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.

Relative comparison of fuzzy numbers plays an important role in the domain of the fuzzy multi-criteria and multi-attribute decision making analysis. For making a comparison between two fuzzy numbers, beyond the determination of their order, the concept of inequality is no longer crisp. That is, an inequality becomes fuzzy in the sense of representing partial belonging or degree of membership when one makes a cardinal comparison between two or more hypothetical fuzzy numbers. It means that we need to calculate m (mÎ[0,1]) in the fuzzy inequalities £m and ³m among two normal fuzzy numbers. In this paper we propose a method for capturing the membership degree of fuzzy inequalities through discretizing the membership axis (m) into equidistant intervals. In this method, the two discretized fuzzy numbers are compared point by point and at each point the degree of preferences is identified. To show its validity, this method is examined against the essential properties of fuzzy number ordering methods in [Wang, X. and E.E. Kerre, Reasonable properties for the ordering of fuzzy quantities (I). fuzzy Sets and Systems, 2001. 118(3): p. 375-385]. Furthermore it is compared numerically with some of the celebrated fuzzy number ranking methods. The results, which provides promising outcomes, may come in useful in the domain of fuzzy multi-criteria or multi-attribute decision making analysis, and more importantly, for fuzzy mathematical programming with fuzzy inequality constraints.

Stock portfolio problems are one of the most relevant real-world problems. In this study, we discuss the portfolio's risk amount, rate of risk-return, and expected return rate under a Fermatean fuzzy environment. A linear programming problem is used to formulate a Fermatean fuzzy portfolio. The Fermatean fuzzy portfolio is converted to a deterministic form using the score function. Lingo software is used to solve these deterministic portfolio problems. The main feature of this model is that investors can select a risk coefficient to enhance predicted returns and customize their strategies according to their circumstances. An example is offered that illustrates the effectiveness and dependability of the proposed approach.

In this paper, we consider the width invariant trapezoidal and triangular approximations of fuzzy numbers. The presented methods avoid the effortful computation of Karush-Kuhn-Tucker Theorem. Some properties of the new approximation methods are presented and the applicability of the methods is illustrated by examples. In addition, we show that the proposed approximations of fuzzy numbers preserve the expected value too.

The operations in the set of fuzzy numbers are usually obtained by the Zadeh extension principle. But these definitions can have some disadvantages for the applications both by an algebraic point of view and by practical aspects.  In fact the Zadeh multiplication is not distributive with respect to the addition, the shape of fuzzy numbers is not preserved by multiplication, and the indeterminateness of the sum is too increasing. Then, for the applications in the Natural and Social Sciences it is important to individuate some suitable variants of the classical addition and multiplication of fuzzy numbers that have not the previous disadvantage. Here, some possible alternatives to the Zadeh operations are studied.