Due to its reasonable properties, the Kulisch and Miranker binary relation on the family of all closed and bounded real intervals has attracted the attention of many researchers, especially in the field of Computation. However, it is not total, so there are intervals that are not comparable. To face this problem, Bustince et al. introduced the notion of admissible order , which is coherent to the Kulisch and Miranker binary relation. Due to its technical construction, most of the examples of admissible orders are defined by only employing the extremes of such intervals. In this paper we introduce a non-countable family of admissible orders in the set of all closed and bounded subintervals contained in a concrete closed and bounded real interval. The approach is novel in two senses: on the one hand, due to the mathematical objects that are involved (a dense sequence and a family of continuous functions); and, on the other hand, we do not handle the intervals through their extremes, but only by their interior points.

Random generation of fuzzy measures is an important computational task in applied problems related to fuzzy integrals such as the Choquet, Sugeno or Shilkret integrals. In general, a desirable property is the uniformity over the set of fuzzy measures. However, testing this property is not an easy task. In this paper, properties of uniform random fuzzy measures are derived. Special attention is being paid to the families of balanced fuzzy measures, belief measures and possibilities measures. Then, based of such properties, statistical tests for the uniformity of random fuzzy measures are developed. Finally, the uniformity of the most used algorithms is tested using the proposed methods.

In recent decades, the significance of multi-objective problems has grown substantially. One of the most popular methods for solving these problems involves the construction of Pareto sets . Pareto sets are exponentially sized relative to the input size of the problem, and so the need to reduce, or at least order, them arises. This work proposes the order of Pareto solutions by Borda count, an approach that makes use of ranking methods to establish preferences among the optimal solutions. To evaluate the proposed approach, a comparative study is conducted, evaluating its performance in comparison to other widely used and well-established methods within this domain. Finally, a case study with real-world data is used to show how the methodology works.

Aggregation functions are widely used to fuse information from different sources in a unique value. In many cases, the aggregated information is related to some experimental measure or random sampling of a population. In this direction, it is reasonable to consider aggregation of random elements. In this paper, the concept of aggregation functions of random elements over bounded lattices, which are measurable functions from a probability space to a bounded lattice, is presented. In particular, starting from a partially ordered set, a measurable space is constructed. Random elements are considered to be measurable functions from a probability space to the measurable space. The concept of aggregation of random elements over bounded lattices is defined by generalizing the monotonicity and the boundary conditions in terms of stochastic orders. Several types, such as the induced, random and degenerated aggregations of random elements over bounded lattices are defined and some coherence properties are studied. Particular examples regarding the aggregation of random variables, random graphs and random semi-positive matrices are provided.

In aggregation theory, there exists a large number of aggregation functions that are defined in terms of rearrangments in increasing order of the arguments. Prominent examples are the Ordered Weighted Operator and the Choquet and Sugeno integrals. Following a probability approach, ordering random variables by means of stochastic orders can be also a way to define aggregations of random variables. However, stochastic orders are not total orders, thus pairs of incomparable distributions can appear. This paper is focused on the definition of aggregations of random variables that take into account the stochastic ordination of the components of the input random vectors. Three alternatives are presented, the first one by using expected values and admissible permutations, then a modification for multivariate Gaussian random vectors and a third one that involves a transformation of the initial random vectors in new ones whose components are ordered with respect to the usual stochastic order. A deep theoretical study of the properties of all the proposals is made. A practical example regarding temperature prediction is provided

Aggregation theory is devoted to the fusing of several values into a unique output that summarizes the given information. Typically, the aggregation process is formalized in terms of an increasing mathematical function that maps the input values to the result, fulfilling some boundary conditions. However, this formalization can be too restrictive for some scenarios. In some cases, the inputs can be seen as observations of random variables, the aggregation result being also a random variable. In others, the aggregation process can be identified as a program that performs the aggregation rather than a mathematical function. In this direction, the concepts of aggregation of random variables and computable aggregation have been defined in the literature. This paper is devoted to the definition of computable aggregation of random variables, which are computer programs, not functions, that aggregate random variables, not numbers. Special attention is given to different possible alternatives to modelize random variables and monotonicity. The implementation of some examples is also provided.

Interval-valued fuzzy sets are a generalization of classical fuzzy sets where the membership values are intervals. The epistemic interpretation of interval-valued fuzzy sets assumes that there is one real-valued membership degree of an element within the membership interval of possible membership degrees. Considering this epistemic interpretation, we propose a new measure, called IV-embedding, to compare the precision of two interval-valued fuzzy sets. An axiomatic definition for this concept as well as a construction method are provided. The construction method is based on aggregation operators and the concept of interval embedding, which is also introduced and deeply studied.

Choquet integral is a widely used aggregation operator on 1-D and interval-valued information, since it is able to take into account the possible interaction among data. However, there are many cases where the information taken into account is vectorial, such as long short-term memories (LSTM). LSTM units are a kind of recurrent neural networks that have become one of the most powerful tools to deal with sequential information since they have the power of controlling the information flow. In this article, we first generalize the standard Choquet integral to admit an input composed by$n$-dimensional vectors, which produces an$n$-dimensional vector output. We study several properties and construction methods of vector Choquet integrals (VCIs). Then, we use this integral in the place of the summation operator, introducing in this way the new VCI-LSTM architecture. Finally, we use the proposed VCI-LSTM to deal with two problems: 1) sequential image classification; 2) text classification.

Starting from the notion of a multidistance, we formalize, through a suitable system of axioms, the concept of an inequality measure defined on a nonempty set with no additional structure implemented a priori. Among inequality measures, apart from multidistances we pay special attention to dispersions, and study their main features. Classical concepts will be generalized to this abstract setting. Multidistances are then revisited, and some new methods to generate them are implemented. A wide spectrum of interdisciplinary applications is outlined in the final section.

The notion of an orness measure for aggregation functions has been a relevant study subject whose history can be traced back to the early works of Dujmović in 1973. Intuitively, an orness measure quantifies the similarity of an aggregation function to the “or” function and results in an essential tool for decision engineering, field in which the choice of aggregation function is sometimes restricted to a desired value of orness (orness-directed aggregation). In 1988, Yager presented a particular example of orness measure for ordered weighted averaging (OWA) functions and initiated a series of contributions aiming at proposing an axiomatic definition of orness measure for OWA functions. In this paper, we go much further and present an axiomatic definition of orness measure for the whole family of aggregation functions. We end by proposing two natural construction methods for an orness measure for aggregation functions. The particular examples of the (discrete) Choquet integral and uninorms are studied in detail.

Dissimilarities are a very usual way to compare two fuzzy sets and also two interval-valued fuzzy sets. In both cases, the dissimilarity between two sets is a number. In this work, we introduce a generalization of the notion of dissimilarity for interval-valued fuzzy sets such that it assumes values on the set of subintervals instead of the set of numbers. This seems to be more realistic taking into account the available information. We also investigate its relationship with the classical notions of dissimilarity between fuzzy sets and we obtain that the new class is richer than the existing one.

Hesitant fuzzy sets represent a useful tool in many areas such as decision making or image processing. Finite interval-valued hesitant fuzzy sets are a particular kind of hesitant fuzzy sets that generalize fuzzy sets, interval-valued fuzzy sets or Atanassov’s intuitionistic fuzzy sets, among others. Partitioning is a long-standing open problem due to its remarkable importance in many areas such as clustering. Thus, many different partitioning approaches have been developed for crisp and fuzzy sets. This work presents a partitioning method for the so-called finite interval-valued hesitant fuzzy sets. The definition of this partitioning method involves a definition of an ordering relation for finite interval-valued fuzzy sets membership degrees, i.e, finitely generated sets, as well as the definitions of t-norm and t-conorm for these kinds of sets.

The comparison of Atanassov intuitionistic fuzzy sets (AIF-sets) is a topic that has been widely studied due to its several applications in image segmentation or decision making, among other fields. Divergences for AIF-sets (AIF-divergences) were introduced as an adequate measure of comparison for AIF-sets. This study investigates a family of AIF-divergences that satisfies a local property. Such a property allows us to compute the divergence between AIF-sets pointwise. A characterization of those AIF-divergences satisfying the local property is provided. Several interesting properties of local divergence are also discussed. Some applications of these AIF-divergences in pattern recognition and decision making illustrate their utility.

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.

Fuzzy choice functions obtained from preference relations have been recently used to develop automated negotiation systems. This contribution contains a theoretical study of coherence conditions in the process of creating a fuzzy choice function from a preference relation. In particular, the role of the acyclicity property of fuzzy preference relations is studied in the framework of rationality of fuzzy choice functions. Two different definitions of fuzzy acyclicity are compared. The two classical ways of constructing a fuzzy choice function from a given fuzzy preference relation are considered and properties such as acyclicity and completeness are proved to be sufficient conditions to ensure that the constructed function is a rational choice function. Special attention has been paid to the choice of the t-norm too. The results obtained are also compared to the classical results on rationality in the theory of crisp choice functions.

Different definitions of the concept of a fuzzy semiorder are compared. It is proved that their α-cuts are crisp binary relations that may fail to be Ferrers and semitransitive, in general. Consequently, we analyze the preservation of semiorders when coming back from the fuzzy to the crisp setting using α-cuts. In the final sections, a discussion is developed about the extension to the fuzzy setting of the concept of a threshold of utility discrimination, and its corresponding numerical representability of fuzzy semiorders by means of the representability of their α-cuts as crisp binary relations.

Classically, the comparison of random variables have been done by means of a crisp order, which is known as stochastic dominance. In the last years, the classical stochastic dominance have been extended to a graded version by means of a probabilistic relation. In this work we propose different ways of measuring the gradual order among random variables by using fuzzy relations instead of probabilistic relations. The connection between the cycle-transitivity of the probabilistic relation and the T-transitivity of the associated fuzzy weak preference relation is characterized in the particular case of the minimum t-norm.

In some cases the fitness value of a knowledge base is not completely determined, but just bounded in an interval. In this case the fitness value is modelled by a random variable. Thus the comparison of random variables allows to compare the fitness values when they are not completely determined. In this contribution we consider a quite new proposal in stochastic comparison: statistical preference. We recall the advantages of this method with respect to (the classical) stochastic dominance. We also order by statistical preference two fitness values modelled by beta distributions with some special parameters.

The concept of (classical) complete preorder can be characterized in several ways. In previous works we have studied whether complete fuzzy preorders can be characterized by the same properties as in the crisp case. We have proven that this is not usually the case. We have studied five possible characterizations and we have proven that only one still characterizes a fuzzy preorder. In this work we study those properties for additive fuzzy preference structures without incomparability. Despite they do not characterize complete fuzzy preorders, they can be related among them. In this contribution we show their connection when the preference structure does not admit incomparable alternatives.

The representation of the degree of difference between two fuzzy subsets by means of a real number has been proposed in previous papers, and it seems to be useful in some situations. However, the requirement of assigning a precise number may lead us to the loss of essential information about this difference. Thus, (crisp) divergence measures studied in previous papers may not distinguish whether the differences between two fuzzy subsets are in low or high membership degrees. In this paper we propose a way of measuring these differences by means of a fuzzy valued function which we will call fuzzy divergence measure. We formulate a list of natural axioms that these measures should satisfy. We derive additional properties from these axioms, some of them are related to the properties required to crisp divergence measures. We finish the paper by establishing a one-to-one correspondence between families of crisp and fuzzy divergence measures. This result provides us with a method to build a fuzzy divergence measure from a crisp valued one.

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

Some aspects of the relationship between Goodman and Nguyen's one-point coverage interpretation of a fuzzy set and Zadeh's 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.