Chebyshev distances associated to the second members of systems of Max-product/Lukasiewicz Fuzzy relational equations

Abstract

In this article, we study the inconsistency of a system of $\max$-product fuzzy relational equations and of a system of $\max$-Lukasiewicz fuzzy relational equations. For a system of $\max -\min$ fuzzy relational equations $A{\square }_{\min }^{\max }x=b$ and using the $L_{\infty }$ norm, (Baaj, 2023) showed that the Chebyshev distance $\Delta ={\inf }_{c\in \mathcal{C}}\Vert b-c\Vert$, where $\mathcal{C}$ is the set of second members of consistent systems defined with the same matrix $A$, can be computed by an explicit analytical formula according to the components of the matrix $A$ and its second member $b$. In this article, we give analytical formulas analogous to that of (Baaj, 2023) to compute the Chebyshev distance associated to the second member of a system of $\max$-product fuzzy relational equations and that associated to the second member of a system of $\max$-Lukasiewicz fuzzy relational equations.

arXiv admin note: text overlap with arXiv:2301.06141