System Of Fuzzy Relation Equations Research Articles

On solvability of systems of partial fuzzy relational equations

Systems of fuzzy relational equations belong to key mathematical problems related to the correct behavior of fuzzy systems, especially of fuzzy inference systems. Indeed, if we find a fuzzy relation that solves a given system of fuzzy relational equations, we found a model of a fuzzy rule base related to the given system that will behave correctly in the sense that it will preserve the fundamental modus ponens property. This problem with numerous results becomes absolutely unexplored as soon as we allow partiality in the system. Partial fuzzy sets have only partially defined membership degrees, i.e., for some elements, the membership degree is undefined, i.e., partial propositions have only partially defined truth-values. There are many origins from which we may obtain propositions with undefined truth-values, e.g., non-denoting or irrelevant terms, inconsistent (both true and false) propositions, or truth-values are simply unknown. Partial logics belong to classical topics in logic so, the extensions to partial fuzzy logic in recent years were not surprising. However, we should not dare to deal with partial fuzzy sets in fuzzy systems without any guarantee of the preservation of fundamental properties such as modus ponens. This article is focusing on filling this gap.

On the solvability of weakly linear systems of fuzzy relation equations☆

Systems of fuzzy relation equations and inequalities in which an unknown fuzzy relation is on the one side of the equation or inequality are linear systems. They are the most studied ones, and a vast literature on linear systems focuses on finding solutions and solvability criteria for such systems. The situation is quite different with the so-called weakly linear systems, in which an unknown fuzzy relation is on both sides of the equation or inequality. Precisely, the scholars have only given the characterization of the set of exact solutions to such systems. This paper describes the set of fuzzy relations that solve weakly linear systems to a certain degree and provides ways to compute them. We pay special attention to developing the algorithms for computing fuzzy preorders and fuzzy equivalences that are solutions to some extent to weakly linear systems. We establish additional properties for the set of such approximate solutions over some particular types of complete residuated lattices. We demonstrate the advantage of this approach via many examples that arise from the problem of aggregation of fuzzy networks.

A GIS-based framework using fuzzy relation equation system solutions in urban planning

Several fuzzy decision models are proposed in literature to solve urban planning problems. In this research we present a novel GIS-based framework to solve decision problems in urban planning based on a System of Fuzzy Relation Equations in which the unknowns represent characteristics affecting observable facts constituting the input variables. Aim of this research is to partition the urban study area into subzones, each of which identifies a sub-area of the study area within which the set of analyzed characteristics are homogeneous. The study area is initially decomposed in atomic urban areas called microzones; for each microzone are calculated the greatest and lowest solutions of a System of Fuzzy Relation Equations by using the Universal solution Algorithm and are calculated and fuzzified the values of the output variables. Spatially adjoining microzones with same output variables are dissolved forming homogeneous urban areas with reference to the problem analyzed, called Urban Contexts. For each output variable a thematic map is constructed; in addition, a thematic map of its reliability is created. This framework is tested on a study area given by the district of Ponticelli in the municipality of Naples (Italy); comparison tests performed with respect to a previous GIS-based framework based on a System of Fuzzy Relation Equations show that our method provides a more detailed knowledge of the characteristics of the urban study area with reference to the problem dealt with.

Resolution of bipolar fuzzy relation equations with max-Łukasiewicz composition

The bipolar system of fuzzy relation equations with the max-Łukasiewicz composition is investigated in this work. A bipolar path approach is proposed for such a system. It is found that the complete solution set of the bipolar system is fully determined by its conservative bipolar paths, which are finite. Our proposed resolution approach is performed using the so-called path-based algorithm, step by step, and illustrated with numerical examples. Moreover, the global and local minimal (or maximal) solutions are discussed in this paper with a comparison to those of a classical unipolar system.

On pre-solution matrices of fuzzy relation equations over complete Brouwerian lattices

Each solution of a system of fuzzy relation equations over a complete Brouwerian lattice can be decomposed into a matrix which we call pre-solution matrix. Every pre-solution matrix can induce at least one solution. In this paper, we discuss connections between solutions and pre-solution matrices of a system of fuzzy relation equations over a complete Brouwerian lattice. Then, we explore properties of pre-solution matrices and characterize lower pre-solution matrices. Moreover, we discuss constructions of pre-solution matrices.

An efficient genetic algorithm for solving nonlinear optimization problems defined with fuzzy relational equations and max-Lukasiewicz composition

We study a nonlinear optimization problem with a system of fuzzy relational equations as its constraints. We firstly investigate the resolution of the feasible region when it is defined with max-Lukasiewicz composition and present some necessary and sufficient conditions for the feasibility and some procedures for simplifying the problem. Since the feasible solution set of the fuzzy relational equations (FRE) is non-convex and the finding of all minimal solutions is an NP-hard problem, conventional nonlinear programming methods may involve high computational complexity. Based on the theoretical properties of the problem, a genetic algorithm (GA) is presented, which preserves the feasibility of new generated solutions. The proposed GA does not need to initially find the minimal solutions. Also, it does not need to check the feasibility after generating the new solutions. Moreover, we present a method to generate feasible max-Lukasiewicz FREs as test problems for evaluating the performance of our algorithm. The proposed method has been compared with some related works. The obtained results confirm the high performance of the proposed method in solving such nonlinear problems.

Nonlinear optimization problem subjected to fuzzy relational equations defined by Dubois-Prade family of t-norms

Abstract In fuzzy set theory, triangular norms (t-norm for short) and triangular co-norms (t-conorm for short) play a key role by providing generic models for intersection and union operations on fuzzy sets. Various continuous and discontinuous t-norms have been proposed by many authors. Despite variation in the t-norms, most of the well-known continuous t-norms are Archimedean (for example, Frank, Yager, Hamacher, Sugeno-Weber and Schweizer-Sklar family). An interesting family of non-Archimedean continuous t-norms was introduced by Dubois and Prade. This paper is an attemp to study a nonlinear optimization problem whose constraints are formed as a special system of fuzzy relational equations (FRE). In this type of constraint, FREs are defined with max-Dubois-Prade composition. Firstly, we investigate the resolution of the feasible solutions set. Then, some necessary and sufficient conditions are presented to determine the feasibility or infeasibility of the solutions set. Also, some procedures are introduced for simplifying the problem. Since the feasible solutions sets of FREs are non-convex, conventional nonlinear programming methods may not be directly employed to solve the problem. Therefore, in order to overcome this difficulty, a genetic algorithm (GA) is designed based on some theoretical properties of the problem. It is shown that the proposed algorithm preserves the feasibility of new generated solutions. Moreover, a method is presented to generate feasible max-Dubois-Prade FREs as test problems. These test problems are used to evaluate the performance of our algorithm. Finally, the algorithm are compared with some related works. The obtained results confirm the high performance of the proposed algorithm in solving such nonlinear problems.

Computation of the greatest right and left invariant fuzzy quasi-orders and fuzzy equivalences

Right invariant fuzzy quasi-orders for fuzzy automata are broadly studied in the recent literature, as they arise as solutions to particular systems of fuzzy relation equations and inequalities. Some of their applications include determinization and state reduction procedures, as well as simulations and bisimulations for fuzzy automata. In this paper we provide a procedure for computing the greatest right invariant fuzzy quasi-order for a given fuzzy automaton over a complete residuated lattice. The proposed procedure terminates in a finite number of steps whenever the underlying structure of the fuzzy automaton is locally finite. When the previous condition is not satisfied, we show that the greatest right invariant fuzzy quasi-order can be obtained by taking the limit value of the convergent array of fuzzy quasi-orders for fuzzy automata over BL-algebras on the real unit interval [0,1]. Analogous procedures for computing the greatest left invariant fuzzy quasi-order, as well as the greatest right and left invariant fuzzy equivalences for a fuzzy automaton are also presented. In addition, the faster algorithm for computing the greatest right invariant equivalence on a nondeterministic automaton is also presented.

Fuzzy relation equations and inequalities with two unknowns and their applications

In this paper we study several types of systems of fuzzy relation equations and inequalities composed of a given family of fuzzy relations between two sets and two unknown fuzzy relations on these sets. Solutions to these systems are pairs of fuzzy relations on the underlying sets and can be ordered coordinatewise. We show that solutions to each of these systems form a complete lattice, and provide procedures for computing the greatest solution which is less than or equal to a given pair of fuzzy relations. We also demonstrate the application of solutions to these systems in data reduction.

A Set Covering-Based Diagnostic Expert System to Economic and Financial Applications

This paper considers the identification of problems which generate anomalies at firms through the observed symptoms on the basis of fuzzy relations and Zadeh's compositional rule of inference. A procedure for determining the fuzzy cause vector of an economic and financial diagnosis problem is proposed, which consists of the design of fuzzy relational matrix and the resolution of a system of fuzzy relational equations. An efficient algorithm for solving fuzzy relational equations in terms of the associated set covering problem is introduced. It utilizes a back-tracking method to generate each minimal covering, where no duplicate or non-minimal coverings exist. A numerical example of firms' insolvency causes diagnosis is also included.

The Set of Optimal Solutions of Geometric Programming Problem with Max-Product Fuzzy Relational Equations Constraints

In this paper, we consider how to get the set of optimal solutions of geometric programming problem with single-term exponents subject to a system of fuzzy relational equations about max-product composition. The feasible domain of this problem is nonconvex. Firstly, we propose some algorithms to illustrate how to get the set of optimal solutions based on three cases. Secondly, we show that the set of all optimal solutions is fully determined by one maximum optimal solution and a finite number of optimal lower solutions. If we know the optimal value, the maximum optimal solution can be easily computed. However, solving all optimal lower solutions remains as a challenging problem, since finding all the potential minimal solutions of max-product fuzzy relational equations is an NP-hard problem. Finally, four numerical examples are provided to illustrate validity of the proposed method.

Neural-network approach to structural tuning of classification rules based on fuzzy relational equations

An adaptive approach to structural tuning of classification rules based on solving fuzzy relational equations, allowing to avoid the fuzzy knowledge base redundancy is proposed. The approach lies in building and training the neural-fuzzy backward chaining network, the isomorphic system of fuzzy relational equations, which allows to adjust the structure of fuzzy rules as new experimental data appear. Because of the lack of effective selection methods, there is no single methodological standard for structural tuning of fuzzy rules. Modern neural-fuzzy systems use heuristic selection methods to reduce the number of rules without losing the distinctive ability of the network. It was found that using the neural-fuzzy backward chaining network allows to avoid the knowledge base redundancy while preserving the inference precision. The number of rules in the class is equal to the number of solutions, and the form of membership functions of fuzzy terms is defined by intervals of values of input variables in each solution. This approach is an alternative to the classical approach, based on selection of rules from a set of rules-candidates that prevents obtaining compact fuzzy knowledge bases.

Remarks on minimal solutions of a system of fuzzy relation equations over complete infinitely distributive lattices

In Section 3.6 of Fuzzy relation equations and their applications to knowledge engineering. Kluwer Academic Publishers, Boston (1989), Di Nola et al. presented a procedure to find a minimal solution from a fixed solution of a system of fuzzy relation equations over complete infinitely distributive lattices, and put the question: is the minimal solution found by the procedure unique or not? In this paper, we give a negative answer to the question and make some further remarks. We not only give a necessary and sufficient condition for the uniqueness of such minimal solutions, but also characterize the existence of the least solution and a unique solution of a system of fuzzy relation equations over complete infinitely distributive lattices.

Matrix elementary transformations in solving systems of fuzzy relation equations

In this paper, we consider a method using matrix elementary transformations, which is similar to the method in linear algebraic systems, for solving systems of fuzzy relation equations. The solution sets of a system of fuzzy relation equations before and after performing some elementary transformations are compared. We also give some necessary and sufficient conditions for some elementary transformations which do not change the solution sets.

Fuzzy classification knowledge base construction based on trend rules and inverse inference

In this paper, an approach to fuzzy classification rules construction within the framework of fuzzy relation equations is proposed. At the same time, the system of fuzzy trend rules serves as a carrier of expert information and generator of rules - solutions of fuzzy relation equations. The system of fuzzy classification rules can be rearranged as a set of linguistic solutions of fuzzy relation equations using the composite system of fuzzy terms, e.g. “significant rise”, “essential drop” etc., where causes and effects significance measures are described by fuzzy quantifiers. The problem of inverse logical inference, which lies in restoring the coordinates of the maximum of the fuzzy input terms membership functions for each output class is reduced to solving the system of fuzzy relation equations using a genetic algorithm. The proposed approach allows to avoid the alternative rule selection. The aim of the rule selection methods is to reduce the system complexity by removing inefficient and redundant rules and improve the system accuracy by introducing alternative rules into the final rule base. Using expert knowledge cannot guarantee the optimal cooperation activity among rules. The rule selection problem is still relevant since there is currently no single methodical standard for the optimal structural adjustment of fuzzy classification knowledge bases. Solving fuzzy relation equations using the genetic algorithm ensures the optimal number of fuzzy rules for each output term and optimal form of the membership functions of the fuzzy input terms for each linguistic solution. Consecutive solution of the optimization problems provides complexity reduction of the problem of fuzzy classification knowledge bases generation.

Linear fractional multi-objective optimization problems subject to fuzzy relational equations with the max-average composition

‎In this paper‎, ‎linear fractional multi-objective optimization problems subject to a system of fuzzy relational equations (FRE) using the max-average composition are considered‎. ‎First‎, ‎some theorems and results are presented to thoroughly identify and reduce the feasible set of the fuzzy relation equations‎. ‎Next‎, ‎the linear fractional multi-objective optimization problem is converted to a linear one using Nykowski and Zolkiewski's approach‎. ‎Then‎, ‎the efficient solutions are obtained by applying the improved e-constraint method‎. ‎‎Finally‎, ‎the proposed method is effectively tested by solving a consistent test problem‎.

Fuzzy relational inequalities and equations, fuzzy quasi-orders, closures and openings of fuzzy sets

The paper deals with systems of fuzzy relation equations known in the literature as eigen fuzzy set equations, as well as with systems of related fuzzy relation inequalities. Our main results are algorithms for computing the greatest and the smallest solutions to the considered systems of fuzzy relation inequalities, and algorithms for computing the greatest solutions to the considered systems of fuzzy relation equations. The systems are studied in the framework of complete residuated lattices as the structure of membership values, by means of fuzzy quasi-orders and closures and openings on a collection of fuzzy sets.

Reduction of fuzzy automata by means of fuzzy quasi-orders

In our recent paper we have established close relationships between state reduction of a fuzzy automaton and resolution of a particular system of fuzzy relation equations. In that paper we have also studied reductions by means of those solutions which are fuzzy equivalences. In this paper we will see that in some cases better reductions can be obtained using the solutions of this system that are fuzzy quasi-orders. Generally, fuzzy quasi-orders and fuzzy equivalences are equally good in the state reduction, but we show that right and left invariant fuzzy quasi-orders give better reductions than right and left invariant fuzzy equivalences. We also show that alternate reductions by means of fuzzy quasi-orders give better results than alternate reductions by means of fuzzy equivalences. Furthermore we study a more general type of fuzzy quasi-orders, weakly right and left invariant ones, and we show that they are closely related to determinization of fuzzy automatons. We also demonstrate some applications of weakly left invariant fuzzy quasi-orders in conflict analysis of fuzzy discrete event systems.

A Divide-and-Conquer Approach for Solving Fuzzy Max-Archimedeant-Norm Relational Equations

A system of fuzzy relational equations with the max-Archimedeant-norm composition was considered. The relevant literature indicated that this problem can be reduced to the problem of finding all the irredundant coverings of a binary matrix. A divide-and-conquer approach is proposed to solve this problem and, subsequently, to solve the original problem. This approach was used to analyze the binary matrix and then decompose the matrix into several submatrices such that the irredundant coverings of the original matrix could be constructed using the irredundant coverings of each of these submatrices. This step was performed recursively for each of these submatrices to obtain the irredundant coverings. Finally, once all the irredundant coverings of the original matrix were found, they were easily converted into the minimal solutions of the fuzzy relational equations. Experiments on binary matrices, with the number of irredundant coverings ranging from 24 to 9680, were also performed. The results indicated that, for test matrices that could initially be partitioned into more than one submatrix, this approach reduced the execution time by more than three orders of magnitude. For the other test matrices, this approach was still useful because certain submatrices could be partitioned into more than one submatrix.

Linear fractional multi-objective optimization problems subject to fuzzy relational equations with a continuous Archimedean triangular norm

In this paper, linear fractional multi-objective optimization problems subject to a system of fuzzy relational equations (FREs) using the max-Archimedean triangular norm composition are considered. First, some theorems and results are presented to simplify systems of fuzzy relational equations. Next, the feasible set of the optimization problem is reduced. Then, the linear fractional multi-objective optimization problem is converted to a linear one using Nykowski and Zolkiewski’s approach. Furthermore, the efficient solutions are obtained by applying the improved ε-constraint method.In addition, we design an algorithm to generate consistent systems of fuzzy relational equations, randomly, using the covering problem. Finally, the proposed method is effectively tested by solving a randomly generated consistent test problem.