Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications

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Topic Information

Dear Colleagues,

With the rapid development of new technologies such as artificial intelligence and big date, as the mathematical basis of artificial intelligence, fuzzy set theory has always been concerned by scholars. And fuzzy analysis theory is an important part of it. As the basic concept of fuzzy analysis, fuzzy number, fuzzy difference, fuzzy differential, etc. involve many related important theories, such as: fuzzy difference equation, fuzzy differential equation, fuzzy optimization, fuzzy game, fuzzy decision-making, fuzzy control, etc. Therefore, it is necessary to study the fuzzy number, fuzzy difference, fuzzy differential and related theories and applications.

This Topic, “Fuzzy Number, Fuzzy Difference, Fuzzy Differential：Theory and  Applications”, invites papers on theoretical and applied issues, including, but not limited to, the following:

• Theories and applications of fuzzy Number.
• Theories and applications of fuzzy difference equation.
• Theories and applications of fuzzy differential equation. 
• Techniques and algorithms of fuzzy optimization.
• Strategy of fuzzy game.
• Methods and analysis fuzzy decision-making.  
• Techniques and methods of fuzzy control.
• This Topic will present the results of research describing recent advances in fuzzy number, fuzzy difference, and fuzzy differential.

Prof. Dr. Changyou Wang
Prof. Dr. Dong Qiu
Dr. Yonghong Shen
Topic Editors

Keywords

Participating Journals

Journal Name	Impact Factor	CiteScore	Launched Year	First Decision (median)	APC
Axioms axioms	1.9	-	2012	22.8 Days	CHF 2400
Information information	2.4	6.9	2010	16.4 Days	CHF 1600
Mathematics mathematics	2.3	4.0	2013	18.3 Days	CHF 2600
Symmetry symmetry	2.2	5.4	2009	17.3 Days	CHF 2400

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8 pages, 176 KiB  

Open AccessEditorial

Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications

by Changyou Wang, Dong Qiu and Yonghong Shen

Axioms 2025, 14(4), 254; https://doi.org/10.3390/axioms14040254 - 28 Mar 2025

Viewed by 268

Abstract

This Topic Issue comprises sixteen pioneering academic papers that delve deeply into the realm of fuzzy reasoning, optimization, decision making, and control [...] Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

24 pages, 683 KiB  

Open AccessArticle

Fuzzy Reasoning Symmetric Quintuple-Implication Method for Mixed Information and Its Application

by Ning Yao, Hao Chen, Ruirui Zhao and Minxia Luo

Symmetry 2025, 17(3), 369; https://doi.org/10.3390/sym17030369 - 28 Feb 2025

Viewed by 405

Abstract

Rule-based reasoning with different kinds of uncertain information has been identified in numerous applications within the real world. Any reasoning method must be able to coherently obtain the inference result by composing the given if–then rule with the assertion of the given input. [...] Read more.

Rule-based reasoning with different kinds of uncertain information has been identified in numerous applications within the real world. Any reasoning method must be able to coherently obtain the inference result by composing the given if–then rule with the assertion of the given input. The symmetric quintuple-implication principle was established by introducing symmetry into the five implication operators included. For example, the first, third and fifth implication operators exhibit symmetric properties, i.e., the three implication operators are taken as the same kind of operator and the second and fourth implication operators satisfy symmetry, that is, the two implication operators take the same kind of operator. So, the reasoning method induced by this principle possesses significant advantages in terms of its logical foundation and reductivity. This paper derives and studies reasoning methods for the mixture of fuzzy information and intuitionistic fuzzy information based on the symmetric quintuple-implication principle where all five implication operators satisfy symmetry (also called the quintuple-implication principle). Such reasoning methods are based on the ideas that the input and the given if–then rule can be combined for calculation only when their information representations exhibit consistency. An inconsistent given if–then rule with two different representations should be regarded as the composition of two different consistent given if–then rules with their own unique representations. This paper then elaborates on the methods by employing the possibility and necessity operators and the quintuple-implication principle from the perspective of whether the representation of rule antecedent and rule consequent is consistent or not. The reductivity of all the proposed reasoning methods is also analyzed in detail. This paper mainly contributes to the development of a novel mixed information reasoning framework, along with the introduction of the quintuple-implication principle into reasoning with mixed information. The proposed methods have also been applied to pattern recognition, and several experiments demonstrate that the mixed information reasoning methods based on the quintuple-implication principle are superior to the corresponding methods based on the triple I principle. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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23 pages, 374 KiB  

Open AccessArticle

General Trapezoidal-Type Inequalities in Fuzzy Settings

by Muhammad Amer Latif

Mathematics 2024, 12(19), 3112; https://doi.org/10.3390/math12193112 - 4 Oct 2024

Viewed by 936

Abstract

In this study, trapezoidal-type inequalities in fuzzy settings have been investigated. The theory of fuzzy analysis has been discussed in detail. The integration by parts formula of analysis of fuzzy mathematics has been employed to establish an equality. Trapezoidal-type inequality for functions with [...] Read more.

In this study, trapezoidal-type inequalities in fuzzy settings have been investigated. The theory of fuzzy analysis has been discussed in detail. The integration by parts formula of analysis of fuzzy mathematics has been employed to establish an equality. Trapezoidal-type inequality for functions with values in the fuzzy number-valued space is proven by applying the proven equality together with the properties of a metric defined on the set of fuzzy number-valued space and Höler’s inequality. The results proved in this research provide generalizations of the results from earlier existing results in the field of mathematical inequalities. An example is designed by defining a function that has values in fuzzy number-valued space and validated the results numerically using the software Mathematica (latest v. 14.1). The p-levels of the defined fuzzy number-valued mapping have been shown graphically for different values of $p\in \left[0,1\right]$. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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23 pages, 350 KiB  

Open AccessArticle

Introducing Fixed-Point Theorems and Applications in Fuzzy Bipolar b-Metric Spaces with ψ_α- and ϝ_η-Contractive Maps

by Salam Alnabulsi, Wael Mahmoud Mohammad Salameh and Mohammad H. M. Rashid

Symmetry 2024, 16(6), 777; https://doi.org/10.3390/sym16060777 - 20 Jun 2024

Cited by 1 | Viewed by 1371

Abstract

In this study, we introduce novel concepts within the framework of fuzzy bipolar b-metric spaces, focusing on various mappings such as ${\psi }_{\alpha }$-contractive and ${\mathcal{ϝ}}_{\eta }$-contractive mappings, which are essential for quantifying distances between dissimilar elements. We establish fixed-point theorems [...] Read more.

In this study, we introduce novel concepts within the framework of fuzzy bipolar b-metric spaces, focusing on various mappings such as ${\psi }_{\alpha }$-contractive and ${\mathcal{ϝ}}_{\eta }$-contractive mappings, which are essential for quantifying distances between dissimilar elements. We establish fixed-point theorems for these mappings, demonstrating the existence of invariant points under certain conditions. To enhance the credibility and applicability of our findings, we provide illustrative examples that support these theorems and expand the existing knowledge in this field. Furthermore, we explore practical applications of our research, particularly in solving integral equations and fractional differential equations, showcasing the robustness and utility of our theoretical advancements. Symmetry, both in its traditional sense and within the fuzzy context, is fundamental to our study of fuzzy bipolar b-metric spaces. The introduced contractive mappings and fixed-point theorems expand the theoretical framework and offer robust tools for addressing practical problems where symmetry is significant. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

20 pages, 633 KiB  

Open AccessArticle

Intuitionistic Fuzzy Granular Matrix: Novel Calculation Approaches for Intuitionistic Fuzzy Covering-Based Rough Sets

by Jingqian Wang and Xiaohong Zhang

Axioms 2024, 13(6), 411; https://doi.org/10.3390/axioms13060411 - 18 Jun 2024

Cited by 15 | Viewed by 1255

Abstract

Intuitionistic fuzzy (IF) $\beta$-minimal description operators can deal with noise data in the IF covering-based rough set theory. That is to say, they can be used to find data that we need in IF environments. For an IF $\beta$-covering approximation space [...] Read more.

Intuitionistic fuzzy (IF) $\beta$-minimal description operators can deal with noise data in the IF covering-based rough set theory. That is to say, they can be used to find data that we need in IF environments. For an IF $\beta$-covering approximation space (i.e., an IF environment) with a high cardinality, it would be tedious and complicated to use IF set representations to calculate them. Therefore, it is necessary to find a quick method to obtain them. In this paper, we present the notion of IF $\beta$-maximal description based on the definition of IF $\beta$-minimal description, along with the concepts of IF granular matrix and IF reduction. Moreover, we propose matrix calculation methods for IF covering-based rough sets, such as IF $\beta$-minimal descriptions, IF $\beta$-maximal descriptions, and IF reductions. Firstly, the notion of an IF granular matrix is presented, which is used to calculate IF $\beta$-minimal description. Secondly, inspired by IF $\beta$-minimal description, we give the notion of IF $\beta$-maximal description. Furthermore, the matrix representations of IF $\beta$-maximal descriptions are presented. Next, two types of reductions for IF $\beta$-covering approximation spaces via IF $\beta$-minimal and fuzzy $\beta$-minimal descriptions are presented, along with their matrix representations. Finally, the new calculation methods are compared with corresponding set representations by carrying out several experiments. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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19 pages, 848 KiB  

Open AccessArticle

Enhancing Integer Time Series Model Estimations through Neural Network-Based Fuzzy Time Series Analysis

by Mohammed H. El-Menshawy, Mohamed S. Eliwa, Laila A. Al-Essa, Mahmoud El-Morshedy and Rashad M. EL-Sagheer

Symmetry 2024, 16(6), 660; https://doi.org/10.3390/sym16060660 - 27 May 2024

Cited by 2 | Viewed by 1237

Abstract

This investigation explores the effects of applying fuzzy time series (FTSs) based on neural network models for estimating a variety of spectral functions in integer time series models. The focus is particularly on the skew integer autoregressive of order one (NSINAR(1)) model. To [...] Read more.

This investigation explores the effects of applying fuzzy time series (FTSs) based on neural network models for estimating a variety of spectral functions in integer time series models. The focus is particularly on the skew integer autoregressive of order one (NSINAR(1)) model. To support this estimation, a dataset consisting of NSINAR(1) realizations with a sample size of n = 1000 is created. These input values are then subjected to fuzzification via fuzzy logic. The prowess of artificial neural networks in pinpointing fuzzy relationships is harnessed to improve prediction accuracy by generating output values. The study meticulously analyzes the enhancement in smoothing of spectral function estimators for NSINAR(1) by utilizing both input and output values. The effectiveness of the output value estimates is evaluated by comparing them to input value estimates using a mean-squared error (MSE) analysis, which shows how much better the output value estimates perform. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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24 pages, 522 KiB  

Open AccessArticle

The Enumeration of (⊙,∨)-Multiderivations on a Finite MV-Chain

by Xueting Zhao, Kai Duo, Aiping Gan and Yichuan Yang

Axioms 2024, 13(4), 250; https://doi.org/10.3390/axioms13040250 - 10 Apr 2024

Viewed by 972

Abstract

In this paper, $(\odot ,\vee )$-multiderivations on an MV-algebra A are introduced, the relations between $(\odot ,\vee )$-multiderivations and $(\odot ,\vee )$-derivations are discussed. The set $\mathrm{MD}\left(A\right)$ of  [...] Read more.

In this paper, $(\odot ,\vee )$-multiderivations on an MV-algebra A are introduced, the relations between $(\odot ,\vee )$-multiderivations and $(\odot ,\vee )$-derivations are discussed. The set $\mathrm{MD}\left(A\right)$ of $(\odot ,\vee )$-multiderivations on A can be equipped with a preorder, and $\left(\mathrm{MD}\right(A)/\sim ,\preccurlyeq )$ can be made into a partially ordered set with respect to some equivalence relation ∼. In particular, for any finite MV-chain $L_n$, $(\mathrm{MD}\left(L_n\right)/\sim ,\preccurlyeq )$ becomes a complete lattice. Finally, a counting principle is built to obtain the enumeration of $\mathrm{MD}\left(L_n\right)$. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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18 pages, 1844 KiB  

Open AccessArticle

Two Extensions of the Sugeno Class and a Novel Constructed Method of Strong Fuzzy Negation for the Generation of Non-Symmetric Fuzzy Implications

by Maria N. Rapti, Avrilia Konguetsof and Basil K. Papadopoulos

Symmetry 2024, 16(3), 317; https://doi.org/10.3390/sym16030317 - 6 Mar 2024

Viewed by 1048

Abstract

In this paper, we present two new classes of fuzzy negations. They are an extension of a well-known class of fuzzy negations, the Sugeno Class. We use it as a base for our work for the first two construction methods. The first method [...] Read more.

In this paper, we present two new classes of fuzzy negations. They are an extension of a well-known class of fuzzy negations, the Sugeno Class. We use it as a base for our work for the first two construction methods. The first method generates rational fuzzy negations, where we use a second-degree polynomial with two parameters. We investigate which of these two conditions must be satisfied to be a fuzzy negation. In the second method, we use an increasing function instead of the parameter δ of the Sugeno class. In this method, using an arbitrary increasing function with specific conditions, fuzzy negations are produced, not just rational ones. Moreover, we compare the equilibrium points of the produced fuzzy negation of the first method and the Sugeno class. We use the equilibrium point to present a novel method which produces strong fuzzy negations by using two decreasing functions which satisfy specific conditions. We also investigate the convexity of the new fuzzy negation. We give some conditions that coefficients of fuzzy negation of the first method must satisfy in order to be convex. We present some examples of the new fuzzy negations, and we use them to generate new non-symmetric fuzzy implications by using well-known production methods of non-symmetric fuzzy implications. We use convex fuzzy negations as decreasing functions to construct an Archimedean copula. Finally, we investigate the quadratic form of the copula and the conditions that the coefficients of the first method and the increasing function of the second method must satisfy in order to generate new copulas of this form. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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20 pages, 1847 KiB  

Open AccessArticle

Special Discrete Fuzzy Numbers on Countable Sets and Their Applications

by Na Qin and Zengtai Gong

Symmetry 2024, 16(3), 264; https://doi.org/10.3390/sym16030264 - 21 Feb 2024

Viewed by 1257

Abstract

There are some drawbacks to arithmetic and logic operations of general discrete fuzzy numbers, which limit their application. For example, the result of the addition operation of general discrete fuzzy numbers defined by the Zadeh’s extension principle may not satisfy the condition of [...] Read more.

There are some drawbacks to arithmetic and logic operations of general discrete fuzzy numbers, which limit their application. For example, the result of the addition operation of general discrete fuzzy numbers defined by the Zadeh’s extension principle may not satisfy the condition of becoming a discrete fuzzy number. In order to solve these problems, special discrete fuzzy numbers on countable sets are investigated in this paper. Since the representation theorem of fuzzy numbers is the basic tool of fuzzy analysis, two kinds of representation theorems of special discrete fuzzy numbers on countable sets are studied first. Then, the metrics of special discrete fuzzy numbers on countable sets are defined, and the relationship between these metrics and the uniform Hausdorff metric (i.e., supremum metric) of general fuzzy numbers is discussed. In addition, the triangular norm and triangular conorm operations (t-norm and t-conorm for short) of special discrete fuzzy numbers on countable sets are presented, and the properties of these two operators are proven. We also prove that these two operators satisfy the basic conditions for closure of operation and present some examples. Finally, the applications of special discrete fuzzy numbers on countable sets in image fusion and aggregation of subjective evaluation are proposed. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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12 pages, 1442 KiB  

Open AccessArticle

Simulation-Enhanced MQAM Modulation Identification in Communication Systems: A Subtractive Clustering-Based PSO-FCM Algorithm Study

by Zhi Quan, Hailong Zhang, Jiyu Luo and Haijun Sun

Information 2024, 15(1), 42; https://doi.org/10.3390/info15010042 - 12 Jan 2024

Viewed by 1740

Abstract

Signal modulation recognition is often reliant on clustering algorithms. The fuzzy c-means (FCM) algorithm, which is commonly used for such tasks, often converges to local optima. This presents a challenge, particularly in low-signal-to-noise-ratio (SNR) environments. We propose an enhanced FCM algorithm that incorporates [...] Read more.

Signal modulation recognition is often reliant on clustering algorithms. The fuzzy c-means (FCM) algorithm, which is commonly used for such tasks, often converges to local optima. This presents a challenge, particularly in low-signal-to-noise-ratio (SNR) environments. We propose an enhanced FCM algorithm that incorporates particle swarm optimization (PSO) to improve the accuracy of recognizing M-ary quadrature amplitude modulation (MQAM) signal orders. The process is a two-step clustering process. First, the constellation diagram of the received signal is used by a subtractive clustering algorithm based on SNR to figure out the initial number of clustering centers. The PSO-FCM algorithm then refines these centers to improve precision. Accurate signal classification and identification are achieved by evaluating the relative sizes of the radii around the cluster centers within the MQAM constellation diagram and determining the modulation order. The results indicate that the SC-based PSO-FCM algorithm outperforms the conventional FCM in clustering effectiveness, notably enhancing modulation recognition rates in low-SNR conditions, when evaluated against a variety of QAM signals ranging from 4QAM to 64QAM. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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18 pages, 371 KiB  

Open AccessArticle

S-Embedding of Lie Superalgebras and Its Implications for Fuzzy Lie Algebras

by Abdullah Assiry, Sabeur Mansour and Amir Baklouti

Axioms 2024, 13(1), 2; https://doi.org/10.3390/axioms13010002 - 19 Dec 2023

Viewed by 1521

Abstract

This paper performed an investigation into the s-embedding of the Lie superalgebra $\overrightarrow{(}{S}^{1\mid 1})$, a representation of smooth vector fields on a (1,1)-dimensional super-circle. Our primary objective was to establish a precise definition of the s-embedding, effectively [...] Read more.

This paper performed an investigation into the s-embedding of the Lie superalgebra $\overrightarrow{(}{S}^{1\mid 1})$, a representation of smooth vector fields on a (1,1)-dimensional super-circle. Our primary objective was to establish a precise definition of the s-embedding, effectively dissecting the Lie superalgebra into the superalgebra of super-pseudodifferential operators ( $S\psi D\odot$) residing on the super-circle $S^{1|1}$. We also introduce and rigorously define the central charge within the framework of $\overrightarrow{(}{S}^{1\mid 1})$, leveraging the canonical central extension of $S\psi D\odot$. Moreover, we expanded the scope of our inquiry to encompass the domain of fuzzy Lie algebras, seeking to elucidate potential connections and parallels between these ostensibly distinct mathematical constructs. Our exploration spanned various facets, including non-commutative structures, representation theory, central extensions, and central charges, as we aimed to bridge the gap between Lie superalgebras and fuzzy Lie algebras. To summarize, this paper is a pioneering work with two pivotal contributions. Initially, a meticulous definition of the s-embedding of the Lie superalgebra $\overrightarrow{(}{S}^{1|1})$ is provided, emphasizing the representationof smooth vector fields on the (1,1)-dimensional super-circle, thereby enriching a fundamental comprehension of the topic. Moreover, an investigation of the realm of fuzzy Lie algebras was undertaken, probing associations with conventional Lie superalgebras. Capitalizing on these discoveries, we expound upon the nexus between central extensions and provide a novel deformed representation of the central charge. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

17 pages, 341 KiB  

Open AccessArticle

Multi-Attribute Group Decision-Making Methods Based on Entropy Weights with q-Rung Picture Uncertain Linguistic Fuzzy Information

by Mengran Sun, Yushui Geng and Jing Zhao

Symmetry 2023, 15(11), 2027; https://doi.org/10.3390/sym15112027 - 8 Nov 2023

Viewed by 1755

Abstract

This paper introduces a new concept called q-rung picture uncertain linguistic fuzzy sets (q-RPULSs). These sets provide a reliable and comprehensive method for describing complex and uncertain decision-making information. In addition, q-RPULSs help to integrate the decision maker’s quantitative assessment ideas with qualitative [...] Read more.

This paper introduces a new concept called q-rung picture uncertain linguistic fuzzy sets (q-RPULSs). These sets provide a reliable and comprehensive method for describing complex and uncertain decision-making information. In addition, q-RPULSs help to integrate the decision maker’s quantitative assessment ideas with qualitative assessment information. For the q-RPUL multi-attribute group decision-making problem with unknown weight information, an entropy-based fuzzy set method for q-rung picture uncertainty language is proposed. The method considers the interrelationships among attributes and builds a q-rung picture uncertain language model. In addition, the q-RPULMSM operator and its related properties are discussed in this paper. This operator enables the fusion of q-RPULSs and helps to reach consensus in decision-making scenarios. To demonstrate the validity of the methodology, we provide a real case study involving commodity selection. Based on this case study, the reasonableness and superiority of the method are evaluated, highlighting the practical advantages and applicability of q-RPULSs in decision-making processes. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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29 pages, 1203 KiB  

Open AccessArticle

Defuzzification of Non-Linear Pentagonal Intuitionistic Fuzzy Numbers and Application in the Minimum Spanning Tree Problem

by Ali Mert

Symmetry 2023, 15(10), 1853; https://doi.org/10.3390/sym15101853 - 2 Oct 2023

Cited by 6 | Viewed by 1842

Abstract

In recent years, with the variety of digital objects around us becoming a source of information, the fields of artificial intelligence (AI) and machine learning (ML) have experienced very rapid development. Processing and converting the information around us into data within the framework [...] Read more.

In recent years, with the variety of digital objects around us becoming a source of information, the fields of artificial intelligence (AI) and machine learning (ML) have experienced very rapid development. Processing and converting the information around us into data within the framework of the information processing theory is important, as AI and ML techniques need large amounts of reliable data in the training and validation stages. Even though information naturally contains uncertainty, information must still be modeled and converted into data without neglecting this uncertainty. Mathematical techniques, such as the fuzzy theory and the intuitionistic fuzzy theory, are used for this purpose. In the intuitionistic fuzzy theory, membership and non-membership functions are employed to describe intuitionistic fuzzy sets and intuitionistic fuzzy numbers (IFNs). IFNs are characterized by the mathematical statements of these two functions. A more general and inclusive definition of IFN is always a requirement in AI technologies, as the uncertainty introduced by various information sources needs to be transformed into similar IFNs without neglecting the variety of uncertainty. In this paper, we proposed a general and inclusive mathematical definition for IFN and called this IFN a non-linear pentagonal intuitionistic fuzzy number (NLPIFN), which allows its users to maintain variety in uncertainty. We know that AI technology implementations are performed in computerized environments, so we need to transform the IFN into a crisp number to make such IFNs available in such environments. Techniques used in transformation are called defuzzification methods. In this paper, we proposed a short-cut formula for the defuzzification of a NLPIFN using the intuitionistic fuzzy weighted averaging based on levels (IF-WABL) method. We also implemented our findings in the minimum spanning tree problem by taking weights as NLPIFNs to determine the uncertainty in the process more precisely. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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20 pages, 330 KiB  

Open AccessArticle

Interval-Valued Topology on Soft Sets

by Sadi Bayramov, Çiğdem Gündüz Aras and Ljubiša D. R. Kočinac

Axioms 2023, 12(7), 692; https://doi.org/10.3390/axioms12070692 - 16 Jul 2023

Cited by 1 | Viewed by 1372

Abstract

In this paper, we study the concept of interval-valued fuzzy set on the family $\mathsf{SS}\left(X,E\right)$ of all soft sets over X with the set of parameters E and examine its basic properties. Later, we define the concept of interval-valued fuzzy [...] Read more.

In this paper, we study the concept of interval-valued fuzzy set on the family $\mathsf{SS}\left(X,E\right)$ of all soft sets over X with the set of parameters E and examine its basic properties. Later, we define the concept of interval-valued fuzzy topology (cotopology) $\tau$ on $\mathsf{SS}\left(X,E\right)$. We obtain that each interval-valued fuzzy topology is a descending family of soft topologies. In addition, we study some topological structures such as interval-valued fuzzy neighborhood system of a soft point, base and subbase of $\tau$ and investigate some relationships among them. Finally, we give some concepts such as direct sum, open mapping and continuous mapping and consider connections between them. A few examples support the presented results. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

16 pages, 313 KiB  

Open AccessArticle

Multiple-Attribute Decision Making Based on the Probabilistic Dominance Relationship with Fuzzy Algebras

by Amir Baklouti

Symmetry 2023, 15(6), 1188; https://doi.org/10.3390/sym15061188 - 2 Jun 2023

Cited by 4 | Viewed by 1465

Abstract

In multiple-attribute decision-making (MADM) problems, ranking the alternatives is an important step for making the best decision. Intuitionistic fuzzy numbers (IFNs) are a powerful tool for expressing uncertainty and vagueness in MADM problems. However, existing ranking methods for IFNs do not consider the [...] Read more.

In multiple-attribute decision-making (MADM) problems, ranking the alternatives is an important step for making the best decision. Intuitionistic fuzzy numbers (IFNs) are a powerful tool for expressing uncertainty and vagueness in MADM problems. However, existing ranking methods for IFNs do not consider the probabilistic dominance relationship between alternatives, which can lead to inconsistent and inaccurate rankings. In this paper, we propose a new ranking method for IFNs based on the probabilistic dominance relationship and fuzzy algebras. The proposed method is able to handle incomplete and uncertain information and can generate consistent and accurate rankings. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

17 pages, 326 KiB  

Open AccessArticle

New Arithmetic Operations of Non-Normal Fuzzy Sets Using Compatibility

by Hsien-Chung Wu

Axioms 2023, 12(3), 277; https://doi.org/10.3390/axioms12030277 - 7 Mar 2023

Cited by 2 | Viewed by 1642

Abstract

The new arithmetic operations of non-normal fuzzy sets are studied in this paper by using the extension principle and considering the general aggregation function. Usually, the aggregation functions are taken to be the minimum function or t-norms. In this paper, we considered a [...] Read more.

The new arithmetic operations of non-normal fuzzy sets are studied in this paper by using the extension principle and considering the general aggregation function. Usually, the aggregation functions are taken to be the minimum function or t-norms. In this paper, we considered a general aggregation function to set up the arithmetic operations of non-normal fuzzy sets. In applications, the arithmetic operations of fuzzy sets are always transferred to the arithmetic operations of their corresponding $\alpha$-level sets. When the aggregation function is taken to be the minimum function, this transformation is clearly realized. Since the general aggregation function was adopted in this paper, the concept of compatibility with $\alpha$-level sets is needed and is proposed, which can cover the conventional case using minimum functions as the special case. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

9 pages, 273 KiB  

Open AccessArticle

Best Approximation Results for Fuzzy-Number-Valued Continuous Functions

by Juan J. Font and Sergio Macario

Axioms 2023, 12(2), 192; https://doi.org/10.3390/axioms12020192 - 11 Feb 2023

Viewed by 1428

Abstract

In this paper, we study the best approximation of a fixed fuzzy-number-valued continuous function to a subset of fuzzy-number-valued continuous functions. We also introduce a method to measure the distance between a fuzzy-number-valued continuous function and a real-valued one. Then, we prove the [...] Read more.

In this paper, we study the best approximation of a fixed fuzzy-number-valued continuous function to a subset of fuzzy-number-valued continuous functions. We also introduce a method to measure the distance between a fuzzy-number-valued continuous function and a real-valued one. Then, we prove the existence of the best approximation of a fuzzy-number-valued continuous function to the space of real-valued continuous functions by using the well-known Michael selection theorem. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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