Susana Montes

Transitivity Bounds in Additive Fuzzy Preference Structures

Transitivity plays a crucial role in preference modeling and related fields. In this paper, we discuss this property in the general context of additive fuzzy preference structures. Of particular interest is the decomposition of a large preference relation R in its symmetric part I (indifference relation) and its asymmetric part P (strict preference relation) by means of a so-called (indifference) generator i. Given the type of transitivity of a large preference relation R (w.r.t. a conjunctor) and a generator, we establish basic lower bounds and general upper bounds on the transitivity of P and I. These bounds are due to the careful design of generic counterexamples. Moreover, we identify the situations in which these bounds are effectively reached, thereby establishing connections with interesting properties such as dominance, bisymmetry, the 1-Lipschitz property and rotation invariance

Divergence measure between fuzzy sets

In this paper we propose a way of measuring the difference between two fuzzy sets by means of a function which we will call divergence. We define this concept by means of a group of natural axioms and we study in detail the most important classes of such measures, those which have the local property.

Additive decomposition of fuzzy pre-orders

Fuzzy pre-orders (reflexive and min-transitive fuzzy relations) constitute an important class of fuzzy relations. By means of an indifference generator, a fuzzy pre-order can be decomposed additively into two parts: an indifference relation and a strict preference relation. When using a Frank t-norm as indifference generator, we fully characterize the transitivity of these parts. Only in case the minimum operator is used as generator, both parts are min-transitive. The transitivity of the indifference relation is determined by the Frank t-norms, while the transitivity of the strict preference relation is determined by transforms of the nilpotent minimum t-norm.

The necessity of the strong a-cuts of a fuzzy set

Some aspects of the relationship between Goodman and Nguyens one-point coverage interpretation of a fuzzy set and Zadehs possibilistic interpretation are discussed. As a result of this, we derive a new interpretation of the strong α-cut of a normalized fuzzy set, namely that of being the most precise set we are sure to contain an unknown parameter with probability greater than or equal to 1-α.

Divergence Measures for Intuitionistic Fuzzy Sets

Characterization of dissimilarity/divergence between intuitionistic fuzzy sets (IFSs) is important as it has applications in different areas including image segmentation and decision making. This study deals with the problem of comparison of intuitionistic fuzzy sets. An axiomatic definition of divergence measures for IFSs is presented, which are particular cases of dissimilarities between IFSs. The relationships among IF-divergences, IF-dissimilarities, and IF-distances are studied. Finally, we propose a very general framework for comparison of IFSs, where depending on the conditions imposed on a particular function, we can realize measures of distance, dissimilarity, and divergence for IFSs. Some methods for building divergence measures for IFSs are also introduced, as well as some examples of IF-divergences. In particular, we have proved some results that can be used to generate measures of divergence for fuzzy sets as well as for intuitionistic fuzzy sets.

General results on the decomposition of transitive fuzzy relations

We study the transitivity of fuzzy preference relations, often considered as a fundamental property providing coherence to a decision process. We consider the transitivity of fuzzy relations w.r.t. conjunctors, a general class of binary operations on the unit interval encompassing the class of triangular norms usually considered for this purpose. Having fixed the transitivity of a large preference relation w.r.t. such a conjunctor, we investigate the transitivity of the strict preference and indifference relations of any fuzzy preference structure generated from this large preference relation by means of an (indifference) generator. This study leads to the discovery of two families of conjunctors providing a full characterization of this transitivity. Although the expressions of these conjunctors appear to be quite cumbersome, they reduce to more readily used analytical expressions when we focus our attention on the particular case when the transitivity of the large preference relation is expressed w.r.t. one of the three basic triangular norms (the minimum, the product and the Łukasiewicz triangular norm) while at the same time the generator used for decomposing this large preference relation is also one of these triangular norms. During our discourse, we pay ample attention to the Frank family of triangular norms/copulas.

An entropy measure definition for finite interval-valued hesitant fuzzy sets

In this work, a definition of entropy is studied in an interval-valued hesitant fuzzy environment, instead of the classical fuzzy logic or the interval-valued one. As the properties of this kind of sets are more complex, the entropy is built by three different functions, where each one represents a different measure: fuzziness, lack of knowledge and hesitance. Using all, an entropy measure for interval-valued hesitant fuzzy sets is obtained, quantifying various types of uncertainty.From this definition, several results have been developed for each mapping that shapes the entropy measure in order to get such functions with ease, and as a consequence, allowing to obtain this new entropy in a simpler way.

Second order possibility measure induced by a fuzzy random variable

Random sets and fuzzy random variables are commonly used to model situations where two different types of uncertainty (imprecision/vagueness and randomness) appear simultaneously. In this context, the meaning of random sets is clear. The same does not happen for the case of fuzzy random variables. The meaning depends on the particular interpretation of fuzzy sets chosen.

Stochastic dominance with imprecise information

Stochastic dominance, which is based on the comparison of distribution functions, is one of the most popular preference measures. However, its use is limited to the case where the goal is to compare pairs of distribution functions, whereas in many cases it is interesting to compare sets of distribution functions: this may be the case for instance when the available information does not allow to fully elicitate the probability distributions of the random variables. To deal with these situations, a number of generalisations of the notion of stochastic dominance are proposed; their connection with an equivalent p-box representation of the sets of distribution functions is studied; a number of particular cases, such as sets of distributions associated to possibility measures, are investigated; and an application to the comparison of the Lorenz curves of countries within the same region is presented.

Consistent models of transitivity for reciprocal preferences on a finite ordinal scale

In this paper we consider a decision maker who shows his/her preferences for different alternatives through a finite set of ordinal values. We analyze the problem of consistency taking into account some transitivity properties within this framework. These properties are based on the very general class of conjunctors on the set of ordinal values. Each reciprocal preference relation on a finite ordinal scale has both a crisp preference and a crisp indifference relation associated to it in a natural way. Taking this into account, we have started by analyzing the problem of propagating transitivity from the preference relation on a finite ordinal scale to the crisp preference and indifference relations. After that, we carried out the analysis in the opposite direction. We provide some necessary and sufficient conditions for that propagation, and therefore, we characterize the consistent class of conjunctors in each direction.