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# Special Issue "Fuzzy Sets Theory and Its Applications"

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (31 October 2017).

## Special Issue Editor

Prof. Dr. Hsien-Chung Wu
Website
Guest Editor
Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan
Interests: fuzzy optimization; fuzzy real analysis; fuzzy statistical analysis; operations research; computational intelligence; soft computing; fixed point theory; applied functional analysis
Special Issues and Collections in MDPI journals

## Special Issue Information

Dear Colleagues,

The concept of the fuzzy set, introduced by L.A. Zadeh in 1965, tried to extend the classical set theory. It is well-known that a classical set corresponds to an indicator function, of which values are only taken to be 0 and 1. With the aid of the membership function associated with a fuzzy set, each element in a set allows to take any values between 0 and 1 that can be regarded as the degree of membership. This kind of imprecision draws forth a number of applications. This Special Issue will focus on the original research that reflects the theoretical developments and applicable results.

## Keywords

• Fuzzy Sets
• Fuzzy Neural Networks
• Fuzzy Systems
• Computational Intelligence
• Machine Learning
• Soft Computing

## Published Papers (14 papers)

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# Research

Open AccessArticle
Diagonally Implicit Multistep Block Method of Order Four for Solving Fuzzy Differential Equations Using Seikkala Derivatives
Symmetry 2018, 10(2), 42; https://doi.org/10.3390/sym10020042 - 08 Feb 2018
Cited by 2
Abstract
In this paper, the solution of fuzzy differential equations is approximated numerically using diagonally implicit multistep block method of order four. The multistep block method is well known as an efficient and accurate method for solving ordinary differential equations, hence in this paper [...] Read more.
In this paper, the solution of fuzzy differential equations is approximated numerically using diagonally implicit multistep block method of order four. The multistep block method is well known as an efficient and accurate method for solving ordinary differential equations, hence in this paper the method will be used to solve the fuzzy initial value problems where the initial value is a symmetric triangular fuzzy interval. The triangular fuzzy number is not necessarily symmetric, however by imposing symmetry the definition of a triangular fuzzy number can be simplified. The symmetric triangular fuzzy interval is a triangular fuzzy interval that has same left and right width of membership function from the center. Due to this, the parametric form of symmetric triangular fuzzy number is simple and the performing arithmetic operations become easier. In order to interpret the fuzzy problems, Seikkala’s derivative approach is implemented. Characterization theorem is then used to translate the problems into a system of ordinary differential equations. The convergence of the introduced method is also proved. Numerical examples are given to investigate the performance of the proposed method. It is clearly shown in the results that the proposed method is comparable and reliable in solving fuzzy differential equations. Full article
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
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Open AccessArticle
Continuity of Fuzzified Functions Using the Generalized Extension Principle
Symmetry 2017, 9(12), 299; https://doi.org/10.3390/sym9120299 - 01 Dec 2017
Cited by 4
Abstract
To fuzzify the crisp functions, the extension principle has been widely used for performing this fuzzification. The purpose of this paper is to investigate the continuity of fuzzified function using the more generalized extension principle. The Hausdorff metric will be invoked to study [...] Read more.
To fuzzify the crisp functions, the extension principle has been widely used for performing this fuzzification. The purpose of this paper is to investigate the continuity of fuzzified function using the more generalized extension principle. The Hausdorff metric will be invoked to study the continuity of fuzzified function. We also apply the principle of continuity of fuzzified function to the fuzzy topological vector space. Full article
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
Open AccessArticle
Granular Structure of Type-2 Fuzzy Rough Sets over Two Universes
Symmetry 2017, 9(11), 284; https://doi.org/10.3390/sym9110284 - 21 Nov 2017
Cited by 1
Abstract
Granular structure plays a very important role in the model construction, theoretical analysis and algorithm design of a granular computing method. The granular structures of classical rough sets and fuzzy rough sets have been proven to be clear. In classical rough set theory, [...] Read more.
Granular structure plays a very important role in the model construction, theoretical analysis and algorithm design of a granular computing method. The granular structures of classical rough sets and fuzzy rough sets have been proven to be clear. In classical rough set theory, equivalence classes are basic granules, and the lower and upper approximations of a set can be computed by those basic granules. In the theory of fuzzy rough set, granular fuzzy sets can be used to describe the lower and upper approximations of a fuzzy set. This paper discusses the granular structure of type-2 fuzzy rough sets over two universes. Definitions of type-2 fuzzy rough sets over two universes are given based on a wavy-slice representation of type-2 fuzzy sets. Two granular type-2 fuzzy sets are deduced and then proven to be basic granules of type-2 fuzzy rough sets over two universes. Then, the properties of lower and upper approximation operators and these two granular type-2 fuzzy sets are investigated. At last, several examples are given to show the applications of type-2 fuzzy rough sets over two universes. Full article
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
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Open AccessArticle
A Hybrid Fuzzy DEA/AHP Methodology for Ranking Units in a Fuzzy Environment
Symmetry 2017, 9(11), 273; https://doi.org/10.3390/sym9110273 - 14 Nov 2017
Cited by 6
Abstract
In this paper, a novel approach combining fuzzy data envelopment analysis (DEA) and the analytical hierarchical process (AHP) is proposed to rank units with multiple fuzzy criteria. The hybrid fuzzy DEA/AHP approach derives the AHP pairwise comparisons by fuzzy DEA and utilizes AHP [...] Read more.
In this paper, a novel approach combining fuzzy data envelopment analysis (DEA) and the analytical hierarchical process (AHP) is proposed to rank units with multiple fuzzy criteria. The hybrid fuzzy DEA/AHP approach derives the AHP pairwise comparisons by fuzzy DEA and utilizes AHP to fully rank units. It shows that the proposed approach generates a logical ranking of units that has perfect compatibility with fuzzy DEA ranking and there is no any form of subjective analysis engaged within the methodology. A study on the facility layout design in manufacturing systems is provided to illustrate the superiority of the proposed approach and show the compatibility between the proposed approach and fuzzy DEA ranking. Full article
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
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Open AccessArticle
A Two-Factor Autoregressive Moving Average Model Based on Fuzzy Fluctuation Logical Relationships
Symmetry 2017, 9(10), 207; https://doi.org/10.3390/sym9100207 - 01 Oct 2017
Cited by 12
Abstract
Many of the existing autoregressive moving average (ARMA) forecast models are based on one main factor. In this paper, we proposed a new two-factor first-order ARMA forecast model based on fuzzy fluctuation logical relationships of both a main factor and a secondary factor [...] Read more.
Many of the existing autoregressive moving average (ARMA) forecast models are based on one main factor. In this paper, we proposed a new two-factor first-order ARMA forecast model based on fuzzy fluctuation logical relationships of both a main factor and a secondary factor of a historical training time series. Firstly, we generated a fluctuation time series (FTS) for two factors by calculating the difference of each data point with its previous day, then finding the absolute means of the two FTSs. We then constructed a fuzzy fluctuation time series (FFTS) according to the defined linguistic sets. The next step was establishing fuzzy fluctuation logical relation groups (FFLRGs) for a two-factor first-order autoregressive (AR(1)) model and forecasting the training data with the AR(1) model. Then we built FFLRGs for a two-factor first-order autoregressive moving average (ARMA(1,m)) model. Lastly, we forecasted test data with the ARMA(1,m) model. To illustrate the performance of our model, we used real Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) and Dow Jones datasets as a secondary factor to forecast TAIEX. The experiment results indicate that the proposed two-factor fluctuation ARMA method outperformed the one-factor method based on real historic data. The secondary factor may have some effects on the main factor and thereby impact the forecasting results. Using fuzzified fluctuations rather than fuzzified real data could avoid the influence of extreme values in historic data, which performs negatively while forecasting. To verify the accuracy and effectiveness of the model, we also employed our method to forecast the Shanghai Stock Exchange Composite Index (SHSECI) from 2001 to 2015 and the international gold price from 2000 to 2010. Full article
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
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Open AccessArticle
A Generalization of Trapezoidal Fuzzy Numbers Based on Modal Interval Theory
Symmetry 2017, 9(10), 198; https://doi.org/10.3390/sym9100198 - 21 Sep 2017
Cited by 4
Abstract
We propose a generalization of trapezoidal fuzzy numbers based on modal interval theory, which we name “modal interval trapezoidal fuzzy numbers”. In this generalization, we accept that the alpha cuts associated with a trapezoidal fuzzy number can be modal intervals, also allowing that [...] Read more.
We propose a generalization of trapezoidal fuzzy numbers based on modal interval theory, which we name “modal interval trapezoidal fuzzy numbers”. In this generalization, we accept that the alpha cuts associated with a trapezoidal fuzzy number can be modal intervals, also allowing that two interval modalities can be associated with a trapezoidal fuzzy number. In this context, it is difficult to maintain the traditional graphic representation of trapezoidal fuzzy numbers and we must use the interval plane in order to represent our modal interval trapezoidal fuzzy numbers graphically. Using this representation, we can correctly reflect the modality of the alpha cuts. We define some concepts from modal interval analysis and we study some of the related properties and structures, proving, among other things, that the inclusion relation provides a lattice structure on this set. We will also provide a semantic interpretation deduced from the modal interval extensions of real continuous functions and the semantic modal interval theorem. The application of modal intervals in the field of fuzzy numbers also provides a new perspective on and new applications of fuzzy numbers. Full article
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
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Open AccessArticle
Sandor Type Fuzzy Inequality Based on the (s,m)-Convex Function in the Second Sense
Symmetry 2017, 9(9), 181; https://doi.org/10.3390/sym9090181 - 04 Sep 2017
Cited by 1
Abstract
Integral inequalities play critical roles in measure theory and probability theory. Given recent profound discoveries in the field of fuzzy set theory, fuzzy inequality has become a hot research topic in recent years. For classical Sandor type inequality, this paper intends to extend [...] Read more.
Integral inequalities play critical roles in measure theory and probability theory. Given recent profound discoveries in the field of fuzzy set theory, fuzzy inequality has become a hot research topic in recent years. For classical Sandor type inequality, this paper intends to extend the Sugeno integral. Based on the (s,m)-convex function in the second sense, a new Sandor type inequality is proposed for the Sugeno integral. Examples are given to verify the conclusion of this paper. Full article
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
Open AccessArticle
The Orthogonality between Complex Fuzzy Sets and Its Application to Signal Detection
Symmetry 2017, 9(9), 175; https://doi.org/10.3390/sym9090175 - 31 Aug 2017
Cited by 16
Abstract
A complex fuzzy set is a set whose membership values are vectors in the unit circle in the complex plane. This paper establishes the orthogonality relation of complex fuzzy sets. Two complex fuzzy sets are said to be orthogonal if their membership vectors [...] Read more.
A complex fuzzy set is a set whose membership values are vectors in the unit circle in the complex plane. This paper establishes the orthogonality relation of complex fuzzy sets. Two complex fuzzy sets are said to be orthogonal if their membership vectors are perpendicular. We present the basic properties of orthogonality of complex fuzzy sets and various results on orthogonality with respect to complex fuzzy complement, complex fuzzy union, complex fuzzy intersection, and complex fuzzy inference methods. Finally, an example application of signal detection demonstrates the utility of the orthogonality of complex fuzzy sets. Full article
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
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Open AccessArticle
On Extended Representable Uninorms and Their Extended Fuzzy Implications (Coimplications)
Symmetry 2017, 9(8), 160; https://doi.org/10.3390/sym9080160 - 18 Aug 2017
Cited by 1
Abstract
In this work, by Zadeh’s extension principle, we extend representable uninorms and their fuzzy implications (coimplications) to type-2 fuzzy sets. Emphatically, we investigate in which algebras of fuzzy truth values the extended operations are type-2 uninorms and type-2 fuzzy implications (coimplications), respectively. Full article
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
Open AccessArticle
Risk Evaluation in Failure Mode and Effects Analysis Using Fuzzy Measure and Fuzzy Integral
Symmetry 2017, 9(8), 162; https://doi.org/10.3390/sym9080162 - 17 Aug 2017
Cited by 21
Abstract
Failure mode and effects analysis (FMEA) is a popular and useful approach applied to examine potential failures in different products, designs, processes, and services. As a vital index, the risk priority number (RPN) can determine the risk priorities of failure modes by some [...] Read more.
Failure mode and effects analysis (FMEA) is a popular and useful approach applied to examine potential failures in different products, designs, processes, and services. As a vital index, the risk priority number (RPN) can determine the risk priorities of failure modes by some risk factors such as occurrence (O), severity (S), and detection (D). However, in FMEA, the traditional risk priority number approach has some shortcomings, especially in setting the weight of risk factors. This paper presents an improved risk priority number approach based on a fuzzy measure and fuzzy integral. A fuzzy measure is used to reflect the importance of the individual indicators and the indicator set and a fuzzy integral is a nonlinear function defined on the basis of fuzzy measure. The weights of risk factors given by domain experts are seen as fuzzy densities to generate a $λ$ -fuzzy measure which can reflect the weights’ difference and relevance about risk factors. Then, the Choquet integral is used to fuse every value of risk factors about failure modes so as to obtain the comprehensive evaluation result. The result can reflect the comprehensive risk level, so it has a definite physical significance. Finally, an illustrative example and a comparison with another approach are given to show the effectiveness of the proposed approach in the paper. Full article
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
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Open AccessArticle
Intuitionistic-Fuzzy Goals in Zero-Sum Multi Criteria Matrix Games
Symmetry 2017, 9(8), 158; https://doi.org/10.3390/sym9080158 - 15 Aug 2017
Cited by 9
Abstract
The classical matrix theory is deficient to express the vagueness of the real life. The fuzzy set theory has been successfully applied to bridge this gap. Much work has already been done on a two-person zero sum matrix game with fuzzy goals. In [...] Read more.
The classical matrix theory is deficient to express the vagueness of the real life. The fuzzy set theory has been successfully applied to bridge this gap. Much work has already been done on a two-person zero sum matrix game with fuzzy goals. In continuation, this paper is dedicated to define and study a multi-criteria two-person zero sum game with intuitionistic fuzzy goals. It is shown that solving such games is equivalent to solving two crisp multi object linear programming problems. Our work generalizes the previous study on a multi-criteria game with fuzzy goals by adopting the approach of linear programming with intuitionistic fuzzy sets. Finally, an illustrative numerical example is provided to elaborate the proposed approach. Full article
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
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Open AccessArticle
Solving Multi-Objective Matrix Games with Fuzzy Payoffs through the Lower Limit of the Possibility Degree
Symmetry 2017, 9(8), 130; https://doi.org/10.3390/sym9080130 - 25 Jul 2017
Cited by 6
Abstract
In this article, we put forward the multi-objective matrix game model based on fuzzy payoffs. In order to solve the game model, we first discuss the relationship of two fuzzy numbers via the lower limit $- 1 2$ of the possibility degree. Then, [...] Read more.
In this article, we put forward the multi-objective matrix game model based on fuzzy payoffs. In order to solve the game model, we first discuss the relationship of two fuzzy numbers via the lower limit $- 1 2$ of the possibility degree. Then, utilizing this relationship, we conclude that the equilibrium solution of this game model and the optimal solution of multicriteria linear optimization problems are of equal value. Finally, to illustrate the effectiveness and correctness of the obtained model, an example is provided. Full article
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
Open AccessArticle
Forecasting Based on High-Order Fuzzy-Fluctuation Trends and Particle Swarm Optimization Machine Learning
Symmetry 2017, 9(7), 124; https://doi.org/10.3390/sym9070124 - 21 Jul 2017
Cited by 12
Abstract
Most existing fuzzy forecasting models partition historical training time series into fuzzy time series and build fuzzy-trend logical relationship groups to generate forecasting rules. The determination process of intervals is complex and uncertain. In this paper, we present a novel fuzzy forecasting model [...] Read more.
Most existing fuzzy forecasting models partition historical training time series into fuzzy time series and build fuzzy-trend logical relationship groups to generate forecasting rules. The determination process of intervals is complex and uncertain. In this paper, we present a novel fuzzy forecasting model based on high-order fuzzy-fluctuation trends and the fuzzy-fluctuation logical relationships of the training time series. Firstly, we compare each piece of data with the data of theprevious day in a historical training time series to generate a new fluctuation trend time series (FTTS). Then, we fuzzify the FTTS into a fuzzy-fluctuation time series (FFTS) according to the up, equal, or down range and orientation of the fluctuations. Since the relationship between historical FFTS and the fluctuation trend of the future is nonlinear, a particle swarm optimization (PSO) algorithm is employed to estimate the proportions for the lagged variables of the fuzzy AR (n) model. Finally, we use the acquired parameters to forecast future fluctuations. In order to compare the performance of the proposed model with that of the other models, we apply the proposed method to forecast the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) time series datasets. The experimental results and the comparison results show that the proposed method can be successfully applied in stock market forecasting or similarkinds of time series. We also apply the proposed method to forecast Shanghai Stock Exchange Composite Index (SHSECI) and DAX30 index to verify its effectiveness and universality. Full article
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
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Open AccessArticle
The Fuzzy u-Chart for Sustainable Manufacturing in the Vietnam Textile Dyeing Industry
Symmetry 2017, 9(7), 116; https://doi.org/10.3390/sym9070116 - 12 Jul 2017
Cited by 1
Abstract
The inevitability of measurement errors and/or humans of subjectivity in data collection processes make accumulated data imprecise, and are thus called fuzzy data. To adapt to this fuzzy domain in a manufacturing process, a traditional u control chart for monitoring the average number [...] Read more.
The inevitability of measurement errors and/or humans of subjectivity in data collection processes make accumulated data imprecise, and are thus called fuzzy data. To adapt to this fuzzy domain in a manufacturing process, a traditional u control chart for monitoring the average number of nonconformities per unit is required to extend. In this paper, we first generalize the u chart, named fuzzy u-chart, whose control limits are built on the basis of resolution identity, which is a well-known fuzzy set theory. Then, an approach to fuzzy-logic reasoning, incorporating the decision-maker’s varying levels of optimism towards the online process, is proposed to categorize the manufacturing conditions. In addition, we further develop a condition-based classification mechanism, where the process conditions can be discriminated into intermittent states between in-control and out-of-control. As anomalous conditions are monitored to some extent, this condition-based classification mechanism can provide the critical information to deliberate the cost of process intervention with respect to the gain of quality improvement. Finally, the proposed fuzzy u-chart is implemented in the Vietnam textile dyeing industry to replace its conventional u-chart. The results demonstrate that the industry can effectively evade unnecessary adjustments to its current processes; thus, the industry can substantially reduce its operational cost and potential loss. Full article
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
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