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18 pages, 5241 KiB

Open AccessArticle

Complex Uncertainty of Surface Data Modeling via the Type-2 Fuzzy B-Spline Model

by Rozaimi Zakaria, Abd. Fatah Wahab, Isfarita Ismail and Mohammad Izat Emir Zulkifly

Mathematics 2021, 9(9), 1054; https://doi.org/10.3390/math9091054 - 7 May 2021

Cited by 12 | Viewed by 2020

Abstract

This paper discusses the construction of a type-2 fuzzy B-spline model to model complex uncertainty of surface data. To construct this model, the type-2 fuzzy set theory, which includes type-2 fuzzy number concepts and type-2 fuzzy relation, is used to define the complex [...] Read more.

This paper discusses the construction of a type-2 fuzzy B-spline model to model complex uncertainty of surface data. To construct this model, the type-2 fuzzy set theory, which includes type-2 fuzzy number concepts and type-2 fuzzy relation, is used to define the complex uncertainty of surface data in type-2 fuzzy data/control points. These type-2 fuzzy data/control points are blended with the B-spline surface function to produce the proposed model, which can be visualized and analyzed further. Various processes, namely fuzzification, type-reduction and defuzzification are defined to achieve a crisp, type-2 fuzzy B-spline surface, representing uncertainty complex surface data. This paper ends with a numerical example of terrain modeling, which shows the effectiveness of handling the uncertainty complex data. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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16 pages, 672 KiB

Open AccessArticle

Fuzzy Governance Model

by Enriqueta Mancilla-Rendón, Carmen Lozano and Enrique Torres-Esteva

Mathematics 2021, 9(5), 481; https://doi.org/10.3390/math9050481 - 26 Feb 2021

Cited by 2 | Viewed by 2081

Abstract

This article aims to analyze the functions of corporate governance agents as a key part of the study and evaluation of the internal control by the independent auditor to propose a governance fuzzy model based on legality. This is a descriptive–hermeneutical study based [...] Read more.

This article aims to analyze the functions of corporate governance agents as a key part of the study and evaluation of the internal control by the independent auditor to propose a governance fuzzy model based on legality. This is a descriptive–hermeneutical study based on mercantile-securities law, the code of best practice of corporate governance, and auditing standards. The research design is cross-sectional and uses fuzzy logic theory as an alternative tool in contrast to classical mathematical models. The results suggest that corporate governance agents strongly influence the application of a management system. Evidence is given regarding the positive relationship between the functions of corporate governance agents as a management system. Additionally, the importance of an internal control management system as an inherent mechanism for governance is proven. The scientific value of this work lies in showing how the interaction between the application of mathematical models based on fuzzy set theory and the qualitative attributes of internal control policies and practices. It is a tool to evaluate governance as a management system for decision making. This work emphasizes that a model based on fuzzy sets is useful to evaluate a management system of internal control policies and procedures necessary to improve corporate governance. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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27 pages, 6191 KiB

Open AccessEditor’s ChoiceArticle

Application of Hexagonal Fuzzy MCDM Methodology for Site Selection of Electric Vehicle Charging Station

by Arijit Ghosh, Neha Ghorui, Sankar Prasad Mondal, Suchitra Kumari, Biraj Kanti Mondal, Aditya Das and Mahananda Sen Gupta

Mathematics 2021, 9(4), 393; https://doi.org/10.3390/math9040393 - 16 Feb 2021

Cited by 43 | Viewed by 4994

Abstract

In this paper, the application of hexagonal fuzzy multiple-criteria decision-making (MCDM) methodology for the site selection of electric vehicle charging stations is considered. In this regard, four factors and thirteen sub-factors have been taken into consideration for E-vehicle charging site selection. In this [...] Read more.

In this paper, the application of hexagonal fuzzy multiple-criteria decision-making (MCDM) methodology for the site selection of electric vehicle charging stations is considered. In this regard, four factors and thirteen sub-factors have been taken into consideration for E-vehicle charging site selection. In this research, the geographic information system (GIS) has been incorporated with MCDM techniques. The fuzzy analytic hierarchy process (FAHP) is used to obtain a fuzzy weight of factors and sub-factors. MCDM tools fuzzy technique for order of preference by similarity to ideal solution (FTOPSIS) and fuzzy complex proportional assessment (FCOPRAS) have been used to rank the selected sites. A centroid-based method for defuzzification and distance measure between two hexagonal fuzzy numbers (HFN) has been developed for this paper. A practical example in Howrah, India, is considered to show the applicability and usefulness of the model. The results depict the suitability of the proposed research. Comparative and sensitivity analyses have been demonstrated to check the reliability, robustness and effectiveness of the proposed method. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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20 pages, 352 KiB

Open AccessArticle

Statistical Parameters Based on Fuzzy Measures

by Fernando Reche, María Morales and Antonio Salmerón

Mathematics 2020, 8(11), 2015; https://doi.org/10.3390/math8112015 - 12 Nov 2020

Cited by 2 | Viewed by 1711

Abstract

In this paper, we study the problem of defining statistical parameters when the uncertainty is expressed using a fuzzy measure. We extend the concept of monotone expectation in order to define a monotone variance and monotone moments. We also study parameters that allow [...] Read more.

In this paper, we study the problem of defining statistical parameters when the uncertainty is expressed using a fuzzy measure. We extend the concept of monotone expectation in order to define a monotone variance and monotone moments. We also study parameters that allow the joint analysis of two functions defined over the same reference set. Finally, we propose some parameters over product spaces, considering the case in which a function over the product space is available and also the case in which such function is obtained by combining those in the marginal spaces. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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9 pages, 3377 KiB

Open AccessArticle

Eigen Fuzzy Sets and their Application to Evaluate the Effectiveness of Actions in Decision Problems

by Ferdinando Di Martino and Salvatore Sessa

Mathematics 2020, 8(11), 1999; https://doi.org/10.3390/math8111999 - 9 Nov 2020

Cited by 2 | Viewed by 2041

Abstract

We propose a new method based on the greatest (resp., smallest) eigen fuzzy set (GEFS, resp., SEFS) of a fuzzy relation R with respect to the max–min (resp., min–max) composition in order to implement the actions of a decisor. Using information derived from [...] Read more.

We propose a new method based on the greatest (resp., smallest) eigen fuzzy set (GEFS, resp., SEFS) of a fuzzy relation R with respect to the max–min (resp., min–max) composition in order to implement the actions of a decisor. Using information derived from judgments of the evaluators on how much a characteristic is improved with respect to others, we construct the fuzzy relations, RMAX (resp., RMIN), where any entry RMAX_ijj (resp., RMIN_ij) expresses how much the efficacy produced on the ith characteristic is equal to or greater (resp., lesser) than that one produced by the jth characteristic. The GEFS of RMAX (resp., SEFS of RMIN) are calculated in order to improve the performances of each characteristic. In the wake of previous applications based on GEFS and SEFS, we propose a method to evaluate the tourism enhancement policies in the historical center of an important Italian city. This method is new and different from those known in the literature so far. It is applied to evaluate benefits brought about by locals in order to enhance tourism in a historical center Comparison tests show that the results obtained are consistent with those expressed by the tourists interviewed Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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15 pages, 349 KiB

Open AccessArticle

A New Continuous-Discrete Fuzzy Model and Its Application in Finance

by Hoang Viet Long, Haifa Bin Jebreen and Y. Chalco-Cano

Mathematics 2020, 8(10), 1808; https://doi.org/10.3390/math8101808 - 16 Oct 2020

Cited by 2 | Viewed by 2096

Abstract

In this paper, we propose a fuzzy differential-difference equation for modeling of mixed continuous-discrete phenomena. In the special case, we present the general solution of linear fuzzy differential-difference equations. The dynamical process in the intervals is presented by the corresponding fuzzy differential equation [...] Read more.

In this paper, we propose a fuzzy differential-difference equation for modeling of mixed continuous-discrete phenomena. In the special case, we present the general solution of linear fuzzy differential-difference equations. The dynamical process in the intervals is presented by the corresponding fuzzy differential equation and with impulsive jumps in some points. We illustrate the applicability of the model to study the time value of money. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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16 pages, 295 KiB

Open AccessArticle

by Güzide Şenel, Jeong-Gon Lee and Kul Hur

Mathematics 2020, 8(10), 1690; https://doi.org/10.3390/math8101690 - 1 Oct 2020

Cited by 7 | Viewed by 1850

Abstract

In this paper, in order to apply the concept of octahedron sets to multi-criteria group decision-making problems, we define several similarity and distance measures for octahedron sets. We present a multi-criteria group decision-making method with linguistic variables in octahedron set environment. We give [...] Read more.

In this paper, in order to apply the concept of octahedron sets to multi-criteria group decision-making problems, we define several similarity and distance measures for octahedron sets. We present a multi-criteria group decision-making method with linguistic variables in octahedron set environment. We give a numerical example for multi-criteria group decision-making problems. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

15 pages, 283 KiB

Open AccessArticle

On the Generalized Cross-Law of Importation in Fuzzy Logic

by Yifan Zhao and Kai Li

Mathematics 2020, 8(10), 1681; https://doi.org/10.3390/math8101681 - 1 Oct 2020

Viewed by 1652

Abstract

Recently, Baczyński et al. introduced two pexider-type generalisations of the law of importation in fuzzy logic, i.e.,

I (C (x, α), y) = I (x, J (α, y)

[...] Read more.

Recently, Baczyński et al. introduced two pexider-type generalisations of the law of importation in fuzzy logic, i.e.,

I (C (x, α), y) = I (x, J (α, y)

(GLI) and

I (C (x, α), y) = J (x, I (α, y)

(CLI), where C is a fuzzy conjunction and I, J are fuzzy implications. However, (CLI) has not been adequately investigated so far. In this paper, we firstly show that (CLI) can be derived from the α-migrativity of an R-implication obtained from an α-migrative t-norm. Secondly, the relationships between the satisfaction of the law of importation (LI) by the pairs (C, I) or (C, J) and the satisfaction of (CLI) by the triple (C, I, J) are studied. Moreover, some necessary conditions of (CLI) are given. Finally, we study (CLI) under three different perspectives. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

18 pages, 322 KiB

Open AccessArticle

Schauder-Type Fixed Point Theorem in Generalized Fuzzy Normed Linear Spaces

by S. Chatterjee, T. Bag and Jeong-Gon Lee

Mathematics 2020, 8(10), 1643; https://doi.org/10.3390/math8101643 - 23 Sep 2020

Cited by 2 | Viewed by 2311

Abstract

In the present article, the Schauder-type fixed point theorem for the class of fuzzy continuous, as well as fuzzy compact operators is established in a fuzzy normed linear space (fnls) whose underlying t-norm is left-continuous at

(1, 1)

. [...] Read more.

In the present article, the Schauder-type fixed point theorem for the class of fuzzy continuous, as well as fuzzy compact operators is established in a fuzzy normed linear space (fnls) whose underlying t-norm is left-continuous at

(1, 1)

. In the fuzzy setting, the concept of the measure of non-compactness is introduced, and some basic properties of the measure of non-compactness are investigated. Darbo’s generalization of the Schauder-type fixed point theorem is developed for the class of

ψ

-set contractions. This theorem is proven by using the idea of the measure of non-compactness. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

18 pages, 378 KiB

Open AccessArticle

Construction of Fuzzy Measures over Product Spaces

by Fernando Reche, María Morales and Antonio Salmerón

Mathematics 2020, 8(9), 1605; https://doi.org/10.3390/math8091605 - 17 Sep 2020

Cited by 5 | Viewed by 3598

Abstract

In this paper, we study the problem of constructing a fuzzy measure over a product space when fuzzy measures over the marginal spaces are available. We propose a definition of independence of fuzzy measures and introduce different ways of constructing product measures, analyzing [...] Read more.

In this paper, we study the problem of constructing a fuzzy measure over a product space when fuzzy measures over the marginal spaces are available. We propose a definition of independence of fuzzy measures and introduce different ways of constructing product measures, analyzing their properties. We derive bounds for the measure on the product space and show that it is possible to construct a single product measure when the marginal measures are capacities of order 2. We also study the combination of real functions over the marginal spaces in order to produce a joint function over the product space, compatible with the concept of marginalization, paving the way for the definition of statistical indices based on fuzzy measures. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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13 pages, 432 KiB

Open AccessArticle

Which Alternative for Solving Dual Fuzzy Nonlinear Equations Is More Precise?

by Joanna Kołodziejczyk, Andrzej Piegat and Wojciech Sałabun

Mathematics 2020, 8(9), 1507; https://doi.org/10.3390/math8091507 - 4 Sep 2020

Cited by 7 | Viewed by 1730

Abstract

To answer the question stated in the title, we present and compare two approaches: first, a standard approach for solving dual fuzzy nonlinear systems (DFN-systems) based on Newton’s method, which uses 2D FN representation and second, the new approach, based on multidimensional fuzzy [...] Read more.

To answer the question stated in the title, we present and compare two approaches: first, a standard approach for solving dual fuzzy nonlinear systems (DFN-systems) based on Newton’s method, which uses 2D FN representation and second, the new approach, based on multidimensional fuzzy arithmetic (MF-arithmetic). We use a numerical example to explain how the proposed MF-arithmetic solves the DFN-system. To analyze results from the standard and the new approaches, we introduce an imprecision measure. We discuss the reasons why imprecision varies between both methods. The imprecision of the standard approach results (roots) is significant, which means that many possible values are excluded. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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19 pages, 300 KiB

Open AccessArticle

Strong Tolerance and Strong Universality of Interval Eigenvectors in a Max-Łukasiewicz Algebra

by Martin Gavalec, Zuzana Němcová and Ján Plavka

Mathematics 2020, 8(9), 1504; https://doi.org/10.3390/math8091504 - 4 Sep 2020

Cited by 1 | Viewed by 1404

Abstract

The Łukasiewicz conjunction (sometimes also considered to be a logic of absolute comparison), which is used in multivalued logic and in fuzzy set theory, is one of the most important t-norms. In combination with the binary operation ‘maximum’, the Łukasiewicz t-norm forms the [...] Read more.

The Łukasiewicz conjunction (sometimes also considered to be a logic of absolute comparison), which is used in multivalued logic and in fuzzy set theory, is one of the most important t-norms. In combination with the binary operation ‘maximum’, the Łukasiewicz t-norm forms the basis for the so-called max-Łuk algebra, with applications to the investigation of systems working in discrete steps (discrete events systems; DES, in short). Similar algebras describing the work of DES’s are based on other pairs of operations, such as max-min algebra, max-plus algebra, or max-T algebra (with a given t-norm, T). The investigation of the steady states in a DES leads to the study of the eigenvectors of the transition matrix in the corresponding max-algebra. In real systems, the input values are usually taken to be in some interval. Various types of interval eigenvectors of interval matrices in max-min and max-plus algebras have been described. This paper is oriented to the investigation of strong, strongly tolerable, and strongly universal interval eigenvectors in a max-Łuk algebra. The main method used in this paper is based on max-Ł linear combinations of matrices and vectors. Necessary and sufficient conditions for the recognition of strong, strongly tolerable, and strongly universal eigenvectors have been found. The theoretical results are illustrated by numerical examples. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

32 pages, 425 KiB

Open AccessArticle

Octahedron Subgroups and Subrings

by Jeong-Gon Lee, Young Bae Jun and Kul Hur

Mathematics 2020, 8(9), 1444; https://doi.org/10.3390/math8091444 - 28 Aug 2020

Cited by 1 | Viewed by 1730

Abstract

In this paper, we define the notions of i-octahedron groupoid and i-OLI [resp., i-ORI and i-OI], and study some of their properties and give some examples. Also we deal with some properties for the image and the preimage of [...] Read more.

In this paper, we define the notions of i-octahedron groupoid and i-OLI [resp., i-ORI and i-OI], and study some of their properties and give some examples. Also we deal with some properties for the image and the preimage of i-octahedron groupoids [resp., i-OLI, i-ORI and i-OI] under a groupoid homomorphism. Next, we introduce the concepts of i-octahedron subgroup and normal subgroup of a group and investigate some of their properties. In particular, we obtain a characterization of an i-octahedron subgroup of a group. Finally, we define an i-octahedron subring [resp., i-OLI, i-ORI and i-OI] of a ring and find some of their properties. In particular, we obtain two characterizations of i-OLI [resp., i-ORI and i-OI] of a ring and a skew field, respectively. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

22 pages, 3836 KiB

Open AccessEditor’s ChoiceArticle

AHP-TOPSIS Inspired Shopping Mall Site Selection Problem with Fuzzy Data

by Neha Ghorui, Arijit Ghosh, Ebrahem A. Algehyne, Sankar Prasad Mondal and Apu Kumar Saha

Mathematics 2020, 8(8), 1380; https://doi.org/10.3390/math8081380 - 17 Aug 2020

Cited by 46 | Viewed by 5988

Abstract

In the consumerist world, there is an ever-increasing demand for consumption in urban life. Thus, the demand for shopping malls is growing. For a developer, site selection is an important issue as the optimal selection involves several complex factors and sub-factors for a [...] Read more.

In the consumerist world, there is an ever-increasing demand for consumption in urban life. Thus, the demand for shopping malls is growing. For a developer, site selection is an important issue as the optimal selection involves several complex factors and sub-factors for a successful investment venture. Thus, these tangible and intangible factors can be best solved by the Multi Criteria Decision Making (MCDM) models. In this study, optimal site selection has been done out of multiple alternative locations in and around the city of Kolkata, West Bengal, India. The Fuzzy Analytic Hierarchy Process (FAHP) and Fuzzy Technique for Order of Preference by Similarity to Ideal Solution (FTOPSIS) has been applied for shopping mall site selection. The AHP is used to obtain the crispified weight of factors. Imprecise linguistic terms used by the decision-maker are converted to Triangular Fuzzy Numbers (TFNs). This research used integrated sub-factors fuzzy weights using FAHP to FTOPSIS for ranking of the alternatives. Hardly any research is done with the use of sub-factors. In this study, seven factors and seventeen sub-factors are considered, the authors collected data from different locations with the help of municipal authorities and architects. This work further provides useful guidelines for shopping mall selection in different states and countries. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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15 pages, 355 KiB

Open AccessArticle

by Sabeena Begam S, Vimala J, Ganeshsree Selvachandran, Tran Thi Ngan and Rohit Sharma

Mathematics 2020, 8(8), 1255; https://doi.org/10.3390/math8081255 - 31 Jul 2020

Cited by 52 | Viewed by 3065

Abstract

Many effective tools in fuzzy soft set theory have been proposed to handle various complicated problems in different fields of our real life, especially in decision making. Molodtsov’s soft set theory has been regarded as a newly emerging mathematical tool to deal with [...] Read more.

Many effective tools in fuzzy soft set theory have been proposed to handle various complicated problems in different fields of our real life, especially in decision making. Molodtsov’s soft set theory has been regarded as a newly emerging mathematical tool to deal with uncertainty and vagueness. Lattice ordered multi-fuzzy soft set (

L M F S S

) has been applied in forecasting process. However, similarity measure is not used in this application. In our research, similarity measure of

L M F S S

is proposed to calculate the similarity between two

L M F S S s

. Moreover, some of its properties are introduced and proved. Finally, an application of

L M F S S

in decision making using similarity measure is analysed. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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16 pages, 273 KiB

Open AccessArticle

Solvability of a Bounded Parametric System in Max-Łukasiewicz Algebra

by Martin Gavalec and Zuzana Němcová

Mathematics 2020, 8(6), 1026; https://doi.org/10.3390/math8061026 - 23 Jun 2020

Cited by 1 | Viewed by 1694

Abstract

The max-Łukasiewicz algebra describes fuzzy systems working in discrete time which are based on two binary operations: the maximum and the Łukasiewicz triangular norm. The behavior of such a system in time depends on the solvability of the corresponding bounded parametric max-linear system. [...] Read more.

The max-Łukasiewicz algebra describes fuzzy systems working in discrete time which are based on two binary operations: the maximum and the Łukasiewicz triangular norm. The behavior of such a system in time depends on the solvability of the corresponding bounded parametric max-linear system. The aim of this study is to describe an algorithm recognizing for which values of the parameter the given bounded parametric max-linear system has a solution—represented by an appropriate state of the fuzzy system in consideration. Necessary and sufficient conditions of the solvability have been found and a polynomial recognition algorithm has been described. The correctness of the algorithm has been verified. The presented polynomial algorithm consists of three parts depending on the entries of the transition matrix and the required state vector. The results are illustrated by numerical examples. The presented results can be also applied in the study of the max-Łukasiewicz systems with interval coefficients. Furthermore, Łukasiewicz arithmetical conjunction can be used in various types of models, for example, in cash-flow system. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

23 pages, 768 KiB

Open AccessFeature PaperArticle

An Extension of Fuzzy Competition Graph and Its Uses in Manufacturing Industries

by Tarasankar Pramanik, G. Muhiuddin, Abdulaziz M. Alanazi and Madhumangal Pal

Mathematics 2020, 8(6), 1008; https://doi.org/10.3390/math8061008 - 19 Jun 2020

Cited by 23 | Viewed by 3123

Abstract

Competition graph is a graph which constitutes from a directed graph (digraph) with an edge between two vertices if they have some common preys in the digraph. Moreover, Fuzzy competition graph (briefly, FCG) is the higher extension of the crisp competition graph by [...] Read more.

Competition graph is a graph which constitutes from a directed graph (digraph) with an edge between two vertices if they have some common preys in the digraph. Moreover, Fuzzy competition graph (briefly, FCG) is the higher extension of the crisp competition graph by assigning fuzzy value to each vertex and edge. Also, Interval-valued FCG (briefly, IVFCG) is another higher extension of fuzzy competition graph by taking each fuzzy value as a sub-interval of the interval

[0, 1]

. This graph arises in many real world systems; one of them is discussed as follows: Each and every species in nature basically needs ecological balance to survive. The existing species depends on one another for food. If there happens any extinction of any species, there must be a crisis of food among those species which depend on that extinct species. The height of food crisis among those species varies according to their ecological status, environment and encompassing atmosphere. So, the prey to prey relationship among the species cannot be assessed exactly. Therefore, the assessment of competition of species is vague or shadowy. Motivated from this idea, in this paper IVFCG is introduced and several properties of IVFCG and its two variants interval-valued fuzzy k-competition graphs (briefly, IVFKCG) and interval-valued fuzzy m-step competition graphs (briefly, IVFMCG) are presented. The work is helpful to assess the strength of competition among competitors in the field of competitive network system. Furthermore, homomorphic and isomorphic properties of IVFCG are also discussed. Finally, an appropriate application of IVFCG in the competition among the production companies in market is presented to highlight the relevance of IVFCG. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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15 pages, 272 KiB

Open AccessArticle

Applying Fixed Point Techniques to Stability Problems in Intuitionistic Fuzzy Banach Spaces

by P. Saha, T. K. Samanta, Pratap Mondal, B. S. Choudhury and Manuel De La Sen

Mathematics 2020, 8(6), 974; https://doi.org/10.3390/math8060974 - 15 Jun 2020

Cited by 7 | Viewed by 2020

Abstract

In this paper we investigate Hyers-Ulam-Rassias stability of certain nonlinear functional equations. Considerations of such stabilities in different branches of mathematics have been very extensive. Again the fuzzy concepts along with their several extensions have appeared in almost all branches of mathematics. Here [...] Read more.

In this paper we investigate Hyers-Ulam-Rassias stability of certain nonlinear functional equations. Considerations of such stabilities in different branches of mathematics have been very extensive. Again the fuzzy concepts along with their several extensions have appeared in almost all branches of mathematics. Here we work on intuitionistic fuzzy real Banach spaces, which is obtained by combining together the concepts of fuzzy Banach spaces with intuitionistic fuzzy sets. We establish that pexiderized quadratic functional equations defined on such spaces are stable in the sense of Hyers-Ulam-Rassias stability. We adopt a fixed point approach to the problem. Precisely, we use a generxalized contraction mapping principle. The result is illustrated with an example. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

17 pages, 1059 KiB

Open AccessArticle

Entropy Measures for Plithogenic Sets and Applications in Multi-Attribute Decision Making

by Shio Gai Quek, Ganeshsree Selvachandran, Florentin Smarandache, J. Vimala, Son Hoang Le, Quang-Thinh Bui and Vassilis C. Gerogiannis

Mathematics 2020, 8(6), 965; https://doi.org/10.3390/math8060965 - 12 Jun 2020

Cited by 19 | Viewed by 3926

Abstract

Plithogenic set is an extension of the crisp set, fuzzy set, intuitionistic fuzzy set, and neutrosophic sets, whose elements are characterized by one or more attributes, and each attribute can assume many values. Each attribute has a corresponding degree of appurtenance of the [...] Read more.

Plithogenic set is an extension of the crisp set, fuzzy set, intuitionistic fuzzy set, and neutrosophic sets, whose elements are characterized by one or more attributes, and each attribute can assume many values. Each attribute has a corresponding degree of appurtenance of the element to the set with respect to the given criteria. In order to obtain a better accuracy and for a more exact exclusion (partial order), a contradiction or dissimilarity degree is defined between each attribute value and the dominant attribute value. In this paper, entropy measures for plithogenic sets have been introduced. The requirements for any function to be an entropy measure of plithogenic sets are outlined in the axiomatic definition of the plithogenic entropy using the axiomatic requirements of neutrosophic entropy. Several new formulae for the entropy measure of plithogenic sets are also introduced. The newly introduced entropy measures are then applied to a multi-attribute decision making problem related to the selection of locations. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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15 pages, 2624 KiB

Open AccessArticle

A Method of Generating Fuzzy Implications from n Increasing Functions and n + 1 Negations

by Maria N. Rapti and Basil K. Papadopoulos

Mathematics 2020, 8(6), 886; https://doi.org/10.3390/math8060886 - 1 Jun 2020

Cited by 4 | Viewed by 2112

Abstract

In this paper, we introduce a new construction method of a fuzzy implication from n increasing functions

g_{i} : [0, 1] \to [0, \infty), (g (0) = 0)

( [...] Read more.

In this paper, we introduce a new construction method of a fuzzy implication from n increasing functions

g_{i} : [0, 1] \to [0, \infty), (g (0) = 0)

(

i = 1, 2, \dots, n, n \in ℕ

) and

n + 1

fuzzy negations

N_{i}

(

i = 1, 2, \dots, n + 1, n \in ℕ

). Imagine that there are plenty of combinations between

n

increasing functions

g_{i}

and

n + 1

fuzzy negations

N_{i}

in order to produce new fuzzy implications. This method allows us to use at least two fuzzy negations

N_{i}

and one increasing function

g

in order to generate a new fuzzy implication. Choosing the appropriate negations, we can prove that some basic properties such as the exchange principle (EP), the ordering property (OP), and the law of contraposition with respect to

N

are satisfied. The worth of generating new implications is valuable in the sciences such as artificial intelligence and robotics. In this paper, we have found a novel method of generating families of implications. Therefore, we would like to believe that we have added to the literature one more source from which we could choose the most appropriate implication concerning a specific application. It should be emphasized that this production is based on a generalization of an important form of Yager’s implications. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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18 pages, 440 KiB

Open AccessArticle

A Novel Technique to Solve the Fuzzy System of Equations

by Nasser Mikaeilvand, Zahra Noeiaghdam, Samad Noeiaghdam and Juan J. Nieto

Mathematics 2020, 8(5), 850; https://doi.org/10.3390/math8050850 - 24 May 2020

Cited by 13 | Viewed by 2791

Abstract

The aim of this research is to apply a novel technique based on the embedding method to solve the

n \times n

fuzzy system of linear equations (FSLEs). By using this method, the strong fuzzy number solutions of FSLEs can be obtained in [...] Read more.

The aim of this research is to apply a novel technique based on the embedding method to solve the

n \times n

fuzzy system of linear equations (FSLEs). By using this method, the strong fuzzy number solutions of FSLEs can be obtained in two steps. In the first step, if the created

n \times n

crisp linear system has a non-negative solution, the fuzzy linear system will have a fuzzy number vector solution that will be found in the second step by solving another created

n \times n

crisp linear system. Several theorems have been proved to show that the number of operations by the presented method are less than the number of operations by Friedman and Ezzati’s methods. To show the advantages of this scheme, two applicable algorithms and flowcharts are presented and several numerical examples are solved by applying them. Furthermore, some graphs of the obtained results are demonstrated that show the solutions are fuzzy number vectors. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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19 pages, 4400 KiB

Open AccessArticle

An Evolving Partial Consensus Fuzzy Collaborative Forecasting Approach

by Tin-Chih Toly Chen, Yu-Cheng Wang and Chin-Hau Huang

Mathematics 2020, 8(4), 554; https://doi.org/10.3390/math8040554 - 10 Apr 2020

Cited by 5 | Viewed by 1754

Abstract

Current fuzzy collaborative forecasting methods have rarely considered how to determine the appropriate number of experts to optimize forecasting performance. Therefore, this study proposes an evolving partial-consensus fuzzy collaborative forecasting approach to address this issue. In the proposed approach, experts apply various fuzzy [...] Read more.

Current fuzzy collaborative forecasting methods have rarely considered how to determine the appropriate number of experts to optimize forecasting performance. Therefore, this study proposes an evolving partial-consensus fuzzy collaborative forecasting approach to address this issue. In the proposed approach, experts apply various fuzzy forecasting methods to forecast the same target, and the partial consensus fuzzy intersection operator, rather than the prevalent fuzzy intersection operator, is applied to aggregate the fuzzy forecasts by experts. Meaningful information can be determined by observing partial consensus fuzzy intersection changes as the number of experts varies, including the appropriate number of experts. We applied the evolving partial-consensus fuzzy collaborative forecasting approach to forecasting dynamic random access memory product yield with real data. The proposed approach forecasting performance surpassed current fuzzy collaborative forecasting that considered overall consensus, and it increased forecasting accuracy 13% in terms of mean absolute percentage error. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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12 pages, 298 KiB

Open AccessArticle

On Bipolar Fuzzy Gradation of Openness

by Subhadip Roy, Jeong-Gon Lee, Syamal Kumar Samanta, Anita Pal and Ganeshsree Selvachandran

Mathematics 2020, 8(4), 510; https://doi.org/10.3390/math8040510 - 2 Apr 2020

Cited by 3 | Viewed by 1719

Abstract

The concept of bipolar fuzziness is of relatively recent origin where in addition to the presence of a property, which is done in fuzzy theory, the presence of its counter-property is also taken into consideration. This seems to be much natural and realistic. [...] Read more.

The concept of bipolar fuzziness is of relatively recent origin where in addition to the presence of a property, which is done in fuzzy theory, the presence of its counter-property is also taken into consideration. This seems to be much natural and realistic. In this paper, an attempt has been made to incorporate this bipolar fuzziness in topological perspective. This is done by introducing a notion of bipolar gradation of openness and to redefine the bipolar fuzzy topology. Furthermore, a notion of bipolar gradation preserving map is given. A concept of bipolar fuzzy closure operator is also introduced and its characteristic properties are studied. A decomposition theorem involving our bipolar gradation of openness and Chang type bipolar fuzzy topology is established. Finally, some categorical results of bipolar fuzzy topology (both Chang type and in our sense) are proved. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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15 pages, 2054 KiB

Open AccessReview

A Philosophical Treatise on the Connection of Scientific Reasoning with Fuzzy Logic

by Evangelos Athanassopoulos and Michael Gr. Voskoglou

Mathematics 2020, 8(6), 875; https://doi.org/10.3390/math8060875 - 1 Jun 2020

Cited by 10 | Viewed by 4238

Abstract

The present article studies the connection of scientific reasoning with fuzzy logic. Induction and deduction are the two main types of human reasoning. Although deduction is the basis of the scientific method, almost all the scientific progress (with pure mathematics being probably the [...] Read more.

The present article studies the connection of scientific reasoning with fuzzy logic. Induction and deduction are the two main types of human reasoning. Although deduction is the basis of the scientific method, almost all the scientific progress (with pure mathematics being probably the unique exception) has its roots to inductive reasoning. Fuzzy logic gives to the disdainful by the classical/bivalent logic induction its proper place and importance as a fundamental component of the scientific reasoning. The error of induction is transferred to deductive reasoning through its premises. Consequently, although deduction is always a valid process, it is not an infallible method. Thus, there is a need of quantifying the degree of truth not only of the inductive, but also of the deductive arguments. In the former case, probability and statistics and of course fuzzy logic in cases of imprecision are the tools available for this purpose. In the latter case, the Bayesian probabilities play a dominant role. As many specialists argue nowadays, the whole science could be viewed as a Bayesian process. A timely example, concerning the validity of the viruses’ tests, is presented, illustrating the importance of the Bayesian processes for scientific reasoning. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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