Fuzzy Complex Numbers Research Articles

Numerical Method for the Solution of Algebraic Fuzzy Complex Equations

In this paper, the numerical solution of an algebraic complex fuzzy equation of degree ${n}$, based on the parametric fuzzy numbers, is discussed. The unknown variable and right-hand side of the equation are considered as fuzzy complex numbers, whereas, the coefficients of the equation, are considered to be real crisp numbers. The given method is a numerical method and proposed based on the separation of the real and imaginary parts of the equation and using the parametric forms of the fuzzy numbers in the form of polynomials of degree at most ${m}$. In this case, a system of nonlinear equations achieved. To get the solutions of the system, we used the Gauss-Newton iterative method. We also very briefly explain the conjugate of the solution of such equations. Finally, the efficiency and quality of the given method are tested by applying it to some numerical examples.

Data envelopment analysis with fuzzy complex numbers with an empirical case on power plants of iran

Using Data Envelopment Analysis (DEA) in complex environment is an idea that has recently presented for measuring the relative efficiencies of a set of Decision Making Units (DMUs) with complex inputs and outputs. The values of the input and output data in real-world problems appear sometimes as fuzzy complex number. For dealing with these types of data in DEA, we need to design a new model. This paper proposes a DEA model with triangular fuzzy complex numbers and solve it by using the concept of the data size and the α-level approach. This method transforms DEA model with fuzzy complex data to a linear programing problem with crisp data. In the following, a ranking model is also developed using the above approach to rank the efficient DMUs. The proposed method is presented for the first time by the authors and there is no similar method. Finally, we present a case study in the generators of the steam power plants to demonstrate the applicability of the proposed methods in the power industry.

Fuzzy complex numbers: Representations, operations, and their analysis

As a key theoretical basis of fuzzy complex analysis, a new representation of fuzzy complex numbers is proposed in this paper which unifies the exponential form, trigonometric form and algebraic form of fuzzy complex numbers, and some arithmetic operations are investigated. The concepts of strong sum and strong difference are defined in order to simplify the addition and subtraction operations of complex fuzzy numbers and establish the calculus theory of fuzzy complex valued functions. The metric between two fuzzy complex numbers, the derivative, differentiability and analyticity of fuzzy complex-valued functions are defined and characterized. The necessary and sufficient conditions of the analyticity of fuzzy complex-valued functions are investigated. Furthermore, the integral of fuzzy complex-valued functions and Cauchy integral theorem are given and discussed.

Classification of Complex Fuzzy Numbers and Fuzzy Inner Products

The paper is concerned with complex fuzzy numbers and complex fuzzy inner product spaces. In the classical complex number set, a complex number can be expressed using the Cartesian form or polar form. Both expressions are needed because one expression is better than the other depending on the situation. Likewise, the Cartesian form and the polar form can be defined in a complex fuzzy number set. First, the complex fuzzy numbers (CFNs) are categorized into two types, the polar form and the Cartesian form, as type I and type II. The properties of the complex fuzzy number set of those two expressions are discussed, and how the expressions can be used practically is shown through an example. Second, we study the complex fuzzy inner product structure in each category and find the non-existence of an inner product on CFNs of type I. Several properties of the fuzzy inner product space for type II are proposed from the modulus that is newly defined. Specfically, the Cauchy-Schwartz inequality for type II is proven in a compact way, not only the one for fuzzy real numbers. In fact, it was already discussed by Hasanhani et al; however, they proved every case in a very complicated way. In this paper, we prove the Cauchy-Schwartz inequality in a much simpler way from a general point of view. Finally, we introduce a complex fuzzy scalar product for the generalization of a complex fuzzy inner product and propose to study the condition for its existence on CFNs of type I.

On the rectangular fuzzy complex linear systems

In this paper, we focus on generalized fuzzy complex numbers and provide several examples of membership functions and graphical representations of these numbers. Then, a specific type of fuzzy complex linear systems, called the rectangular fuzzy complex linear system, is considered and its algebraic and general solutions are defined. Finally an approach based on restricting the general solution is presented to solve a rectangular fuzzy complex linear system. It is illustrated that the proposed method gives a unique algebraic solution to a rectangular fuzzy complex linear system if it exists. To show the ability and efficiency of the method and for more illustration, three numerical examples including a real application in electrical engineering are modeled and solved.

Gradual Complex Numbers and Their Application for Performance Evaluation Classifiers

Usually, the evaluation of the classifiers performance is not an easy task to be performed, mainly when we analyze different criteria (output parameters). In this evaluation process, we can use quantitative measures (accuracy, specificity, among others), however, when the output values are very close and we have several criteria, the results are difficult to be interpreted by users. This paper aims to propose a new linguistic model to evaluate the performance of several classifiers. It is based on the notion of gradual complex numbers (GCN), proposed in [18] . In this paper, we present the theoretical basis of GCNs for classifier evaluator and we assess the performance of the proposed model (GCN) through an empirical study. In addition, the performance of GCN is compared with that of fuzzy complex numbers [6] , and it reveals gains.

Mehar and Keerat method for solving system of fuzzy complex linear equations

In this paper, it is pointed out that in the existing methods (Sarangam Majumdar, Numerical solutions of fuzzy complex system of linear equations, German Journal of Advanced Mathematical Sciences 1 (2013), 20-26) for solving fuzzy complex system of linear equations in which all the parameters except variables are represented by triangular/trapezoidal fuzzy complex numbers as well as the fuzzy complex system of linear equations in which all the parameters except coefficients are represented by triangular/trapezoidal fuzzy complex numbers, Majumdar has assumed that on subtracting a fuzzy number from itself, the obtained value is a zero fuzzy number as well as on dividing a fuzzy number by itself, the obtained value is a unit fuzzy number. To resolve these flaws of the existing methods, two new methods (named as Mehar method and Keerat method) are proposed for solving the same systems of fuzzy complex linear equations.

Functional sequences and Taylor series with a fuzzy complex number as an argument

This article considers functional sequences ƒn(A) with fuzzy complex number A for an argument. The convergences limn→∞ƒ'n(z)=ƒ(z) and limn→∞ƒ'n(x)=ƒ'(x) are assumed to be uniform inside each circle supp A. Due to analyticity, the conditions of point-wise convergence of derivatives and finiteness of the number of solutions for equation ƒ(z)=w with respect to z for each w inside each circle supp A are satisfied. The paper proposes the sufficient conditions for the convergence ƒn(A) in the sense that the sequence of membership functions μƒn(A)(w) converges point-wise. The convergence limn→∞μƒn(A)(w)=μƒ(A)(w) is proved for all points w∈X, except such w=ƒ(z), that z is a discontinuity point of μA(z), or ƒ'(z)=0. As a particular case of a sequence ƒn(A), the generalization of Taylor series ƒ(z)=∑i=0∞ƒ(i)(z0)/i!(z-z0)i is considered for an analytical function ƒ(z) for the case of fuzzy complex argument z=A. The convergence of the series is considered in the sense of point-wise convergence of the partial sum μSn(A)(w), where Sn(z)=∑i=0∞ƒ(i)(z0)/i!(z-z0)i.

Multi-Fuzzy Complex Numbers and Multi-Fuzzy Complex Sets

This paper presents a method to construct more general fuzzy complex numbers and sets from ordinary fuzzy complex numbers by introduced the ordered sequences of membership functions. It is concluded that multi-fuzzy complex set is an extension of Buckley fuzzy complex set. Also, two types of multi-fuzzy complex numbers based on the forms and are investigated.

Solving fuzzy complex system of linear equations

This paper proposes a new and simple method for solving general fuzzy complex system of linear equations. In the original system, the elements of unknown variable vector and right hand side vector are considered as complex fuzzy number. Initially the general system is solved by adding and subtracting the left and right bounds of the fuzzy complex unknown and right hand side fuzzy complex vector respectively. Subsequently above obtained solutions are used to get the final solution of the general fuzzy complex system of linear equations. Two theorems are stated and proved related to the proposed method. A mathematical example and an application problem based on electrical circuit are solved using the present method. Results obtained are compared with the known results and are found to be in good agreement.

Complex fuzzy set-valued complex fuzzy measures and their properties.

Let F*(K) be the set of all fuzzy complex numbers. In this paper some classical and measure-theoretical notions are extended to the case of complex fuzzy sets. They are fuzzy complex number-valued distance on F*(K), fuzzy complex number-valued measure on F*(K), and some related notions, such as null-additivity, pseudo-null-additivity, null-subtraction, pseudo-null-subtraction, autocontionuous from above, autocontionuous from below, and autocontinuity of the defined fuzzy complex number-valued measures. Properties of fuzzy complex number-valued measures are studied in detail.

Axiomatic of Fuzzy Complex Numbers

Fuzzy numbers are fuzzy subsets of the set of real numbers satisfying some additional conditions. Fuzzy numbers allow us to model very difficult uncertainties in a very easy way. Arithmetic operations on fuzzy numbers have also been developed, and are based mainly on the crucial Extension Principle. When operating with fuzzy numbers, the results of our calculations strongly depend on the shape of the membership functions of these numbers. Logically, less regular membership functions may lead to very complicated calculi. Moreover, fuzzy numbers with a simpler shape of membership functions often have more intuitive and more natural interpretations. But not only must we apply the concept and the use of fuzzy sets, and its particular case of fuzzy number, but also the new and interesting mathematical construct designed by Fuzzy Complex Numbers, which is much more than a correlate of Complex Numbers in Mathematical Analysis. The selected perspective attempts here that of advancing through axiomatic descriptions.

On Fuzzy Complex Derivatives

In this paper, we define and study the concepts of bounded closed complex complement normalized fuzzy numbers, and generalized rectangular valued bounded closed complex complement normalized fuzzy numbers, so that some basic properties and some characterizations are presented. Some important theorems of a fuzzy derivative for fuzzy complex functions which map a regular complex numbers into bounded closed complex complement normalized fuzzy numbers are proved. All this may be a foundation for researching fuzzy complex analysis. Keywords− Fuzzy numbers, Fuzzy complex numbers, Fuzzy derivatives.

Fuzzy complex numbers and their application for classifiers performance evaluation

There are a variety of measures to describe classification performance with respect to different criteria and they are often represented by numerical values. Psychologists have commented that human beings can only reasonably manage to process seven or-so items of information at any one time. Hence, selecting the best classifier amongst a number of alternatives whose performances are represented by similar numerical values is a difficult problem faced by end users. To alleviate such difficulty, this paper presents a new method of linguistic evaluation of classifiers performance. In particular, an innovative notion of fuzzy complex numbers (FCNs) is developed in an effort to represent and aggregate different evaluation measures conjunctively without necessarily integrating them. Such an approach well maintains the underlying semantics of different evaluation measures, thereby ensuring that the resulting ranking scores are readily interpretable and the inference easily explainable. The utility and applicability of this research are illustrated by means of an experiment which evaluates the performance of 16 classifiers using different benchmark datasets. The effectiveness of the proposed approach is compared to conventional statistical approach. Experimental results show that the FCN-based performance evaluation provides an intuitively reliable and consistent means in assisting end users to make informed choices of available classifiers.

Fuzzy Compositional Modeling

Automated modeling refers to automatic (re-)formulation of alternative system models that embody the simplification, abstraction, and approximation of knowledge and data for a given task. This technique is highly desirable for effective problem solving in many application domains. Over the past two decades, compositional modeling (CM) has established itself as a leading approach in automated modeling. CM is a framework to construct system models by composing generic and reusable model fragments (MFs) selected from a knowledge base. However, the existing work mainly concerns the knowledge and data that are represented by crisp and precise information. Little work has been carried out to explore its potential to deal with uncertain environments. This paper presents an innovative framework of fuzzy compositional modeling (FCM) to develop such work. The proposed approach is capable of representing and reasoning with a wide range of inexact information. An innovative notion of fuzzy complex numbers (FCNs) is developed in an effort to enable synthesis of consistent scenario descriptions from imprecise MFs. This paper also introduces the modulus of FCNs to constrain the resulting scenario descriptions. The usefulness of this study is illustrated by means of an example to construct possible scenario descriptions from given evidence, which is in support of crime investigation.

The theoretical fundamentals of learning theory based on fuzzy complex random samples

Statistical learning theory based on real-valued random samples has been regarded as one of the influential developments for small samples statistical estimation and learning. The key theorem of learning theory and the bounds on the rate of convergence of learning process are the most important theoretical fundamentals of the statistical learning theory. In this paper, we discuss a statistical learning theory based on fuzzy complex random samples. Firstly, the definition of fuzzy complex numbers is introduced and the fuzzy complex random variables along with their numeric characteristic are investigated. Secondly, we carry out further research focused on a special type of fuzzy complex number, namely rectangular fuzzy complex number and establish some properties and develop important theorems. We also prove the strong law of large numbers based on fuzzy complex random variables. Thirdly, the definitions of the fuzzy complex expected risk functional, the fuzzy complex empirical risk functional, the fuzzy complex empirical risk minimization principle and the consistency are provided and discussed. Finally, the key theorem of learning theory and the bounds on the rate of convergence of learning process based on fuzzy complex random samples are discussed.

Notes on “On the restudy of fuzzy complex analysis: Part I and Part II”

In this note, we show by counterexamples that some results of Qiu and colleagues [On the restudy of fuzzy complex analysis: Part I. The sequence and series of fuzzy complex numbers and their convergences, Fuzzy Sets and Systems 115 (2000) 445–450; On the restudy of fuzzy complex analysis: Part II. The continuity and differentiation of fuzzy complex functions, Fuzzy Sets and Systems 120 (2001) 517–521.] concerning the convergence of the series of fuzzy complex numbers and the differentiation of fuzzy complex functions are incorrect, and offer some of their modified versions. We also show that the derivative introduced by them is inappropriate for the functions mapping complex numbers into fuzzy complex numbers.

Optimal capacitor switching with fuzzy load model for radial distribution systems

An approach which uses fuzzy variables to solve the capacitor-switching (CS) problem in radial distribution systems is proposed. CS is an important measure for loss minimisation of distribution systems via an optimal capacitor dispatch schedule. Uncertainty of forecasted load values are represented as trapezoidal fuzzy numbers via fuzzy sets. The principles of fuzzy set theory are applied to form a methodology suitable for calculations of fuzzy complex numbers while maintaining a fast computational speed. An approach based on successive linear integer programming with fuzzy variables is exploited to optimise the on-off status of each switchable capacitor bank. A computer program has been developed according to this approach, and numerical studies on a radial distribution system are verified in detail.

Generalized Lebesgue integrals of fuzzy complex valued functions

In this paper, on the basis of the theory of fuzzy complex numbers introduced by Buckley, we define fuzzy complex valued measurable functions, fuzzy complex valued fuzzy measures and corresponding generalized Lebesgue integrals of the fuzzy complex valued functions. Furthermore, some basic properties and convergence theorems with respect to this kind of Lebesgue integrals will be discussed.

Complex fuzzy sets

The objective of this paper is to investigate the innovative concept of complex fuzzy sets. The novelty of the complex fuzzy set lies in the range of values its membership function may attain. In contrast to a traditional fuzzy membership function, this range is not limited to [0, 1], but extended to the unit circle in the complex plane. Thus, the complex fuzzy set provides a mathematical framework for describing membership in a set in terms of a complex number. The inherent difficulty in acquiring intuition for the concept of complex-valued membership presents a significant obstacle to the realization of its full potential. Consequently, a major part of this work is dedicated to a discussion of the intuitive interpretation of complex-valued grades of membership. Examples of possible applications, which demonstrate the new concept, include a complex fuzzy representation of solar activity (via measurements of the sunspot number), and a signal processing application. A comprehensive study of the mathematical properties of the complex fuzzy set is presented. Basic set theoretic operations on complex fuzzy sets, such as complex fuzzy complement, union, and intersection, are discussed at length. Two novel operations, namely set rotation and set reflection, are introduced. Complex fuzzy relations are also considered. Index Terms-Complex fuzzy intersection, complex fuzzy relations, complex fuzzy sets, complex fuzzy union, complex-valued grades of membership, fuzzy complex numbers.