Comparison between the linearly correlated difference and the generalized Hukuhara difference of fuzzy numbers

Abstract

This paper establishes the relationship between the linearly correlated difference and the generalized Hukuhara difference of fuzzy numbers. Specifically, these two types of differences are slightly different when the basic fuzzy number is non-symmetric. But they are completely coincident when the basic fuzzy number is symmetric. The main difference of two types of differences is that the linearly correlated difference always exists in the space of linearly correlated fuzzy numbers, while the generalized Hukuhara difference does not necessarily exist. Furthermore, the relationship between the linearly correlated derivative and the generalized Hukuhara derivative is also examined with the help of the relationship between two types of differences. It is interesting that, under certain appropriate conditions, the linearly correlated differentiability and the generalized Hukuhara differentiability are equivalent for a linearly correlated fuzzy number-valued function regardless of whether the basic fuzzy number is symmetric or not. Compared with the generalized Hukuhara difference and the generalized Hukuhara derivative, the calculation of the linearly correlated difference and the linearly correlated derivative is easier by using the corresponding representation functions.