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 e ectively 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.

Purpose: In classical data envelopment analysis models, a production system for measuring performance is considered as a black box and no attention is paid to the internal structure of decision-making units in the process of evaluation performance. However, it is important to consider the internal structure of the units to identify sources of inefficiency to calculate efficiency. On the other hands, the observed values of input and output data in real world problems are sometimes imprecise and vague. Therefore, in this paper, the network data envelopment analysis model is used in order to evaluate the performance of decision-making units with a two-stage structure in a Fuzzy environment in which input and output values are displayed in terms of triangular Fuzzy numbers. Methodology: To solve the Fuzzy two-stage data envelopment analysis model, the Fuzzy Arithmetic approach is used and a lexicographic optimization method for calculating the Fuzzy efficiency of processes and the Fuzzy efficiency of the system is proposed. Findings: The main advantage of the proposed approach over the exsiting approaches is that it solves fewer models for finding Fuzzy efficiency. Originality/Value: The application of the proposed model is explained by evaluating the performance of 24 insurance companies.

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.

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. We define certain notions, including a bipolar Fuzzy number and bipolar Fuzzy Arithmetic. In this paper we propose the new Fuzzy Arithmetic operations based on on bipolar Fuzzy numbers from [1, 2]. We define 2-dipole Fuzzy numbers and cutting them. The properties of these propose operations and their fundamental qualities are discussed in detail. Several illustrative examples were given to show the accomplishment and ability of the proposed method. At the end it is shown that the solution of the proposed method in comparison to other methods of solving Fuzzy equations are more realistic, that is, they have smaller support. Further, we analyze our new approach to find the solutions of a bipolar Fuzzy linear equations. In this paper, in addition to familiarity with bipolar Fuzzy number operations based on transmission-average and presenting practical solutions for calculations in specific cases, this problem is identified.

In this paper, we have studied the basic Arithmetic operations for developed parabolic Fuzzy numbers by using the concept of the transmission average, which was already implied in [F. Abbasi et al., A new attitude coupled with Fuzzy thinking to Fuzzy rings and  elds, Journal of Intelligent and Fuzzy Systems, 2015] in its rudimentary form and was  nally presented in its fully-edged form in [F. Abbasi et al., A new and e cient method for elementary Fuzzy Arithmetic operations on pseudo-geometric Fuzzy numbers, Journal of Fuzzy Set Valued Analysis, 2016]. The major advantage of these operations is that they  ndings are closer to reality than extension principle-based Fuzzy Arithmetic operations (in the domain of the membership function) or interval Arithmetic (in the domain of -cuts). A technical example is given to illustrate applying the method. The proposed method can model and analyze the Fuzzy system reliability in a more exible and intelligent manner in comparison with the other methods.

In this paper, the numerical solution of the improper Fuzzy integral ~I=ò+¥~a ~¦(~x)~dx is proposed where, ~¦(~x) is a bounded and closed Fuzzy valued function defined on the closed Fuzzy real number system and a is ~a Fuzzy number with triangular or bell-shape membership function. For this purpose, the a -level sets of the Fuzzy number ~I, -Ia and -Ia, 0£a£1, are evaluated whose end points are the crisp improper integrals. Then, a reliable method is introduced to estimate -Ia and -Ia by using the stochastic Arithmetic. In this case, an algorithm is presented which evaluates these integrals by using the Simpson rule. By using the CESTAC method, we find the optimal natural numbers -ma and -ma such that these improper integrals are estimated by definite integrals. Also, for a given r, we evaluate the approximate value of the membership function m~I (r) and determine the accuracy of the results. At last, two Fuzzy integrals are computed using the given algorithm to show the results of the research.

A hesitant Fuzzy number (HFN) is important as a generalization of the Fuzzy number for hesitant Fuzzy analysis and takes some applications that were discussed in recent literature. In this paper, we develop the hesitant Fuzzy Arithmetic, which is based on the extension principle for hesitant Fuzzy sets. Employing this principle, standard Arithmetic operations on Fuzzy numbers are extended to HFNs and we show that the outcome of these operations on two HFNs are an HFN. Also we use the extension principle in HFSs for the ranking of HFNs, which may be an interesting topic. In this paper, we show that the HFNs can be ordered in a natural way. To introduce a meaningful ordering of HFNs, we use a new lattice operation on HFNs based upon extension principle and de ning the Hamming distance on them. Finally, the applications of them are explained on optimization and decision-making problems.