fuzzy arithmetic

Fuzzy field and arithmetic operators based on transmission average (TA) and neutral member are dealt by  Abbasi et al. \cite{2}. Many examples are exist such that based on their definition $ \tilde{A}.\tilde{0}_{\tilde{A}} \tilde{0}_{\tilde{A}}.\tilde{A}) $ is not equal to $ \tilde{0}_{\tilde{A}} $. Therefore, we investigate the conditions that $ \tilde{A}.\tilde{0}_{\tilde{A}}(=\tilde{0}_{\tilde{A}}.\tilde{A})=\tilde{0}_{\tilde{A}} $. Numerical examples show the applicability of theorems and mentioned problems.

Systems of simulations linear equations play major role in various areas such as mathematics, statistics, and social sciences. Since in many applications, at least some of the system’s parameters and measurements are represented by fuzzy rather than crisp numbers, therefore, it is important to develop mathematical models and numerical procedures that would appropriately treat general fuzzy linear systems and solve them. In this paper, a new method based on fuzzy operations approach for solving Fuzzy Linear System (FLS) is introduced. The related theorems are proved in details. Finally, the proposed method is illustrated by solving two numerical examples.

Solving fuzzy linear systems has been widely studied during the last decades‎. ‎However‎, ‎there are still many challenges to solving fuzzy linear equations‎, ‎as most of the studies have used the principle of extension‎, ‎which suffers from shortcomings such as the lack of solution‎, ‎achieving solutions under very strong conditions‎, ‎large support of the obtained solutions‎, ‎inaccurate or even incorrect solutions due to not utilizing all the available information‎, complicated process and high computational load‎. ‎These problems motivated us to present a fuzzy Cramer method for solving fuzzy linear equations‎, ‎which uses arithmetic operations based on the Transmission Average (TA)‎. ‎In this study‎, ‎fully fuzzy linear systems in the form of $ \tilde{A}\tilde{X}=\tilde{B} $‎, ‎and dual fuzzy linear systems in the form of $ \tilde{A}\tilde{X}+\tilde{B}=\tilde{C}\tilde{X}+\tilde{D} $ are solved using the proposed fuzzy Cramer method‎, ‎and numerical examples are provided to confirm the effectiveness and applicability of the proposed method.

Currently fuzzy set theory has a wide range to model real life problems with incomplete or vague information which perfectly suits the reality and its application is theatrically increasing. This work explored the basic fuzzy operations with the Gaussian Membership using the α-cut method. As it is known that, the Gaussian membership function has a great role in modelling the fuzzy problems this is what impelled to explore its operation which can further be used in analysis of fuzzy problems. Primarily the basic operations which has been discussed here are addition, subtraction, multiplication, division, reciprocal, exponential, logarithmic and nth power.

Fuzzy arithmetic with standard methods such as the extension principle and$alpha $-cut lead to restricted possibilities for solving fuzzy equations.The procedures to find a solution to a fuzzy equality with these methodsrequire strong assumptions and high computation costs. Among severalapproaches dealing with this restrictions this paper focuses on theTransmission Average (TA) fuzzy arithmetic. The shape preservation ofthe TA arithmetic operations on L-R fuzzy numbers is proven. Theseproperties together with some other algebraic properties investigatedin the paper are applied to solve fuzzy polynomial equations as well as systems of linear fuzzy equations in general form. Several examples in the paper present the advantages of TA arithmetic in solving fuzzy equations.  It is shown that the results in this paper support the fact that TA arithmetic is an easy to implement approach in fuzzy modeling.

Fuzzy arithmetic performed with the product t-norm is the focus of this paper. The subject is handled from both practical and theoretical perspectives. Explicit formulas for product-sum and product-multiplication of triangular fuzzy numbers are obtained. These formulas can effectively replace the computational methods proposed so far. The issue that these operations are not shape preserving is solved by the presentation of appropriate approximations. Finally, the product arithmetic is compared in detail to the arithmetic performed with the boundary t-norms, namely the minimum and drastic sum.

The article presents a new, multidimensional arithmetic of type 2 fuzzy numbers (M-IT2-F arithmetic) in which the result is a multidimensional fuzzy set. This arithmetic increases the accuracy of calculations and the scope of problems solved in relation to the currently used interval type 2 standard fuzzy arithmetic (IT2-SF arithmetic). The proposed M-IT2-F arithmetic has mathematical properties that IT2-SF arithmetic does not have. Thanks to these properties, it provides accurate calculation results that are not over- or under-estimated in terms of uncertainty. The paper contains comparisons of both types of arithmetic in application to two problems. Fuzzy arithmetic is not a finished work and is in a phase of continuous improvement and development. M-IT2-F arithmetic is a higher form of M-IT2 (non-fuzzy) arithmetic.

Fuzzy control is one of the most important parts of fuzzy theory for which several approaches exist. Mamdani uses $\alpha$-cuts and builds the union of the membership functions which is called the aggregated consequence function. The resulting function is the starting point of the defuzzification process. In this article, we define a more natural way to calculate the aggregated consequence function via arithmetical operators. Defuzzification is the optimum value of the resultant membership function. The left and right hand sides of the membership function will be handled separately. Here, we present a new ABFC (Arithmetic Based Fuzzy Control) algorithm based on arithmetic operations which use a new defuzzification approach. The solution is much smoother, more accurate, and much faster than the classical Mamdani controller.

Ordering on fuzzy quantities have been attracted a wide domains of studies in fuzzy sets theory in the two last decades. In many practical situations as well as fuzzy mathematical programming, it is necessary to the decision makers consider L-R fuzzy numbers according to their aims. But in the most of methods which are presented to order fuzzy numbers, the authors have been considered a special kind of fuzzy numbers such as triangular fuzzy numbers, trapezoidal fuzzy numbers and etc. But as we know the L-R fuzzy numbers as a general kind of these numbers have not been discussed. Hence in this paper, we focus on a general L-R fuzzy number and propose a new approach to order them as an extension of the method which is given by Nasseri in [14]. For validity of the proposed method, we will illustrate this method based on a convenient examples which is appeared in the literature of fuzzy ordering. Furthermore, we emphasize that the proposed method will be useful for evaluating the optimality conditions in the fuzzy primal simplex algorithms and the other related algorithms such as the fuzzy dual simplex algorithm and the fuzzy two phase simplex algorithm, fuzzy transportation models, fuzzy interval linear programming and etc.

Data obtained in association with many real-life random experiments from different fields cannot be perfectly/exactly quantified.hspace{.1cm}Often the underlying imprecision can be suitably described in terms of fuzzy numbers/\values. For these random experiments, the scale of fuzzy numbers/values enables to capture more variability and subjectivity than that of categorical data, and more accuracy and expressiveness than that of numerical/vectorial data. On the other hand, random fuzzy numbers/sets model the random mechanisms generating experimental fuzzy data, and they are soundly formalized within the probabilistic setting.This paper aims to review a significant part of the recent literature concerning the statistical data analysis with fuzzy data and being developed around the concept of random fuzzy numbers/sets.