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.

This paper first improves Chen and Hsieh’ s definition of generalized fuzzy numbers, which makes it the generalization of definition of fuzzy numbers. Secondly, in terms of the generalized fuzzy numbers set, we introduce two different kinds of orders and arithmetic operations and metrics based on the -cutting sets or generalized -cutting sets, so that the generalized fuzzy numbers are integrated into a partial ordering set, a semi-ring and a complete non-separable metric space. Again, through two isomorphism theorem, the relationship between generalized fuzzy number space and fuzzy number space is established. Finally, as an application of generalized fuzzy numbers, the concept of continuous generalized fuzzy number-valued functions is introduced. And it is pointed out that many results of trapezoidal generalized fuzzy numbers can also be extended to generalized fuzzy numbers.

Background & Purpose: Evaluating employees and managers as the most important assets of the organization requires a method which considers the uncertainty and complexity in the judgment and decision-making process. Thus, in this research, an attempt has been made to improve the quality and validity of human capital evaluation results with the help of new evaluation methods. Methodology: In the past studies, triangular and trapezoidal fuzzy numbers have generally been used in multi-criteria decision-making methods, but in recent years, higher-order fuzzy numbers have also been introduced in the multi-criteria decision-making literature. In this study, the fuzzy TOPSIS method with hexagonal numbers is proposed to evaluate managers. Findings: The use of the fuzzy TOPSIS approach with hexagonal numbers in the evaluation and development centers of managers' competencies improves the quality of converting the subjective judgments of evaluators into quantitative numbers and while reducing the ambiguity and complexity of the evaluation process, it makes it possible to perform calculations with more accuracy and quality. Conclusion: The results suggest that in complex and important processes such as the evaluation of managers, which has a great impact on other human resource systems and organizational productivity, the use of high-order fuzzy numbers along with multi-criteria decision-making methods can increase the validity of the evaluation results and promote meritocracy in the organization.

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.