Some Algebraic Aspects of Generalized Hukuhara Differentiability of Fuzzy-valued Functions

Moloudzadeh, Saeid; Hussein Othman, Farhad

doi:10.311581/JAMDA.2509.1011.1.2.3

Some Algebraic Aspects of Generalized Hukuhara Differentiability of Fuzzy-valued Functions

Document Type : Research Article

Authors

Saeid Moloudzadeh ¹

Farhad Hussein Othman ²

¹ Department of Mathematics, Faculty of Education, Soran University, Soran, Iraq

² Department of Mathematics, Faculty of Science, Soran University, Iraq

10.311581/JAMDA.2509.1011.1.2.3

Abstract

This paper investigates some algebraic aspects of the generalized Hukuhara differentiability (gHD) of fuzzy-valued functions (FVFs) based on the generalized Hukuhara difference (gH-difference) using α-cuts. It starts by describing some operations on real compact intervals and fuzzy numbers (FNs) through their α-cuts when the α-cuts method converts FNs to real compact intervals. And establishing the necessary conditions for the existence of the gH-difference between two FNs. The paper examines two distinct FNs: trapezoidal fuzzy numbers (TrFNs) and triangular fuzzy numbers (TFNs). The concept of the gH-difference is then extended to FVFs, thereby facilitating the definition of gHD for such functions. The paper investigates whether the gH-difference, summation, subtraction and scalar multiplication of gHD FVFs are themselves gHD. Furthermore, it provides practical methods for deriving the gHD of these FVFs.

Keywords

Generalized Hukuhara differentiability

Fuzzy-valued functions

&alpha

-cut set

Operations on real compact intervals

Fuzzy numbers