Solutions Of Fuzzy Relation Equations Research Articles

Minimal solutions of fuzzy relation equations via maximal independent elements

Minimal solutions of fuzzy relation equations via maximal independent elements

Inverse inference based on interpretable constrained solutions of fuzzy relational equations with extended max–min composition

Inverse inference based on interpretable constrained solutions of fuzzy relational equations with extended max–min composition

Television Rating Control in the Multichannel
 Environment Using Trend Fuzzy Knowledge Bases
 and Monitoring Results †

The purpose of this study is to control the ratio of programs of different genres whenforming the broadcast grid in order to increase and maintain the rating of a channel. In themultichannel environment, television rating controls consist of selecting content, the ratings ofwhich are completely restored after advertising. The hybrid approach to rule set refinement basedon fuzzy relational calculus simplifies the process of expert recommendation systems construction.By analogy with the problem of the inverted pendulum control, the managerial actions aim to retainthe balance between the fuzzy demand and supply. The increase or decrease trends of the demandand supply are described by primary fuzzy relations. The rule-based solutions of fuzzy relationalequations connect significance measures of the primary fuzzy terms. Program set refinement bysolving fuzzy relational equations allows avoiding procedures of content-based selective filtering.The solution set generation corresponds to the granulation of television time, where each solutionrepresents the time slot and the granulated rating of the content. In automated media planning,generation of the weekly TV program in the form of the granular solution provides the decrease oftime needed for the programming of the channel broadcast grid.

Using max-continuous T fuzzy relation equations to determine the vector

Solving fuzzy relation equations (FRQs) can be used to determine the weight vector in fuzzy synthetic evaluations, etc. However, the solutions of FRQs derived by real problems usually are infinite, so the vector cannot be determined. Many real problems need use FRQs to determine an objective and unknown vector, a particular solution. This paper discusses this problem for FRQs with max-continuous t-norm composition, which contain the two most frequently used types, max-min and max-product FRQs. It further analyzes the traits that FRQs have a unique solution, gives a sufficient condition that any vector can be determined by FRQs, and points out that it does for max-continuous T FRQs. Generally, when the FRQs cannot determine the vector, we need do some new tests and add those new equations into the system so as to the new system can determine it. But the new equations derived by blind testing often are redundant, or have little value, so the vector still cannot be determined, even if we add a lot. To solve this problem, the paper investigates how to extract valuable information from the old system and then to do new tests purposely so as to let the new system efficiently determine the vector. A procedure to realize it is thus given. Besides, three examples are also given to show the study content.

Classification rule hierarchical tuning with linguistic modification based on solving fuzzy relational equations

The common problem with the hierarchical tuning methods is the lack of conditions for modification of the primary rules. The incremental approach accelerates the generation of candidate rules, but complicates the selection of the primary and modified rules. In the paper, the approach that combines semantic training, granular partition and solution of fuzzy relational equations for constructing accurate and interpretable rules is developed. The composite fuzzy model of direct logic inference based on the primary rules with granular parameters is proposed. The method of hierarchical tuning with the linguistic modification based on solving fuzzy relational equations is developed, which allows reducing the training time. It is shown that the weights of the primary rules, which are subject to modification, as well as the hedging threshold of the primary terms, are solutions of the primary system of fuzzy logic equations with the hierarchical max-min/min-max composition, which solves the problem of the hierarchical selection of the primary and modified rules for the given output classes. The genetic-neural approach was used for tuning the primary rules and solving the system of equations, as well as tuning the composite rules. The effectiveness of the approach is illustrated by the example of tuning and interpreting the solutions to the technological process quality control problem for the specified productivity classes. The primary model with granular parameters allows reducing the tuning error by 25 % compared to the primary relational model. The solution of the hierarchical selection problem allows reducing the tuning time by half.

Optimization of fuzzy classification knowledge bases using improving transformations

The method of fuzzy classification knowledge base optimization using improving transformations in the form of solutions of fuzzy relational equations is proposed. The logic-algorithmic models of improving transformations are developed, on the basis of which the genetic algorithm of fuzzy knowledge base optimization is proposed. Methods of rule generation and selection differ in computational complexity due to the redundancy of the initial model. The methods of candidate rule generation do not guarantee the optimum number of rules and optimum granularity of input variables. The selection process becomes more complicated with increasing number of criteria, in particular, when taking into account the rule length. Improving transformations are: transition to a composite model for selecting output classes and rules; transition to a relational model for selecting input terms. The min-max clustering problem is solved by generating composite rules in the form of interval solutions of the trend system of equations. The number of rules in a class is determined by the number of solutions, and the granularity is determined by intervals of values of input variables in rules. The set of minimum solutions provides the minimum rule length. Linguistic interpretation of the solutions obtained is reduced to solving the relational clustering problem. The level of detail and the density of coverage are determined by the “input terms – output classes” relational matrix, and the dimensions of hyperboxes are tuned using triangular membership functions. Improving transformations allow formalizing the process of fuzzy knowledge base generation and selection. Each improving transformation is related to the control variables (the number of terms, classes, rules) that affect the accuracy and complexity of the model. At the same time, consistent use of composite and relational improving transformations provides tuning process simplification.

Do We Need Minimal Solutions of Fuzzy Relational Equations in Advance?

Minimal solutions play a crucial role in description of all solutions to a fuzzy relational equation. The reason is that all solutions form a convex set with respect to (fuzzy) set inclusion; therefore, having all extremal solutions, we can represent the entire solution set as a union of intervals bounded from above by the greatest solution and from below by the minimal solutions. However, when computing the intervals, we obtain many duplicate solutions. The obvious question is as follows: Is there another way of representing the solution set, for instance, without the need of having all the minimal solutions in advance? We provide the positive answer to this question.

Uniqueness of solutions to fuzzy relational equations regarding <i>Max</i>-<i>av</i> composition and strong regularity of the matrices in <i>Max</i>-<i>av</i> algebra

The problem of solving a fuzzy relational equation plays an important role in fuzzy systems. In this paper, we investigate the uniqueness of solutions of fuzzy relational equations regarding Max-av composition through the relationship between minimal solutions and minimal coverage. A method for verifying the strong regularity of matrices in fuzzy Max-av algebra is proposed in the paper.

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.

Fuzzy goal programming applied to multi-objective programming problem with FREs as constraints

Article history: Received February 9, 2015 Received in revised format: May 12, 2015 Accepted June 12, 2015 Available online June 15 2015 This paper presents an alternate technique based on fuzzy goal programming (FGP) approach to solve multi-objective programming problem with fuzzy relational equations (FREs) as constraints. The proposed technique is more efficient and requires less computational work than that of algorithm suggested by Jain and Lachhwani (2009) [Jain, & Lachhwani (2009). Multiobjective programming problem with fuzzy relational equations. International Journal of Operations Research, 6(2), 55−63.]. In FGP formulation, objectives are transformed into the fuzzy goals using maximum and minimal solutions elements of FREs feasible solution set. A pseudo code computer algorithm is developed for computation of maximum solution of FREs. Suitable linear membership function is defined for each objective function. Then achievement of the highest membership value of each of the fuzzy goals is formulated by minimizing the sum of negative deviational variables. The aim of this paper is to present a simple and efficient solution procedure to obtain compromise optimal solution of multiobjective optimization problem with FREs as constraints. A comparative analysis is also carried out between two methodologies based on numerical examples. All rights reserved. Growing Science Ltd. 5 © 201

Fuzzy relation equations in pseudo BL-algebras

Bandler and Kohout investigated the solvability of fuzzy relation equations with inf-implication compositions in complete lattices. Perfilieva and Noskova investigated the solvability of fuzzy relation equations with inf-implication compositions in BL-algebras. In this paper, we investigate various solutions of fuzzy relation equations with inf-implication compositions in pseudo BL-algebras.

Minimal join decompositions and their applications to fuzzy relation equations over complete Brouwerian lattices

This paper deals with the existence of minimal solutions of fuzzy relation equations over complete Brouwerian lattices from the viewpoint of join decompositions. First, the concept of a minimal join decomposition is introduced. Then the existence conditions and properties of such decomposition are discussed. As their applications, the conditions for the existence of minimal solutions of fuzzy relation equations over complete Brouwerian lattices and complete lattices are given, respectively.

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

Abstract A 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.

Bivalent and other solutions of fuzzy relational equations via linguistic hedges

We show that the well-known results regarding solutions of fuzzy relational equations and their systems can easily be generalized to obtain criteria regarding constrained solutions such as solutions which are crisp relations. When the constraint is empty, constrained solutions are ordinary solutions. The generalization is obtained by employing intensifying and relaxing linguistic hedges, conceived in this paper as certain unary functions on the scale of truth degrees. One aim of the paper is to highlight the problem of constrained solutions and to demonstrate that this problem naturally appears when identifying unknown relations. The other is to emphasize the role of linguistic hedges as constraints.

Numerical solution of fuzzy relational equations based on smooth fuzzy norms

In this paper, we study and formulate a BP learning algorithm for fuzzy relational neural networks based on smooth fuzzy norms for functions approximation. To elaborate the model behavior more, we have used different fuzzy norms led to a new pair of fuzzy norms. An important practical case in fuzzy relational equations (FREs) is the identification problem which is studied in this work. In this work we employ a neuro-based approach to numerically solve the set of FREs and focus on generalized neurons that use smooth s-norms and t-norms as fuzzy compositional operators.

Solving nonlinear optimization problems subjected to fuzzy relation equation constraints with max–average composition using a modified genetic algorithm

In this paper a nonlinear objective optimization model subject to a system of fuzzy relation equations with max-average composition are presented. When the set of solutions of fuzzy relation equations is not empty, it is in general a non-convex set and so the conventional nonlinear programming methods are not ideal for solving such a problem. In order to solve this problem, a modified genetic algorithm is reviewed and some of its components are changed to solve the problem. The construction of test problems is also developed to evaluate the performance of the proposed algorithm.

Solutions of fuzzy relation equations based on continuous t-norms

This study is concerned with fuzzy relation equations with continuous t-norms in the form ATR = B, where A and B are the fuzzy subsets of X and Y, respectively; R ⊂ X × Y is a fuzzy relation, and T is a continuous t-norm. The problem is how to determine A from R and B. First, an “if and only if” condition of being solvable is presented. Novel algorithms are then presented for determining minimal solutions when X and Y are finite. The proposed algorithms generate all minimal solutions for the equations, making them efficient solving procedures.

ON A LOSSY IMAGE COMPRESSION/RECONSTRUCTION METHOD BASED ON FUZZY RELATIONAL EQUATIONS

The pioneer work of image compression/reconstruction based on fuzzy relational equations (ICF) and the related works are introduced. The ICF regards an original image as a fuzzy relation by embedding the bright- ness level into (0,1). The compression/reconstruction of ICF correspond to the composition/solving inverse problem formulated on fuzzy relational equations. Optimizations of ICF can be consequently deduced based on fuzzy relational calculus, i.e., computation time reduction/improvement of reconstructed im- age quality are correspond to a fast solving method/finding an approximate solution of fuzzy relational equations, respectively. Through the experiments using test images extracted from Standard Image DataBAse (SIDBA), the eectiveness of the ICF and its optimizations are shown.

Approximate solutions of fuzzy relational equations and a characterization of t-norms that define metrics for fuzzy sets

Abstract If one considers the membership degrees of fuzzy sets as truth values of a many-valued logic there is a canonical way to define a fuzzified equality for fuzzy sets. This generalized equality relation can be used to define a degree to which some value of the unknown variable is a solution of the fuzzy equation as well as a degree of solvability of the whole equation. Furthermore there is an intimate relationship between such a degree to which some fuzzy set is a solution and the “quality” to which it is an approximate solution. The background is that the negation of the fuzzified equality measures a kind of distinctness of fuzzy sets. Formally, the definition of this fuzzified equality contains as parameter some t-norm which acts as truth functions of a conjunction connective. For some such t-norms this negation of the fuzzified equality really has the properties of a metric in the mathematical sense of the word. The paper gives a necessary and sufficient condition which characterizes all those t-norms which in that way yield a metric for fuzzy sets.

Equations in classes of fuzzy relations

Abstract Particular solutions of fuzzy relation equations are examined in diverse classes of fuzzy relations. The main results concern the existence of solutions and the description of the family of all desired solutions.