Axioms

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15 pages, 292 KiB

Open AccessArticle

Modeling Directional Monotonicity in Sequence with Copulas

by José Juan Quesada-Molina and Manuel Úbeda-Flores

Axioms 2024, 13(11), 785; https://doi.org/10.3390/axioms13110785 - 14 Nov 2024

Viewed by 249

In this paper, we present the concept of being monotonic in sequence according to a specific direction for a collection of random variables. This concept broadens the existing notions of multivariate dependence, such as sequential left-tail and right-tail dependence. Furthermore, we explore connections [...] Read more.

In this paper, we present the concept of being monotonic in sequence according to a specific direction for a collection of random variables. This concept broadens the existing notions of multivariate dependence, such as sequential left-tail and right-tail dependence. Furthermore, we explore connections with other multivariate dependence concepts, highlight key properties, and analyze the new concept within the framework of copulas. Several examples are provided to demonstrate our findings. Full article

(This article belongs to the Special Issue New Perspectives in Fuzzy Sets and Their Applications)

15 pages, 297 KiB

Open AccessArticle

Robust Semi-Infinite Interval Equilibrium Problem Involving Data Uncertainty: Optimality Conditions and Duality

by Gabriel Ruiz-Garzón, Rafaela Osuna-Gómez, Antonio Rufián-Lizana and Antonio Beato-Moreno

Axioms 2024, 13(11), 781; https://doi.org/10.3390/axioms13110781 - 13 Nov 2024

Viewed by 344

Abstract

In this paper, we model uncertainty in both the objective function and the constraints for the robust semi-infinite interval equilibrium problem involving data uncertainty. We particularize these conditions for the robust semi-infinite mathematical programming problem with interval-valued functions by extending the results from [...] Read more.

In this paper, we model uncertainty in both the objective function and the constraints for the robust semi-infinite interval equilibrium problem involving data uncertainty. We particularize these conditions for the robust semi-infinite mathematical programming problem with interval-valued functions by extending the results from the literature. We introduce the dual robust version of the above problem, prove the Mond–Weir-type weak and strong duality theorems, and illustrate our results with an example. Full article

(This article belongs to the Special Issue New Perspectives in Fuzzy Sets and Their Applications)

31 pages, 753 KiB

Open AccessArticle

Divergence and Similarity Characteristics for Two Fuzzy Measures Based on Associated Probabilities

by Gia Sirbiladze, Bidzina Midodashvili and Teimuraz Manjafarashvili

Axioms 2024, 13(11), 776; https://doi.org/10.3390/axioms13110776 - 9 Nov 2024

Viewed by 538

Abstract

The article deals with the definitions of the distance, divergence, and similarity characteristics between two finite fuzzy measures, which are generalizations of the same definitions between two finite probability distributions. As is known, a fuzzy measure can be uniquely represented by the so-called [...] Read more.

The article deals with the definitions of the distance, divergence, and similarity characteristics between two finite fuzzy measures, which are generalizations of the same definitions between two finite probability distributions. As is known, a fuzzy measure can be uniquely represented by the so-called its associated probability class (APC). The idea of generalization is that new definitions of distance, divergence, and similarity between fuzzy measures are reduced to the definitions of distance, divergence, and similarity between the APCs of fuzzy measures. These definitions are based on the concept of distance generator. The proof of the correctness of generalizations is provided. Constructed distance, similarity, and divergence relations can be used in such applied problems as: determining the difference between Dempster-Shafer belief structures; Constructions of collaborative filtering similarity relations; non-additive and interactive parameters of machine learning in phase space metrics definition, object clustering, classification and other tasks. In this work, a new concept is used in the fuzzy measure identification problem for a certain multi-attribute decision-making (MADM) environment. For this, a conditional optimization problem with one objective function representing the distance, divergence or similarity index is formulated. Numerical examples are discussed and a comparative analysis of the obtained results is presented. Full article

(This article belongs to the Special Issue New Perspectives in Fuzzy Sets and Their Applications)

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33 pages, 3475 KiB

Open AccessArticle

Adding a Degree of Certainty to Deductions in a Fuzzy Temporal Constraint Prolog: FTCProlog

by María-Antonia Cárdenas-Viedma

Axioms 2024, 13(7), 472; https://doi.org/10.3390/axioms13070472 - 12 Jul 2024

Viewed by 738

Abstract

The management of time is essential in most AI-related applications. In addition, we know that temporal information is often not precise. In fact, in most cases, it is necessary to deal with imprecision and/or uncertainty. On the other hand, there is the need [...] Read more.

The management of time is essential in most AI-related applications. In addition, we know that temporal information is often not precise. In fact, in most cases, it is necessary to deal with imprecision and/or uncertainty. On the other hand, there is the need to handle the implicit common-sense information present in many temporal statements. In this paper, we present FTCProlog, a logic programming language capable of handling fuzzy temporal constraints soundly and efficiently. The main difference of FTCProlog with respect to its predecessor, PROLogic, is its ability to associate a certainty index with deductions obtained through SLD-resolution. This resolution is based on a proposal within the theoretical logical framework FTCLogic. This model integrates a first-order logic based on possibilistic logic with the Fuzzy Temporal Constraint Networks (FTCNs) that allow efficient time management. The calculation of the certainty index can be useful in applications where one wants to verify the extent to which the times elapsed between certain events follow a given temporal pattern. In this paper, we demonstrate that the calculation of this index respects the properties of the theoretical model regarding its semantics. FTCProlog is implemented in Haskell. Full article

(This article belongs to the Special Issue New Perspectives in Fuzzy Sets and Their Applications)

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14 pages, 287 KiB

Open AccessArticle

On Proximity Spaces Constructed on Rough Sets

by Jong Il Baek, S. E. Abbas, Kul Hur and Ismail Ibedou

Axioms 2024, 13(3), 199; https://doi.org/10.3390/axioms13030199 - 15 Mar 2024

Viewed by 1012

Abstract

Based on equivalence relation R on X, equivalence class

[x]

of a point and equivalence class

[A]

of a subset represent the neighborhoods of x and A, respectively. These neighborhoods play the main role in defining separation [...] Read more.

Based on equivalence relation R on X, equivalence class

[x]

of a point and equivalence class

[A]

of a subset represent the neighborhoods of x and A, respectively. These neighborhoods play the main role in defining separation axioms, metric spaces, proximity relations and uniformity structures on an approximation space

(X, R)

depending on the lower approximation and the upper approximation of rough sets. The properties and the possible implications of these definitions are studied. The generated approximation topology

τ_{R}

on X is equivalent to the generated topologies associated with metric d, proximity

δ

and uniformity

U

on X. Separated metric spaces, separated proximity spaces and separated uniform spaces are defined and it is proven that both are associating exactly discrete topology

τ_{R}

on X. Full article

(This article belongs to the Special Issue New Perspectives in Fuzzy Sets and Their Applications)

28 pages, 1770 KiB

Open AccessArticle

Fitting Insurance Claim Reserves with Two-Way ANOVA and Intuitionistic Fuzzy Regression

by Jorge De Andrés-Sánchez

Axioms 2024, 13(3), 184; https://doi.org/10.3390/axioms13030184 - 11 Mar 2024

Cited by 1 | Viewed by 1275

Abstract

A highly relevant topic in the actuarial literature is so-called “claim reserving” or “loss reserving”, which involves estimating reserves to be provisioned for pending claims, as they can be deferred over various periods. This explains the proliferation of methods that aim to estimate [...] Read more.

A highly relevant topic in the actuarial literature is so-called “claim reserving” or “loss reserving”, which involves estimating reserves to be provisioned for pending claims, as they can be deferred over various periods. This explains the proliferation of methods that aim to estimate these reserves and their variability. Regression methods are widely used in this setting. If we model error terms as random variables, the variability of provisions can consequently be modelled stochastically. The use of fuzzy regression methods also allows modelling uncertainty for reserve values using tools from the theory of fuzzy subsets. This study follows this second approach and proposes projecting claim reserves using a generalization of fuzzy numbers (FNs), so-called intuitionistic fuzzy numbers (IFNs), through the use of intuitionistic fuzzy regression. While FNs allow epistemic uncertainty to be considered in variable estimation, IFNs add bipolarity to the analysis by incorporating both positive and negative information regarding actuarial variables. Our analysis is grounded in the ANOVA two-way framework, which is adapted to the use of intuitionistic regression. Similarly, we compare our results with those obtained using deterministic and stochastic chain-ladder methods and those obtained using two-way statistical ANOVA. Full article

(This article belongs to the Special Issue New Perspectives in Fuzzy Sets and Their Applications)

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14 pages, 592 KiB

Open AccessArticle

On Equivalence Operators Derived from Overlap and Grouping Functions

by Lei Du, Yingying Xu, Haifeng Song and Songsong Dai

Axioms 2024, 13(2), 123; https://doi.org/10.3390/axioms13020123 - 17 Feb 2024

Cited by 1 | Viewed by 1035

Abstract

This paper introduces the concept of equivalence operators based on overlap and grouping functions where the associativity property is not strongly required. Overlap functions and grouping functions are weaker than positive and continuous t-norms and t-conorms, respectively. Therefore, these equivalence operators do not [...] Read more.

This paper introduces the concept of equivalence operators based on overlap and grouping functions where the associativity property is not strongly required. Overlap functions and grouping functions are weaker than positive and continuous t-norms and t-conorms, respectively. Therefore, these equivalence operators do not necessarily satisfy certain properties, such as associativity and the neutrality principle. In this paper, two models of fuzzy equivalence operators are obtained by the composition of overlap functions, grouping functions and fuzzy negations. Their main properties are also studied. Full article

(This article belongs to the Special Issue New Perspectives in Fuzzy Sets and Their Applications)

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18 pages, 1158 KiB

Open AccessArticle

Integrating the Coupled Markov Chain and Fuzzy Analytic Hierarchy Process Model for Dynamic Decision Making

by Jih-Jeng Huang and Chin-Yi Chen

Axioms 2024, 13(2), 95; https://doi.org/10.3390/axioms13020095 - 30 Jan 2024

Cited by 2 | Viewed by 1301

Abstract

This paper introduces a pioneering model that merges coupled Markov chains (CMC) with the fuzzy analytic hierarchy process (FAHP) to enhance multi-criteria decision making (MCDM) amidst the dynamic interplay of criteria. Traditional MCDM frameworks often lack the granularity to manage the intricate and [...] Read more.

This paper introduces a pioneering model that merges coupled Markov chains (CMC) with the fuzzy analytic hierarchy process (FAHP) to enhance multi-criteria decision making (MCDM) amidst the dynamic interplay of criteria. Traditional MCDM frameworks often lack the granularity to manage the intricate and changing relationships among criteria. Our model addresses this gap by integrating fuzzy numbers into AHP, providing a nuanced means to handle the inherent uncertainty of decision criteria. The application of the Kronecker product in CMC enriches our approach, offering a data-driven analysis while mitigating excessive dependence on subjective expert opinion. A comprehensive numerical example underlines the model’s improved decision-making accuracy and efficiency, marking a substantial advancement in MCDM methodologies. This research contributes to the field by presenting a sophisticated yet practical framework for dynamic decision analysis in complex uncertain environments. Full article

(This article belongs to the Special Issue New Perspectives in Fuzzy Sets and Their Applications)

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18 pages, 825 KiB

Open AccessArticle

Ranking Alternatives Using a Fuzzy Preference Relation-Based Fuzzy VIKOR Method

by Hanh-Thao Le and Ta-Chung Chu

Axioms 2023, 12(12), 1079; https://doi.org/10.3390/axioms12121079 - 24 Nov 2023

Viewed by 1179

Abstract

The process of evaluating and ranking alternatives, including the aggregation of various qualitative and quantitative criteria and weights of criteria, can be recognized as a fuzzy multiple criteria decision-making (MCDM) problem. In fuzzy MCDM problems, qualitative criteria and criteria weights are usually indicated [...] Read more.

The process of evaluating and ranking alternatives, including the aggregation of various qualitative and quantitative criteria and weights of criteria, can be recognized as a fuzzy multiple criteria decision-making (MCDM) problem. In fuzzy MCDM problems, qualitative criteria and criteria weights are usually indicated in linguistic values expressed in terms of fuzzy numbers, and values under quantitative criteria are usually crisp numbers. How to properly aggregate them for evaluating and selecting alternatives has been an important research issue. To help decision-makers make the most suitable selection, this paper proposes a fuzzy preference relation-based fuzzy VIKOR method. VIKOR is a compromise ranking method to solve discrete MCDM problems in complex systems. In this study, the F-preference relation is applied to compare fuzzy numbers with their means to produce a single index of a dominance level while still maintaining fuzzy meaning of the original linguistic values. The inverse function is applied to obtain the defuzzification values of Beta 1–4 to assist in the completion of the proposed method, and formulas can be clearly derived to facilitate the ranking procedure. Introducing fuzzy preference relation into fuzzy VIKOR can simplify the calculation procedure for more efficient decision-making. The proposed method is new and has never been seen before. A numerical example and a comparison of the proposed method are conducted to show and verify its expedience and advantage. Full article

(This article belongs to the Special Issue New Perspectives in Fuzzy Sets and Their Applications)

33 pages, 367 KiB

Open AccessArticle

Normed Space of Fuzzy Intervals and Its Topological Structure

by Hsien-Chung Wu

Axioms 2023, 12(10), 996; https://doi.org/10.3390/axioms12100996 - 22 Oct 2023

Viewed by 1117

Abstract

The space,

Ƒ_{c c} (R)

, of all fuzzy intervals in

R

cannot form a vector space. However, the space

Ƒ_{c c} (R)

maintains a vector structure by treating the addition of fuzzy intervals as a vector [...] Read more.

The space,

Ƒ_{c c} (R)

, of all fuzzy intervals in

R

cannot form a vector space. However, the space

Ƒ_{c c} (R)

maintains a vector structure by treating the addition of fuzzy intervals as a vector addition and treating the scalar multiplication of fuzzy intervals as a scalar multiplication of vectors. The only difficulty in taking care of

Ƒ_{c c} (R)

is missing the additive inverse element. This means that each fuzzy interval that is subtracted from itself cannot be a zero element in

Ƒ_{c c} (R)

. Although

Ƒ_{c c} (R)

cannot form a vector space, we still can endow a norm on the space

Ƒ_{c c} (R)

by following its vector structure. Under this setting, many different types of open sets can be proposed by using the different types of open balls. The purpose of this paper is to study the topologies generated by these different types of open sets. Full article

(This article belongs to the Special Issue New Perspectives in Fuzzy Sets and Their Applications)

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