Max-min Composition Research Articles

Solution algorithms for fuzzy relational equations with max-product composition

The conditions for the existence of an inverse solution to the max-min composition of fuzzy relational equations have been well documented since the original work by Sanchez [30, 31]. These same existence theorems have been extended to the t- norm composition of relational equations, in which the max-product composition is a member [5,13,26]. Several studies [8,15, 24, 33, 34, 38] have shown that the max-min operator may not always be the most desirable fuzzy relational composition and in fact the max-product operator was superior in these instances. This paper reviews the algorithms necessary to determine the complete solution of the inverse for fuzzy relational equations with max-product composition.

The min-max composition rule and its superiority over the usual max-min composition rule

A close analysis of the Syllogism inference rule shows that if one uses Zadeh's notion of fuzzy if-then, then the proper way of combining the membership values of two fuzzy rules r 1: “if A, then B” and r 2: “if B, then C”is not by the usual max-min composition rule, but by the following min-max rule; τ ij = min {max( μ ik , ν kj ): all j}, where τ ij = m A ( χ i ) → m c ( z j ), μ ik = m A ( χ i ) → m b ( y k ), and v kj = m B ( y k ) → m c ( z j ). The min-max value gives an upper bound on τ ik . The min-max rule results in a new notion of transitivity and a corresponding notion of a fuzzy equivalence relation. We demonstrate the superiority of the min-max rule in terms of the properties of this equivalence relation. In particular, we argue that the new form of transitivity is particularly suitable for studying non-logical ( ≠ “↔”) fuzzy equivalence relationships.

Fuzzy Concepts Applied to Preemptive Real-Time Tasks Scheduling

Good advances to task scheduling can be reached by the using both of existing scheduling algorithms and of fuzzy rules.In literature we find different number of tasks scheduling algorithms. Usually, each one of these algorithms is different fronl the others because they are based on different concepts. In this paper, we shall use fuzzy concepts to define the rules that schedule a set of tasks in real-time. These rules lead to a successful implementations of fuzzy based algorithms to schedule independent and preemptive tasks in a multiprocessor environment.The rules are of two types: constraints rules and scheduling rules. This rules are based on fuzzy interpretations of the task constraints and on well known existing algorithms in the real-time scheduling literature.The scheduling is done with the use of the max-min composition, so that it determines which task has to be scheduled at every instant. A scheduling time not too long is finally obtained by means of the preceding considerations.

Application of fuzzy reasoning in an expert system for ultrasonography.

To apply fuzzy reasoning in a computer-assisted diagnostic system for ultrasonography and to evaluate its clinical utility as a diagnostic aid for the unskilled clinician. A computer-assisted diagnostic system for ultrasonography was constructed to differentiate metastatic from inflammatory lymph nodes. The pattern of a skilled clinician's considerations was simulated with the fuzzy expert system. Three fuzzy production rules were set up according to the diagnostic criteria for lymphadenopathy. Each rule was transformed into a membership function. The max-min composition method was used for inference and the centroid method for defuzzification. The system was tested with 20 clinicians who were one to three years after graduation and inexperienced in ultrasonography. Accuracy, sensitivity and specificity were evaluated. The average increase in accuracy was 8.5%, and the sensitivity and specificity 10.7% and 6.4% respectively. These scores were significant for accuracy (P = 0.005), sensitivity (P = 0.02) and specificity (P = 0.036). Application of fuzzy reasoning in an expert system for ultrasonography improves the diagnostic performance of inexperienced clinicians.

On fuzzy difunctional relations

Difunctional relations have proved to play an important role in software design and in database theory. On the other hand, the concept of fuzzy set has received increasing attention from researchers in a wide range of scientific areas, especially in computer science. This paper extends difunctional relations in the framework of fuzzy relations with max-min composition for the purpose of gaining a better understanding of their properties, their structure, and their behavior. One motivation for this is to study fuzzy difunctional dependencies in the framework of the fuzzy relational data model.

Forecasting enrollments based on fuzzy time series

This paper presents a new method to forecast university enrollments based on fuzzy time series. The data of historical enrollments of the University of Alabama shown in Song and Chissom (1993a, 1994) are adopted to illustrate the forecasting process of the proposed method. The robustness of the proposed method is also tested. The proposed method not only can make good forecasts of the university enrollments, but also can make robust forecasts when the historical data are not accurate. The proposed method is more efficient than the one presented in Song and Chissom (1993a) due to the fact that the proposed method uses simplified arithmetic operations rather than the complicated max-min composition operations presented in Song and Chissom (1993a).

Convergence, eigen fuzzy sets and stability analysis of relational matrices

If stable control applications are formulated with a max-product composition, then instability in the relational matrix under any conditions is undesirable. This paper reviews the stability analysis of relational matrices combined with the max-min composition [19] and then presents an analysis of the stability of relational matrices combined with the max-product composition. This analysis includes results defining the convergence properties of the relational matrix, determination of the eigen fuzzy sets of the stable matrices and algorithmic solutions for several important cases. A method is also presented to stabilize unstable relational matrices for some control applications.

On fuzzy bilinear equations

When A 1 and A 2 are m × n fuzzy matrices, X is an n × 1 fuzzy matrix and · is the max-min composition, the equations A 1 · X = A 2 · X are called fuzzy bilinear equations. The purpose of this present note is to solve these equations and determine the greatest solution.

On the min-max composition of fuzzy matrices

We give some results on fuzzy matrices under the min-max composition. We also show how to construct an idempotent fuzzy matrix from a given one through the min-max composition of fuzzy matrices.

An application of fuzzy set theory to differentiating the effectiveness of drugs in treatment of inflammation of genital organs

Abstract This paper is an attempt to apply fuzzy set theory to differentiate the effectiveness of action of drugs in a group of patients. The method presented here should give an answer to the question whether it is possible to find the degree of effectiveness of the drug on some determined symptoms which are characteristics of a sample of patients. It can be concluded that the final (minimal and maximal) level of drug action, found theoretically, does not change even if the patients take the drugs for a long time. Such a conclusion is a result of finding eigen-fuzzy sets associated with given fuzzy relations where the basic operation is a max-min composition of them. The theoretical model was tested in a group of women with trichomoniasis. Three drugs (metronidazole, tinidazole, ornidazole) were taken by the patients and the mathematical results obtained were similar when compared with the physicians' observations.

The complete resolution of Cartesian products of fuzzy sets

A complete solution is given to the Cartesian problem: Given the fuzzy sets A ⊂ U, B ⊂ V, and B′ ⊂ V, find all fuzzy sets A′ ⊂ U such that A′ ° ( A × B) = B′ , where ° denotes the max-min composition. This work is an extension of the work by Pappis [ Fuzzy Sets and Systems 26 (1988) 387–391] and Kim et al. [ Fuzzy Sets and Systems 41 (1991) 293–297].

The inverse problem for distributive lattices

Abstract The inverse problem: givenA in VL(n), B in VL(m) andB′ in VL(m) find all A′ in VL(n) such that A′ ∘ (A × B) = B′, where L is distributive lattice, ∘ denotes max-min composition, VL(n) denotes the set of all vector with n member on L and x: (a1,a2,…,an)×(b1,b2,…,bn)=(ai∧bj)n×m, is investigated. The necessary and sufficient condition for the existence of a solution is shown and an analytical solution is given.

Estimation of tool wear length in finish milling using a fuzzy inference algorithm

The geometric accuracy and surface roughness are mainly affected by the flank wear at the minor cutting edge in finish machining. A fuzzy estimator obtained by a fuzzy inference algorithm with a max-min composition rule to evaluate the minor flank wear length in finish milling is introduced. The features sensitive to minor flank wear are extracted from the dispersion analysis of a time series AR model of the feed directional acceleration of the spindle housing. Linguistic rules for fuzzy estimation are constructed using these features, and then fuzzy inferences are carried out with test data sets under various cutting conditions. The proposed system turns out to be effective for estimating minor flank wear length, and its mean error is less than 12%.

Estimation of fuzzy relational matrix by using probabilistic descent method

Abstract Fuzzy relation (and fuzzy relational matrices, in particular) and a class of fuzzy relational systems become important in many areas of applications like diagnosis of complex systems, pattern classification or control. The essential task emerging there requires determination of the numerical values of the fuzzy relations. In this paper we will look at this as a certain optimization problem and examine the use of a probabilistic descent method. Especially we will be concerned with the max-min composition and its numerical representation that legitimizes the use of the abovementioned algorithm. We will propose several approximations to the max and min functions that will allow us to cope with their nondifferentiable characters. These modifications pertain both to the functions as well as their derivatives. The results of extensive numerical studies are also reported.

Study on stability of fuzzy closed-loop control systems

The problem of stability of fuzzy closed-loop control systems has long been unsolved. Through analyzing the max-min composition operator in fuzzy systems, this paper presents a reasonable analytical model of fuzzy closed-loop systems, and then proposes a method to analyze the stability of control systems by the relational matrix of fuzzy systems. An example is given to show the method available.

Controlling a Graphic Human Body Model by the Fuzzy Inference.

A three dimensional model to represent a human body in motion is investigated as a new trial in the design of garments. The body model generated by computer graphics are composed of many elements such as a head, forearms, upper arms, thighs, etc., Dynamic posture of the model is simulated by a method based on the fuzzy inference. The motion of each element is described in terms of the max-min composition of two membership functions which specify the present position and the displacement required for the change from an arbitrary posture to another. This simulation system provides the interactive method to construct the body model which assist in designing garments comfortable not only in motionless posture but also in violent motion.

A software routine to solve the generalized inverse problem of fuzzy systems

Abstract A software routine to produce the solution of the generalized inverse problem of fuzzy relational equations is presented. Given the fuzzy matrices R and B defined on U × V and W × V, respectively, the routine produces all fuzzy matrices A defined on the product space W × U which satisfy A ∘ R = B , where ∘ denotes maxmin composition.

The generalized modus ponens considered as a fuzzy relation

The generalized modus ponens (GMP) is an inference rule in fuzzy logic, with the following scheme: if S 1 is P 1 then S 2 is P 2, but S 1 is Q 1, hence S 2 is Q 2. The objective is to determine Q 2, given P 1 P 2 and Q 1. Thepredicate P 1 in the antecedent of the if-then rule is fuzzy, hence the condition S 1 is P 1 to derive S 2 is P 2 is not strict. Therefore, it should be possible to render a reasonable approximation of Q 2 when Q 1 diverges slightly from P 1. A number of criteria for this inference rule will be presented. In literature there are three basic assumptions for dealing with the GMP, namely: (1) the if-then rule is implemented by an implication rule from some multivalued logic, i.e. the if-then rule is represented by a fuzzy relation, (2) the premiss P 1 can be strengthened or weakened to obtain Q 1 by taking a power of the membership function of P 1, and (3) the conclusion Q 2 is derived by means of the max-min composition rule. It will be shown by a number of theorems that these three items are incompatible with the criteria. Furthermore, it will be shown that it is arduous to construct an implication rule to meet the conditions (2) and (3). A way to restore this is to alter condition (2). Instead of taking a power of the membership function, it is also possible to change the parameters of this function. But this also causes complications. Hence, requirement (3) should be modified into some functional relation. It will be shown how this can be accomplished in combination with the functions with parameters changed.

Resolution of the Cartesian product of fuzzy sets

The problem “given A in V ( n ), B in V ( m ) and B ′ in V ( m ) find all A ′ in V ( n ) such that A ′ · ( A × B ) = B ′”, where · denotes max-min composition, x denotes Cartesian product and V ( n ) denotes the set of all fuzzy vectors with n members, is investigated. The condition for the existence of a solution is shown and an analytical solution is given.

A computer algorithm for the solution of the inverse problem of fuzzy systems

A computer algorithm based on a method for dealing with the so-called ‘inverse problem’ is presented. The algorithm produces the solution semi-lattice formed by the fuzzy subsets A ⊂ U satisfying the fuzzy relational equation A ⊗ R = B where ⊗ denotes maxmin composition, given a fuzzy relation R⊂ U × V and a fuzzy subset B ⊂ V.