Witold Kosiński

On Algebraic Operations on Fuzzy Numbers

New definition of the fuzzy counterpart of real number is presented. An extra feature, called the orientation of the membership curve is introduced. It leads to a novel concept of an ordered fuzzy number, represented by the ordered pair of real continuous functions. Four algebraic operations on ordered fuzzy numbers are defined; they enable to avoid some drawbacks of the classical approach.

On Algebraic Operations on Fuzzy Reals

Fuzzy counterpart of real numbers is investigated in order to algorithmise algebraic operations on fuzzy reals. Fuzzy membership functions satisfying conditions similar to the quasi-convexity are discussed. An extra feature, called the orientation of their graph, is added to the definition. Two operations: addition and subtraction between fuzzy numbers are proposed. They are programmed and implemented in the Delphi language for selected types of membership functions.

On Defuzzyfication of Ordered Fuzzy Numbers

Ordered fuzzy number is an ordered pair of continuous real functions defined on the interval [0, 1]. Such numbers have been introduced by the author and his co-workers as an enlargement of classical fuzzy numbers by requiring a membership relation. It was done in order to define four algebraic operations between them, i.e. addition, subtraction, multiplication and division, in a way that renders them an algebra. Further, a normed topology is introduced which makes them a Banach space, and even more, a Banach algebra with unity. General form of linear functional on this space is presented which makes possible to define a large family of defuzzification methods of that class of numbers.

Defuzzification Functionals of Ordered Fuzzy Numbers

Defuzzification functionals, which play the main role when dealing with fuzzy controllers and fuzzy inference systems, for convex as well for ordered fuzzy numbers, are discussed. Three characteristic conditions for them are formulated. It is shown that most of the known defuzzification functionals satisfy them. Motivations for introducing the extended class of convex fuzzy numbers are presented, together with operations on them.

Fuzziness – Representation of Dynamic Changes by Ordered Fuzzy Numbers

In our daily life there are many cases when observations of objects in a population are fuzzy, inaccurate. Fuzzy concepts have been introduced in order to model such vague terms as observed values of some physical or economical terms. Measured physical fields or observed economical parameters may be inaccurate, noisy or difficult to measure and to observe with an appropriate precision because of technical reasons.

Fuzzy Reals with Algebraic Operations: Algorithmic Approach

Fuzzy counterpart of real numbers, called fuzzy numbers (reals), are investigated. Their membership functions satisfy conditions similar to quasi-convexity. In order to operate on them in a similar way to real numbers revised algebraic operations are introduced. At first four operations between fuzzy and real numbers are in use in a form suitable for their algorithmisations. Two operations: addition and subtraction between fuzzy numbers are proposed to omit some drawbacks of the corresponding operations originally defined by L. A. Zadeh with the help of his extension principle.

Evolutionary algorithm determining defuzzyfication operators

The space of ordered fuzzy numbers (OFN) that make possible to deal with fuzzy inputs quantitatively, exactly in the same way as with real numbers, is shortly presented. For defuzzyfication operators which play the main role in dealing with fuzzy controllers and fuzzy inference systems, an approximation formula is given and then a dedicated evolutionary algorithm is presented.

Defuzzification and implication within ordered fuzzy numbers

Ordered fuzzy numbers (OFN) invented by the first author and his two coworkers in 2002 make possible to utilize the fuzzy arithmetic and to construct the Abelian group of fuzzy numbers and then an ordered ring. This new model uses the extension of the parametric representation of convex fuzzy numbers and overcomes known drawbacks of the classical approach. Addition and multiplication by positive scalar are the same as in convex number case if numbers have the same orientation. However, subtraction, multiplication and division give quit different results. Several new defuzzification operations are introduced together with fuzzy implications which can play a new role in evolutionary fuzzy systems.

Fuzzy Calculator – Useful Tool for Programming with Fuzzy Algebra

Process of implementing operations’algorithms for ordered fuzzy numbers (OFN’s)are presented. First version of the program in the Delphi environment is created that uses algorithms dedicated to trapezoidal-type membership relations (functions). More useful implementation is a Fuzzy Calculator which allows counting with OFN’s of general type membership relations and is equipped with a graphical shell.

On lattice structure and implications on ordered fuzzy numbers

Ordered fuzzy numbers (OFN) invented by the second author and his two coworkers in 2002 make possible to utilize the fuzzy arithmetic and to construct the Abelian group of fuzzy numbers and then an ordered ring. The definition of OFN uses the extension of the parametric representation of convex fuzzy numbers. Fuzzy implication is proposed with the help of algebraic operations and a lattice structure defined on OFN.