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We study the differentiability, integral, and calculus of linear fuzzy-number-valued functions. Special emphasis is placed on the linear fuzzy-number-valued function F ˜ ( t ) = u ˜ 1 f 1 ( t ) + u ˜ 2 f 2 ( t ) + ⋯ + u ˜ m f m ( t ) , where u ˜ 1 , u ˜ 2 , … , u ˜ m denote fuzzy n-cell numbers and f 1 , f 2 , … , f m represent real functions of a real variable. The concepts of the limit and continuity of fuzzy n-cell-number-valued functions are defined, which are the basis for studying the calculus of linear fuzzy-number-valued functions. Using the fuzzy generalized difference introduced by Gomes and Barros (2015) [8] , we define a generalized difference for fuzzy n-cell numbers. Then a generalized differentiability of fuzzy n-cell-number-valued functions is proposed, and a sufficient condition for generalized differentiability of linear fuzzy-number-valued functions is given by means of real function theory. Furthermore, a Riemann integral of fuzzy n-cell-number-valued functions is introduced, and some properties of the integral are discussed. Finally, the relationship between generalized differentiability and the Riemann integral of the linear fuzzy-number-valued function F ˜ ( t ) = u ˜ 1 f 1 ( t ) + u ˜ 2 f 2 ( t ) + ⋯ + u ˜ m f m ( t ) is derived.

Up to now, there have been many research results about approximations of fuzzy numbers by specific fuzzy numbers under preservation of some operators. However, there are few investigations about approximations of n-dimensional fuzzy numbers. The main reason is that the structure of n-dimensional fuzzy numbers are more complex. This paper devotes to seek an approximation operator that produces a special kind of 2-dimensional fuzzy number which is the nearest one to the given 2-dimensional fuzzy number along the direction k. The concept of four edges approximation for 2-dimensional fuzzy numbers is defined, which is a special kind of 2-dimensional fuzzy numbers. At the same time, the concept of fuzzy 2-cell prismoid numbers is introduced, and the nearest fuzzy 2-cell prismoid numbers of 2-dimensional fuzzy numbers which preserve the centroid of the core for the 2-dimensional fuzzy numbers with respect to the weighted pseudometric are discussed. Finally, with respect to the criteria: scale invariance, identity criterion and nearness criterion, some properties of the approximation operator are presented.

In this paper, we present a characteristic theorem for the generalized difference concept of n-dimensional fuzzy-numbers by means of support functions. The concepts of differentiability for n-dimensional fuzzy-number-valued functions are presented and discussed using the new generalized difference. The relationships of g-differentiability, s-differentiability, and l-differentiability are characterized. Furthermore, optimization problems with s-convex and s-differentiable fuzzy-number-valued functions are described.

We devote to study the arithmetic operator of fuzzy regular prismoid numbers as well as the degree of similarity between fuzzy regular prismoid numbers, and then the arithmetic operator and the degree of similarity are applied in risk analysis. Firstly, the arithmetic operator of fuzzy regular prismoid numbers are researched, and some properties of the arithmetic operators are discussed. At the same time, the fuzzy regular prismoid numbers approximation of 2-dimensional fuzzy numbers is discussed as an indispensable part to study the fuzzy risk analysis. Then, the degree of similarity between fuzzy regular prismoid numbers are deeply researched, which are the basic work of studying the fuzzy risk analysis. Finally, the proposed results are applied in risk analysis of a company.

Taking into account that interval-valued fuzzy numbers can provide more flexibility to represent the imprecise information and interval-valued trapezoidal fuzzy numbers are widely used in practice, this paper devotes to seek an approximation operator that produces an interval-valued trapezoidal fuzzy number which is the nearest one to the given interval-valued fuzzy number, and the approximation operator preserves the core of the original interval-valued fuzzy number with respect to the weighted distance. As an application, we use the interval-valued trapezoidal approximation to handle fuzzy risk analysis problems, which overcome the drawback of existing fuzzy risk analysis methods.