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Mathematics
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Aggregation Functions and Indistinguishability Operators: Applications
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Aggregation Functions and Indistinguishability Operators: Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Fuzzy Sets, Systems and Decision Making".

Deadline for manuscript submissions: closed (31 December 2022) | Viewed by 10785

Special Issue Editors

Prof. Dr. Jesús Rodríguez López

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Guest Editor
Department of Applied Mathematics, Universitat Politècnica de València, Valencia, Spain
Interests: aggregation functions; fuzzy metric spaces; fuzzy uniform spaces; hypertopologies; topological spaces with richer structures (quasi-metrics, quasi-uniformities, and quasi-proximities)
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Óscar Valero Sierra

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Guest Editor
Department of Mathematics and Computer Science, Universitat de les Illes Balears, 07122 Palma, Spain
Interests: information fusion; aggregation functions; decision making methods; generalized metric structures; similarity measures; fixed point theory; applications to engineering, economics and medicine
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Radko Mesiar

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Guest Editor
Faculty of Civil Engineering, Department of Mathematics and Descriptive Geometry, Slovak University of Technology in Bratislava, Radlinského 11, 810 05 Bratislava, Slovakia
Interests: aggregation operators and related operators; triangular norms; copulas; fuzzy sets and fuzzy logic; uncertainty modeling; measure theory; intelligent computing
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Tomasa Calvo-Sánchez

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Guest Editor
Department of Computer Science, Universidad de Alcalá, Madrid, Spain
Interests: aggregation operators; uncertainty modeling; fuzzy logic; preference structures; non-additive measure; integral theory

Special Issue Information

Dear Colleagues,

Aggregation functions have become an important area of research. The necessity of merging information contained in a collection of pieces of information into a single one for making decisions in applied sciences, has led to a growing interest in studying numerical functions that allow this aggregation. These functions are usually called aggregation functions and they have been successfully applied to different fields such as artificial intelligence, pattern recognition, data fusion, etc.

Indistinguishability operators are a fundamental tool in fuzzy logic. When traditional binary equivalence relations are not enough to deal with uncertainty, indistinguishability operators come into play.

This Special Issue will publish high-quality mathematical papers collecting recent advances in the area of aggregation functions, indistinguishability operators, and related topics. Potential topics of this Special Issue include, but are not limited to, the following: aggregation functions; nonadditive integrals; construction of aggregation functions; means and averages; decision-making; implication operators; indistinguishability operators; similarities; approximate reasoning; etc.

Prof. Dr. Jesús López
Prof. Dr. Óscar Valero Sierra
Prof. Dr. Radko Mesiar
Prof. Dr. Tomasa Calvo-Sánchez
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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Published Papers (6 papers)

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Research

21 pages, 362 KiB  
Article
Some Characterizations of Complete Hausdorff KM-Fuzzy Quasi-Metric Spaces
by Salvador Romaguera
Mathematics 2023, 11(2), 381; https://doi.org/10.3390/math11020381 - 11 Jan 2023
Cited by 1 | Viewed by 1283
Abstract
Gregori and Romaguera introduced, in 2004, the notion of a KM-fuzzy quasi-metric space as a natural asymmetric generalization of the concept of fuzzy metric space in the sense of Kramosil and Michalek. Ever since, various authors have discussed several aspects of such spaces, [...] Read more.
Gregori and Romaguera introduced, in 2004, the notion of a KM-fuzzy quasi-metric space as a natural asymmetric generalization of the concept of fuzzy metric space in the sense of Kramosil and Michalek. Ever since, various authors have discussed several aspects of such spaces, including their topological and (quasi-)metric properties as well as their connections with domain theory and their relationship with other fuzzy structures. In particular, the development of the fixed point theory for these spaces and other related ones, such as fuzzy partial metric spaces, has received remarkable attention in the last 15 years. Continuing this line of research, we here establish general fixed point theorems for left and right complete Hausdorff KM-fuzzy quasi-metric spaces, which are applied to deduce characterizations of these distinguished kinds of fuzzy quasi-metric completeness. Our approach, which mixes conditions of Suzuki-type with contractions of α−ϕ-type in the well-known proposal of Samet et al., allows us to extend and improve some recent theorems on complete fuzzy metric spaces. The obtained results are accompanied by illustrative and clarifying examples. Full article
(This article belongs to the Special Issue Aggregation Functions and Indistinguishability Operators: Applications)
14 pages, 290 KiB  
Article
An Application of Ordered Weighted Averaging Operators to Customer Classification in Hotels
by Pere Josep Pons-Vives, Mateu Morro-Ribot, Carles Mulet-Forteza and Oscar Valero
Mathematics 2022, 10(12), 1987; https://doi.org/10.3390/math10121987 - 9 Jun 2022
Cited by 5 | Viewed by 1905
Abstract
An algorithm widely used in hotel companies for demand analysis is the so-called K-means. The aforementioned algorithm is based on the use of the Euclidean distance as a dissimilarity measure and this fact can cause a main handicap. Concretely, the Euclidean distance provides [...] Read more.
An algorithm widely used in hotel companies for demand analysis is the so-called K-means. The aforementioned algorithm is based on the use of the Euclidean distance as a dissimilarity measure and this fact can cause a main handicap. Concretely, the Euclidean distance provides a global difference measure between the values of the descriptive variables that can blur the relative differences in each component separately and, hence, the cluster algorithm can assign a custom to an incorrect cluster. In order to avoid this drawback, this paper proposes an application of the use of Ordered Weighted Averaging (OWA) operators and an OWA-based K-means for clustering customers staying at a real five-star hotel, located in a mature sun-and-beach area, according to their propensity to spend. It must be pointed out that OWA-based distance calculates relative distances and it is sensitive to the differences in each component separately. All experiments show that the use of the OWA operator improves the performance of the classical K-means up to 21.6% and reduces the number of convergence iterations up to 48.46%. Such an improvement has been tested through a ground truth, designed by the marketing department of the firm, which states the cluster to which each tourist belongs. Moreover, the customer classification is achieved regardless of the season in which the customer stays at the hotel. All these facts confirm that the OWA-based K-means could be used as an appropriate tool for classifying tourists in purely exploratory and predictive stages. Furthermore, the new methodology can be implemented without requiring radical changes in the implementation of the classical methodology and in data processing which is crucial so that it can be incorporated into the control panel of a real hotel without additional implementation costs. Full article
(This article belongs to the Special Issue Aggregation Functions and Indistinguishability Operators: Applications)
19 pages, 376 KiB  
Article
New Results on the Aggregation of Norms
by Tatiana Pedraza and Jesús Rodríguez-López
Mathematics 2021, 9(18), 2291; https://doi.org/10.3390/math9182291 - 17 Sep 2021
Cited by 2 | Viewed by 1457
Abstract
It is a natural question if a Cartesian product of objects produces an object of the same type. For example, it is well known that a countable Cartesian product of metrizable topological spaces is metrizable. Related to this question, Borsík and Doboš characterized [...] Read more.
It is a natural question if a Cartesian product of objects produces an object of the same type. For example, it is well known that a countable Cartesian product of metrizable topological spaces is metrizable. Related to this question, Borsík and Doboš characterized those functions that allow obtaining a metric in the Cartesian product of metric spaces by means of the aggregation of the metrics of each factor space. This question was also studied for norms by Herburt and Moszyńska. This aggregation procedure can be modified in order to construct a metric or a norm on a certain set by means of a family of metrics or norms, respectively. In this paper, we characterize the functions that allow merging an arbitrary collection of (asymmetric) norms defined over a vector space into a single norm (aggregation on sets). We see that these functions are different from those that allow the construction of a norm in a Cartesian product (aggregation on products). Moreover, we study a related topological problem that was considered in the context of metric spaces by Borsík and Doboš. Concretely, we analyze under which conditions the aggregated norm is compatible with the product topology or the supremum topology in each case. Full article
(This article belongs to the Special Issue Aggregation Functions and Indistinguishability Operators: Applications)
12 pages, 273 KiB  
Article
Aggregation of Indistinguishability Fuzzy Relations Revisited
by Juan-De-Dios González-Hedström, Juan-José Miñana and Oscar Valero
Mathematics 2021, 9(12), 1441; https://doi.org/10.3390/math9121441 - 21 Jun 2021
Cited by 4 | Viewed by 1849
Abstract
Indistinguishability fuzzy relations were introduced with the aim of providing a fuzzy notion of equivalence relation. Many works have explored their relation to metrics, since they can be interpreted as a kind of measure of similarity and this is, in fact, a dual [...] Read more.
Indistinguishability fuzzy relations were introduced with the aim of providing a fuzzy notion of equivalence relation. Many works have explored their relation to metrics, since they can be interpreted as a kind of measure of similarity and this is, in fact, a dual notion to dissimilarity. Moreover, the problem of how to construct new indistinguishability fuzzy relations by means of aggregation has been explored in the literature. In this paper, we provide new characterizations of those functions that allow us to merge a collection of indistinguishability fuzzy relations into a new one in terms of triangular triplets and, in addition, we explore the relationship between such functions and those that aggregate extended pseudo-metrics, which are the natural distances associated to indistinguishability fuzzy relations. Our new results extend some already known characterizations which involve only bounded pseudo-metrics. In addition, we provide a completely new description of those indistinguishability fuzzy relations that separate points, and we show that both differ a lot. Full article
(This article belongs to the Special Issue Aggregation Functions and Indistinguishability Operators: Applications)
12 pages, 409 KiB  
Article
Directional Shift-Stable Functions
by Radko Mesiar and Andrea Stupňanová
Mathematics 2021, 9(10), 1077; https://doi.org/10.3390/math9101077 - 11 May 2021
Cited by 2 | Viewed by 1308
Abstract
Recently, some new types of monotonicity—in particular, weak monotonicity and directional monotonicity of an n-ary real function—were introduced and successfully applied. Inspired by these generalizations of monotonicity, we introduce a new notion for n-ary functions acting on [...] Read more.
Recently, some new types of monotonicity—in particular, weak monotonicity and directional monotonicity of an n-ary real function—were introduced and successfully applied. Inspired by these generalizations of monotonicity, we introduce a new notion for n-ary functions acting on [0,1]n, namely, the directional shift stability. This new property extends the standard shift invariantness (difference scale invariantness), which can be seen as a particular directional shift stability. The newly proposed property can also be seen as a particular kind of local linearity. Several examples and a complete characterization for the case of n=2 of directionally shift-stable aggregation and pre-aggregation functions are also given. Full article
(This article belongs to the Special Issue Aggregation Functions and Indistinguishability Operators: Applications)
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12 pages, 306 KiB  
Article
Sub-Additive Aggregation Functions and Their Applications in Construction of Coherent Upper Previsions
by Serena Doria, Radko Mesiar and Adam Šeliga
Mathematics 2021, 9(1), 2; https://doi.org/10.3390/math9010002 - 22 Dec 2020
Cited by 4 | Viewed by 1667
Abstract
In this paper, we explore the use of aggregation functions in the construction of coherent upper previsions. Sub-additivity is one of the defining properties of a coherent upper prevision defined on a linear space of random variables and thus we introduce a new [...] Read more.
In this paper, we explore the use of aggregation functions in the construction of coherent upper previsions. Sub-additivity is one of the defining properties of a coherent upper prevision defined on a linear space of random variables and thus we introduce a new sub-additive transformation of aggregation functions, called a revenue transformation, whose output is a sub-additive aggregation function bounded below by the transformed aggregation function, if it exists. Method of constructing coherent upper previsions by means of shift-invariant, positively homogeneous and sub-additive aggregation functions is given and a full characterization of shift-invariant, positively homogeneous and idempotent aggregation functions on [0,∞[n is presented. Lastly, some concluding remarks are added. Full article
(This article belongs to the Special Issue Aggregation Functions and Indistinguishability Operators: Applications)
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