Approximate matrices of systems of max-min fuzzy relational equations
Ismaïl Baaj
TL;DR
This work addresses the inconsistency of max-min fuzzy relational equations by minimally altering the governing matrix $A$ while preserving the right-hand side $b$, introducing a set $\mathcal{T}$ of consistent matrices and a Chebyshev distance $\mathring{\Delta}_{p}$ to quantify closeness under $L_p$ norms. The core contribution is the construction of auxiliary matrices $A^{(i,j)}$ and their iterative generalizations $A^{(\vec{i},\vec{j})}$ that yield consistent systems with the same $b$ and minimal matrix perturbations, together with explicit formulas for the $L_\infty$ distance and a finite set of closest matrices. The authors also extend these results to min-max systems via a dual transformation, obtaining analogous distance measures and constructive procedures. Collectively, the paper provides exact, computationally tractable methods to repair inconsistent fuzzy-relational systems, with potential impact on fuzzy learning, reasoning, and diagnostic applications where system consistency is critical.
Abstract
In this article, we address the inconsistency of a system of max-min fuzzy relational equations by minimally modifying the matrix governing the system in order to achieve consistency. Our method yields consistent systems that approximate the original inconsistent system in the following sense: the right-hand side vector of each consistent system is that of the inconsistent system, and the coefficients of the matrix governing each consistent system are obtained by modifying, exactly and minimally, the entries of the original matrix that must be corrected to achieve consistency, while leaving all other entries unchanged. To obtain a consistent system that closely approximates the considered inconsistent system, we study the distance (in terms of a norm among $L_1$, $L_2$ or $L_\infty$) between the matrix of the inconsistent system and the set formed by the matrices of consistent systems that use the same right-hand side vector as the inconsistent system. We show that our method allows us to directly compute matrices of consistent systems that use the same right-hand side vector as the inconsistent system whose distance in terms of $L_\infty$ norm to the matrix of the inconsistent system is minimal (the computational costs are higher when using $L_1$ norm or $L_2$ norm). We also give an explicit analytical formula for computing this minimal $L_\infty$ distance. Finally, we translate our results for systems of min-max fuzzy relational equations and present some potential applications.