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.

Managing big data traffic plays an important role in contemporary communication and is essential for efficiently handling an unprecedented volume of information. In the globalized context of the internet, the ability to measure and predict this traffic is a strategically valuable resource that requires a deep understanding of historical data. This article proposes a predictor based on a generalization of the Choquet integral, which aggregates data, reducing the complexity and dimensionality of traffic predictions. The approach is assessed using real data, demonstrating that the Choquet integral achieves higher accuracy with the appropriate$\alpha$parameter. Considering the worst-case scenario in terms of wasted time, and given that the overall algorithm achieved a satisfactory error rate, we can conclude that the least efficient algorithm was the brute force search. In comparison, binary search and random binary search demonstrated a time efficiency improvement of 56.75 % and 59.49 %, respectively. Among the integrals evaluated, the Choquet (a) integral yielded the smallest errors.

Abstract In this work, we explore the impact of adaptive fuzzy measures on edge detection, aiming to enhance how computers interpret images by identifying edges more accurately. Traditional methods rely on analysing changes in image brightness to locate edges, but they often use fixed rules that do not account for the unique characteristics of each image. Our approach differs by adjusting fuzzy measures based on the information within specific areas of an image under a sliding window approach, utilizing a variety of fusion functions and generalizations of the Choquet integral to analyse and combine pixel data. The proposed method is flexible, allowing for the adaptation of measures in response to the image's local features. We put our method to the test against the well‐established Canny edge detector to evaluate its effectiveness. Our experimental results suggest that by adapting fuzzy measures for each image section, we can improve edge detection results.

In data stream (DS) learning, the system has to extract knowledge from data generated continuously, usually at high speed and in large volumes, making it impossible to store the entire set of data to be processed in batch mode. Hence, machine learning models must be built incrementally by processing the incoming examples, as data arrive, while updating the model to be compatible with the current data. In fuzzy DS clustering, the model can either absorb incoming data into existing clusters or initiate a new cluster. As the volume of data increases, there is a possibility that the clusters will overlap to the point where it is convenient to merge two or more clusters into one. Then, a cluster comparison measure (CM) should be applied, to decide whether such clusters should be combined, also in an incremental manner. This defines an incremental fusion process based on aggregation functions that can aggregate the incoming inputs without storing all the previous inputs. The objective of this article is to solve the fuzzy DS clustering problem of incrementally comparing fuzzy clusters on a formal basis. First, we formalize and operationalize incremental fusion processes of fuzzy clusters by introducing recursively extendable (RE) aggregation functions, studying construction methods and different classes of such functions. Second, we propose two approaches to compare clusters: 1) similarity and 2) overlapping between clusters, based on RE aggregation functions. Finally, we analyze the effect of those incremental CMs on the online and offline phases of the well-known fuzzy clustering algorithm d-FuzzStream, showing that our new approach outperforms the original algorithm and presents better or comparable performance to other state-of-the-art DS clustering algorithms found in the literature.

Ordered Weighted Averaging (OWA) operators are some of the most widely used aggregation functions in classic literature, but their application to fuzzy numbers has been limited due to the complexity of defining a total order in fuzzy contexts. However, the recent notion of admissible order for fuzzy numbers provides an effective method to totally order them by refining a given partial order. Therefore, this paper is devoted to defining OWA operators for fuzzy numbers with respect to admissible orders and investigating their properties. Firstly, we define the OWA operators associated with such admissible orders and then we show their main properties. Afterward, an example is presented to illustrate the applicability of these AOWA operators in linguistic decision-making. In this regard, we also develop an admissible order for trapezoidal fuzzy numbers that can be efficiently applied in practice.

Social network analysis is a popular tool to understand the relationships between interacting agents by studying the structural properties of their connections. However, this kind of analysis can miss some of the domain-specific knowledge available in the original information domain and its propagation through the associated network. In this work, we develop an extension of classical social network analysis to incorporate external information from the original source of the network. With this extension we propose a new centrality measure, the semantic value, and a new affinity function, the semantic affinity, that establishes fuzzy-like relationships between the different actors in the network. We also propose a new heuristic algorithm based on the shortest capacity problem to compute this new function. As an illustrative case study, we use the novel proposals to analyze and compare the gods and heroes from three different classical mythologies: 1) Greek; 2) Celtic; and 3) Nordic. We study the relationships of each individual mythology and those of the common structure that is formed when we fuse the three of them. We also compare our results with those obtained using other existing centrality measures and embedding approaches. In addition, we test the proposed measures on a classical social network, the Reuters terror news network, as well as in a Twitter network related to the COVID-19 pandemic. We found that the novel method obtains more meaningful comparisons and results than previous existing approaches in every case.

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.

Automatic image detection is one of the most important areas in computing due to its potential application in numerous real-world scenarios. One important tool to deal with that is calledoverlap indices. They were introduced as a procedure to provide the maximum lack of knowledge when comparing two fuzzy objects. They have been successfully applied in the following fields: image processing, fuzzy rule-based systems, decision making, and computational brain interfaces. This notion ofoverlap indicesis also necessary for applications in which type-2 fuzzy sets are required. In this article, we introduce the notion oftype-$(2,k)$overlap index($k \in \lbrace 0,1,2\rbrace$) in the setting of type-2 fuzzy sets. We describe both the reasons that have led to this notion and the relationships that naturally arise among the algebraic underlying structures. Finally, we illustrate how type-$(2,k)$overlap indices can be employed in the setting of fuzzy rule-based systems when the involved objects are type-2 fuzzy sets.

The Choquet integral (CI) is an averaging aggregation function that has been used, e.g., in the fuzzy reasoning method (FRM) of fuzzy rule-based classification systems (FRBCSs) and in multicriteria decision making in order to take into account the interactions among data/criteria. Several generalizations of the CI have been proposed in the literature in order to improve the performance of FRBCSs and also to provide more flexibility in the different models by relaxing both the monotonicity requirement and averaging conditions of aggregation functions. An important generalization is the$C_{F}$-integrals, which are preaggregation functions that may present interesting nonaveraging behavior depending on the function$F$adopted in the construction and, in this case, offering competitive results in classification. Recently, the concept of d-Choquet integrals was introduced as a generalization of the CI by restricted dissimilarity functions (RDFs), improving the usability of CIs, as when comparing inputs by the usual difference may not be viable. The objective of this article is to introduce the concept of$dC_{F}$-integrals, which is a generalization of$C_{F}$-integrals by RDFs. The aim is to analyze whether the usage of$dC_{F}$-integrals in the FRM of FRBCSs represents a good alternative toward the standard$C_{F}$-integrals that just consider the difference as a dissimilarity measure. For that, we consider six RDFs combined with five fuzzy measures, applied with more than 20 functions$F$. The analysis of the results is based on statistical tests, demonstrating their efficiency. Additionally, comparing the applicability of$dC_{F}$-integrals versus$C_{F}$-integrals, the range of the good generalizations of the former is much larger than that of the latter.

Aggregation functions have been extensively studied and applied in several practical problems involving some sort of fuzzy modeling, by enacting the fusion process of data from the unit interval. T-norms and t-conorms, as well as overlap and grouping functions, are examples of pairs of aggregation functions that are related through the duality property, which is associated with some definition of fuzzy negation. By constructing pairs of dual aggregation functions and applying them in some practical problem, one can analyze which type of behaviour (conjunctive or disjunctive, for example) of the aggregation operator can benefit the whole system. However, when dealing with applications that do not involve fuzzy modeling, such as classification via convolutional neural networks, the data to be aggregated do not necessarily comes from the unit interval. Recently, a framework for defining classes aggregation functions on an arbitrary closed real interval (namely, (a, b)-aggregation functions) based on core known classes of aggregation functions have been introduced, but the study of negations and duality in this context is yet to be developed. Thus, in this paper we introduce and study the concept of negations defined on a arbitrary closed real intervals, called (a, b)-negations, presenting a construction method for them based core fuzzy negations. From that, we develop the concept of duality between (a, b)-aggregation functions, showing that the duality property is preserved when constructing (a, b)-aggregation functions from dual aggregation functions.

Grouping functions and their dual counterpart, overlap functions, have drawn the attention of many authors, mainly because they constitute a richer class of operators compared to other types of aggregation functions. Grouping functions are a useful theoretical tool to be applied in various problems, like decision making based on fuzzy preference relations. In pairwise comparisons, for instance, those functions allow one to convey the measure of the amount of evidence in favor of either of two given alternatives. Recently, some generalizations of grouping functions were proposed, such as (i) the n-dimensional grouping functions and the more flexible general grouping functions, which allowed their application in n-dimensional problems, and (ii) n-dimensional and general interval-valued grouping functions, in order to handle uncertainty on the definition of the membership functions in real-life problems. Taking into account the importance of interval-valued fuzzy implication functions in several application problems under uncertainty, such as fuzzy inference mechanisms, this paper aims at introducing a new class of interval-valued fuzzy material implication functions. We study their properties, characterizations, construction methods and provide examples.

Several generalizations of the Choquet integral have been applied in the Fuzzy Reasoning Method (FRM) of Fuzzy Rule-Based Classification Systems (FRBCS’s) to improve its performance. Additionally, to achieve that goal, researchers have searched for new ways to provide more flexibility to those generalizations, by restricting the requirements of the functions being used in their constructions and relaxing the monotonicity of the integral. This is the case of CT-integrals, CC-integrals, CF-integrals, CF1F2-integrals and dCF-integrals, which obtained good performance in classification algorithms, more specifically, in the fuzzy association rule-based classification method for high-dimensional problems (FARC-HD). Thereafter, with the introduction of Choquet integrals based on restricted dissimilarity functions (RDFs) in place of the standard difference, a new generalization was made possible: the d-XChoquet (d-XC) integrals, which are ordered directional increasing functions and, depending on the adopted RDF, may also be a pre-aggregation function. Those integrals were applied in multi-criteria decision making problems and also in a motor-imagery brain computer interface framework. In the present paper, we introduce a new FRM based on the d-XC integral family, analyzing its performance by applying it to 33 different datasets from the literature.

Overlap functions are a class of aggregation functions that measure the overlapping degree between two values. They have been successfully applied as a fuzzy conjunction operation in several problems in which associativity is not required, such as image processing and classification. Interval-valued overlap functions were defined as an extension to express the overlapping of interval-valued data, and they have been usually applied when there is uncertainty regarding the assignment of membership degrees, as in interval-valued fuzzy rule-based classification systems. In this context, the choice of a total order for intervals can be significant, which motivated the recent developments on interval-valued aggregation functions and interval-valued overlap functions that are increasing to a given admissible order, that is, a total order that refines the usual partial order for intervals. Also, width preservation has been considered on these recent works, in an intent to avoid the uncertainty increase and guarantee the information quality, but no deeper study was made regarding the relation between the widths of the input intervals and the output interval, when applying interval-valued functions, or how one can control such uncertainty propagation based on this relation. Thus, in this paper we: (i) introduce and develop the concepts of width-limited interval-valued functions and width limiting functions, presenting a theoretical approach to analyze the relation between the widths of the input and output intervals of bivariate interval-valued functions, with special attention to interval-valued aggregation functions; (ii) introduce the concept of (a,b)-ultramodular aggregation functions, a less restrictive extension of one-dimension convexity for bivariate aggregation functions, which have an important predictable behaviour with respect to the width when extended to the interval-valued context; (iii) define width-limited interval-valued overlap functions, taking into account a function that controls the width of the output interval and a new notion of increasingness with respect to a pair of partial orders (≤1,≤2); (iv) present and compare three construction methods for these width-limited interval-valued overlap functions, considering a pair of orders (≤1,≤2), which may be admissible or not, showcasing the adaptability of our developments.

Traditionally, Convolutional Neural Networks make use of the maximum or arithmetic mean in order to reduce the features extracted by convolutional layers in a downsampling process known as pooling. However, there is no strong argument to settle upon one of the two functions and, in practice, this selection turns to be problem dependent. Further, both of these options ignore possible dependencies among the data. We believe that a combination of both of these functions, as well as of additional ones which may retain different information, can benefit the feature extraction process. In this work, we replace traditional pooling by several alternative functions. In particular, we consider linear combinations of order statistics and generalizations of the Sugeno integral, extending the latter's domain to the whole real line and setting the theoretical base for their application. We present an alternative pooling layer based on this strategy which we name "CombPool" layer. We replace the pooling layers of three different architectures of increasing complexity by CombPool layers, and empirically prove over multiple datasets that linear combinations outperform traditional pooling functions in most cases. Further, combinations with either the Sugeno integral or one of its generalizations usually yield the best results, proving a strong candidate to apply in most architectures.

Overlap functions are a type of aggregation functions that are not required to be associative, generally used to indicate the overlapping degree between two values. They have been successfully used as a conjunction operator in several practical problems, such as fuzzy-rule-based classification systems (FRBCSs) and image processing. Some extensions of overlap functions were recently proposed, such as general overlap functions and, in the interval-valued context,n-dimensional interval-valued overlap functions. The latter allow them to be applied inn-dimensional problems with interval-valued inputs, such as interval-valued classification problems, where one can apply interval-valued FRBCSs (IV-FRBCSs). In this case, the choice of an appropriate total order for intervals, such as an admissible order, can play an important role. However, neither the relationship between the interval order and then-dimensional interval-valued overlap function (which may or may not be increasing for that order) nor the impact of this relationship in the classification process have been studied in the literature. Moreover, there is not a clear preferredn-dimensional interval-valued overlap function to be applied in an IV-FRBCS. Hence, in this article, we: first, present some new results on admissible orders, which allow us to introduce the concept ofn-dimensional admissibly ordered interval-valued overlap functions, that is,n-dimensional interval-valued overlap functions that are increasing with respect to an admissible order; second, develop a width-preserving construction method for this kind of function, derived from an admissible order and ann-dimensional overlap function, discussing some of its features; finally, analyze the behavior of several combinations of admissible orders andn-dimensional (admissibly ordered) interval-valued overlap functions when applied in IV-FRBCSs. All in all, the contribution of this article resides in pointing out the effect of admissible orders andn-dimensional admissibly ordered interval-valued overlap functions, both from a theoretical and applied points of view, the latter when considering classification problems.

In this article, we propose a new generalization for the notion of homogeneous functions. We show some properties and how it appears in some scenarios. Finally, we show how this generalization can be used in order to provide a new paradigm for decision-making theory calledconsistent influenced/disturbed decision making. In order to illustrate the applicability of this new paradigm, we provide a toy example.

Restricted dissimilarity functions (RDFs) were introduced to overcome problems resulting from the adoption of the standard difference. Based on those RDFs, Bustinceet al.introduced a generalization of the Choquet integral (CI), called d-Choquet integral, where the authors replaced standard differences with RDFs, providing interesting theoretical results. Motivated by such worthy properties, joint with the excellent performance in applications of other generalizations of the CI (using its expanded form, mainly), this article introduces a generalization of the expanded form of the standard Choquet integral (X-CI) based on RDFs, which we named d-XC integrals. We present not only relevant theoretical results but also two examples of applications. We apply d-XC integrals in two problems in decision making, namely a supplier selection problem (which is a multicriteria decision-making problem) and a classification problem in signal processing, based on motor-imagery brain-computer interface (MI-BCI). We found that two d-XC integrals provided better results when compared to the original CI in the supplier selection problem. Besides that, one of the d-XC integrals performed better than any previous MI-BCI results obtained with this framework in the considered signal processing problem.

This work aims to evaluate explainable classification methods for the detection of fish species from hydroacoustic data acquired by echo sounders at a region near the coastline of south and southeastern Brazil. Decision trees and fuzzy rule-based methods were adopted. The fitted models were evaluated by quality measures based on the performance of the classifiers and also by an expert which analyzed the usefulness of the rules on describing the schools. The models learned by the algorithms performed well for the available data and were able to represent the documented behavior of the species considered in the studied region, according to the literature.

Brain-Computer Interfaces based on the analysis of ElectroEncephaloGraphy (EEG) are composed of several elements to process and classify brain input signals. A relevant phase of these systems is the decision making module, in which often the outputs from different classifiers are fused into a single one. In this work, the use of weighted-moderate deviation based functions is proposed to improve the Enhanced-Multimodal Fusion BCI Framework (EMF) decision making phase. Moderate Deviation-based aggregation functions (MDs) allow us to choose the best value to aggregate a vector of points involving a moderate deviation function. Using a weighted MD, the relative importance of each dimension in the multi-dimensional aggregated data set can also be taken into account. By applying these functions in the EMF, each one of the different brain signals can be weighted according to their importance. Moreover, using automatic differentiation, it is possible to optimize them for the present problem.

It is known that the human visual system performs a hierarchical information process in which early vision cues (or primitives) are fused in the visual cortex to compose complex shapes and descriptors. While different aspects of the process have been extensively studied, such as lens adaptation or feature detection, some other aspects, such as feature fusion, have been mostly left aside. In this work, we elaborate on the fusion of early vision primitives using generalizations of the Choquet integral, and novel aggregation operators that have been extensively studied in recent years. We propose to use generalizations of the Choquet integral to sensibly fuse elementary edge cues, in an attempt to model the behaviour of neurons in the early visual cortex. Our proposal leads to a fully-framed edge detection algorithm whose performance is put to the test in state-of-the-art edge detection datasets.

Grouping functions are aggregation functions used in decision making based on fuzzy preference relations in order to express the measure of the amount of evidence in favor of either of the two alternatives when performing pairwise comparisons. They have been also used as a disjunction operator in some important problems, such as image thresholding and the construction of a class of implication functions for the generation of fuzzy subsethood and entropy measures. Some generalizations of this concept were recently proposed, such as n-dimensional and general grouping functions, which allowed their application in ndimensional problems, such as fuzzy community detection. Also the concept of interval-valued overlap functions was presented, in order to deal with the uncertainty when defining membership functions. The aim of this paper is to introduce the concepts of n-dimensional interval-valued grouping functions and general interval-valued grouping functions, studying representability, characterization and construction methods.

In Multi-Criteria Decision Making, one of the most used algorithm designed to deal with decision making is the Technical Order by Preference to Ideal Solution (TOPSIS), which is based on finding a solution that is close to the best possible solution and distant from the worst possible solution. The Group Modular Choquet Random TOPSIS (GMC-RTOPSIS) is a generalization of the TOPSIS method capable of dealing with multiple and heterogeneous data types and interaction among criteria by means of the discrete Choquet integral. On the other hand, CT-integrals are a generalization of the Choquet integral using t-norms, which are more flexible than the standard Choquet integral. CT-integrals are pre-aggregation functions, which means that we do not require them to be monotonic in the whole domain, just in some specific directions, that is, they are directionally monotonic. Due to the excellent performance of CT-integrals in classification and multimodal fuzzy fusion decision problems, the objective of this paper is to generalize the GMC-RTOPSIS by using CT-integrals and to analyze the results provided by the use of five different t-norms in an example of a decision making problem.

It is common sense the relevance of overlap and grouping functions, specially in applications in which associativity is not required. In this work, based on previous investigations concerning classes of implication functions derived from overlap functions O, grouping functions G and fuzzy negations N (namely, (G, N)-implications, RO-implications and QL-implications constructed from (O, G, N)), a deep study on D-implications constructed from grouping functions G is carried out. Such fuzzy implications, also known as Dishkant implications, are obtained from D-operations derived from (O, G, N), which are used for the generalization of the implication p → q ≡ q(¬p∧¬q) of orthomodular lattices. We investigate under which conditions such D-operations are fuzzy implication functions, presenting a general form for obtaining D-implications. We also provide a comparative study of D-implications derived from grouping functions and other classes of fuzzy implications constructed from N, O and G, analysing the intersections among such classes.

Fuzzy Rule Based Classification Systems are an useful tool to deal with classification problems. In these systems, one fundamental point is the manner of how the available information about the problem is aggregated. The mechanism responsible to perform the aggregation is the Fuzzy Reasoning Method (FRM). Recently, some FRMs using generalizations of the Choquet integral to perform the aggregation were proposed in the literature. Since these generalizations are defined considering a specific fuzzy measure, in this paper we apply different fuzzy measures in the generalizations that presented the best performance in each study in the literature. We analyze how the performance is affected according to each fuzzy measure.

In this work we propose a new axiomatic definition of distance between interval-valued fuzzy sets which takes into account the width of the membership intervals and linear orders. We discuss some construction methods using aggregation functions which are defined in terms of admissible orders.

A key component of fuzzy rule-based classification systems (FRBCS) is the fuzzy reasoning method (FRM) since it infers the class predicted for new examples. A crucial stage in any FRM is the way in which the information given by the fired rules during the inference process is aggregated. A widely used FRM is the winning rule, which applies the maximum to accomplish this aggregation. The maximum is an averaging operator, which means that its result is within the range delimited by the minimum and the maximum of the aggregated values. Recently, new averaging operators based on generalizations of the Choquet integral have been proposed to perform this aggregation process. However, the most accurate FRBCSs use the FRM known as additive combination that considers the normalized sum as the aggregation operator, which is nonaveraging. For this reason, this paper is aimed at introducing a new nonaveraging operator named CF1F2-integral, which is a generalization of the Choquet-like Copula-based integral (CC-integral). CF1F2-integrals present the desired properties of an aggregation-like operator since they satisfy appropriate boundary conditions and have some kind of increasingness property. We show that CF1F2-integrals, when used to cope with classification problems, enhance the results of the previous averaging generalizations of the Choquet integral and provide competitive results (even better) when compared with state-of-the-art FRBCSs.

Douro region, in Portugal, is worldwide famous for its Port wine, having a huge economic importance in the region. Among the enological parameters that define the grape maturity, the sugar level is considered one of the most relevant ones. Then, in this study we intend to classify their sugar level based on reflectance measurements obtained by hyperspectral images from grapes collected in this region. Recently, to cope with classification problems, we have applied some generalizations of the Choquet integral in the fuzzy reasoning method of a fuzzy rule-based classification system. These generalizations produced different aggregation and/or pre-aggregation functions, presenting good results. For this reason, in this study, we have compared the performance of different generalizations among themselves and versus classical classifiers found in the specialized literature.

In this paper, we introduce the notion of ordered directionally monotone function as a type of function which allows monotonicity along different directions in different points. In particular, these functions take into account the ordinal size of the coordinates of the inputs in order to fuse them. We show several examples of these functions and we study their properties. Finally, we present an illustrative example of an application which justifies the introduction and the study of the concept of ordered directional monotonicity.

The aim of this paper is to introduce the concept of pre-t-conorms, based on the notion of pre-aggregation function, which was introduced by Lucca et al. as an “aggregation” concept that it is not monotonic in all its domain. We also study the concept of light pre-t-conorms, which are non necessarily associative commutative functions with neutral element e = 0. We present some properties of (light) pre-t-conorms and the classes of (light) pre-t-conorms, showing interesting examples. Finally, we present an application of pre-conorms and pre-negations for defining directional fuzzy implication functions. We notice that the introduction of pre-t-conorms allows applications where the full monotonicity is not required (as in classification problems) and the light pre-t-conorms can be used in applications that do not require the associativity property (as in image processing and decision making).

The main drawback of Fuzzy Rule-Based Classification Systems (FRBCSs) when they are applied in Big Data problems is the lack of scalability. Previously proposed approaches consist in concurrently fitting several Chi et al. FRBCSs whose rule bases are then aggregated to obtain the final model. This methodology is seriously affected by the degree of parallelism used for the execution of the algorithm, showing a significant decrease in classification performance as the degree of parallelism increases. This work focuses on the design of a new FRBCS for Big Data classification problems (CHI-BD) that generates exactly the same rule base regardless of the degree of parallelism. Our approach recovers the model that would be built by the original Chi et al. algorithm if it was able to deal with Big Data problems.

In a recent paper, it was introduced the concept of Choquet-like Copula-based integral (CC-integral for short). This kind of function extends the standard Choquet integral and generalizes it by copula functions. These functions were applied in the Fuzzy Reasoning Method (FRM) of a Fuzzy Rule-Based Classification System (FRBCS), presenting an example where the CC-integral based on the minimum t-norm had different behaviors according to the values being aggregated. Therefore, the resulting FRM is theoretically more flexible than those associated with classical aggregation functions like the maximum. In this work, we present a methodology to study the flexibility of the aggregation function used in the FRM. Specifically, we conduct an analysis of 3 different methods to aggregate values in the FRM, namely, the CC-integral based on the minimum t-norm, the standard Choquet integral and the maximum (classical FRM of the winning rule - WR). We prove that the CC-integral behaves in different ways according to the values to be aggregated, whereas the Choquet integral offers an averaging behavior and the WR presents an strict behavior, since it considers only the rule having the maximum compatibility with the example.

Penalty functions are widely used to measure disagreement or consensus. On the other hand, subsethood measures have been applied in several areas. In this paper, we introduce a method for constructing penalty functions by means of QL fuzzy subsethood measures, introduced by Dimuro et al., which are built from QL-operations derived from tuples (O, G, N), for overlap functions O, grouping functions G and fuzzy negations N.

Game theory is an important basis to simulate several situations where multiple agents interact strategically for decision making and support. In many applications, such as auctions, frequently used for resource management involving two or more agents competing for the resources, the interacting agents only know their own characteristics and must make decisions while having to estimate the characteristics of the others. When probabilities are assigned for the different types of the interacting agents, this kind of strategic interaction constitutes a Bayesian game. In cases in which it is very difficult to characterize the private information of each agent, the payoffs can be given by approximate (not probabilistic) values, but the concept of Bayesian Nash equilibrium cannot be applied in this context. Fuzzy set theory is an excellent basis for studying this type of game, where the payoffs are represented by fuzzy numbers. When it is the case that there is also uncertainty about such fuzzy numbers, the use of interval fuzzy numbers appears as a good modeling alternative. This paper introduces an approach for interval-based fuzzy Bayesian games, based on interval-valued fuzzy probabilities for modeling the types of agents involved in the interaction. We present two different case studies, namely the (Interval) Fuzzy Bayesian Hiring Game and (Interval) Fuzzy Bayesian Prisoner's Dilemma with Moral Standards, comparing the results obtained with the crisp, fuzzy and interval fuzzy approaches, highlighting a particular case in which the interval fuzzy approach presents a solution although the two other do not.

In this work we introduce the definition of pre-aggregation functions. These functions generalize aggregation functions, in the sense that they fulfill the same boundary conditions as the latter, but only directional monotonicity is demanded to them. We discuss a construction method which is inspired from the way the discrete Choquet integral is built and it replaces the product by other appropriate functions. Finally, we propose to apply them in fuzzy rule-based classification systems.

In this paper, we introduce the notion of preaggregation function. Such a function satisfies the same boundary conditions as an aggregation function, but, instead of requiring monotonicity, only monotonicity along some fixed direction (directional monotonicity) is required. We present some examples of such functions. We propose three different methods to build preaggregation functions. We experimentally show that in fuzzy rule-based classification systems, when we use one of these methods, namely, the one based on the use of the Choquet integral replacing the product by other aggregation functions, if we consider the minimum or the Hamacher product t-norms for such construction, we improve the results obtained when applying the fuzzy reasoning methods obtained using two classical averaging operators such as the maximum and the Choquet integral.

Overlap functions are aggregation operators used in the overlap problem or when the associativity is not required. Residual implications derived from them (RO-implications) preserve the residuation property, and any overlap function O and the respective RO-implication form an adjoint pair, which is important in many applications. RO-implications are weaker than R-implications constructed from positive and continuous t-norms, since RO-implications do not necessarily satisfy certain properties, but only weaker versions of these properties. However, in general, such properties are not demanded for many applications. In this paper, we analyze the laws of contraposition for RO-implications with respect to a fuzzy negation.

In this paper, we introduce a family of Choquet-based non-associative aggregation functions for application in the fuzzy reasoning method proposed by Barrenechea et al. for fuzzy rule-based classification systems. The family is constructed manipulating the standard definition of Choquet Integral and substituting the product operator by the general definition of the family of overlap functions Cα(x; y) = xy(1 + α(1 - x)(1 - y)), for α ∈ [-1; 0[∪]0; 1], resulting in non-associative aggregation functions. A comparative study considering different values for α and the power measure (whose exponent is learned genetically to adapt it to each class) is presented. The approach is tested in seventeen numerical dataset selected from the KEEL dataset repository. We compare the obtained results with the work presented by Barrenechea et al., showing that the proposed approach can offer also good performance, so providing more flexibility to that proposal, enlarging its scope of applications.

An open problem in social simulation and MAS applications is the self-regulation of social exchange processes, aiming at the achievement/maintenance of equilibrated exchanges by the agents themselves, providing the continuation of the interactions in time. This paper faces this problem through an approach based on the proposed spatial and evolutionary Game of Self-Regulation of Social Exchange Processes. The agents, adopting different social exchange strategies, which take into account both the short and long-term aspects of interactions, evolve such strategies by themselves in time, in order to maximize their respective strategy-based fitness functions. In consequence, the agents happen to perform more equilibrated and fair interactions, increasing the number of successful exchanges.

This paper describes an extended Genetic Algorithm Approach for path planning of multiple mobile robots with obstacle detection and avoidance in static and dynamic scenarios. Through the software Netlogo, used in simulations of multi-agent applications, a model was developed for the given problem. The model, which contains multiple robots and a scenario with several dynamic and static obstacles, is responsible for determining the best path used by the robots to achieve the goal state in a shorter number of steps and avoiding collisions. Additionally, a performance evaluation of this model in comparison with A* algorithm is presented.

The main contribution of this paper is the introduction of an intrinsic definition of the connective “fuzzy exclusive or” E (f-Xor E), based only on the properties of boundary conditions, commutativity and partial isotonicity-antitonicity on the the end-points of the unit interval U = [0,1], in a way that the classical definition of the boolean Xor is preserved. We show three classes of the f-Xor E that can be also obtained from the composition of fuzzy connectives, namely, triangular norms, triangular conorms and fuzzy negations. A discussion about extra properties satisfied by the f-Xor E is presented. Additionally, the paper introduces a class of fuzzy equivalences that generalizes the Fodor and Roubens's fuzzy equivalence, and four classes of fuzzy implications induced by the f-Xor E, discussing their main properties. The relationships between those classes of fuzzy implications and automorphisms are explored. The action of automorphisms on f-Xor E is analyzed.

The aim of this work is to analyze interval fuzzy S-implications and interval automorphisms. Starting from any fuzzy S-implication, it is shown how to obtain an interval fuzzy S-implication canonically. We proved that such interval fuzzy S-implications meet the optimality property and preserve the same properties satisfied by fuzzy S-implications. In addition, commutative diagrams are used in order to relate fuzzy S-implications to interval fuzzy S-implications, and to understand how interval automorphisms act on interval S-implications, generating other interval fuzzy S-implications.