Chi-Tsuen Yeh

Trapezoidal and triangular approximations preserving the expected interval

In this paper, an improvement of the nearest trapezoidal approximation operator preserving the expected interval is studied, which is proposed by Grzegorzewski and Mrowka. A formula for computing the improved approximation is presented. Moreover, the nearest triangular approximation operator preserving the expected interval is also investigated.

On improving trapezoidal and triangular approximations of fuzzy numbers

Recently, various researchers have proved that the approximations of fuzzy numbers may fail to be fuzzy numbers, such as the trapezoidal approximations of fuzzy numbers. In this paper, we show by an example that the weighted triangular approximation of fuzzy numbers, proposed by Zeng and Li, may lead to the same result. For filling the gap, improvements of trapezoidal and triangular approximations are proposed. The formulas for computing the two improved approximations are provided. Some properties of the two improved approximations are also proved.

A note on trapezoidal approximations of fuzzy numbers

In this paper, we propose some generalized and new properties of the trapezoidal approximations of fuzzy numbers. First, we claim that the trapezoidal approximation operator is linear with respect to the operations of fuzzy numbers. Second, we show a new property concerning the centroid points of fuzzy numbers. Furthermore, a comment for computing the trapezoidal approximations is provided. Finally, some properties about the distance between a fuzzy number and its trapezoidal approximation are proved.

Weighted trapezoidal and triangular approximations of fuzzy numbers

In 2007, Zeng and Li proposed a weighted triangular approximation of a fuzzy number. Unfortunately, this approximation may fail to be a fuzzy number. In this paper, we improve this approximation and propose a generalization by the name of weighted trapezoidal approximation. Their algorithms are also presented. Finally, some examples and relevant properties are discussed.

On the minimal solutions of max--min fuzzy relational equations

In this paper, the minimal solutions of max-min fuzzy relational equations are investigated. A sufficient and necessary condition, for discriminating whether a given solution is minimal or not, is shown. Furthermore, we propose a new algorithm for computing all minimal solutions less than or equal to a given one.

Approximations by LR-type fuzzy numbers

Recently, many scholars investigated interval, triangular, and trapezoidal approximations of fuzzy numbers. These publications can be grouped into two classes: Euclidean distance class and non-Euclidean distance class. Most approximations in Euclidean distance class can be calculated by formulas, but calculating approximations in the other class is more complicated. Furthermore, approximations in Euclidean distance class can be divided into two subclasses. One is to study approximations of fuzzy numbers without constraints, the other one is to study approximations preserving some attributes. In this paper, we use LR-type fuzzy numbers to approximate fuzzy numbers. The proposed approximations will generalize all recent approximations without constraints in Euclidean class. Also, an efficient formula is provided.

Reduction to Least-Squares Estimates in Multiple Fuzzy Regression Analysis

In this paper, we deal with the problem of least-squares multiple regression with fuzzy data. The regression coefficients are assumed to be real (crisp). A formula for solving the regression coefficients in one-variable models is derived. If each independent variable is effective (i.e., its corresponding regression coefficient is nonzero), the multiple regression problem can be replaced with a 0-1 programming problem. Its optimal solution is easily computed. Finally, we also propose effective algorithms to compute the regression coefficients in a general case.

Existence of interval, triangular, and trapezoidal approximations of fuzzy numbers under a general condition

Abstract Recently, many scholars studied approximations of fuzzy numbers by specific fuzzy numbers under preservation of some operators. In fact, these approximations may not exist for some linear operators. The purpose of this paper is to study necessary and sufficient conditions of linear operators which are preserved by interval, triangular, symmetric triangular, trapezoidal, or symmetric trapezoidal approximations of fuzzy numbers. In addition, an effective method for solving such problems is proposed.