Reasonings of fuzzy arithmetical operations with Second Function Principle under generalized trapezoidal fuzzy numbers

The main purpose of this paper is to find the reasons why we use the Second Function Principle as the fuzzy arithmetical operations in the field of trapezoidal fuzzy numbers. Generalized trapezoidal fuzzy numbers are applied on data analysis, artificial intelligence, and decision making widely. Until now, if we want to find the optimization of a fuzzy equality, usually we have to defuzzify the operation result of the fuzzy equation. Therefore, we have to find a nice defuzzifing method. Here we use the graded mean integration representation method for representing a generalized trapezoidal fuzzy number. We also have proved some representing properties of generalized trapezoidal fuzzy numbers after the fuzzy arithmetical operations with Second Function Principle. The results show that this representation method is better than other methods. Moreover, we use this representation value to define the ranking, distance, similarity absolute value, and fuzzy distance as a excellence ideas.