Fuzzy Number Vector Research Articles

The general algebraic solution of dual fuzzy linear systems and fuzzy Stein matrix equations

In this paper, we present a new method for solving a dual fuzzy linear system (DFLS), AX˜+C˜=BX˜+D˜, where the coefficient matrices A and B are arbitrary real m×n matrices and C˜ and D˜ are given fuzzy number vectors. A necessary and sufficient condition for the R-consistency of the associated system of linear equations is obtained. The straightforward method for solving m×n DFLS based on an arbitrary {1}-inverse of A−B is introduced. Also, as an application, we present the first algorithm for solving the fuzzy Stein matrix equations, based on {1}-inverses. Finally, these results are illustrated by examples.

ОПТИМИЗАЦИЯ КОНЦЕНТРАЦИИ ИНГРЕДИЕНТОВ ПОЛИМЕРНОЙ КОМПОЗИЦИИ В УСЛОВИЯХ НЕЧЕТКО ЗАДАННОГО ВЗАИМОДЕЙСТВИЯ АКТИВНЫХ ДОБАВОК

It is proposed to use the solution of the fuzzy programming problem to calculate the optimal concentration of the ingredients of the polymer composition in order to best display the specified properties. The analysis of the latest achievements of the world science showed a rather high relevance of the problem under consideration. It is proposed to solve the task of fuzzy programming based on the decomposition of fuzzy numbers into alpha levels. As a result of such a decomposition, a complex of real quadratic programming problems is obtained, the solutions of which give the alpha levels of the original fuzzy programming problem. These alpha levels are approximated by a membership function, and the result is a vector of fuzzy numbers: a solution to a fuzzy programming problem. The proposed approach is illustrated by a computational experiment.

USING THE ANALYTIC HIERARCHY PROCESS WITH FUZZY LOGIC ELEMENTS TO OPTIMIZE THE DATABASE STRUCTURE

Context. Informational systems are very common and use databases to store information that users need. Many different data models can be used but the relational model is still relevant. The last decade show tendency of using distributed databases while working with relational data model and this approach requires a specially designed module to synchronize data of all separate databases. Considering optimizing the database structure, researchers didn’t pay much attention to the potential of users’ SQL-queries history. The optimal structure of all the distributed nodes could reduce the necessity of synchronization while the data access speed and its actuality would remain stable. The object of the research is the process of optimizing the structure of the distributed database of corporate information systems, which are based on the relational database’s model. Objective. The research aims at improving the accuracy of the data representation marker’s value on the distributed corporate information system’s (DCIS) node, obtained using the analytic hierarchy process by applying the fuzzy logic elements while processing the alternatives’ global priority vector. Method. The research’s authors in the set of their previous works emphasize the potential of using the collected history of users’ SQL queries. Firstly presented technology of users’ queries parsing. Then, the idea of using the multidimensional database for analyzing users’ queries by slices of workstation type, application, user, and his/her position was considered. Finally, the authors gave the full-scaled mathematical model for formalizing database and query models, and criteria of database structure’s optimality.The current research continues the given sequence and tries to increase the efficiency of the decision support system, by introducing elements of fuzzy logic to the analytic hierarchy process algorithm. The approach’s main idea is in presenting the global priorities vector in the form of a series of fuzzy sets of one variable with subsequent transformation to the exact value. This approach made it possible to maintain the accuracy of the obtained result while decreasing the number of solution alternatives. For new tuples added to the database’s tables after all calculations had been performed, the problem was formalized. After obtaining the probability of a tuple belonging to the class “needed” and performing the normalization of the value, it is taken as the level of the representation marker. Accordingly, the data is loaded onto the node if this value is greater than the optimal level of the representation marker for the DCIS node. Results. After calculating and obtaining the alternatives global priorities’ vector in order to improve the accuracy of the obtained result, the apparatus of fuzzy sets was used. The obtained vector of global priorities was presented as a vector of fuzzy digits for the data representation marker with subsequent transformation to the exact value. This approach made it possible to maintain the accuracy of the obtained result while decreasing the number of solution alternatives. Conclusions. While working on the research, the concept of a data representation marker on the DCIS node for the elements of the SQL query model was introduced. An aggregation function has been developed that allows determining the level of need for attributes and tuples in the database’s relation for the DCIS node based on the statistics of SQL queries. A model of the dependence of the database structure’s optimality criteria on the value of the data representation marker is built. Received further development method of analytic hierarchy process. The initialization of the alternatives’ pairwise comparisons matrix can be performed automatically according to the obtained mathematical models. Representation of the obtained result in the form of the vector of fuzzy numbers with the reduction to the exact value allows increasing the accuracy of the obtained results.

Bases and dimension of interval vector space

Linear independence of interval vectors is one of the most important issue in analyzing the controllability or observability of systems under uncertainty. From the viewpoint of quasilinear space, this paper aims to determine the linear independence of interval vectors by the conjunctive and disjunctive views of sets, respectively. We first show the concept of linear independence of interval vectors as conjunctive sets. And then, the dimension of $$\langle {\mathbb {I}}{\mathbb {R}}^n, +,\cdot \rangle $$ is investigated. Second, viewed the interval as a disjunctive set, we introduce the weak, strong, tolerable, and controllable linear independence of interval vectors, respectively. The method and algorithms are developed to check the weak, strong, tolerable, and controllable linear independence of a set of interval vectors. Moreover, the linear independence of fuzzy number vectors is studied, too. Finally, four numerical examples are provided to illustrate and substantiate our theoretical developments and established algorithms.

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.

Weighted Moving Averages for a Series of Fuzzy Numbers Based on Nonadditive Measures with σ − λ Rules and Choquet Integral of Fuzzy-Number-Valued Function

The aim of this study is to generalize moving average by means of Choquet integral. First, by employing nonadditive measures with δ − λ rules, the calculation of the moving average for a series of fuzzy numbers can be transformed into Choquet integration of fuzzy-number-valued function under discrete case. Meanwhile, the Choquet integral of fuzzy number and Choquet integral of fuzzy number vector are defined. Finally, some properties are investigated by means of convolution formula of Choquet integral. It shows that the results obtained in this paper extend the previous conclusions.

Type-2 Fuzzy Contingency Analysis Based on Order of Two-Dimensional Fuzzy Number Vectors

Type-2 Fuzzy Contingency Analysis Based on Order of Two-Dimensional Fuzzy Number Vectors

A Novel Weak Fuzzy Solution for Fuzzy Linear System

This article proposes a novel weak fuzzy solution for the fuzzy linear system. As a matter of fact, we define the right-hand side column of the fuzzy linear system as a piecewise fuzzy function to overcome the related shortcoming, which exists in the previous findings. The strong point of this proposal is that the weak fuzzy solution is always a fuzzy number vector. Two complex and non-complex linear systems under uncertainty are tested to validate the effectiveness and correctness of the presented method.

M-Preconditioner for Solving Fuzzy Linear Systems

An M-preconditioner is provided for solving fuzzy linear systems whose coefficient matrices are crisp M-matrices and the right-hand side columns are arbitrary fuzzy number vectors. The iterative algorithm is given for the M-preconditioned conjugate gradient (MPCG) method. The convergence is analyzed with convergence theorems. Numerical examples are given to illustrate the procedure and show the effectiveness and efficiency of the method.

On the global solution of a fuzzy linear system

The global solution of a fuzzy linear system contains the crisp vector solution of a real linear system. So discussion about the global solution of a n ×n fuzzy linear system A˜ x = ˜ b with a fuzzy number vector b in the right hand side and crisp a coefficient matrix A is considered. The advantage of the paper is developing a new algorithm to find the solution of such system by considering a global solution based upon the concept of a convex fuzzy numbers. At first the existence and uniqueness of the solution are introduced and then the related theorems and properties about the solution are proved in details. Finally the method is illustrated by solving some numerical examples.

Uzawa method for fuzzy linear system

An Uzawa method is presented for solving fuzzy linear systems whose coefficient matrix is crisp and the right-hand side column is arbitrary fuzzy number vector. The explicit iterative scheme is given. The convergence is analyzed with convergence theorems and the optimal parameter is obtained. Numerical examples are given to illustrate the procedure and show the effectiveness and efficiency of the method.

A new metric for L–R fuzzy numbers and its application in fuzzy linear systems

In this paper, a metric based on modified Euclidean metric on interval numbers, for L---R fuzzy numbers with fixed $$L(\cdot)$$ and $$R(\cdot)$$ is introduced. Then, it is applied for solving L---R fuzzy linear system (L---R-FLS) with fuzzy right-hand side, so that L---R-FLS is transformed to the minimization problem. The solution of the mentioned non-linear programming problem is our favorite fuzzy number vector solution. Two constructive Algorithms are proposed in detail and the method is illustrated by solving several numerical examples.

Simulation and evaluation of dual fully fuzzy linear systems by fuzzy neural network

In this paper, a novel hybrid method based on fuzzy neural network for approximate solution of fuzzy linear systems of the form Ax = Bx + d, where A and B are two square matrices of fuzzy coefficients, x and d are two fuzzy number vectors, is presented. Here a neural network is considered as a part of a large field called neural computing or soft computing. Moreover, in order to find the approximate solution, a simple and fast algorithm from the cost function of the fuzzy neural network is proposed. Finally, we illustrate our approach by some numerical examples.

Genetic Algorithm for Fuzzy Neural Networks using Locally Crossover

Fuzzy feed-forward (FFNR) and fuzzy recurrent networks (FRNN) proved to be solutions for "real world problems". In the most cases, the learning algorithms are based on gradient techniques adapted for fuzzy logic with heuristic rules in the case of fuzzy numbers. In this paper we propose a learning mechanism based on genetic algorithms (GA) with locally crossover that can be applied to various topologies of fuzzy neural networks with fuzzy numbers. The mechanism is applied to FFNR and FRNN with L-R fuzzy numbers as inputs, outputs and weights and fuzzy arithmetic as forward signal propagation. The α-cuts and fuzzy biases are also taken into account. The effectiveness of the proposed method is proven in two applications: the mapping a vector of triangular fuzzy numbers into another vector of triangular fuzzy numbers for FFNR and the dynamic capture of fuzzy sinusoidal oscillations for FRNN.

A note on “Fuzzy linear systems”

In this note, we show by a counterexample that the so-called weak solution of a fuzzy linear system, defined by Friedman et al. [Fuzzy linear systems, Fuzzy Sets and Systems 96 (1998) 201–209], is not always a fuzzy number vector.

Numerical solution of fully fuzzy linear systems by fuzzy neural network

In this paper, a new hybrid method based on fuzzy neural network (FNN) for approximate solution of fuzzy linear systems of the form $$Ax=d,$$ where $$A$$ is a square matrix of fuzzy coefficients, $$x$$ and $$d$$ are fuzzy number vectors, is presented. Here a neural network is considered as a part of a large field called neural computing or soft computing. Moreover, in order to find the approximate solution of an $$n\times n$$ system of fuzzy linear equations that supposedly has a unique fuzzy solution, a simple algorithm from the cost function of the FNN is proposed. Finally, we illustrate our approach by some numerical examples.

SOLVING A TYPE OF FUZZY LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS

In this study, a type of fuzzy linear systems of differential equations (FLSDE) is analyzed. The dynamics of the system is definite (crisp) but initial value is a fuzzy set the alpha-cuts of which are ellipsoids. Many practice problems are modeled with FLSDE naturally. For example, the motion equation of a satellite, a free-falling body or a charged particle is definite, but its initial position and velocity are usually uncertain. In the researches made up to now, FLSDE the initial value of which is given with a vector of fuzzy numbers is analyzed. It means that each component of the initial position vector has independent uncertainty. Such an assumption simplifies soft calculations, but it causes either serious data loss or excessive data handling in fuzzy model. For example, in two-dimensional case, approximation of a circle with the most appropriate square (or, in the more general case, approximation of an ellipse with the most appropriate rectangle) will leave out the regions that are necessary to examine or vice versa. The initial value given as a fuzzy ellipsoid makes solution of FLSDE significantly difficult and impossible to apply the methods proposed in the previous researches. In this study, a method of solution for such type of FLSDE is suggested. The method is based on the fact that a linear transformation maps an ellipsoid to another ellipsoid. At any time the system's solution constitutes a fuzzy set, alpha-cuts of which are nested ellipsoids. The suggested approach and method of solution were explained on different examples.

Decision-making in a single-period inventory environment with fuzzy demand

This paper first defines the profitability to be the probability of achieving a target profit under the optimal ordering policy, and introduces a new index (achievable capacity index; I A ) which can briefly analyze the profitability for newsboy-type product with normally distributed demand. Note that since the level of profitability depends on the demand mean μ and the demand standard deviation σ if the related costs, selling price, and target profit are given, the index I A is a function of μ and σ. Then, we assess level performance which examines if the profitability meets designated requirement. The results can determine whether the product is still desirable to order/manufacture. However, μ and σ are always unknown, and the demand quantity is common to be imprecise, especially for new product. To tackle these problems, a constructive approach combining the vector of fuzzy numbers is introduced to establish the membership function of the fuzzy estimator of I A . Furthermore, a three-decision testing rule and step-by-step procedure are developed to assess level performance based on fuzzy critical values and fuzzy p-values.

Maximal- and minimal symmetric solutions of fully fuzzy linear systems

In this paper, we shall propose a new method to obtain symmetric solutions of a fully fuzzy linear system (FFLS) based on a 1-cut expansion. To this end, we solve the 1-cut of a FFLS (in the present paper, we assumed that the 1-cut of a FFLS is a crisp linear system or equivalently, the matrix coefficient and right hand side have triangular shapes), then some unknown symmetric spreads are allocated to each row of a 1-cut of a FFLS. So, after some manipulations, the original FFLS is transformed to solving 2n linear equations to find the symmetric spreads. However, our method always give us a fuzzy number vector solution. Moreover, using the proposed method leads to determining the maximal- and minimal symmetric solutions of the FFLS which are placed in a Tolerable Solution Set and a Controllable Solution Set, respectively. However, the obtained solutions could be interpreted as bounded symmetric solutions of the FFLS which are useful for a large number of multiplications existing between two fuzzy numbers. Finally, some numerical examples are given to illustrate the ability of the proposed method.

Solution to General Fuzzy Linear System and Its Necessary and Sufficient Condition

This paper investigates general fuzzy linear systems of the form Ax = y and general dual fuzzy linear systems of the form Ax + y = Bx + z with A, B matrices of crisp coefficients and x, y fuzzy number vectors. The aim of this paper is twofold. First, by the unique least Euclidean norm solution we solve the systems with non-full rank matrices A, B. Second, we give the new necessary and sufficient condition for a strong fuzzy solution existence. Moreover, some numerical examples are designed.