Fuzzy Function Research Articles

Fuzzification using the extension principle and the expression in decomposition theorem

Fuzzifying the crisp functions is a well-known methodology to study the problems under fuzzy uncertainty. The traditional way for fuzzifying the crisp functions is based on the extension principle. In this paper, by referring to the expression in decomposition theorem, we present a new methodology to fuzzify the crisp functions for non-normal fuzzy sets. The main purpose of this paper is to establish the equivalence between the fuzzy functions obtained from the extension principle and the expression in decomposition theorem under the proposed concept of compatibility. The practical cases regarding the equivalences are also studied in order to be used for the practical problems.

Fixed-time solution of inequality constrained time-varying linear systems via zeroing neural networks

Zeroing neural network (ZNN) has been explored and applied in a variety of time-varying problems solving. However, ZNN models for inequality constrained time-varying linear equation (ICTVLE) solving are rarely reported. Generally, numerous practical problems can be mathematically modeled as ICTVLE problems. Motivated by the above mentioned issue, a nonlinear tunable activation function activated ZNN (NTAF-ZNN) model is constructed to effectively solve ICTVLE problems. To further improve the performances of NTAF-ZNN model, a fuzzy nonlinear tunable activation function (FNTAF) is also designed. Based on the FNTAF, a fuzzy nonlinear tunable activation function-based ZNN (FNTAF-ZNN) model is realized. Firstly, the fixed-time convergence property and disturbance rejection ability of the NTAF-ZNN and FNTAF-ZNN models are proved by theoretical analysis. Then, comparative simulation results of the two models with other existing ZNN models for solving the ICTVLE problem are provided to further verify their robustness and effectiveness. Additionally, the two models and other existing ZNN models are also employed to realize inequality constrained manipulator trajectory tracking under ideal and noisy environments for further comparisons. Finally, the superior performances of the FNTAF-ZNN model for manipulator trajectory tracking are further verified by physical experiment results.

Applications of L_{gfs}-closed sets in mixed fuzzy soft ideal topological spaces

The primary goal of this treatise is to introduce a new kind of generalized closed sets termed as Lgfs-closed sets in the light of fuzzy soft local function in a mixed fuzzy soft ideal topological space. Wealso define the notion of fuzzy soft ∗-separated set. In addition, we procure the idea of fuzzy soft regularity,normality and fuzzy soft compactness. We also study their behavior in terms of Lgfs-closed sets.

The Stochastic Transportation Problem with Imprecise Data using Lomax Distribution

The stochastic transportation problem with imprecise data is a probabilistic chance-constrained programming (CCP) problem in which the objective function is fuzzy and the supply and demand are random. Three models for the STP with ID with mixed-type restrictions that follow the Lomax distribution (LD) are created in this research. Optimizing the transportation cost in FTP under probabilistic mixed constraints is the goal of the research project. To do this, the probabilistic mixed constraints are transformed into deterministic form using the LD, and the cost coefficient of the fuzzy objective function is changed with alpha cut representation. Numerical examples are presented to demonstrate the suggested models.

Using Genetic Algorithms and Core Values of Cooperative Games to Solve Fuzzy Multiobjective Optimization Problems

A new methodology for solving the fuzzy multiobjective optimization problems is proposed in this paper by considering the fusion of cooperative game theory and genetic algorithm. The original fuzzy multiobjective optimization problem needs to be transformed into a scalar optimization problem, which is a conventional optimization problem. Usually, the assignments of suitable coefficients to the corresponding scalar optimization problem are subjectively determined by the decision makers. However, these assignments may cause some biases by their subjectivity. Therefore, this paper proposes a mechanical procedure to avoid this subjective biases. We are going to formulate a cooperative game using the α-level functions of the multiple fuzzy objective functions. Under this setting, the suitable coefficients can be determined mechanically by involving the core values of the cooperative game, which is formulated using the multiple fuzzy objective functions. We shall prove that the optimal solutions of the transformed scalar optimization problem are indeed the nondominated solutions of fuzzy multiobjective optimization problem. Since the core-nondominated solutions will depend on the coefficients that are determined by the core values of cooperative game, there will be a lot of core-nondominated solutions that will also depend on the corresponding coefficients. In order to obtain the best core-nondominated solution, we shall invoke the genetic algorithms by evolving the coefficients.

Projection depth and Lr-type depths for fuzzy random variables

Statistical depth functions are a standard tool in nonparametric statistics to extend order-based univariate methods to the multivariate setting. Since there is no universally accepted total order for fuzzy data (even in the univariate case) and there is a lack of parametric models, a fuzzy extension of depth-based methods is very interesting. In this paper, we adapt the multivariate depths projection depth and Lr-type depth functions to the fuzzy setting, proposing different generalizations for the Lr-type depths. We prove that the proposed fuzzy depth functions have very good properties, obtaining that the fuzzy projection depth is the second example in the literature to satisfy simultaneously the notion of semilinear and of geometric depth. This implies that the fuzzy projection depth is extremely well behave, to order fuzzy sets with respect to fuzzy random variables. Furthermore, we illustrate the good empirical behavior of the proposed fuzzy depth functions with a real data example of trapezoidal fuzzy sets and the used of fuzzy depths in depth-based classification procedures. Finally, as trapezoidal fuzzy sets can be represented by elements of R4, we justify our proposals by also showing empirically the superiority of the fuzzy depths over the multivariate projection depth applied to fuzzy sets.

An entropy - based fuzzy QFD model on used product remanufacturing design

Compared with new products, consumers put forward higher requirements on remanufactured products. Aimed at effectively transforming customer requirements (CRs) to technical characteristics (TCs) of remanufactured product design, an entropy-based fuzzy quality function deployment (QFD) model is proposed. In this model, triangular fuzzy numbers are suggested to express the vagueness information of CRs first, and then the importance of CRs is determined by an entropy-based algorithm, and the relationship between CRs and TCs is well explained by fuzzy QFD; finally, the importance of TCs is determined to clarify the designing emphasis. A remanufacturing engine is conducted to illuminate the practicability of this model. The results show that “Restorability” and “Reliability” will be treated as the key TCs to control, with the importance degree of 0.16 and 0.15. The entropy-based fuzzy QFD model can also be applied to similar product designs.

Research on Fault Diagnosis Algorithm of Power Cable Based on Deep Learning

The self-learning method is used to realize the online monitoring of optical cable after the fault occurs. According to the fuzzy characteristic of cable fault, a fuzzy basis function network is constructed which is similar to nonlinear function. Thus, the dynamic characteristics of the cable system are obtained. Then, this paper iterates each additional sample to realize the learning of the center point and radial of the network. A normal multivariable fuzzy distribution is used to determine the expected output of the new sample, and recursive operations are performed on the matrix and covariance matrix of the original sample. In this way, all failure probability distributions at the sampling points are obtained. Finally, the hardware module of the system is designed from two aspects: the distributed short-circuit fault collecting and processing equipment and the on-line temperature measuring equipment for grounding fault. In this way, the functions of software module reset, storage and communication are realized.

A robust multi-view knowledge transfer-based rough fuzzy C-means clustering algorithm

Rough fuzzy clustering algorithms have received extensive attention due to the excellent ability to handle overlapping and uncertainty of data. However, existing rough fuzzy clustering algorithms generally consider single view clustering, which neglects the clustering requirements of multiple views and results in the failure to identify diverse data structures in practical applications. In addition, rough fuzzy clustering algorithms are always sensitive to the initialized cluster centers and easily fall into local optimum. To solve the above problems, the multi-view and transfer learning are introduced into rough fuzzy clustering and a robust multi-view knowledge transfer-based rough fuzzy c-means clustering algorithm (MKT-RFCCA) is proposed in this paper. First, multiple distance metrics are adopted as multiple views to effectively recognize different data structures, and thus positively contribute to clustering. Second, a novel multi-view transfer-based rough fuzzy clustering objective function is constructed by using fuzzy memberships as transfer knowledge. This objective function can fully explore and utilize the potential information between multiple views and characterize the uncertainty information. Then, combining the statistical information of color histograms, an initialized centroids selection strategy is presented for image segmentation to overcome the instability and sensitivity caused by the random distribution of the initialized cluster centers. Finally, to reduce manual intervention, a distance-based adaptive threshold determination mechanism is designed to determine the threshold parameter for dividing the lower approximation and boundary region of rough fuzzy clusters during the iteration process. Experiments on synthetic datasets, real-world datasets, and noise-contaminated Berkeley and Weizmann images show that MKT-RFCCA obtains favorable clustering results. Especially, it provides satisfactory segmentation results on images with different types of noise and preserves more specific detail information of images.

A fuzzy Gaussian process regression function approach for forecasting problem

A fuzzy regression function approach is a fuzzy inference system method whose rules cannot be determined based on expert opinion, unlike a classical fuzzy inference system. In a fuzzy regression function approach, an input matrix consists of memberships obtained by the fuzzy clustering method and lagged variables of the time series. In the fuzzy regression function approach, the output vector corresponding to this input matrix is also created and the parameter estimation for the method is carried out with the ordinary least square method. As it is known, the ordinary least square method assumes that the data are linear. In addition, although it is very useful to include a priori information describing the formation of the data in the model, in most cases this information is not available. It is also inappropriate to use a model that does not accurately characterize the data. However, it is not appropriate to estimate parameters for nonlinear data using the ordinary least square method. One of the methods to be used in such a situation is the Gaussian process regression method. While the parameters of a selected basis function are fitted in the ordinary least squares regression method, how all measured data are related is determined in the Gaussian process regression. Besides, Gaussian process regression is a Bayesian approach, it can provide uncertainty measurements on forecasts. In this study, a fuzzy Gaussian process regression function is proposed. The contribution of this paper is to propose a new fuzzy inference system that can be used to solve nonlinear data by proposing a fuzzy Gaussian process regression function. The performance of the newly proposed method is evaluated based on the closing values of the Bitcoin and Crude oil time series. The performance comparison of the proposed method is evaluated with many different forecasting methods and it is concluded that the proposed method has superior forecasting performance.

Hierarchical fuzzy regression functions for mixed predictors and an application to real estate price prediction

Categorical features appear in datasets from almost every practice area, including real estate datasets. One of the most critical handicaps of machine learning algorithms is that they are not designed to capture the qualitative nature of the categorical features, leading to sub-optimal predictions for the datasets with categorical observations. This study focuses on a new fuzzy regression functions framework, namely hierarchical fuzzy regression functions, that can handle categorical features properly for the regression task. The proposed framework is benchmarked with linear regression, support vector machines, deep neural networks, and adaptive neuro-fuzzy inference systems with real estate data having categorical features from six markets. It is observed that the proposed method produces better prediction performance for real estate price prediction than the benchmark methods in a wide variety of real estate markets. Since we provide all the required software codes to implement the proposed hierarchical fuzzy regression functions framework, our approach offers practitioners a readily applicable, high-performing tool for real estate price prediction and other regression problems involving categorical independent features.

The Investigation of Some Essential Concepts of Extended Fuzzy-Valued Convex Functions and Their Applications

In this paper, we are thus motivated to define and introduce the extended fuzzy-valued convex functions that can take the singleton fuzzy values −∞˜ and +∞˜ at some points. Such functions can be characterized using the notions of effective domain and epigraph. In this way, we study important concepts such as fuzzy indicator function and fuzzy infimal convolution for extended fuzzy-valued functions. Finally, we introduce the concept of directional generalized derivative for extended above functions and its properties. Eventually, we give a practical example that will illustrate well the directional g-derivative for the extended fuzzy-valued convex function.

Modulated Lacunary 𝒫-Statistical Convergence of Order γ of Sequences of Fuzzy Functions

The central objective of this paper is to investigate the lacunary statistical convergence of order [Formula: see text] and strong lacunary summability of order [Formula: see text] for sequences of fuzzy functions by using the modulus function and a regular matrix. We examine the relevant inclusion relations between these spaces under specific conditions. Additionally, we study properties related to these spaces and establish several interesting results.

Picture Fuzzy Aggregation Operator-Based Integrated MEREC-WASPAS Technique for Video Conferencing Tool Selection

The COVID-19 epidemic has drastically altered the global landscape and thus organizations are finding it difficult to provide a modern and productive work environment to their stakeholders. As a result, personnel have shifted from in-person interactions to working remotely via video conferencing tools (VCTs). Owing to the onset of COVID-19 outbreak, usage of VTCs has gained great interest among people, especially student fraternity and employees as it reduces the hassle of direct interaction. But selecting the right platform for virtual interaction can be challenging, as it involves a number of factors in order to reduce expenses and maximize productivity. The present treatise has focused on devising a mathematical approach to identify the best VCT by structurizing the problem as a multi-criteria group decision making (MCGDM) model. To carry out the research, the notions of two popular techniques, namely Method based on the Removal Effects of Criteria (MEREC) and Weighted Aggregated Sum Product Assessment (WASPAS) have been extended under Picture fuzzy environment with the aid of Picture fuzzy weighted averaging operator to deal with MCGDM situation. Furthermore, to overcome limitations of existing Picture fuzzy score function (PFSF), a novel PFSF has been developed and its significant properties are analyzed through theoretical and numerical justifications, showing its superiority over the existing one. The proposed PFSF can not only overcome the limitations of the existing SF, but also can rank any type of Picture fuzzy numbers irrespective of their membership and non-membership grades. Thereafter, the extended MCGDM techniques have been integrated with the aid of the proposed novel PFSF and applied to assess the best VCT. The proposed technique has identified Google Meet as the best VCT for virtual communication. Consistency and robustness of proposed technique have been validated through comparative and sensitivity analysis.

VCH: A Velocity Measurement Method Combining HFM Signals

During sonar detection, the fuzzy function of hyperbolic frequency modulation (HFM) in the waveform of active signal is in the shape of a “oblique blade,” which leads to the coupling characteristics of its fuzzy function in the time dimension and frequency dimension. Doppler causes a frequency shift in the frequency dimension and a time delay in the time dimension, which makes it impossible for a single HFM signal to measure speed accurately. However, the time delay caused by the same moving target to HFM signals with different frequency bands and different pulse widths is also different. A velocity measurement method combining HFM signals (VCH) is proposed, which employs the time delay correlation between the combined HFM signals to achieve the target distance and speed. In the VCH method, the echo signal is no longer matched and filtered as a whole but is divided into two channels and matched and filtered separately. And then, the combined HFM signals are used to obtain the distance and speed of the target. Extensive simulation results show that the proposed method can estimate the distance and speed of moving targets accurately, and it has reference value for engineering application.

Reliability analysis and optimization design of magnetic fluid dynamic seal shell structure under thermal/mechanical load

In order to study the reliability of Magnetic fluid dynamic seal shell structure under thermal/mechanical load, the improving cooling structure, calculating fuzzy fatigue reliability, and collaborative optimization design of fatigue reliability and lightweight are studied in this paper. The temperature model of the shell structure under four temperature conditions is established to analyze the influence of the thermal deformation caused by excessive local temperature. The cooling structure is improved, and the maximum temperature under the maximum temperature condition is reduced by 40.154 °C. For the sake of calculating the reliability of the new shell structure, a strength analysis model under thermal/mechanical load is built, and the fuzzy fatigue reliability function is obtained based on Latin hypercube sampling and the second-order response surface method. Four kinds of reliability are acquired by coding the functional functions and four kinds of membership functions with MATLAB, but all of them are lower than 90 % of the industrial requirements. To improve the reliability of the new shell structure and carry out the anti-fatigue lightweight design at the same time, an approximate model was constructed based on the interval analysis method and second-order response surface method for analysis. The accuracy of the approximate model was verified by re-sampling and finite element analysis. Based on the non-dominated sorting genetic algorithm, the optimal solution is determined. The optimization results show that the stress target value is reduced by 19.3 %, the weight target value is reduced by 3 %, and the fuzzy fatigue reliability reaches 94.65 %, 94.58 %, 93.89 %, and 96.31 % respectively. A new cooling structure of the Magnetic fluid dynamic seal is obtained and the maximum temperature is reduced. Moreover, the reliability of the new shell structure under thermal/mechanical load coupling is improved.

Max-min fuzzy bi-level programming: resource sharing system with application

This study delves into the application of min-max fuzzy relation systems, specifically focusing on equality and inequality, in the examination of educational institution networks and their resource-sharing dynamics, akin to a P2P network structure. Expanding upon this framework, this article intricately explores a mathematical system featuring three or more terminals within an education resource system, each comprising distinct sharing capabilities. The terminals exchange information on various educational facets, encompassing variations and expenditures per share data, within the framework of a programming-type fuzzy objective function, constrained by the parameters of an educational information system. In these scenarios, the transfer of resources from one terminal to another is interlinked, introducing complexities that warrant a comprehensive examination. Notably, the consideration of greater downloading resources in specific cases is proposed for enhanced practicality. The primary objective of this article is to minimize network congestion within different educational networks, given fixed priority grades assigned to the terminals. To achieve this, the concept of a Lexicographic minimum solution is introduced to address max-min fuzzy relation inequalities, aligning with the defined objectives. The article presents a detailed and effective scenario for implementing the Lexicographic minimal solution. For validation purposes, the proposed scenario is supported by illustrative examples, offering tangible insights into its applicability and efficacy.

Applying the fuzzy homotopy analysis method to solve fuzzy initial value problems with variable coefficients

In this study, we applied the fuzzy homotopy analysis method to find an approximate but analytical solution to initial value problems that have fuzzy variable coefficients. These coefficients were represented by triangular fuzzy functions. This approach lets us solve the fuzzy differential equations as an endless series of fuzzy numbers, where each fuzzy component can be determined without much difficulty. Based on the numerical data we analyzed, the series solutions we arrived at are both precise and closely match the exact analytical solutions for fuzzy problems.

Diagnosis of a rotor imbalance in a wind turbine based on support vector machine

Rotor imbalances in wind turbines present safety risks and lead to economic losses, and a method to diagnose rotor imbalances is urgently needed. A diagnostic method for rotor imbalances is proposed in this paper. First, a signal reconstruction method is proposed, and a novel index is used to determine the number of components used in signal decomposition in order to effectively address the interference by noise on the sensor. Second, an entropy calculation method is proposed, and the Gaussian kernel function is used to replace the fuzzy functions. The results indicate significant differences for different types of rotor imbalances. Moreover, it exhibits good noise robustness and a low dependence on the data length. Third, a support vector machine with multiscale kernels is proposed, and kernel functions with various characteristics and scales are combined. It has a well-distributed hyperplane and better classification performance, and it is robust to wind conditions. Finally, the method is tested and verified with varying levels of noise and turbulence. The results demonstrate satisfactory performance because the proposed method can effectively identify rotor imbalances under different noise and wind conditions.

On Structure Fuzzy Fibration

In this paper, we introduced concept of fuzzy homotopy lifting property, fuzzy fiber structure is introduced. Also, the definition of the fuzzy lifting function and some theorems in a fuzzy topological space are discussed.