Fuzzy Eigenvalue Research Articles

A Non-Probabilistic Strategy to Investigate Imprecisely Defined Nonlinear Eigenvalue for Structural Dynamic Problems

Structural dynamic problems are influenced by many parameters which are uncertain in nature. As such, for better estimation, the problems can be considered with epistemic-type uncertainties. Due to the epistemic type of uncertainties, the modeled system appears in the form of an imprecise nonlinear eigenvalue problem. Therefore, this paper proposes a novel nonprobabilistic approach to analyze nonlinear eigenvalue problems in structural dynamics, particularly when dealing with imprecisely defined parameters. In order to solve the fuzzy nonlinear eigenvalue models, the same is to be transformed into a fuzzy eigenvalue problem. Traditional methods often rely on probabilistic frameworks, which may not adequately capture uncertainties inherent in real-world structural systems. The proposed approach leverages concepts from fuzzy set theory to handle imprecise data, providing a more robust and realistic analysis. Then, the developed method is used to solve a case study of structural dynamic problem. Here, a spring-mass system is demonstrated through the developed technique to quantify the uncertain field variables. Based on the imprecise system parameters, different fuzzy models are discussed in various cases and results are reported here. This framework offers valuable insights for engineers and researchers working on dynamic analysis of complex structures, paving the way for more reliable and informed decision-making in structural engineering applications.

New methods for computing fuzzy eigenvalues and fuzzy eigenvectors of fuzzy matrices using nonlinear programming approach

In this paper, we propose a new method to obtain the eigenvalues and fuzzy triangular eigenvectors of a fuzzy triangular matrix $$\left( {\tilde{A}} \right)$$ , where the elements of the fuzzy triangular matrix are given. For this purpose, we solve 1-cut of a fuzzy triangular matrix $$\left( {\tilde{A}} \right)$$ to obtain 1-cut of eigenvalues and eigenvectors. Considering the interval system $$\left[ {\tilde{A}} \right]_{\alpha } \left[ {\tilde{X}} \right]_{\alpha } = \left[ {\tilde{\lambda }} \right]_{\alpha } \left[ {\tilde{X}} \right]_{\alpha } 0 \le \alpha \le 1$$ as α-cut of the fuzzy system $$\tilde{A}\tilde{X} = \tilde{\lambda }\tilde{X}$$ , to determine the left and right width of eigenvalues $$\left[ {\tilde{\lambda }} \right]_{\alpha }$$ and eigenvector elements $$\left[ {\tilde{X}} \right]_{\alpha } 0 \le \alpha \le 1$$ , we make a system of linear and nonlinear equations and inequalities. And we propose nonlinear programming models to solve the system of linear and nonlinear equations and inequalities and to calculate $$\left[ {\tilde{\lambda }} \right]_{\alpha }$$ and $$\left[ {\tilde{X}} \right]_{\alpha } 0 \le \alpha \le 1$$ . Furthermore, we define three other new eigenvalues (namely, fuzzy escribed eigenvalue, fuzzy peripheral eigenvalue, and fuzzy approximate eigenvalue) for a fuzzy triangular matrix (Ã) that the fuzzy eigenvalue and fuzzy eigenvector cannot be obtained based on interval calculations. Therefore, the fuzzy escribed eigenvalue which is placed in a tolerable fuzzy triangular eigenvalue set, the fuzzy peripheral eigenvalue placed in a controllable fuzzy triangular eigenvalue set, and the fuzzy approximate eigenvalue placed in an approximate fuzzy triangular eigenvalue set is defined in this paper. Finally, numerical examples are presented to illustrate the proposed method.

Enhancing the efficiency and accuracy of existing FAHP decision-making methods

Fuzzy analytic hierarchy process (FAHP) has been extensively applied to multi-criteria decision making (MCDM). However, the computational burden resulting from the calculation of fuzzy eigenvalue and eigenvector is heavy. As a result, a FAHP problem is usually solved using approximation techniques such as fuzzy geometric mean (FGM) and fuzzy extent analysis (FEA) instead of exact methods. Therefore, the FAHP results are subject to considerable inaccuracy. To solve this problem, in this study, a FAHP method based on the combination of α-cut operations (ACO), center-of-gravity (COG) defuzzification and defuzzification convergence mechanism (DCM) is proposed. First, ACO is applied to derive the near-exact fuzzy maximal eigenvalue and fuzzy weights. Subsequently, the α cuts of the fuzzy maximal eigenvalue and fuzzy weights are interpolated to generate samples that are uniformly distributed along the x-axis so that COG can be correctly applied to defuzzify the fuzzy maximal eigenvalue and fuzzy weights. To accelerate the computation process, DCM is applied to terminate the enumeration process if the defuzzified values of fuzzy weights have converged. The ACO–COG–DCM method has been applied to a real case to illustrate its applicability. In addition, a simulation study was also conducted to perform a parametric analysis. According to the experimental results, the proposed ACO–COG–DCM method improved the accuracy of estimating fuzzy weights by up to 56%. Furthermore, the experimental results also showed that the inaccuracy of estimating fuzzy weights was mostly owing to the deficiency of the FAHP method rather than the inconsistency of fuzzy pairwise comparison results.

Eigenvalue of Intuitionistic Fuzzy Matrices Over Distributive Lattice

In this article, the concepts of intuitionistic fuzzy complete and complete distributive lattice are introduced and the relative pseudocomplement relation of intuitionistic fuzzy sets is defined. The concepts of intuitionistic fuzzy eigenvalue and eigenvector of an intuitionistic fuzzy matrixes are presented and proved that the set of intuitionistic fuzzy eigenvectors of a given intuitionistic fuzzy eigenvalue form an intuitionistic fuzzy subspace. Also, the authors obtain an intuitionistic fuzzy maximum matrix of a given intuitionistic fuzzy eigenvalue and eigenvector and give some properties of an intuitionistic fuzzy maximum matrix. Finally, the invariant of an intuitionistic fuzzy matrix over a distributive lattice is given with some properties.

Filtering Algorithm for Real Eigenvalue Bounds of Interval and Fuzzy Generalized Eigenvalue Problems

This paper deals with an interval and fuzzy generalized eigenvalue problem involving uncertain parameters. Based on a sufficient regularity condition for intervals, an interval filtering eigenvalue procedure for generalized eigenvalue problems with interval parameters is proposed, which iteratively eliminates the parts that do not contain an eigenvalue and thus reduces the initial eigenvalue bound to a precise bound. The same iterative procedure has been proposed for generalized fuzzy eigenvalue problems. In general, the solution of dynamic problems of structures using the finite element method (FEM) leads to a generalized eigenvalue problem. Based on the proposed procedures, various structural examples with an interval and fuzzy parameter such as triangular fuzzy number (TFN) are investigated to show the efficiency of the algorithms stated. Finally, fuzzy filtered eigenvalue bounds are depicted by fuzzy plots using the α-cut.

Filtering algorithm for eigenvalue bounds of fuzzy symmetric matrices

Purpose The solution of dynamic problems of structures using finite element method leads to generalised eigenvalue problem. In general, if the material properties are crisp (exact) then we get crisp eigenvalue problem. But in actual practice, instead of crisp material properties we may have only bounds of values as a result of errors in measurements, observations and calculations or it may be due to maintenance induced error etc. Such bounds of values may be considered in terms of interval or fuzzy numbers. The purpose of this paper is to develop a fuzzy filtering procedure for finding real eigenvalue bounds of different structural problems. Design/methodology/approach The proposed fuzzy filtering algorithm has been developed in terms of fuzzy number to solve the fuzzy eigenvalue problem. The initial bounds of fuzzy eigenvalues are filtered to obtain precise eigenvalue bounds which are depicted by fuzzy (Triangular Fuzzy Number) plots using α-cut. Findings Previously, bounds of eigenvalues of interval matrices have been investigated by few authors. But when the structural problem consists of fuzzy material properties, then the interval eigenvalue bounds may be obtained for each interval of the fuzzy number. The proposed algorithm has been applied for standard fuzzy eigenvalue problems which may be extended to generalised fuzzy eigenvalue problems for obtaining filtered fuzzy bounds. Originality/value The developed fuzzy filtering method is found to be efficient for different structural dynamics problems with fuzzy material properties.

A second-order perturbation method for fuzzy eigenvalue problems

Purpose For eigenvalue problems containing uncertain inputs characterized by fuzzy basic parameters, first-order perturbation methods have been developed to extract eigen-solutions, but either the result accuracy or the computational efficiency of these methods is less satisfactory. This paper presents an efficient method for estimation of fuzzy eigenvalues with high accuracy. Design/methodology/approach Based on the first order derivatives of eigenvalues and modes with respect to the fuzzy basic parameters, expressions of the second order derivatives of eigenvalues are formulated. Then a second-order perturbation method is introduced to provide more accurate fuzzy eigenvalue solutions. Only one eigenvalue solution is sought for the perturbed formulation, and quadratic programming is performed to simplify the alpha-level optimization. Findings Fuzzy natural frequencies and buckling loads of some structures are estimated with good accuracy, illustrating the high computational efficiency of the proposed method. Originality/value Up to the second order derivatives of the eigenvalues with respect to the basic parameters are represented in functional forms, which are used to introduce a second-order perturbation method for treatment of fuzzy eigenvalue problems. The corresponding alpha-level optimization is thus simplified into quadratic programming. The proposed method provides much more accurate interval solutions at alpha-cuts for the membership functions of fuzzy eigenvalues. Analogously, third- and higher-order perturbation methods can be developed for more stringent accuracy demands or for the treatment of stronger nonlinearity. The present work can be applied to realistic structural analysis in civil engineering, especially for those structures made of dispersed materials such as concrete and soil.

Efficient solution of the fuzzy eigenvalue problem in structural dynamics

Purpose – Many analysis and design problems in engineering and science involve uncertainty to varying degrees. This paper is concerned with the structural vibration problem involving uncertain material or geometric parameters, specified as fuzzy parameters. The requirement is to propagate the parameter uncertainty to the eigenvalues of the structure, specified as fuzzy eigenvalues. However, the usual approach is to transform the fuzzy problem into several interval eigenvalue problems by using the α-cuts method. Solving the interval problem as a generalized interval eigenvalue problem in interval mathematics will produce conservative bounds on the eigenvalues. The purpose of this paper is to investigate strategies to efficiently solve the fuzzy eigenvalue problem. Design/methodology/approach – Based on the fundamental perturbation principle and vertex theory, an efficient perturbation method is proposed, that gives the exact extrema of the first-order deviation of the structural eigenvalue. The fuzzy eigenvalue approach has also been improved by reusing the interval analysis results from previous α-cuts. Findings – The proposed method was demonstrated on a simple cantilever beam with a pinned support, and produced very accurate fuzzy eigenvalues. The approach was also demonstrated on the model of a highway bridge with a large number of degrees of freedom. Originality/value – This proposed Vertex-Perturbation method is more efficient than the standard perturbation method, and more general than interval arithmetic methods requiring the non-negative decomposition of the mass and stiffness matrices. The new increment method produces highly accurate solutions, even when the membership function for the fuzzy eigenvalues is complex.

On Fuzzy Fractional Laplace Transformation

Fuzzy and fractional differential equations are used to model problems with uncertainty and memory. Using the fractional fuzzy Laplace transformation we have solved the fuzzy fractional eigenvalue differential equation. By illustrative examples we have shown the results.

Analysis of Factors Affecting Safety Management of Construction Site Based on Fuzzy Factor Analysis Model

Thirty-nine influential factors of construction safety are identified in this study, and then five categories of respondents estimate their influential degrees through a questionnaire survey. In order to analyze these factors more accurately, a fuzzy factor analysis model (FFAM) is proposed. After calculating fuzzy eigenvalue, fuzzy correlation coefficients and factor loadings matrix in the model, seven different common factors are extracted. Finally, the author put forward several effective measures for improving construction safety based on these common factors.

A new fuzzy K-EVD orthogonal complement space clustering method

This paper studies the problem of underdetermined blind source separation with the nonstrictly sparse condition. Different from current approaches in literature, we propose a new and more effective algorithm to estimate the mixing matrices resulted from noise output data sets. After we introduce a clustering prototype of orthogonal complement space and give an extension of the normal vector clustering prototype, a new method combing the fuzzy clustering and eigenvalue decomposition technique to estimate the mixing matrix is presented in order to deal with the nonstrictly sparse situation. A convergent algorithm for estimating the mixing matrices is established, and numerical simulations are given to demonstrate the effectiveness of the proposed approach.

Fuzzy finite element method for solving uncertain heat conduction problems

In this article we have presented a unique representation for interval arithmetic. The traditional interval arithmetic is transformed into crisp by symbolic parameterization. Then the proposed interval arithmetic is extended for fuzzy numbers and this fuzzy arithmetic is used as a tool for uncertain finite element method. In general, the fuzzy finite element converts the governing differential equations into fuzzy algebraic equations. Fuzzy algebraic equations either give a fuzzy eigenvalue problem or a fuzzy system of linear equations. The proposed methods have been used to solve a test problem namely heat conduction problem along with fuzzy finite element method to see the efficacy and powerfulness of the methodology. As such a coupled set of fuzzy linear equations are obtained. These coupled fuzzy linear equations have been solved by two techniques such as by fuzzy iteration method and fuzzy eigenvalue method. Obtained results are compared and it has seen that the proposed methods are reliable and may be applicable to other heat conduction problems too.

Correspondence analysis with fuzzy data: The fuzzy eigenvalue problem

This paper constitutes a first step towards an extension of correspondence analysis with fuzzy data (FCA). At this stage, our main objective is to lay down the algebraic foundations for this fuzzy extension of the usual correspondence analysis. A two-step method is introduced to convert the fuzzy eigenvalue problem to an ordinary one. We consider a fuzzy matrix as the set of its cuts. Each such cut is an interval-valued matrix viewed as a line-segment in the matrix space. In this way, line-segments of cut-matrices are transformed into intervals of eigenvalues. Therefore, the two-step method is essentially a reduction of the fuzzy eigenvalue problem to an ordinary one. We illustrate the FCA-fuzzy eigenvalue problem with a simple numerical example. We hope upon the completion of this project in near future, to be able to supply the necessary tools for the end user of the correspondence analysis with fuzzy data.

The fuzzy eigenvalue problem of fuzzy correspondence analysis

In this paper the main problem is the generalization of the fuzzy eigenvalue problem ofFuzzy Correspondence Analysis(FCA).We investigate the algebraic foundation of finding fuzzy eigenvalues-eigenvectors of a FCA-fuzzy matrix in some more complex cases.We continue to utilize the two-step method with triangular fuzzy numbers introduced in [20], to accomplish the said generalization. The results of the proposed method are illustrated with two simple numerical examples.

Solving fuzzy equations using evolutionary algorithms and neural nets

In this paper we use evolutionary algorithms and neural nets to solve fuzzy equations. In Part I we: (1) first introduce our three solution methods for solving the fuzzy linear equation A¯X¯ + B¯= C¯; for X¯ and (2) then survey the results for the fuzzy quadratic equations, fuzzy differential equations, fuzzy difference equations, fuzzy partial differential equations, systems of fuzzy linear equations, and fuzzy integral equations; and (3) apply an evolutionary algorithm to construct one of the solution types for the fuzzy eigenvalue problem. In Part II we: (1) first discuss how to design and train a neural net to solve A¯X¯ + B¯= C¯ for X¯ and (2) then survey the results for systems of fuzzy linear equations and the fuzzy quadratic.

Fuzzy eigenvalues and input-output analysis

We first generalize the Perron-Frobenius results for nonnegative matrices to nonnegative fuzzy matrices which says that these fuzzy matrices have a (dominant) nonnegative fuzzy eigenvalue. This result is then used to extend Leontief's closed input-output analysis to fuzzy economies and obtain a fuzzy model of the world's economy.