Editorial

2 pages, 154 KiB

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Introduction to Special Issue: New Trends in Fuzzy Set Theory and Related Items

by Humberto Bustince, Javier Fernandez and Esteban Induráin

Axioms 2018, 7(2), 37; https://doi.org/10.3390/axioms7020037 - 05 Jun 2018

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We focus on the articles recently published in the special issue of Axioms devoted to “New Trends in Fuzzy Set Theory and Related Items”. Full article

(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)

Research

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7 pages, 213 KiB

Open AccessArticle

Quantiles in Abstract Convex Structures

by Marta Cardin

Axioms 2018, 7(2), 35; https://doi.org/10.3390/axioms7020035 - 28 May 2018

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Abstract

In this short paper, we aim at a qualitative framework for modeling multivariate decision problems where each alternative is characterized by a set of properties. To this extent, we consider convex spaces as underlying universes and make use of lattice operations in convex [...] Read more.

In this short paper, we aim at a qualitative framework for modeling multivariate decision problems where each alternative is characterized by a set of properties. To this extent, we consider convex spaces as underlying universes and make use of lattice operations in convex spaces to formalize the notion of quantiles. We also put in evidence that many important models of decision problems can be viewed as convex spaces-based models. Several properties of aggregation operators are translated into this general setting, and independence and invariance are used to provide axiomatic characterizations of quantiles. Full article

(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)

16 pages, 500 KiB

Open AccessArticle

Graphs in an Intuitionistic Fuzzy Soft Environment

by Sundas Shahzadi and Muhammad Akram

Axioms 2018, 7(2), 20; https://doi.org/10.3390/axioms7020020 - 23 Mar 2018

Cited by 14 | Viewed by 4798

Abstract

In this research article, we present a novel framework for handling intuitionistic fuzzy soft information by combining the theory of intuitionistic fuzzy soft sets with graphs. We introduce the notion of certain types of intuitionistic fuzzy soft graphs including neighbourly edge regular intuitionistic [...] Read more.

In this research article, we present a novel framework for handling intuitionistic fuzzy soft information by combining the theory of intuitionistic fuzzy soft sets with graphs. We introduce the notion of certain types of intuitionistic fuzzy soft graphs including neighbourly edge regular intuitionistic fuzzy soft graphs and strongyl edge irregular intuitionistic fuzzy soft graphs. We illustrate these novel concepts by several examples, and investigate some of their related properties. We present an application of intuitionistic fuzzy soft graph in a decision-making problem and also present our methods as an algorithm that is used in this application. Full article

(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)

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8 pages, 236 KiB

Open AccessArticle

An Abstract Result on Projective Aggregation Functions

by Juan C. Candeal

Axioms 2018, 7(1), 17; https://doi.org/10.3390/axioms7010017 - 20 Mar 2018

Cited by 2 | Viewed by 2922

Abstract

A general characterization result of projective aggregation functions is shown, the proof of which makes use of the celebrated Arrow’s theorem, thus providing a link between aggregation functions theory and social choice theory. The result can be viewed as a generalization of a [...] Read more.

A general characterization result of projective aggregation functions is shown, the proof of which makes use of the celebrated Arrow’s theorem, thus providing a link between aggregation functions theory and social choice theory. The result can be viewed as a generalization of a theorem obtained by Kim (1990) for real-valued aggregation functions defined on the n-dimensional Euclidean space in the context of measurement theory. In addition, two applications of the core theorem of the article are shown. The first is a simple extension of the main result to the context of multi-valued aggregation functions. The second offers a new characterization of projective bijection aggregators, thus connecting aggregation operators theory with social choice. Full article

(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)

16 pages, 892 KiB

Open AccessArticle

Revision of the Kosiński’s Theory of Ordered Fuzzy Numbers

by Krzysztof Piasecki

Axioms 2018, 7(1), 16; https://doi.org/10.3390/axioms7010016 - 02 Mar 2018

Cited by 32 | Viewed by 3175

Abstract

Ordered fuzzy numbers are defined by Kosiński. In this way, he was going to supplement a fuzzy number by orientation. A significant drawback of Kosiński’s theory is that there exist such ordered fuzzy numbers which, in fact, are not fuzzy numbers. For this [...] Read more.

Ordered fuzzy numbers are defined by Kosiński. In this way, he was going to supplement a fuzzy number by orientation. A significant drawback of Kosiński’s theory is that there exist such ordered fuzzy numbers which, in fact, are not fuzzy numbers. For this reason, a fully formalized correct definition of ordered fuzzy numbers is proposed here. Also, the arithmetic proposed by Kosiński has a significant disadvantage. The space of ordered fuzzy numbers is not closed under Kosiński’s addition. On the other hand, many mathematical applications require the considered space be closed under used arithmetic operations. Therefore, the Kosinski’s theory is modified in this way that the space of ordered fuzzy numbers is closed under revised arithmetic operations. In addition, it is shown that the multiple revised sum of finite sequence of ordered fuzzy numbers depends on its summands ordering. Full article

(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)

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7 pages, 214 KiB

Open AccessArticle

Fuzzy Analogues of Sets and Functions Can Be Uniquely Determined from the Corresponding Ordered Category: A Theorem

by Christian Servin, Gerardo D. Muela and Vladik Kreinovich

Axioms 2018, 7(1), 8; https://doi.org/10.3390/axioms7010008 - 23 Jan 2018

Cited by 2 | Viewed by 3080

Abstract

In modern mathematics, many concepts and ideas are described in terms of category theory. From this viewpoint, it is desirable to analyze what can be determined if, instead of the basic category of sets, we consider a similar category of fuzzy sets. In [...] Read more.

In modern mathematics, many concepts and ideas are described in terms of category theory. From this viewpoint, it is desirable to analyze what can be determined if, instead of the basic category of sets, we consider a similar category of fuzzy sets. In this paper, we describe a natural fuzzy analog of the category of sets and functions, and we show that, in this category, fuzzy relations (a natural fuzzy analogue of functions) can be determined in category terms—of course, modulo 1-1 mapping of the corresponding universe of discourse and 1-1 re-scaling of fuzzy degrees. Full article

(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)

17 pages, 303 KiB

Open AccessArticle

Cubic Interval-Valued Intuitionistic Fuzzy Sets and Their Application in BCK/BCI-Algebras

by Young Bae Jun, Seok-Zun Song and Seon Jeong Kim

Axioms 2018, 7(1), 7; https://doi.org/10.3390/axioms7010007 - 23 Jan 2018

Cited by 30 | Viewed by 4005

Abstract

As a new extension of a cubic set, the notion of a cubic interval-valued intuitionistic fuzzy set is introduced, and its application in

B C K / B C I

-algebra is considered. The notions of

α

-internal,

β

-internal,

α

-external and [...] Read more.

As a new extension of a cubic set, the notion of a cubic interval-valued intuitionistic fuzzy set is introduced, and its application in

B C K / B C I

-algebra is considered. The notions of

α

-internal,

β

-internal,

α

-external and

β

-external cubic IVIF set are introduced, and the P-union, P-intersection, R-union and R-intersection of

α

-internal and

α

-external cubic IVIF sets are discussed. The concepts of cubic IVIF subalgebra and ideal in

B C K / B C I

-algebra are introduced, and related properties are investigated. Relations between cubic IVIF subalgebra and cubic IVIF ideal are considered, and characterizations of cubic IVIF subalgebra and cubic IVIF ideal are discussed. Full article

(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)

24 pages, 576 KiB

Open AccessArticle

Managing Interacting Criteria: Application to Environmental Evaluation Practices

by Teresa González-Arteaga, Rocio De Andrés Calle and Luis Martínez

Axioms 2018, 7(1), 4; https://doi.org/10.3390/axioms7010004 - 16 Jan 2018

Cited by 2 | Viewed by 3886

Abstract

The need for organizations to evaluate their environmental practices has been recently increasing. This fact has led to the development of many approaches to appraise such practices. In this paper, a novel decision model to evaluate company’s environmental practices is proposed to improve [...] Read more.

The need for organizations to evaluate their environmental practices has been recently increasing. This fact has led to the development of many approaches to appraise such practices. In this paper, a novel decision model to evaluate company’s environmental practices is proposed to improve traditional evaluation process in different facets. Firstly, different reviewers’ collectives related to the company’s activity are taken into account in the process to increase company internal efficiency and external legitimacy. Secondly, following the standard ISO 14031, two general categories of environmental performance indicators, management and operational, are considered. Thirdly, since the assumption of independence among environmental indicators is rarely verified in environmental context, an aggregation operator to bear in mind the relationship among such indicators in the evaluation results is proposed. Finally, this new model integrates quantitative and qualitative information with different scales using a multi-granular linguistic model that allows to adapt diverse evaluation scales according to appraisers’ knowledge. Full article

(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)

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303 KiB

Open AccessArticle

On Indistinguishability Operators, Fuzzy Metrics and Modular Metrics

by Juan-José Miñana and Oscar Valero

Axioms 2017, 6(4), 34; https://doi.org/10.3390/axioms6040034 - 15 Dec 2017

Cited by 10 | Viewed by 3106

Abstract

The notion of indistinguishability operator was introduced by Trillas, E. in 1982, with the aim of fuzzifying the crisp notion of equivalence relation. Such operators allow for measuring the similarity between objects when there is a limitation on the accuracy of the performed [...] Read more.

The notion of indistinguishability operator was introduced by Trillas, E. in 1982, with the aim of fuzzifying the crisp notion of equivalence relation. Such operators allow for measuring the similarity between objects when there is a limitation on the accuracy of the performed measurement or a certain degree of similarity can be only determined between the objects being compared. Since Trillas introduced such kind of operators, many authors have studied their properties and applications. In particular, an intensive research line is focused on the metric behavior of indistinguishability operators. Specifically, the existence of a duality between metrics and indistinguishability operators has been explored. In this direction, a technique to generate metrics from indistinguishability operators, and vice versa, has been developed by several authors in the literature. Nowadays, such a measurement of similarity is provided by the so-called fuzzy metrics when the degree of similarity between objects is measured relative to a parameter. The main purpose of this paper is to extend the notion of indistinguishability operator in such a way that the measurements of similarity are relative to a parameter and, thus, classical indistinguishability operators and fuzzy metrics can be retrieved as a particular case. Moreover, we discuss the relationship between the new operators and metrics. Concretely, we prove the existence of a duality between them and the so-called modular metrics, which provide a dissimilarity measurement between objects relative to a parameter. The new duality relationship allows us, on the one hand, to introduce a technique for generating the new indistinguishability operators from modular metrics and vice versa and, on the other hand, to derive, as a consequence, a technique for generating fuzzy metrics from modular metrics and vice versa. Furthermore, we yield examples that illustrate the new results. Full article

(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)

262 KiB

Open AccessArticle

Existence of Order-Preserving Functions for Nontotal Fuzzy Preference Relations under Decisiveness

by Paolo Bevilacqua, Gianni Bosi and Magalì Zuanon

Axioms 2017, 6(4), 29; https://doi.org/10.3390/axioms6040029 - 28 Oct 2017

Cited by 4 | Viewed by 3329

Abstract

Looking at decisiveness as crucial, we discuss the existence of an order-preserving function for the nontotal crisp preference relation naturally associated to a nontotal fuzzy preference relation. We further present conditions for the existence of an upper semicontinuous order-preserving function for a fuzzy [...] Read more.

Looking at decisiveness as crucial, we discuss the existence of an order-preserving function for the nontotal crisp preference relation naturally associated to a nontotal fuzzy preference relation. We further present conditions for the existence of an upper semicontinuous order-preserving function for a fuzzy binary relation on a crisp topological space. Full article

(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)

261 KiB

Open AccessArticle

Orness For Idempotent Aggregation Functions

by Leire Legarreta, Inmaculada Lizasoain and Iraide Mardones-Pérez

Axioms 2017, 6(3), 25; https://doi.org/10.3390/axioms6030025 - 20 Sep 2017

Cited by 3 | Viewed by 3551

Abstract

Aggregation functions are mathematical operators that merge given data in order to obtain a global value that preserves the information given by the data as much as possible. In most practical applications, this value is expected to be between the infimum and the [...] Read more.

Aggregation functions are mathematical operators that merge given data in order to obtain a global value that preserves the information given by the data as much as possible. In most practical applications, this value is expected to be between the infimum and the supremum of the given data, which is guaranteed only when the aggregation functions are idempotent. Ordered weighted averaging (OWA) operators are particular cases of this kind of function, with the particularity that the obtained global value depends on neither the source nor the expert that provides each datum, but only on the set of values. They have been classified by means of the orness—a measurement of the proximity of an OWA operator to the OR-operator. In this paper, the concept of orness is extended to the framework of idempotent aggregation functions defined both on the real unit interval and on a complete lattice with a local finiteness condition. Full article

(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)

1403 KiB

Open AccessArticle

New Order on Type 2 Fuzzy Numbers

by Pablo Hernández, Susana Cubillo, Carmen Torres-Blanc and José A. Guerrero

Axioms 2017, 6(3), 22; https://doi.org/10.3390/axioms6030022 - 28 Jul 2017

Cited by 6 | Viewed by 4107

Abstract

Since Lotfi A. Zadeh introduced the concept of fuzzy sets in 1965, many authors have devoted their efforts to the study of these new sets, both from a theoretical and applied point of view. Fuzzy sets were later extended in order to get [...] Read more.

Since Lotfi A. Zadeh introduced the concept of fuzzy sets in 1965, many authors have devoted their efforts to the study of these new sets, both from a theoretical and applied point of view. Fuzzy sets were later extended in order to get more adequate and flexible models of inference processes, where uncertainty, imprecision or vagueness is present. Type 2 fuzzy sets comprise one of such extensions. In this paper, we introduce and study an extension of the fuzzy numbers (type 1), the type 2 generalized fuzzy numbers and type 2 fuzzy numbers. Moreover, we also define a partial order on these sets, which extends into these sets the usual order on real numbers, which undoubtedly becomes a new option to be taken into account in the existing total preorders for ranking interval type 2 fuzzy numbers, which are a subset of type 2 generalized fuzzy numbers. Full article

(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)

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284 KiB

Open AccessArticle

Assigning Numerical Scores to Linguistic Expressions

by María Jesús Campión, Edurne Falcó, José Luis García-Lapresta and Esteban Induráin

Axioms 2017, 6(3), 19; https://doi.org/10.3390/axioms6030019 - 06 Jul 2017

Cited by 2 | Viewed by 4072

Abstract

In this paper, we study different methods of scoring linguistic expressions defined on a finite set, in the search for a linear order that ranks all those possible expressions. Among them, particular attention is paid to the canonical extension, and its representability through [...] Read more.

In this paper, we study different methods of scoring linguistic expressions defined on a finite set, in the search for a linear order that ranks all those possible expressions. Among them, particular attention is paid to the canonical extension, and its representability through distances in a graph plus some suitable penalization of imprecision. The relationship between this setting and the classical problems of numerical representability of orderings, as well as extension of orderings from a set to a superset is also explored. Finally, aggregation procedures of qualitative rankings and scorings are also analyzed. Full article

(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)

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