In this paper, we study a latticized optimization problem with fuzzy relational inequality constraints where the feasible region is formed as the intersection of two inequality fuzzy systems and Sugeno - Weber family of t - norms is considered as fuzzy composition. Sugeno - Weber family of t - norms and t - conorms is one of the most applied one in various fuzzy modelling problems. Thi s family of t - norms and t - conorms was suggested by Weber for modeling intersection and union of fuzzy sets. Also, the t - conorms were suggested as addition rules by Sugeno for so - called - fuzzy measures. The resolution of the feasible region of the problem is firstly investigated when it is defined with max - Sugeno - Weber composition and a necessary and sufficient condition is presented for determining the feasibility. Then, based on some theoretical properties of the problem, an algorithm is presented for sol ving this nonlinear problem. It is proved that the algorithm can find the exact optimal solution and an example is presented to illustrate the proposed algorithm.

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.

In this article, we introduce a method for learning a capacity underlying a Sugeno integral according to training data based on systems of fuzzy relational equations. To the training data, we associate two systems of equations: a $\max-\min$ system and a $\min-\max$ system. By solving these two systems (in the case that they are consistent) using Sanchez's results, we show that we can directly obtain the extremal capacities representing the training data. By reducing the $\max-\min$ (resp. $\min-\max$) system of equations to subsets of criteria of cardinality less than or equal to $q$ (resp. of cardinality greater than or equal to $n-q$), where $n$ is the number of criteria, we give a sufficient condition for deducing, from its potential greatest solution (resp. its potential lowest solution), a $q$-maxitive (resp. $q$-minitive) capacity. Finally, if these two reduced systems of equations are inconsistent, we show how to obtain the greatest approximate $q$-maxitive capacity and the lowest approximate $q$-minitive capacity, using recent results to handle the inconsistency of systems of fuzzy relational equations.

In this article, we study the inconsistency of systems of $\min-\rightarrow$ fuzzy relational equations. We give analytical formulas for computing the Chebyshev distances $\nabla = \inf_{d \in \mathcal{D}} \Vert \beta - d \Vert$ associated to systems of $\min-\rightarrow$ fuzzy relational equations of the form $\Gamma \Box_{\rightarrow}^{\min} x = \beta$, where $\rightarrow$ is a residual implicator among the G\"odel implication $\rightarrow_G$, the Goguen implication $\rightarrow_{GG}$ or Lukasiewicz's implication $\rightarrow_L$ and $\mathcal{D}$ is the set of second members of consistent systems defined with the same matrix $\Gamma$. The main preliminary result that allows us to obtain these formulas is that the Chebyshev distance $\nabla$ is the lower bound of the solutions of a vector inequality, whatever the residual implicator used. Finally, we show that, in the case of the $\min-\rightarrow_{G}$ system, the Chebyshev distance $\nabla$ may be an infimum, while it is always a minimum for $\min-\rightarrow_{GG}$ and $\min-\rightarrow_{L}$ systems.

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 $A \Box_{\min}^{\max} x = b$ 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.

In this article, we study the approximate solutions set $\Lambda_b$ of an inconsistent system of $\max-\min$ fuzzy relational equations $(S): A \Box_{\min}^{\max}x =b$. Using the $L_\infty$ norm, we compute by an explicit analytical formula the Chebyshev distance $\Delta~=~\inf_{c \in \mathcal{C}} \Vert b -c \Vert$, where $\mathcal{C}$ is the set of second members of the consistent systems defined with the same matrix $A$. We study the set $\mathcal{C}_b$ of Chebyshev approximations of the second member $b$ i.e., vectors $c \in \mathcal{C}$ such that $\Vert b -c \Vert = \Delta$, which is associated to the approximate solutions set $\Lambda_b$ in the following sense: an element of the set $\Lambda_b$ is a solution vector $x^\ast$ of a system $A \Box_{\min}^{\max}x =c$ where $c \in \mathcal{C}_b$. As main results, we describe both the structure of the set $\Lambda_b$ and that of the set $\mathcal{C}_b$. We then introduce a paradigm for $\max-\min$ learning weight matrices that relates input and output data from training data. The learning error is expressed in terms of the $L_\infty$ norm. We compute by an explicit formula the minimal value of the learning error according to the training data. We give a method to construct weight matrices whose learning error is minimal, that we call approximate weight matrices. Finally, as an application of our results, we show how to learn approximately the rule parameters of a possibilistic rule-based system according to multiple training data.