János C. Fodor

Fuzzy preference modelling and multicriteria decision support

Note: Index. Bibliogr. : p. 239-251 Reference Record created on 2004-09-07, modified on 2016-08-08

Structure of uninorms

An exhaustive study of uninorm operators is established. These operators are generalizations of t-norms and t-conorms allowing the neutral element lying anywhere in the unit interval. It is shown that uninorms can be built up from t-norms and t-conorms by a construction similar to ordinal sums. De Morgan classes of uninorms are also described. Representability of uninorms is characterized and a general representation theorem is proved. Finally, pseudo-continuous uninorms are defined and completely classified.

Characterization of the ordered weighted averaging operators

This paper deals with the characterization of two classes of monotonic and neutral (MN) aggregation operators. The first class corresponds to (MN) aggregators which are stable for the same positive linear transformations and presents the ordered linkage property. The second class deals with (MN)-idempotent aggregators which are stable for positive linear transformations with the same unit, independent zeroes and ordered values. These two classes correspond to the weighted ordered averaging operator (OWA) introduced by Yager in 1988. It is also shown that the OWA aggregator can be expressed as a Choquet integral. >

Contrapositive symmetry of fuzzy implications

Abstract Contrapositive symmetry of R- and QL-implications defined from t-norms, t-conorms and strong negations is studied. For R-implications, characterizations of contrapositive symmetry are proved when the underlying t-norm satisfies a residuation condition. Contrapositive symmetrization of R-implications not having this property makes it possible to define a conjunction so that the residuation principle is preserved. Cases when this associated conjunction is a t-norm are characterized. As a consequence, a new family of t-norms (called nilpotent minimum) owing several attractive properties is discovered. Concerning QL-implications, contrapositive symmetry is characterized by solving a functional equation. When the underlying t-conorm is continuous and the t-norm is Archimedean, the t-conorm must be isomorphic to the Lukasiewicz one, while the t-norm must be isomorphic to a member from the well-known Frank family of t-norms. Finally, contrapositive symmetry for some new families of fuzzy implications is investigated.

The functional equations of Frank and Alsina for uninorms and nullnorms

The aim of this work is to study the functional equations of Frank and Alsina for two classes of commutative, associative and increasing binary operators. The first one is the class of uninorms introduced by Yager and Rybalov. The second one is the class of nullnorms arising from our study of the Frank equation for uninorms. Both classes contain t-norms and t-conorms as special cases. Moreover, the structure of the other uninorms and nullnorms is closely related to t-norms and t-conorms. These observations are the motivation for studying some generalizations of the Frank and Alsina equations. However, it is shown that all considerations lead back to the already known t-norm and t-conorm solutions. Important consequences in fuzzy preference modelling are pointed out.

On fuzzy implication operators

Abstract There exist several constructions for fuzzy implication operators via conjunctions. In this paper we present a unifying approach to the generation of implications and we prove that for a rather general class of conjuctions the generation process is closed. In addition, all well-known families of fuzzy implications are within our framework. The results support that the class of weak t-norms having the Exchange Property seems to be a good model of the conjunction (or equivalently, of intersection) operator in fuzzy set theory.

Residual operators of uninorms.

Abstract Uninorms are an important generalization of t-norms and t-conorms, having a neutral element lying anywhere in the unit interval. A uninorm shows a typical block structure and is built from a t-norm, a t-conorm and a mean operator. Two important classes of uninorms are characterized, corresponding to the use of the minimum operator (the class Umin) and maximum operator (the class Umax) as mean operator. The characterization of representable uninorms, i.e. uninorms with an additive generator, and of left-continuous and right-continuous idempotent uninorms is recalled. Two residual operators are associated with a uninorm and it is characterized when they yield an implicator and coimplicator. The block structure of the residual implicator of members of the class Umin and of the residual coimplicator of members of the class Umax is investigated. Explicit expressions for the residual implicator and residual coimplicator of representable uninorms and of certain left-continuous or right-continuous idempotent uninorms are given. Additional properties such as contrapositivity are discussed.

A new look at fuzzy connectives

Abstract A new and general approach is suggested to revise definitions and properties of logical connictives for fuzzy sets. This is based on solutions of a functional equation between implications and conjunctions reflecting the residuation principle. Using that equation, properties of R-implications are characterized by corresponding properties of conjunctions. Representation theorems for weak t-norms and strict negations are also obtained.

Fuzzy Set-Theoretic Operators and Quantifiers

This chapter summarizes main ways to extend classical set-theoretic operations (complementation, intersection, union, set-difference) and related concepts (inclusion, quantifiers) for fuzzy sets. Since these extensions are mainly pointwisely defined, we review basic results on the underlying unary or binary operations on the unit interval such as negations, t-norms, t-conorms, implications, coimplications and equivalences. Some strongly related connectives (means, OWA, weighted, and prioritized operations) are also considered, emphasizing the essential differences between these and the formerly investigated operator classes. We also show other operations which have no counterpart in the classical theory but play some important role in fuzzy sets (like symmetric sums, weak t-norms and conorms, compensatory AND).

Evolutionary optimization-based tuning of low-cost fuzzy controllers for servo systems

This paper suggests the optimal tuning of low-cost fuzzy controllers dedicated to a class of servo systems by means of three new evolutionary optimization algorithms: Gravitational Search Algorithm (GSA), Particle Swarm Optimization (PSO) algorithm and Simulated Annealing (SA) algorithm. The processes in these servo systems are characterized by second-order models with an integral component and variable parameters; therefore the objective functions in the optimization problems include the output sensitivity functions of the sensitivity models defined with respect to the parametric variations of the processes. The servo systems are controlled by Takagi-Sugeno proportional-integral-fuzzy controllers (T-S PI-FCs) that consist of two inputs, triangular input membership functions, nine rules in the rule base, the SUM and PROD operators in the inference engine, and the weighted average method in the defuzzification module. The T-S PI-FCs are implemented as low-cost fuzzy controllers because of their simple structure and of the only three tuning parameters because of mapping the parameters of the linear proportional-integral (PI) controllers onto the parameters of the fuzzy ones in terms of the modal equivalence principle and of the Extended Symmetrical Optimum method. The optimization problems are solved by GSA, PSO and SA resulting in fuzzy controllers with a reduced parametric sensitivity. The comparison of the three evolutionary algorithms is carried out in the framework of a case study focused on the optimal tuning of T-S PI-FCs meant for the position control system of a servo system laboratory equipment. Reduced process gain sensitivity is ensured.