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Fuzzy Sets, Fuzzy Logic and Their Applications 2021
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Fuzzy Sets, Fuzzy Logic and Their Applications 2021

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Fuzzy Sets, Systems and Decision Making".

Deadline for manuscript submissions: closed (1 January 2023) | Viewed by 25780

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Dr. Michael Voskoglou

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Guest Editor
Graduate Technological Educational Institute (T.E.I.) of Western Greece, School of Technological Applications, 263 34 Patras, Greece
Interests: fuzzy sets and logic; Markov chains; abstract and linear algebra; artificial intelligence; mathematics education
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

A few years ago, probability theory was a unique tool in hands of the experts dealing with situations of uncertainty appearing in problems of science and in everyday life. However, nowadays, with the development of fuzzy set theory—introduced by Zadeh in 1965—and the extension of fuzzy logic, the situation has changed. In fact, these new mathematical tools provided scientists with the opportunity to model under conditions that are vague or not precisely defined, thus succeeding in mathematically solving problems whose statements are expressed in our natural language. As a result, the spectrum of application has been rapidly extended, covering all of the physical sciences, economics and management, expert systems like financial planners, diagnostic, meteorological, information retrieval, control systems, etc., industry, robotics, decision making, programming, medicine, biology, humanities, education and almost all the other sectors of the human activity, including human reasoning itself. The first major commercial application of fuzzy logic was in cement kiln control (Zadeh, 1983), followed by a navigation system for automatic cars, a fuzzy controller for the automatic operation of trains, laboratory level controllers, controllers for robot vision, graphics, controllers for automated police sketchers and many others. It should be mentioned that fuzzy mathematics has been also significantly developed on the theoretical level, providing important insights even to branches of the classical mathematics, like algebra, analysis, geometry, etc.

The target of the present Special Issue of the MDPI journal Mathematics is to provide the experts in the field (academics, researchers, practitioners, etc.) the opportunity to present recent theoretical advances on fuzzy sets and fuzzy logic and of their extension/generalization (e.g. intuitionistic fuzzy logic, neurosophic sets, etc.) and their applications to all fields of human activity.

Prof. Dr. Michael Voskoglou
Guest Editor

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Fuzzy Sets and their Generalizations 
  • Fuzzy Logic
  • Defuzzification Techniques 
  • Fuzzy Numbers 
  • Uncertainty in Fuzzy Environments

Published Papers (13 papers)

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Research

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23 pages, 1113 KiB  
Article
Some Certain Fuzzy Aumann Integral Inequalities for Generalized Convexity via Fuzzy Number Valued Mappings
by Muhammad Bilal Khan, Hakeem A. Othman, Michael Gr. Voskoglou, Lazim Abdullah and Alia M. Alzubaidi
Mathematics 2023, 11(3), 550; https://0-doi-org.brum.beds.ac.uk/10.3390/math11030550 - 19 Jan 2023
Cited by 7 | Viewed by 1416
Abstract
The topic of convex and nonconvex mapping has many applications in engineering and applied mathematics. The Aumann and fuzzy Aumann integrals are the most significant interval and fuzzy operators that allow the classical theory of integrals to be generalized. This paper considers the [...] Read more.
The topic of convex and nonconvex mapping has many applications in engineering and applied mathematics. The Aumann and fuzzy Aumann integrals are the most significant interval and fuzzy operators that allow the classical theory of integrals to be generalized. This paper considers the well-known fuzzy Hermite–Hadamard (HH) type and associated inequalities. With the help of fuzzy Aumann integrals and the newly introduced fuzzy number valued up and down convexity (UD-convexity), we increase this mileage even further. Additionally, with the help of definitions of lower UD-concave (lower UD-concave) and upper UD-convex (concave) fuzzy number valued mappings (FNVMs), we have gathered a sizable collection of both well-known and new extraordinary cases that act as applications of the main conclusions. We also offer a few examples of fuzzy number valued UD-convexity to further demonstrate the validity of the fuzzy inclusion relations presented in this study. Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2021)
10 pages, 1630 KiB  
Article
Construction of Fuzzy Numbers via Cumulative Distribution Function
by Georgios Souliotis, Yousif Alanazi and Basil Papadopoulos
Mathematics 2022, 10(18), 3350; https://0-doi-org.brum.beds.ac.uk/10.3390/math10183350 - 15 Sep 2022
Cited by 3 | Viewed by 1197
Abstract
The first person to introduce possibility theory was Lotfi A. Zadeh, in 1977. It was, of course, of no coincidence that he directly combined it with the theory of fuzzy sets. Later, several researchers dealt with the mathematical foundations of the theory of [...] Read more.
The first person to introduce possibility theory was Lotfi A. Zadeh, in 1977. It was, of course, of no coincidence that he directly combined it with the theory of fuzzy sets. Later, several researchers dealt with the mathematical foundations of the theory of possibilities. They introduced possibility distribution as a concept, and they directly combined it with fuzzy numbers. A fuzzy number corresponds to a possibility distribution and vice versa. This correspondence gave a key advantage to possibility theory over probability theory. This advantage is the facility of operations. However, there is also a basic: problem how is a possibility distribution generated? In this paper, we introduce a method of constructing a possibility distribution via a cumulative probability function. The advantage of this method is the simplicity of construction, which is nothing more than the construction of a fuzzy triangular or trapezoidal number via a cumulative probability function. This construction introduces a way to determine a fuzzy number without relying on the experience or intuition of the researcher. We should, of course, emphasize that this specific construction is within the framework of a theoretical model. We do not apply it to specific data. We also considered that the theoretical construction model should be presented through the theory of possibilities, thus avoiding the theory of probabilities. Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2021)
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22 pages, 477 KiB  
Article
New MCDM Algorithms with Linear Diophantine Fuzzy Soft TOPSIS, VIKOR and Aggregation Operators
by Ibtesam Alshammari, Mani Parimala, Cenap Ozel, Muhammad Riaz and Rania Kammoun
Mathematics 2022, 10(17), 3080; https://0-doi-org.brum.beds.ac.uk/10.3390/math10173080 - 26 Aug 2022
Cited by 7 | Viewed by 1341
Abstract
In this paper, we focus on several ideas associated with linear Diophantine fuzzy soft sets (LDFSSs) along with its algebraic structure. We provide operations on LDFSSs and their specific features, elaborating them with real-world examples and statistical depictions to construct an inflow of [...] Read more.
In this paper, we focus on several ideas associated with linear Diophantine fuzzy soft sets (LDFSSs) along with its algebraic structure. We provide operations on LDFSSs and their specific features, elaborating them with real-world examples and statistical depictions to construct an inflow of linguistic variables based on linear Diophantine fuzzy soft (LDFSS) information. We offer a study of LDFSSs to the multi-criteria decision-making (MCDM) process of university determination, together with new algorithms and flowcharts. We construct LDFSS-TOPSIS, LDFSS-VIKOR and the LDFSS-AO techniques as robust extensions of TOPSIS (a technique for order preferences through the ideal solution), VIKOR (Vlse Kriterijumska Optimizacija Kompromisno Resenje) and AO (aggregation operator). We use the LDFSS-TOPSIS, LDFSS-VIKOR and LDFSS-AO techniques to solve a real-world agricultural problem. Moreover, we present a small-sized robotic agri-farming to support the proposed technique. A comparison analysis is also performed to examine the symmetry of optimal decision and to analyze the efficiency of the suggested algorithms. Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2021)
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30 pages, 456 KiB  
Article
On the Search for a Measure to Compare Interval-Valued Fuzzy Sets
by Susana Díaz-Vázquez, Emilio Torres-Manzanera, Irene Díaz and Susana Montes
Mathematics 2021, 9(24), 3157; https://0-doi-org.brum.beds.ac.uk/10.3390/math9243157 - 07 Dec 2021
Cited by 4 | Viewed by 1749
Abstract
Multiple definitions have been put forward in the literature to measure the differences between two interval-valued fuzzy sets. However, in most cases, the outcome is just a real value, although an interval could be more appropriate in this environment. This is the starting [...] Read more.
Multiple definitions have been put forward in the literature to measure the differences between two interval-valued fuzzy sets. However, in most cases, the outcome is just a real value, although an interval could be more appropriate in this environment. This is the starting point of this contribution. Thus, we revisit the axioms that a measure of the difference between two interval-valued fuzzy sets should satisfy, paying special attention to the condition of monotonicity in the sense that the closer the intervals are, the smaller the measure of difference between them is. Its formalisation leads to very different concepts: distances, divergences and dissimilarities. We have proven that distances and divergences lead to contradictory properties for this kind of sets. Therefore, we conclude that dissimilarities are the only appropriate measures to measure the difference between two interval-valued fuzzy sets when the outcome is an interval. Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2021)
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19 pages, 14165 KiB  
Article
Computing the Number of Failures for Fuzzy Weibull Hazard Function
by Hennie Husniah and Asep K. Supriatna
Mathematics 2021, 9(22), 2858; https://0-doi-org.brum.beds.ac.uk/10.3390/math9222858 - 10 Nov 2021
Cited by 1 | Viewed by 1306
Abstract
The number of failures plays an important factor in the study of maintenance strategy of a manufacturing system. In the real situation, this number is often affected by some uncertainties. Many of the uncertainties fall into the possibilistic uncertainty, which are different from [...] Read more.
The number of failures plays an important factor in the study of maintenance strategy of a manufacturing system. In the real situation, this number is often affected by some uncertainties. Many of the uncertainties fall into the possibilistic uncertainty, which are different from the probabilistic uncertainty. This uncertainty is commonly modeled by applying the fuzzy theoretical framework. This paper aims to compute the number of failures for a system which has Weibull failure distribution with a fuzzy shape parameter. In this case two different approaches are used to calculate the number. In the first approach, the fuzziness membership of the shape parameter propagates to the number of failures so that they have exactly the same values of the membership. While in the second approach, the membership is computed through the α-cut or α-level of the shape parameter approach in the computation of the formula for the number of failures. Without loss of generality, we use the Triangular Fuzzy Number (TFN) for the Weibull shape parameter. We show that both methods have succeeded in computing the number of failures for the system under investigation. Both methods show that when we consider the function of the number of failures as a function of time then the uncertainty (the fuzziness) of the resulting number of failures becomes larger and larger as the time increases. By using the first method, the resulting number of failures has a TFN form. Meanwhile, the resulting number of failures from the second method does not necessarily have a TFN form, but a TFN-like form. Some comparisons between these two methods are presented using the Generalized Mean Value Defuzzification (GMVD) method. The results show that for certain weighting factor of the GMVD, the cores of these fuzzy numbers of failures are identical. Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2021)
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12 pages, 1133 KiB  
Article
Fuzzy Logical Algebra and Study of the Effectiveness of Medications for COVID-19
by Shuker Khalil, Ahmed Hassan, Haya Alaskar, Wasiq Khan and Abir Hussain
Mathematics 2021, 9(22), 2838; https://0-doi-org.brum.beds.ac.uk/10.3390/math9222838 - 09 Nov 2021
Cited by 18 | Viewed by 2066
Abstract
A fuzzy logical algebra has diverse applications in various domains such as engineering, economics, environment, medicine, and so on. However, the existing techniques in algebra do not apply to delta-algebra. Therefore, the purpose of this paper was to investigate new types of cubic [...] Read more.
A fuzzy logical algebra has diverse applications in various domains such as engineering, economics, environment, medicine, and so on. However, the existing techniques in algebra do not apply to delta-algebra. Therefore, the purpose of this paper was to investigate new types of cubic soft algebras and study their applications, the representation of cubic soft sets with δ-algebras, and new types of cubic soft algebras, such as cubic soft δ-subalgebra based on the parameter λ (λ-CSδ-SA) and cubic soft δ-subalgebra (CSδ-SA) over η. This study explains why the P-union is not really a soft cubic δ-subalgebra of two soft cubic δ-subalgebras. We also reveal that any R/P-cubic soft subsets of (CSδ-SA) is not necessarily (CSδ-SA). Furthermore, we present the required conditions to prove that the R-union of two members is (CSδ-SA) if each one of them is (CSδ-SA). To illustrate our assumptions, the proposed (CSδ-SA) is applied to study the effectiveness of medications for COVID-19 using the python program. Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2021)
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18 pages, 7166 KiB  
Article
GPS Data Correction Based on Fuzzy Logic for Tracking Land Vehicles
by Pedro J. Correa-Caicedo, Horacio Rostro-González, Martin A. Rodriguez-Licea, Óscar Octavio Gutiérrez-Frías, Carlos Alonso Herrera-Ramírez, Iris I. Méndez-Gurrola, Miroslava Cano-Lara and Alejandro I. Barranco-Gutiérrez
Mathematics 2021, 9(21), 2818; https://0-doi-org.brum.beds.ac.uk/10.3390/math9212818 - 06 Nov 2021
Cited by 5 | Viewed by 2713
Abstract
GPS sensors are widely used to know a vehicle’s location and to track its route. Although GPS sensor technology is advancing, they present systematic failures depending on the environmental conditions to which they are subjected. To tackle this problem, we propose an intelligent [...] Read more.
GPS sensors are widely used to know a vehicle’s location and to track its route. Although GPS sensor technology is advancing, they present systematic failures depending on the environmental conditions to which they are subjected. To tackle this problem, we propose an intelligent system based on fuzzy logic, which takes the information from the sensors and correct the vehicle’s absolute position according to its latitude and longitude. This correction is performed by two fuzzy systems, one to correct the latitude and the other to correct the longitude, which are trained using the MATLAB ANFIS tool. The positioning correction system is trained and tested with two different datasets. One of them collected with a Pmod GPS sensor and the other a public dataset, which was taken from routes in Brazil. To compare our proposal, an unscented Kalman filter (UKF) was implemented. The main finding is that the proposed fuzzy systems achieve a performance of 69.2% higher than the UKF. Furthermore, fuzzy systems are suitable to implement in an embedded system such as the Raspberry Pi 4. Another finding is that the logical operations facilitate the creation of non-linear functions because of the ‘if else’ structure. Finally, the existence justification of each fuzzy system section is easy to understand. Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2021)
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19 pages, 3301 KiB  
Article
Fuzzy Bit-Plane-Dependence Region Competition
by Siukai Choy, Tszching Ng, Carisa Yu and Benson Lam
Mathematics 2021, 9(19), 2392; https://0-doi-org.brum.beds.ac.uk/10.3390/math9192392 - 26 Sep 2021
Viewed by 1196
Abstract
This paper presents a novel variational model based on fuzzy region competition and statistical image variation modeling for image segmentation. In the energy functional of the proposed model, each region is characterized by the pixel-level color feature and region-level spatial/frequency information extracted from [...] Read more.
This paper presents a novel variational model based on fuzzy region competition and statistical image variation modeling for image segmentation. In the energy functional of the proposed model, each region is characterized by the pixel-level color feature and region-level spatial/frequency information extracted from various image domains, which are modeled by the windowed bit-plane-dependence probability models. To efficiently minimize the energy functional, we apply an alternating minimization procedure with the use of Chambolle’s fast duality projection algorithm, where the closed-form solutions of the energy functional are obtained. Our method gives soft segmentation result via the fuzzy membership function, and moreover, the use of multi-domain statistical region characterization provides additional information that can enhance the segmentation accuracy. Experimental results indicate that the proposed method has a superior performance and outperforms the current state-of-the-art superpixel-based and deep-learning-based approaches. Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2021)
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41 pages, 6151 KiB  
Article
An Area Coverage Scheme Based on Fuzzy Logic and Shuffled Frog-Leaping Algorithm (SFLA) in Heterogeneous Wireless Sensor Networks
by Amir Masoud Rahmani, Saqib Ali, Mohammad Sadegh Yousefpoor, Efat Yousefpoor, Rizwan Ali Naqvi, Kamran Siddique and Mehdi Hosseinzadeh
Mathematics 2021, 9(18), 2251; https://0-doi-org.brum.beds.ac.uk/10.3390/math9182251 - 14 Sep 2021
Cited by 22 | Viewed by 2474
Abstract
Coverage is a fundamental issue in wireless sensor networks (WSNs). It plays a important role in network efficiency and performance. When sensor nodes are randomly scattered in the network environment, an ON/OFF scheduling mechanism can be designed for these nodes to ensure network [...] Read more.
Coverage is a fundamental issue in wireless sensor networks (WSNs). It plays a important role in network efficiency and performance. When sensor nodes are randomly scattered in the network environment, an ON/OFF scheduling mechanism can be designed for these nodes to ensure network coverage and increase the network lifetime. In this paper, we propose an appropriate and optimal area coverage method. The proposed area coverage scheme includes four phases: (1) Calculating the overlap between the sensing ranges of sensor nodes in the network. In this phase, we present a novel, distributed, and efficient method based on the digital matrix so that each sensor node can estimate the overlap between its sensing range and other neighboring nodes. (2) Designing a fuzzy scheduling mechanism. In this phase, an ON/OFF scheduling mechanism is designed using fuzzy logic. In this fuzzy system, if a sensor node has a high energy level, a low distance to the base station, and a low overlap between its sensing range and other neighboring nodes, then this node will be in the ON state for more time. (3) Predicting the node replacement time. In this phase, we seek to provide a suitable method to estimate the death time of sensor nodes and prevent possible holes in the network, and thus the data transmission process is not disturbed. (4) Reconstructing and covering the holes created in the network. In this phase, the goal is to find the best replacement strategy of mobile nodes to maximize the coverage rate and minimize the number of mobile sensor nodes used for covering the hole. For this purpose, we apply the shuffled frog-leaping algorithm (SFLA) and propose an appropriate multi-objective fitness function. To evaluate the performance of the proposed scheme, we simulate it using NS2 simulator and compare our scheme with three methods, including CCM-RL, CCA, and PCLA. The simulation results show that our proposed scheme outperformed the other methods in terms of the average number of active sensor nodes, coverage rate, energy consumption, and network lifetime. Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2021)
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13 pages, 1955 KiB  
Article
Artificial Intelligence Based Modelling of Adsorption Water Desalination System
by Hesham Alhumade, Hegazy Rezk, Abdulrahim A. Al-Zahrani, Sharif F. Zaman and Ahmed Askalany
Mathematics 2021, 9(14), 1674; https://0-doi-org.brum.beds.ac.uk/10.3390/math9141674 - 16 Jul 2021
Cited by 5 | Viewed by 2184
Abstract
The main target of this research work is to model the output performance of adsorption water desalination system (AWDS) in terms of switching and cycle time using artificial intelligence. The output performance of the ADC system is expressed by the specific daily water [...] Read more.
The main target of this research work is to model the output performance of adsorption water desalination system (AWDS) in terms of switching and cycle time using artificial intelligence. The output performance of the ADC system is expressed by the specific daily water production (SDWP), the coefficient of performance (COP), and specific cooling power (SCP). A robust Adaptive Network-based Fuzzy Inference System (ANFIS) model of SDWP, COP, and SCP was built using the measured data. To demonstrate the superiority of the suggested ANFIS model, the model results were compared with those achieved by Analysis of Variance (ANOVA) based on the maximum coefficient of determination and minimum error between measured and estimated data in addition to the mean square error (MSE). Applying ANOVA, the average coefficient-of-determination values were 0.8872 and 0.8223, respectively, for training and testing. These values are increased to 1.0 and 0.9673, respectively, for training and testing thanks to ANFIS based modeling. In addition, ANFIS modelling decreased the RMSE value of all datasets by 83% compared with ANOVA. In sum, the main findings confirmed the superiority of ANFIS modeling of the output performance of adsorption water desalination system compared with ANOVA. Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2021)
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23 pages, 361 KiB  
Article
Fuzzy Results for Finitely Supported Structures
by Andrei Alexandru and Gabriel Ciobanu
Mathematics 2021, 9(14), 1651; https://0-doi-org.brum.beds.ac.uk/10.3390/math9141651 - 13 Jul 2021
Cited by 1 | Viewed by 1247
Abstract
We present a survey of some results published recently by the authors regarding the fuzzy aspects of finitely supported structures. Considering the notion of finite support, we introduce a new degree of membership association between a crisp set and a finitely supported function [...] Read more.
We present a survey of some results published recently by the authors regarding the fuzzy aspects of finitely supported structures. Considering the notion of finite support, we introduce a new degree of membership association between a crisp set and a finitely supported function modelling a degree of membership for each element in the crisp set. We define and study the notions of invariant set, invariant complete lattices, invariant monoids and invariant strong inductive sets. The finitely supported (fuzzy) subgroups of an invariant group, as well as the L-fuzzy sets on an invariant set (with L being an invariant complete lattice) form invariant complete lattices. We present some fixed point results (particularly some extensions of the classical Tarski theorem, Bourbaki–Witt theorem or Tarski–Kantorovitch theorem) for finitely supported self-functions defined on invariant complete lattices and on invariant strong inductive sets; these results also provide new finiteness properties of infinite fuzzy sets. We show that apparently, large sets do not contain uniformly supported, infinite subsets, and so they are invariant strong inductive sets satisfying finiteness and fixed-point properties. Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2021)
17 pages, 361 KiB  
Article
Resolution of Fuzzy Relational Inequalities with Boolean Semi-Tensor Product Composition
by Shuling Wang and Haitao Li
Mathematics 2021, 9(9), 937; https://0-doi-org.brum.beds.ac.uk/10.3390/math9090937 - 23 Apr 2021
Cited by 5 | Viewed by 1444
Abstract
Resolution of fuzzy relational inequalities (FRIs) plays a significant role in decision-making, image compression and fuzzy control. This paper studies the resolution of a kind of FRIs with Boolean semi-tensor product composition. First, by resorting to the column stacking technique, the equivalent form [...] Read more.
Resolution of fuzzy relational inequalities (FRIs) plays a significant role in decision-making, image compression and fuzzy control. This paper studies the resolution of a kind of FRIs with Boolean semi-tensor product composition. First, by resorting to the column stacking technique, the equivalent form of FRIs with Boolean semi-tensor product composition is obtained, which is a system of FRIs (SFRIs) with max–min composition. Second, based on the semi-tensor product method, all the solutions to FRIs with Boolean semi-tensor product composition are obtained by finding all possible parameter set solutions. Finally, a general procedure is developed for the resolution of FRIs with Boolean semi-tensor product composition. Two illustrative examples are worked out to show the effectiveness of the obtained new results. Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2021)
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Review

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15 pages, 683 KiB  
Review
Fuzziness, Indeterminacy and Soft Sets: Frontiers and Perspectives
by Michael Gr. Voskoglou
Mathematics 2022, 10(20), 3909; https://0-doi-org.brum.beds.ac.uk/10.3390/math10203909 - 21 Oct 2022
Cited by 6 | Viewed by 1234
Abstract
The present paper comes across the main steps that were laid from Zadeh’s fuzziness and Atanassov’s intuitionistic fuzzy sets to Smarandache’s indeterminacy and to Molodstov’s soft sets. Two hybrid methods for assessment and decision making, respectively, under fuzzy conditions are also presented using [...] Read more.
The present paper comes across the main steps that were laid from Zadeh’s fuzziness and Atanassov’s intuitionistic fuzzy sets to Smarandache’s indeterminacy and to Molodstov’s soft sets. Two hybrid methods for assessment and decision making, respectively, under fuzzy conditions are also presented using suitable examples that use soft sets and real intervals as tools. The decision making method improves on an earlier method of Maji et al. Further, it is described how the concept of topological space, the most general category of mathematical spaces, can be extended to fuzzy structures and how to generalize the fundamental mathematical concepts of limit, continuity compactness and Hausdorff space within such kinds of structures. In particular, fuzzy and soft topological spaces are defined and examples are given to illustrate these generalizations. Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2021)
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