System Of Fuzzy Relation Equations Research Articles

MINIMIZING A LINEAR OBJECTIVE FUNCTION SUBJECT TO FUZZY RELATION EQUATIONS CONSTRAINTS WITH MAX-HAMACHER PRODUCT COMPOSITION

In this paper, an optimization model with a linear objective function subject to a system of fuzzy relation equations, using max-Hamacher product composition operator, is presented. Since its nonempty feasible solution set is in general a nonconvex set, conventional linear programming methods are not suitable to solve such a problem, so an efficient solution procedure for such problems is necessary. In this paper, the feasible solution set of this problem is studied at first. Then, one efficient algorithm (i.e. tabular method algorithm) is proposed in order to solve the problem. Some procedures are also presented to reduce the original problem. Then, the reduced problem is decomposed (if possible) into several sub-problems with smaller dimensions, so solving them becomes very easier by the algorithm. By combining the algorithm and these procedures, another more efficient algorithm is suggested in order to obtain the optimal solution of the original problem. Some numerical examples are also given to illustrate the algorithms.

System of fuzzy relation equations with [formula omitted]- [formula omitted] composition: Complete set of solutions

The problem of solvability of a system of equations with inf - → composition is considered on finite universes. Equations are expressed using operations of a BL-algebra. We study complete set of solutions of the respective system in the particular case (one equation) and in the general case. In both cases, various conditions of solvability are found and proved. We characterize all maximal solutions and prove that under certain conditions each solution of the system (an equation) is less than or equal to a respective maximal one. As a result, we are able to characterize the complete set of solutions. Examples of single equations and their systems are considered and complete sets of solutions are found for each.

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.

System of fuzzy relation equations as a continuous model of IF–THEN rules

We claim that a fuzzy relation which models a system of fuzzy IF–THEN rules has the property of continuity only if the underlying system of fuzzy relation equations is solvable. In view of this connection, it is important to consider the problem of solvability of a system of fuzzy relation equations – with respect to an unknown fuzzy relation. This problem is analyzed in the present paper. We propose several new necessary and sufficient criteria for the so-called Mamdani relation to be a solution to the system. We prove that in the general case, these same criteria are sufficient (but not always necessary) for the solvability of the system. A new, easy-to-check criterion is proposed for the solvability of the system with special fuzzy parameters.

Correct models of fuzzy IF–THEN rules are continuous

We show that a system of fuzzy IF–THEN rules being modeled correctly works as a partially given (fuzzy) function. Its behavior is determined by a chosen structure for IF–THEN rules which assigns meaning to fuzzy sets in the IF and THEN parts as well as to basic connectives. This partial function can be extended to a whole domain of fuzzy sets with the help of the Compositional Rule of Inference which plays a role of a mechanism for computing the dependent functional values. We investigate the question whether this function is continuous. Two new notions are introduced: a correct model of fuzzy IF–THEN rules in a structure and a continuous model of fuzzy IF–THEN rules with respect to given data. We prove that correctness of models of fuzzy IF–THEN rules is connected with the problem of solvability of the respective system of fuzzy relation equations. In this and only this case a model of fuzzy IF–THEN rules is continuous. As a side result we have established a new criterion of a solvability of a system of fuzzy relation equations with sup - * -composition.

Fuzzy Relation Equations (II): The Branch-point-solutions and the Categorized Minimal Solutions

This paper presents some novel theoretical results as well as practical algorithms and computational procedures on fuzzy relation equations (FRE). These results refine and improve what has already been reported in a significant manner. In the previous paper, the authors have already proved that the problem of solving the system of fuzzy relation equations is an NP-hard problem. Therefore, it is practically impossible to determine all minimal solutions for a large system if P ≠ NP. In this paper, an existence theorem is proven: there exists a special branch-point-solution that is greater than all minimal solutions and less than the maximum solution. Such branch-point-solution can be calculated based on the solution-base-matrix. Furthermore, a procedure for determining all branch-point-solutions is designed. We also provide efficient algorithms which is capable of determining as well as searching for certain types of minimal solutions. We have thus obtained: (1) a fast algorithm to determine whether a solution is a minimal solution, (2) the algorithm to search for the minimal solutions that has at least a minimum value at a component in the solution vector, and (3) the procedure of determining if a system of fuzzy relation equations has the unique minimal solution. Other properties are also investigated.

Cause and effect analysis by fuzzy relational equations and a genetic algorithm

This paper proposes using a genetic algorithm as a tool to solve the fault diagnosis problem. The fault diagnosis problem is based on a cause and effect analysis which is formally described by fuzzy relations. Fuzzy relations are formed on the basis of expert assessments. Application of expert fuzzy relations to restore and identify the causes through the observed effects requires the solution to a system of fuzzy relational equations. In this study this search for a solution amounts to solving a corresponding optimization problem. An optimization algorithm is based on the application of genetic operations of crossover, mutation and selection. The genetic algorithm suggested here represents an application in expert systems of fault diagnosis and quality control.

An ABS-FRE algorithm for solving systems of fuzzy relation equations

The general scheme of an algorithm, called an ABS-FRE algorithm, for solving systems of fuzzy relation equations (systems of FRE) via the ABS algorithms is presented. As special cases, two particular algorithms for obtaining the greatest and minimal solutions of systems of FRE are described. Several new operations used in this scheme are given, for instance, operators ∧ and ∨ called quasi-inverses of operators ∧ and ∨, respectively, etc.

Fuzzy function as an approximate solution to a system of fuzzy relation equations

This paper is mostly focused on the problem of approximate solvability of a system of fuzzy relation equations. However, we put a new light on this problem connecting it with the interpolation of a fuzzy function. We introduce a notion of fuzzy function and its representation by fuzzy relation. We demonstrate how problems of interpolation and approximation of fuzzy functions are connected with solvability of systems of fuzzy relation equations. First, we explain the general framework and then we prove some results related to the problem of the best approximation. In particular, we have shown that fuzzy relations introduced by Sanchez and Mamdani are the best approximations in certain approximation spaces.

Solvability and approximate solvability of fuzzy relation equations*

We give here a discussion of approximate solvability of a system of fuzzy relation equations. We demonstrate how problems of interpolation and approximation of fuzzy functions are connected with solvability of systems of fuzzy relation equations. First we explain the general framework, and later on we prove some particular results related to the problem of the best approximation.

Fuzzy relation equations (I): the general and specialized solving algorithms

In this article, we develop a new method and an algorithm to solve a system of fuzzy relation equations. We first introduce a solution-base-matrix and then give a tractable mathematical logic representation of all minimal solutions. Next, we design a new universal algorithm to get them. Two simplification rules are found to simplify the solution-base-matrix. We show that a polynomial time algorithm to find all minimal solutions for a general system of fuzzy relation equations simply does not exist with expectation of P=N P. Hence, the problem of solving fuzzy relation equations is an N P-hard problem in terms of computational complexity. Our universal algorithm is still useful when one does not solve a large number of equations. In many real applications the problem of solving fuzzy relation equations can be simplified in polynomial time problems. In this article, we will discuss several cases of practical applications which have such polynomial algorithms.

Fuzzy controllers with conditionally firing rules

Mamdani (1975) controller was successfully used in many applications. One of its interpretations is that it uses a fuzzy relation as an approximation of the desirable input-output correspondence. We analyze mathematical properties of Mamdani controller and notice that it has lower computational complexity when compared to the residuum-based controller. However, we show that in standard situations, both these fuzzy controllers do not represent the rule base properly in the sense of finding a solution to the related system of fuzzy relational equations. First, we consider the premises and consequents as typical inputs and outputs, and we want their correspondence to be kept. Next, we require that each normal input produces an output that bears nontrivial information. These two conditions appear to be almost contradictory to the previous controllers. We suggest a generalization of Mamdani controller which allows us to satisfy these requirements. The theory and experiments suggest that it performs better without any change of rule base and without a substantial increase of complexity.

Solving nonlinear optimization problems with fuzzy relation equation constraints

An optimization model with a nonlinear objective function subject to a system of fuzzy relation equations is presented. Since the solution set of the fuzzy relation equations is in general a non-convex set, when it is not empty, conventional nonlinear programming methods are not ideal for solving such a problem. In this paper, a genetic algorithm (GA) is proposed. This GA is designed to be domain specific by taking advantage of the structure of the solution set of fuzzy relation equations. The individuals from the initial population are chosen from the feasible solution set and are kept within the feasible region during the mutation and crossover operations. The construction of test problems is also developed to evaluate the performance of the proposed algorithm.

Optimization of fuzzy relation equations with max-product composition

An optimization problem with a linear objective function subject to a system of fuzzy relation equations using max-product composition is considered. Since the feasible domain is non-convex, traditional linear programming methods cannot be applied. We study this problem and capture some special characteristics of its feasible domain and the optimal solutions. Some procedures for reducing the original problem are presented. The problem is transformed into a 0–1 integer program which is then solved by the branch-and-bound method. For illustration purpose, an example of the procedures is provided.

Solving fuzzy relation equations with a linear objective function

An optimization model with a linear objective function subject to a system of fuzzy relation equations is presented. Due to the non-convexity of its feasible domain defined by fuzzy relation equations, designing an efficient solution procedure for solving such problems is not a trivial job. In this paper, we first characterize the feasible domain and then convert the problem to an equivalent problem involving 0–1 integer programming with a branch-and-bound solution technique. After presenting our solution procedure, a concrete example is included for illustration purpose.

An efficient solution procedure for fuzzy relation equations with max-product composition

We study a system of fuzzy relation equations with max-product composition and present an efficient solution procedure to characterize the whole solution set by finding the maximum solution as well as the complete set of minimal solutions. Instead of solving the problem combinatorially, the procedure identifies the nonminimal solutions and eliminates them from the set of minimal solutions.

Solving interval-valued fuzzy relation equations

Solving systems of fuzzy relation equations is an important topic in fuzzy set theory. This paper studies the composite interval-valued fuzzy relation equations. After analyzing the properties of its solution set, we convert the fuzzy relation equations into a fuzzy relation inequality system and propose an efficient computational procedure to generate the whole solution set. Examples are included to illustrate the idea and algorithm.

Analytical resolution and numerical identification of fuzzy relational systems

Since Sanchez's seminal paper on fuzzy relational equations, both their theories and applications have been continuously exploited by researchers. However, the solvable conditions of a system of fuzzy relational equations, also known as a fuzzy relational system (FRS), are still poorly established and their relationship with the methods for obtaining approximate solutions are unclear. When the FRS is adopted to model a fuzzy system, most of the existing identification algorithms focus on parameter estimation and less on the structure identification. In this paper, these two issues are addressed. New theoretical understandings on solving a system of fuzzy relational equations exactly and approximately are presented and their implications on the use of FRS to encode fuzzy rulebases are highlighted. Based upon the guided evolutionary simulated annealing (GESA) algorithm, an evolutionary identification formulation called EVIDENT capable for both parameter and structure identifications in fuzzy system models is proposed. As demonstrated by the simulation results, the new algorithm not only is effective in determining the structure of the systems, but also identifies a better parametric solution, as compared with that of the existing FRS identification algorithms.

A new look at solving a system of fuzzy relational equations

Solving a system of fuzzy relational equations exactly has been studied for many years. A usual understanding on the solvable conditions is that the input fuzzy sets must be normal and pairwise disjoint. Such understanding is reexamined in this paper. We show that the usual pairwise disjoint condition is too conservative and a system of max- t fuzzy relational equations can be solved exactly when the input fuzzy sets are semi-overlapped, a condition commonly found in most rule-based system applications. In addition, the boundary condition of solvability with respect to the compactness of the input fuzzy sets is derived. If it cannot be satisfied, we show that the system of equations could still be solved almost exactly by specifying the t-norm as drastic product. The results have been applied to study the capacity of fuzzy relations.

Solvability criteria for systems of fuzzy relation equations

By solving systems of fuzzy relation equations, qualitative process models can be obtained. To give more information on the solving procedure and to help constructing models, solvability criteria for-systems of fuzzy relation equations are necessary. In this article such criteria will be developed. Both sup- τ - and inf- ϱ- composition methods are considered. In addition to some ideas on general t - norms , the minimum- τ - norm is evaluated in detail. Criteria of practical use will be developed. These criteria will limit the variety of premise intersections to guarantee solvability. Nevertheless, they will still allow to model the significant behaviour of the processes.