzzy quantities, the explicit determination of the resulting fuzzy set (or distribution) can be obviated, while the process may be restricted to a standard computation over single values corresponding to defuzzified initial inputs.

Fuzzy logic and its use in geology

The purpose of this paper is to introduce to the reader the methods of fuzzy logic, that are used in DSS (Decision Support Systems), that simplify control system learning methods for modelling the data, its classification, and the display of data. Paper shows the usefulness of fuzzy logic, basic arithmetic of fuzzy numbers and sets, highlights the differences between the »hard« and fuzzy logic. It also shows how the systems within the fuzzy logic work and how to choose the specific membership function, and why. Described methods are being used more and more in different fields of scientific research, including geology.

Weak arithmetics of fuzzy numbers

Computation with fuzzy numbers appears to be a perspective branch of fuzzy set theory namely regarding the data processing applications. A very compact survey of the actual state of art is given in Dubois and Prade (1988). This paper aims to offer, in a brief and concentrated form, a survey of some results completing the classical ones of Dubois and Prade. Most of its content summarizes the approach submitted and discussed by Mares (1994).

Geostatistics utilizing imprecise (fuzzy) information

Fuzzy set theory, specifically fuzzy number arithmetic and the extension principle, is employed to utilize imprecise data in a geostatistical analysis. This formulation is an alternative of soft kriging. Imprecise data are defined as fuzzy numbers and lead to fuzzy experimental and fuzzy theoretical variograms. Then a fuzzy geostatistical operation termed fuzzy kriging type 3, is performed and the kriged estimates are represented as fuzzy numbers. The width of the fuzzy kriged estimate is interpreted as an uncertainty measure which depends on the data and characterizes local inhomogeniety. The estimation variance is also a fuzzy number. The illustrative example refers to permeability field predictions for a soil liner, a typical case where imprecise measurement data may be available and should be utilized.

On generalized fuzzy relational equations and their applications

The paper provides an idea of generalization of fuzzy relational equations where t- and s-norms are introduced. The first part contains an extensive presentation of the resolution of fuzzy relational equations; next the solutions are specified for a list of several triangular norms. Moreover the dual equations are considered. The second part deals with the applicational aspects of these equations in systems analysis, decision-making, and arithmetic of fuzzy numbers.