Multiplication and inverse operations in parametric form of triangular fuzzy number

Abstract

Many authors have given the arithmetic form of triangular fuzzy numbers, especially for addition and subtraction; however, there is not much difference. The differences occur for multiplication, division, and inverse operations. Several authors define the inverse form of triangular fuzzy numbers in parametric form. However, it always does not obtain ã(r) ⨂1/ã(r) ≠ ĩ(r), because we cannot uniquely determine the inverse that obtains the unique identity. We will not be able to directly determine the inverse of any matrix in the form of a triangular fuzzy number. Thus, all problems using the matrix à in the form of a triangular fuzzy number cannot be solved directly by determining Ã−1. In addition, there are various authors who, with various methods, try to determine Ã−1 but still do not produce à ⨂ Ã−1=ĺ. Consequently, the solution of a fully fuzzy linear system will produce an incompatible solution, which results in different authors obtaining different solutions for the same fully fuzzy linear system. This paper will promote an alternative method to determine the inverse of a fuzzy triangular number in parametric form. It begins with the construction of a midpoint (ã) for any triangular fuzzy number ã = (a, α, β), or in parametric form ã(r)=[a(r), a(r)]. Then the multiplication form will be constructed obtaining a unique inverse which produces ã(r) ⨂1/ã(r) ≠ ĩ(r). The multiplication, division, and inverse forms will be proven to satisfy various algebraic properties. Therefore, if a triangular fuzzy number is used, and also a triangular fuzzy number matrix is used, it can be easily directly applied to produce a unique inverse. At the end of this paper, we will give an example of calculating the inverse of a parametric triangular fuzzy number for various cases. It is expected that the reader can easily develop it in the case of a fuzzy matrix in the form of a triangular fuzzy number.