Open Access
Admissible OWA operators for fuzzy numbers
(Elsevier, 2024) García-Zamora, Diego; Cruz, Anderson; Neres, Fernando; Santiago, Regivan; Roldán López de Hierro, Antonio Francisco; Paiva, Rui; Pereira Dimuro, Graçaliz; Martínez López, Luis; Bedregal, Benjamin; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
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.
Open Access
On admissible orders on the set of discrete fuzzy numbers for application in decision making problems
(MDPI, 2021) Riera, Juan Vicente; Massanet, Sebastia; Bustince Sola, Humberto; Fernández Fernández, Francisco Javier; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
The study of orders is a constantly evolving topic, not only for its interest from a theoretical point of view, but also for its possible applications. Recently, one of the hot lines of research has been the construction of admissible orders in different frameworks. Following this direction, this paper presents a new representation theorem in the field of discrete fuzzy numbers that enables the construction of two families of admissible orders in the set of discrete fuzzy numbers whose support is a closed interval of a finite chain, leading to the first admissible orders introduced in this framework.
Open Access
Interval-valued aggregation functions based on moderate deviations applied to motor-imagery-based brain computer interface
(IEEE, 2021) Fumanal Idocin, Javier; Takáč, Zdenko; Fernández Fernández, Francisco Javier; Sanz Delgado, José Antonio; Goyena Baroja, Harkaitz; Lin, Chin-Teng; Wang, Yu-Kai; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
In this work we develop moderate deviation functions to measure similarity and dissimilarity among a set of given interval-valued data to construct interval-valued aggregation functions, and we apply these functions in two MotorImagery Brain Computer Interface (MI-BCI) systems to classify electroencephalography signals. To do so, we introduce the notion of interval-valued moderate deviation function and, in particular, we study those interval-valued moderate deviation functions which preserve the width of the input intervals. In order to apply them in a MI-BCI system, we first use fuzzy implication operators to measure the uncertainty linked to the output of each classifier in the ensemble of the system, and then we perform the decision making phase using the new interval-valued aggregation functions. We have tested the goodness of our proposal in two MI-BCI frameworks, obtaining better results than those obtained using other numerical aggregation and interval-valued OWA operators, and obtaining competitive results versus some non aggregation-based frameworks.
Open Access
Towards interval uncertainty propagation control in bivariate aggregation processes and the introduction of width-limited interval-valued overlap functions
(Elsevier, 2021) Da Cruz Asmus, Tiago; Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Sanz Delgado, José Antonio; Mesiar, Radko; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
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.
Open Access
Admissible orders on fuzzy numbers
(IEEE, 2022) Zumelzu, Nicolás; Bedregal, Benjamin; Mansilla, Edmundo; Bustince Sola, Humberto; Díaz, Roberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
From the more than two hundred partial orders for fuzzy numbers proposed in the literature, only a few are total. In this paper, we introduce the notion of admissible order for fuzzy numbers equipped with a partial order, i.e. a total order which refines the partial order. In particular, it is given special attention to the partial order proposed by Klir and Yuan in 1995. Moreover, we propose a method to construct admissible orders on fuzzy numbers in terms of linear orders defined for intervals considering a strictly increasing upper dense sequence, proving that this order is admissible for a given partial order. Finally, we use admissible orders to ranking the path costs in fuzzy weighted graphs. IEEE