Open Access
Abstract homogeneous functions and consistently influenced/disturbed multi-expert decision making
(IEEE, 2021) Santiago, Regivan; Bedregal, Benjamin; Pereira Dimuro, Graçaliz; Fernández Fernández, Francisco Javier; Bustince Sola, Humberto; Fardoun, Habib; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
In this paper 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 called consistent influenced/disturbed decision making. In order to illustrate the applicability of this new paradigm, we provide a toy example.
Open Access
A historical account of types of fuzzy sets and their relationships
(IEEE, 2016) Bustince Sola, Humberto; Barrenechea Tartas, Edurne; Pagola Barrio, Miguel; Fernández Fernández, Francisco Javier; Xu, Zeshui; Bedregal, Benjamin; Montero, Javier; Hagras, Hani; Herrera, Francisco; Baets, Bernard de; Automatika eta Konputazioa; Institute of Smart Cities - ISC; Automática y Computación
In this work we review the definition and basic properties of the different types of fuzzy sets that have appeared up to now in the literature. We also analyze the relationships between them and enumerate some of the applications in which they have been used.
Open Access
N-dimensional admissibly ordered interval-valued overlap functions and its influence in interval-valued fuzzy rule-based classification systems
(IEEE, 2021) Da Cruz Asmus, Tiago; Sanz Delgado, José Antonio; Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Fernández Fernández, Francisco Javier; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
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 rulebased 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 in n-dimensional problems with interval-valued inputs, like 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, like an admissible order, can play an important role. However, neither the relationship between the interval order and the n-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 preferred n-dimensional interval-valued overlap function to be applied in an IV-FRBCS. Hence, in this paper we: (i) present some new results on admissible orders, which allow us to introduce the concept of n-dimensional admissibly ordered interval-valued overlap functions, that is, n-dimensional interval-valued overlap functions that are increasing with respect to an admissible order; (ii) develop a width-preserving construction method for this kind of function, derived from an admissible order and an n-dimensional overlap function, discussing some of its features; (iii) analyze the behaviour of several combinations of admissible orders and n-dimensional (admissibly ordered) interval-valued overlap functions when applied in IV-FRBCSs. All in all, the contribution of this paper resides in pointing out the effect of admissible orders and n-dimensional admissibly ordered interval-valued overlap functions, both from a theoretical and applied points of view, the latter when considering classification problems.
Open Access
Pseudo overlap functions, fuzzy implications and pseudo grouping functions with applications
(MDPI, 2022) Zhang, Xiaohong; Liang, Rong; Bustince Sola, Humberto; Bedregal, Benjamin; Fernández Fernández, Francisco Javier; Li, Mengyuan; Ou, Qiqi; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
Overlap and grouping functions are important aggregation operators, especially in information fusion, classification and decision-making problems. However, when we do more in-depth application research (for example, non-commutative fuzzy reasoning, complex multi-attribute decision making and image processing), we find overlap functions as well as grouping functions are required to be commutative (or symmetric), which limit their wide applications. For the above reasons, this paper expands the original notions of overlap functions and grouping functions, and the new concepts of pseudo overlap functions and pseudo grouping functions are proposed on the basis of removing the commutativity of the original functions. Some examples and construction methods of pseudo overlap functions and pseudo grouping functions are presented, and the residuated implication (co-implication) operators derived from them are investigated. Not only that, some applications of pseudo overlap (grouping) functions in multi-attribute (group) decision-making, fuzzy mathematical morphology and image processing are discussed. Experimental results show that, in many application fields, pseudo overlap functions and pseudo grouping functions have greater flexibility and practicability.
Open Access
d-Choquet integrals: Choquet integrals based on dissimilarities
(Elsevier, 2020) Bustince Sola, Humberto; Mesiar, Radko; Fernández Fernández, Francisco Javier; Galar Idoate, Mikel; Paternain Dallo, Daniel; Altalhi, A. H.; Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Takáč, Zdenko; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa, PJUPNA13
The paper introduces a new class of functions from [0,1]n to [0,n] called d-Choquet integrals. These functions are a generalization of the 'standard' Choquet integral obtained by replacing the difference in the definition of the usual Choquet integral by a dissimilarity function. In particular, the class of all d-Choquet integrals encompasses the class of all 'standard' Choquet integrals but the use of dissimilarities provides higher flexibility and generality. We show that some d-Choquet integrals are aggregation/pre-aggregation/averaging/functions and some of them are not. The conditions under which this happens are stated and other properties of the d-Choquet integrals are studied.
Open Access
On some classes of nullnorms and h-pseudo homogeneity
(Elsevier, 2022) Lima, Lucélia; Bedregal, Benjamin; Rocha, Marcus; Castillo López, Aitor; Fernández Fernández, Francisco Javier; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
Nullnorms are aggregation functions which generalize t-norms and t-conorms. For each nullnorm there exists an annihilator element such that, below it, the function behaves like a t-conorm, and, above it, like a t-norm. In this paper, we study some classes of nullnorms which naturally arise from well known classes of t-norms and t-conorms, such as idempotent, Archimedean, cancellative, positive and nilpotent t-norms and t-conorms. We also present the concept of h-pseudo-homogeneous nullnorm and we study when these classes of nullnorms fullfill it.
Open Access
Aggregation of individual rankings through fusion functions: criticism and optimality analysis
(IEEE, 2020) Bustince Sola, Humberto; Bedregal, Benjamin; Campión Arrastia, María Jesús; Silva, Ivanoska da; Fernández Fernández, Francisco Javier; Induráin Eraso, Esteban; Raventós Pujol, Armajac; Santiago, Regivan; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
Throughout this paper, our main idea is to analyze from a theoretical and normative point of view different methods to aggregate individual rankings. To do so, first we introduce the concept of a general mean on an abstract set. This new concept conciliates the social choice where well-known impossibility results as the Arrovian ones are encountered and the decision-making approaches where the necessity of fusing rankings is unavoidable. Moreover it gives rise to a reasonable definition of the concept of a ranking fusion function that does indeed satisfy the axioms of a general mean. Then we will introduce some methods to build ranking fusion functions, paying a special attention to the use of score functions, and pointing out the equivalence between ranking and scoring. To conclude, we prove that any ranking fusion function introduces a partial order on rankings implemented on a finite set of alternatives. Therefore, this allows us to compare rankings and different methods of aggregation, so that in practice one should look for the maximal elements with respect to such orders defined on rankings IEEE.
Open Access
Improving the performance of fuzzy rule-based classification systems based on a nonaveraging generalization of CC-integrals named C-F1F2-integrals
(IEEE, 2019) Lucca, Giancarlo; Pereira Dimuro, Graçaliz; Fernández Fernández, Francisco Javier; Bustince Sola, Humberto; Bedregal, Benjamin; Sanz Delgado, José Antonio; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
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 C-F1F2-integral, which is a generalization of the Choquet-like Copula-based integral (CC-integral). C-F1F2-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 C-F1F2 -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.
Open Access
The law of O-conditionality for fuzzy implications constructed from overlap and grouping functions
(Elsevier, 2019) Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Fernández Fernández, Francisco Javier; Sesma Sara, Mikel; Pintor Borobia, Jesús María; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Ingeniaritza; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas; Ingeniería
Overlap and grouping functions are special kinds of non necessarily associative aggregation operators proposed for many applications, mainly when the associativity property is not strongly required. The classes of overlap and grouping functions are richer than the classes of t-norms and t-conorms, respectively, concerning some properties like idempotency, homogeneity, and, mainly, the self-closedness feature with respect to the convex sum and the aggregation by generalized composition of overlap/grouping functions. In previous works, we introduced some classes of fuzzy implications derived by overlap and/or grouping functions, namely, the residual implications R-0-implications, the strong implications (G, N)-implications and the Quantum Logic implications QL-implications, for overlap functions O, grouping functions G and fuzzy negations N. Such implications do not necessarily satisfy certain properties, but only weaker versions of these properties, e.g., the exchange principle. However, in general, such properties are not demanded for many applications. In this paper, we analyze the so-called law of O-Conditionality, O(x, 1(x, y)) <= y, for any fuzzy implication I and overlap function O, and, in particular, for Ro-implications, (G, N)-implications, QL-implications and D-implications derived from tuples (O, G, N), the latter also introduced in this paper. We also study the conditional antecedent boundary condition for such fuzzy implications, since we prove that this property, associated to the left ordering property, is important for the analysis of the O-Conditionality. We show that the use of overlap functions to implement de generalized Modus Ponens, as the scheme enabled by the law of O-Conditionality, provides more generality than the laws of T-conditionality and U-conditionality, for t-norms T and uninorms U, respectively.
Open Access
On fuzzy implications derived from general overlap functions and their relation to other classes
(MDPI, 2023) Pinheiro, Jocivania; Santos, Helida; Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Santiago, Regivan; Fernández Fernández, Francisco Javier; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
There are distinct techniques to generate fuzzy implication functions. Despite most of them using the combination of associative aggregators and fuzzy negations, other connectives such as (general) overlap/grouping functions may be a better strategy. Since these possibly non-associative operators have been successfully used in many applications, such as decision making, classification and image processing, the idea of this work is to continue previous studies related to fuzzy implication functions derived from general overlap functions. In order to obtain a more general and flexible context, we extend the class of implications derived by fuzzy negations and t-norms, replacing the latter by general overlap functions, obtaining the so-called (GO, N)-implication functions. We also investigate their properties, the aggregation of (GO, N)-implication functions, their characterization and the intersections with other classes of fuzzy implication functions.