A Posynomial Geometric Programming Restricted to a System of Fuzzy Relation Equations

AbstractA posynomial geometric optimization problem subjected to a system of max-min fuzzy relational equations (FRE) constraints is considered. The complete solution set of FRE is characterized by unique maximal solution and finite number of minimal solutions. A two stage procedure has been suggested to compute the optimal solution for the problem. Firstly all the minimal solutions of fuzzy relation equations are determined. Then a domain specific evolutionary algorithm (EA) is designed to solve the optimization problems obtained after considering the individual sub-feasible region formed with the help of unique maximum solution and each of the minimal solutions separately as the feasible domain with same objective function. A single optimal solution for the problem is determined after solving these optimization problems. The whole procedure is illustrated with a numerical example

Max-min Learning of Approximate Weight Matrices from Fuzzy Data

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

An exact algorithm for linear optimization problem subject to max-product fuzzy relational inequalities with fuzzy constraints

Fuzzy relational inequalities with fuzzy constraints (FRI-FC) are the generalized form of fuzzy relational inequalities (FRI) in which fuzzy inequality replaces ordinary inequality in the constraints. Fuzzy constraints enable us to attain optimal points (called super-optima) that are better solutions than those resulted from the resolution of the similar problems with ordinary inequality constraints. This paper considers the linear objective function optimization with respect to max-product FRI-FC problems. It is proved that there is a set of optimization problems equivalent to the primal problem. Based on the algebraic structure of the primal problem and its equivalent forms, some simplification operations are presented to convert the main problem into a more simplified one. Finally, by some appropriate mathematical manipulations, the main problem is transformed into an optimization model whose constraints are linear. The proposed linearization method not only provides a super-optimum (that is better solution than ordinary feasible optimal solutions) but also finds the best super-optimum for the main problem. The current approach is compared with our previous work and some well-known heuristic algorithms by applying them to random test problems in different sizes.Comment: 29 pages, 8 figures, 7 table

Max-Min Fuzzy Relation Equations for a Problem of Spatial Analysis

We implement an algorithm that uses a system of max-min fuzzy  relation equations (SFRE) for solving a problem of spatial analysis. We integrate this algorithm in a Geographical information Systems (GIS) tool. We apply our  process to determine the symptoms after that an expert sets the SFRE with the values of the impact coefficients related to some parameters of a geographic zone under study. We also define an index of evaluation about the reliability of the results

On Solution of Min-Max Composition Fuzzy Relational Equation

In this paper, Min-Max composition fuzzy relation equation are studied. This study is a generalization of the works of Ohsato and Sekigushi. The conditions for the existence of solutions are studied, then the resolution of equations is discussed

Super Fuzzy Matrices and Super Fuzzy Models for Social Scientists

This book introduces the concept of fuzzy super matrices and operations on them. This book will be highly useful to social scientists who wish to work with multi-expert models. Super fuzzy models using Fuzzy Cognitive Maps, Fuzzy Relational Maps, Bidirectional Associative Memories and Fuzzy Associative Memories are defined here. The authors introduce 13 multi-expert models using the notion of fuzzy supermatrices. These models are described with illustrative examples. This book has three chapters. In the first chaper, the basic concepts about super matrices and fuzzy super matrices are recalled. Chapter two introduces the notion of fuzzy super matrices adn their properties. The final chapter introduces many super fuzzy multi expert models.Comment: 280 page

Resolution and simplification of Dombi-fuzzy relational equations and latticized optimization programming on Dombi FREs

In this paper, we introduce a type of latticized optimization problem whose objective function is the maximum component function and the feasible region is defined as a system of fuzzy relational equalities (FRE) defined by the Dombi t-norm. Dombi family of t-norms includes a parametric family of continuous strict t-norms, whose members are increasing functions of the parameter. This family of t-norms covers the whole spectrum of t-norms when the parameter is changed from zero to infinity. Since the feasible solutions set of FREs is non-convex and the finding of all minimal solutions is an NP-hard problem, designing an efficient solution procedure for solving such problems is not a trivial job. Some necessary and sufficient conditions are derived to determine the feasibility of the problem. The feasible solution set is characterized in terms of a finite number of closed convex cells. An algorithm is presented for solving 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.Comment: arXiv admin note: text overlap with arXiv:2206.09716, arXiv:2207.0637