Fuzzy System Of Linear Equations Research Articles

Triangular intuitionistic fuzzy linear system of equations with applications: an analytical approach

This study extended an existing semi-analytical technique, the Homotopy Perturbation Method, to the Block Homotopy Modified Perturbation Method by solving two n × n crisp triangular intuitionistic fuzzy (TIF) systems of linear equations. In the original system, the coefficient matrix is considered as real crisp, while the unknown variable vector and right hand side vector are regarded as triangular intuitionistic fuzzy numbers. The Block Homotopy Modified Perturbation Method is found to be efficient and practical to solve n × n TIF linear systems as it only requires the non-singularity of the n × n TIF linear system's coefficient matrix, whereas the point Homotopy Perturbation Method and other classical numerical iterative methods typically require non-zero diagonal entries in the coefficient matrix. A set of theorems relevant to this study are presented and demonstrated. We solve an engineering application, i.e. a current flow circuit problem that is represented in terms of a triangular intuitionistic fuzzy environment, using the suggested method. The unknown current is then obtained as a triangle intuitionistic fuzzy number. The proposed semi-analytic method is used to solve some numerical test problems in order to validate their performance and efficiency in comparison to other existing techniques. The numerical results of the example are displayed on graphs with different degrees of uncertainty. The efficiency and accuracy of the proposed method are further demonstrated by comparisons to block Jacobi, Adomain Decomposition method, Successive Over-Relaxation method and the classical Gauss-Seidel numerical method.

Modified Block Homotopy Perturbation Method for solving triangular linear Diophantine fuzzy system of equations

Numerous real-world applications can be solved using the broadly adopted notions of intuitionistic fuzzy sets, Pythagorean fuzzy sets, and q-rung orthopair fuzzy sets. These theories, however, have their own restrictions in terms of membership and non-membership levels. Because it utilizes benchmark or control parameters relating to membership and non-membership levels, this theory is particularly valuable for modeling uncertainty in real-world problems. We propose the unique concept of linear Diophantine fuzzy set with benchmark parameters to overcome these restrictions. Different numerical, analytical, and semi-analytical techniques are used to solve linear systems of equations with several fuzzy numbers, such as intuitionistic fuzzy number, triangular fuzzy number, bipolar fuzzy number, trapezoidal fuzzy number, and hexagon fuzzy number. The purpose of this research is to solve a fuzzy linear system of equations with the most generalized fuzzy number, such as Triangular linear Diophantine fuzzy number, using an analytical technique called Homotopy Perturbation Method. The linear systems co-efficient are crisp when the right hand side vector is a triangular linear Diophantine fuzzy number. A numerical test examples demonstrates how our newly improved analytical technique surpasses other existing methods in terms of accuracy and CPU time. The triangular linear Diophantine fuzzy systems of equations’ strong and weak visual representations are explored.

ПОСТРОЕНИЕ МАТЕМАТИЧЕСКИХ МОДЕЛЕЙ С ИСПОЛЬЗОВАНИЕМ ПОЛИНОМИАЛЬНЫХ ФУНКЦИЙ И МАТРИЧНЫХ ИСЧИСЛЕНИЙ С ПРИМЕНЕНИЕМ СИСТЕМ НЕЧЕТКИХ ЛИНЕЙНЫХ УРАВНЕНИЙ ПРИ РЕШЕНИИ ЭКОНОМИЧЕСКИХ ЗАДАЧ

A large number of tasks in the economy are aimed at processing little formalized data. These can be tasks of analysis, planning, forecasting in the economy. The solution of these problems is reduced to the formalization of the data flow into a functional dependence. The definition of functional dependence is reduced to the approximation of data by a mathematical function. It is convenient to represent poorly formalized data as fuzzy numbers and reduce the solution of the approximation problem to the calculation of systems of fuzzy linear equations. This is achieved by developing the apparatus of software systems operating with calculations of fuzzy matrices. We have implemented computational approaches using the Zadeh principle and ordered fuzzy numbers.

АППРОКСИМАЦИЯ ПОЛИНОМАМИ НЕЧЕТКИХ ДАННЫХ И РЕШЕНИЕ НЕЧЕТКИХ СИСТЕМ ЛИНЕЙНЫХ УРАВНЕНИЙ

A large number of tasks in the economy are aimed at processing little formalized data. These can be tasks of analysis, planning, forecasting in the economy. The solution of these problems is reduced to the formalization of the data flow into a functional dependence. The definition of functional dependence is reduced to the approximation of data by a mathematical function. It is convenient to represent poorly formalized data as fuzzy numbers and reduce the solution of the approximation problem to the calculation of systems of fuzzy linear equations. This is achieved by developing the apparatus of software systems operating with calculations of fuzzy matrices. We have implemented computational approaches using the Zadeh principle and ordered fuzzy numbers.

Application of Metaheuristic Algorithm for Solving Fully Fuzzy Linear Equations System

A linear equation is an equation in which each term contains a constant with a variable of degree one or single and can be described as a straight line in a Cartesian coordinate system. A Linear equations system is a collection of several linear equations. A system of linear equations whose coefficients and variables are fuzzy numbers is called a fully fuzzy linear equation system. This study aims to apply a metaheuristic algorithm to solve a system of fully fuzzy linear equations. The objective function used is the minimization objective function. At the same time, the metaheuristic algorithms used in this research are Particle Swarm Optimization (PSO), Firefly Algorithm (FA), and Cuckoo Search (CS). The input in this research is a fully fuzzy linear equation system matrix and parameters of the PSO, FA, and CS algorithms. The resulting output is the best objective function and the variable value of the fully fuzzy linear equations system. The work was compared for accuracy with the Gauss-Jordan elimination method from previous studies with the help of the Matlab programming language. The results obtained indicate that the Particle Swarm Optimization (PSO) algorithm is better at solving fully fuzzy linear equation systems than the Firefly Algorithm (FA) and Cuckoo Search (CS). This case can be seen from the value of the resulting objective function close to the value of the Gauss-Jordan elimination methodKeywords: Mathematics, investation

Type‐2 fuzzy linear system of equations with application in static problem of structures

In this paper, a new method is proposed to solve type‐2 fuzzy linear system of equations by converting the system into three interval linear systems and one crisp linear system of equations. In this work, three possible cases of type‐2 fuzzy linear system is considered depending upon uncertainty (as type‐2 fuzzy number) in right‐hand side vector , in co‐efficient matrix , and both in co‐efficient matrix as well as right‐hand side vector . Triangular perfect quasi type‐2 fuzzy numbers are taken to propose the new method. The main significance of the proposed method is that it uses the arithmetic of interval theory, which makes this method easier to handle. Five numerical examples are solved to show the validation of the proposed technique. Further, two application problems related to 6‐bar truss and rectangular sheet are encountered in type‐2 fuzzy environment. Comparison among the results obtained by the present method and previously achieved results is also made in this work, which shows that the results are close enough and, in some cases, better bounds for the solution are obtained.

Fuzzy Number – A New Hypothesis and Solution of Fuzzy Equations

In this paper, a new hypothesis of fuzzy number has been proposed which is more precise and direct. This new proposed approach is considered as an equivalence class on set of real numbers R with its algebraic structure and its properties along with theoretical study and computational results. Newly defined hypothesis provides a well-structured summary that offers both a deeper knowledge about the theory of fuzzy numbers and an extensive view on its algebra. We defined field of newly defined fuzzy numbers which opens new era in future for fuzzy mathematics. It is shown that, by using newly defined fuzzy number and its membership function, we are able to solve fuzzy equations in an uncertain environment. We have illustrated solution of fuzzy linear and quadratic equations using the defined new fuzzy number. This can be extended to higher order polynomial equations in future. The linear fuzzy equations have numerous applications in science and engineering. We may develop some iterative methods for system of fuzzy linear equations in a very simple and ordinary way by using this new methodology. This is an innovative and purposefulness study of fuzzy numbers along with replacement of this newly defined fuzzy number with ordinary fuzzy number.

Hexagonal fuzzy number inadvertences and its applications to MCDM and HFFLS based on complete ranking by score functions

Zadeh introduced fuzzy set theory in the year 1965 to overcome the enigma and obscurity in the real world problems. The underlying power of fuzzy sets is that linguistic / qualitative variables can be used along with quantitative variables to represent continuous imprecise concepts. The continuous anagram concepts need to be modeled by continuous fuzzy numbers instead of intervals and real numbers. So, many researchers developed various fuzzy numbers such as triangular fuzzy numbers, trapezoidal fuzzy numbers, hexagonal fuzzy numbers, octagonal fuzzy numbers and decagonal fuzzy numbers based on shapes in literature. While offering these fuzzy numbers, there are some inadvertences in the existing definition of hexagonal, octagonal and decagonal fuzzy numbers. In this study, some flaws in the concept of hexagonal fuzzy numbers in literature are rectified and a new definition of hexagonal fuzzy numbers in the general form is proposed. Moreover, a complete ranking of hexagonal fuzzy numbers using score functions has also been proposed and is validated through fuzzy Multi Attribute Decision Making (MADM) problem and fuzzy linear system. Further, an algorithm for solving fully fuzzy generalized hexagonal fuzzy linear system of equations as an application of proposed ranking principle is given and is numerically illustrated.

LU Decomposition Scheme for Solving m-Polar Fuzzy System of Linear Equations

This paper presents a new scheme for solving m-polar fuzzy system of linear equations (m-PFSLEs) by using LU decomposition method. We assume the coefficient matrix of the system is symmetric positive definite, and we discuss this point in detail with some numerical examples. Furthermore, we investigate the inconsistent m×nm-polar fuzzy matrix equation (m-PFME) and find the least square solution (LSS) of this system by using generalized inverse matrix theory. Moreover, we discuss the strong solution of m-polar fuzzy LSS of the inconsistent m-PFME. In the end, we present a numerical example to illustrate our approach.

Certain methods to solve bipolar fuzzy linear system of equations

In this article, we present two different analytical methods based on embedding technique and bipolar fuzzy center to solve bipolar fuzzy linear system (BFLS) of equations. In the first method, to solve BFLS of equations, we replace BFLS of equations by a pair of positive $$(*)$$ and negative $$(\bullet )$$ two $$n \times n$$ crisp linear systems. We provide the necessary and sufficient conditions for the solution of BFLS of equations. In the second method, we use the graphical technique and apply bipolar fuzzy center to draw a graph at some specific end points to solve the BFLS of equations. Further, we develop a technique to solve the fully bipolar fuzzy linear system of equations. We present solutions of some numerical examples to show the effectiveness of the proposed techniques.

A Novel Technique to Solve the Fuzzy System of Equations

The aim of this research is to apply a novel technique based on the embedding method to solve the n × n fuzzy system of linear equations (FSLEs). By using this method, the strong fuzzy number solutions of FSLEs can be obtained in two steps. In the first step, if the created n × n crisp linear system has a non-negative solution, the fuzzy linear system will have a fuzzy number vector solution that will be found in the second step by solving another created n × n crisp linear system. Several theorems have been proved to show that the number of operations by the presented method are less than the number of operations by Friedman and Ezzati’s methods. To show the advantages of this scheme, two applicable algorithms and flowcharts are presented and several numerical examples are solved by applying them. Furthermore, some graphs of the obtained results are demonstrated that show the solutions are fuzzy number vectors.

NUMERICAL METHODS FOR FUZZY SYSTEM OF LINEAR EQUATIONS WITH CRISP COEFFICIENTS

In this paper, numerical algorithms for solving a fuzzy system of linear equations with crisp coefficients are presented. We illustrate the efficiency and accuracy of the proposed methods by solving some numerical examples. We also provide a graphical representation of the fuzzy solutions in three-dimension as a visual reference of the solution of the fuzzy system.

Certain efficient iterative methods for bipolar fuzzy system of linear equations

A bipolar fuzzy set model is an extension of fuzzy set model. We develop new iterative methods: generalized Jacobi, generalized Gauss-Seidel, refined Jacobi, refined Gauss-seidel, refined generalized Jacobi and refined generalized Gauss-seidel methods, for solving bipolar fuzzy system of linear equations(BFSLEs). We decompose n × n BFSLEs into 4n × 4n symmetric crisp linear system. We present some results that give the convergence of proposed iterative methods. We solve some BFSLEs to check the validity, efficiency and stability of our proposed iterative schemes. Further, we compute Hausdorff distance between the exact solutions and approximate solution of our proposed schemes. The numerical examples show that some proposed methods converge for the BFSLEs, but Jacobi and Gauss-seidel iterative methods diverge for BFSLEs. Finally, comparison tables show the performance, validity and efficiency of our proposed iterative methods for BFSLEs.

Программное обеспечение решения полностью нечетких систем линейных уравнений с прямоугольной матрицей системы

The article discusses the problem of solving a fully fuzzy linear system of equations with a fuzzy rectangular matrix and a fuzzy right-hand side described by fuzzy triangular numbers in a form of deviations from the mean. A solution algorithm based on finding pseudo-solutions of systems of linear equations and corresponding software is formed. Various examples of created software application for arbitrary fuzzy linear systems are given.

Alternative Determines Positivity of Hexagonal Fuzzy Numbers and Their Alternative Arithmetic

There are quite a lot of arithmetic operations for hexagonal fuzzy numbers, most of them only define positive fuzzy numbers and few are discussing negative fuzzy numbers. And rarely found inverse of a fuzzy hexagonal number. So, often the results obtained in a hexagonal fuzzy linear equation system are not compatible. In this paper, we will discuss arithmetic alternatives on fuzzy hexagonal numbers. In this paper will definitions of positive and negative fuzzy numbers based on the concept of wide area covered by hexagonal fuzzy numbers in quadrant I and in quadrant II (right and left segments called r). From the concept of positivity and negativity the hexagonal fuzzy numbers will be constructed arithmetic alternatives for hexagonal fuzzy numbers. At the final part be given an inverse for a hexagonal fuzzy number so that, so for any fuzzy number there is an inverse hexagonal fuzzy number and its multiplication produces an identity.

Bipolar fuzzy system of linear equations with polynomial parametric form

Bipolar fuzzy system of linear equations with polynomial parametric form

Solving the nondeterministic static governing equations of structures subjected to various forces under fuzzy and interval uncertainty

This paper investigates the static analysis of structures subjected to uncertain loads using the fuzzy and interval finite element method. Uncertainties are considered in terms of interval and fuzzy numbers. Herein, the material and geometric properties of the structures are considered as crisp in nature. For the presented static analysis, the fuzzy and interval finite element method converts the problem into an interval and fuzzy system of linear equations. As such, the coefficient matrix and the right-hand side vector become crisp and uncertain, respectively. To solve the corresponding system, a new method based on fuzzy center and radius has been proposed using fuzzy arithmetic. Existence of solution and convexity analysis has also been investigated here. Along with these, conditions for weak and strong fuzzy solutions have been studied. Numerical examples of beam, fifteen-bar truss, and rectangular sheet with various types of uncertain loads have been considered to illustrate the computational aspects of the proposed method.

Iterative Methods for Solving a System of Linear Equations in a Bipolar Fuzzy Environment

We develop the solution procedures to solve the bipolar fuzzy linear system of equations (BFLSEs) with some iterative methods namely Richardson method, extrapolated Richardson (ER) method, Jacobi method, Jacobi over-relaxation (JOR) method, Gauss–Seidel (GS) method, extrapolated Gauss-Seidel (EGS) method and successive over-relaxation (SOR) method. Moreover, we discuss the properties of convergence of these iterative methods. By showing the validity of these methods, an example having exact solution is described. The numerical computation shows that the SOR method with ω = 1 . 25 is more accurate as compared to the other iterative methods.

Bipolar fuzzy linear system of equations

A bipolar fuzzy set is a powerful tool for depicting fuzziness and uncertainty. This model is more flexible and practical as compared to the fuzzy model. In this paper, we define certain notions, including a bipolar fuzzy number in the parametric form, the distance between two bipolar fuzzy number and bipolar fuzzy arithmetic. We illustrate these concepts with examples. We discuss the system of linear equations and its solution process with the right-hand side as parametric bipolar fuzzy numbers, and obtain the strong solution of the system. Further, we analyze our new approach to find the solutions of a fully bipolar fuzzy linear system of equations (FBFLSE) which is based on $$(-1,1)$$ -cut expansion. Moreover, by utilizing the proposed method, we determine the maximal and minimal symmetric solutions of the FBFLSEs which are based on a tolerable solution set and a controllable solution set, respectively. We also present numerical examples of our proposed FBFLSE.

Connectionist model for solving static structural problems with fuzzy parameters

In this article, a novel approach for solving static structural problems based on Artificial Neural Network (ANN) has been presented. Various numerical methods are available to solve static problems of structures having crisp parameters by converting the problem to algebraic systems. However, sometimes these structures involve uncertainties. In that case, these problems may not be handled by the usual/existing methods. In the present study, uncertainties are considered as fuzzy. As such, one may get fuzzy linear system of equations. In order to handle these type of fuzzy linear systems, the concept of ANN has been developed. Detail procedure with algorithm has been included for solving the titled problems. Here, various example problems have been examined and results have also been compared with the existing ones in special cases.