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.

本文解决了最大-最小模糊关系方程组的非一致性问题，通过最小修改控制矩阵实现一致性。研究表明，所提出的方法能够有效计算与给定非一致系统相近的合成一致系统矩阵，并提供了明确的公式以优化距离测量，具有重要的应用价值。

近似矩阵的最大-最小模糊关系方程组