Fuzzy Underdetermined and Overdetermined Full Rank Linear Systems of Equations using Granular Approach

This paper proposes a granular approach using the Horizontal Membership Function (HMF) for solving fuzzy underdetermined and overdetermined full rank linear systems of equations Ax = b, where A is a crisp full rank matrix and b is fuzzy. The fuzzy system is converted into a crisp parametric system, yielding the least squares solution x = (ATA)¹ATb for overdetermined cases (m>n) and the minimum norm solution x = AT (AAT)-۱b for underdetermined cases (m