Acceptable optimality in linear fractional programming with fuzzy coefficients

Based on the specified grades of satisfaction, we propose two new concepts of (α, β)-acceptable optimal solution and (α, β)-acceptable optimal value of a fuzzy linear fractional programming problem with fuzzy coefficients, and develop a method to compute them. An example is provided to demonstrate the method.     

Some considerations on fuzzy conditional inference

In much of human reasoning, the form of reasoning is approximate rather than exact as in ‘A red apple is ripe and this apple is more or less red. Then this apple is more or less ripe.’L.A. Zadeh and E.H. Mamdani suggested methods for such a fuzzy reasoning as an application of fuzzy set theory. The method involves an inference rule and a conditional proposition which contains fuzzy concepts.In this paper we point out that the consequence inferred by their methods does not always fit our intuitions and we suggest the improved methods which fit our intuitions under several criteria.     

Theoretical methods of constructing fuzzy inference relations

In this paper, a theoretical method is presented to select fuzzy implication operators for the fuzzy inference sentence “if x is A, then y is B”. By applying representation theorems, thirty-two fuzzy implication operators are obtained. It is shown that the obtained operators are generalizations of classical inference rule A → B, A ^c → B, A → B ^c and A ^c → B ^c respectively and can be divided into four classes. By discussion, it is found that thirty of them among 420 fuzzy implication operators presented by Li can be derived by applying representation theorems and another two new ones are obtained by the use of our methods.     

About premise reduction of fuzzy inference algorithm

The comprehensive model with “weighted-objective nearness degree” is introduced in the process of multi-objective decision-making, by which a reduction problem of inference antecedents is studied in traditional fuzzy inference method. Moreover, based on the comprehensive model with “weighted-objective nearness degree”, SMTT fuzzy inference algorithm is proposed. This algorithm not only shows the relative importance of every antecedent component in fuzzy inference, but also considers the influence of nearness degree between every antecedent component’s evaluation and inference objective on inference conclusions. The enactment of inference objective reflects the preference degree of decision maker to every antecedent component’s evaluation. Therefore, it is much fitter for the demands of practical inference.     

Theory of T-norms and fuzzy inference methods

In this paper, the theory of T-norm and T-conorm is reviewed and the T-norm, T-conorm and negation function are defined as a set of T-operators. Some typical T-operators and their mathematical properties are presented. Finally, the T-operators are extended to the conventional fuzzy reasoning methods which are based on the and operators. This extended fuzzy reasoning provides both a general and a flexible method for the design of fuzzy logic controllers and, more generally, for the modelling of any decision-making process.     

Different optimum notions for fuzzy functions and optimality conditions associated

Fuzzy numbers have been applied on decision and optimization problems in uncertain or imprecise environments. In these problems, the necessity to define optimal notions for decision-maker’s preferences as well as to prove necessary and sufficient optimality conditions for these optima are essential steps in the resolution process of the problem. The theoretical developments are illustrated and motivated with several numerical examples.     

Estimation of optimality for multidimensional control systems

Consider a class of functionsY, U on a multidimensional domainA, satisfying a differential equationY=h(x, Y, U) and given restrictions on values. We seek good lower bounds for a multiple integral over such a class. For givenY, U, such lower bounds allow one to estimate the closeness of the integral to its infimum.This research was performed under a Faculty Research Fellowship at Western Michigan University, Kalamazoo, Michigan.     

Triply fuzzy function approximation for hierarchical Bayesian inference

We prove that three independent fuzzy systems can uniformly approximate Bayesian posterior probability density functions by approximating the prior and likelihood probability densities as well as the hyperprior probability densities that underly the priors. This triply fuzzy function approximation extends the recent theorem for uniformly approximating the posterior density by approximating just the prior and likelihood densities. This approximation allows users to state priors and hyper-priors in words or rules as well as to adapt them from sample data. A fuzzy system with just two rules can exactly represent common closed-form probability densities so long as they are bounded. The function approximators can also be neural networks or any other type of uniform function approximator. Iterative fuzzy Bayesian inference can lead to rule explosion. We prove that conjugacy in the if-part set functions for prior, hyperprior, and likelihood fuzzy approximators reduces rule explosion. We also prove that a type of semi-conjugacy of if-part set functions for those fuzzy approximators results in fewer parameters in the fuzzy posterior approximator.     

Necessary conditions of optimality in quasilinear systems with delays

The optimal control problem for the processes described with quasilinear system of differential equations in presence of mixed restrictions on phase and control variables in case of delays on both variables is considered. Using the Lagrange principle of taking restrictions off necessary conditions of optimality for such kind of systems are given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Control problems in fuzzy systems

This paper deals with some problems of the control of fuzzy systems described by means of a relational equation of the type: X_{κ + 1} = U_κ ° X_κ ° R. The basic aspects such as mutual identification and control, stabilization control, control with constraints, generation of control rules of a fuzzy logic controller, important from the theoretical and applicational point of view are discussed and illustrated by means of numerical examples.     

A nondegenerate fuzzy optimality condition for constrained optimization problems without qualification conditions

Fuzzy optimization conditions in terms of the Fréchet subdifferential for reflexive spaces were investigated by Borwein, Treiman and Zhu (1998) in [1]. To achieve the nondegenerate form, it is well known that some qualification conditions should be assumed. In this paper, we are going to prove that the nondegenerate fuzzy optimality condition even holds with no qualification conditions in Asplund spaces (in particular, reflexive spaces) for optimization problems with semi-continuous and continuous data. The results are even new in finite-dimensional frameworks.     

Necessary and sufficient conditions of optimality in the problems of control with fuzzy parameters

We study the problem of high-speed operation for linear control systems with fuzzy right-hand sides. For this problem, we introduce the notion of optimal solution and establish necessary and sufficient conditions of optimality in the form of the maximum principle.     

Adjointness of fuzzy systems

This paper deals with two adjoint fuzzy systems based on

- and

-compositions, which correspond to the normal use of ∀ and ∃ in L-fuzzy logic, respectively. The resolution of composite fuzzy relational inequations for these adjoint fuzzy systems can be skillfully treated by the Gentzen-type sequent calculus as well as by the lattice-theoretical way, owing to the adjointness of

- and

-compositions.     

Near optimality of stochastic control in systems with unknown parameter processes

Stochastic control for systems with an unknown parameter is considered in this paper. The underlying problem is to minimize a functional subject to a system described by a singularly perturbed differential equation with an unknown parameter process driven by fast fluctuating random disturbances. This problem arises in the context of stochastic adaptive control, adaptive signal processing, and failure-prone manufacturing systems. Due to the nature of the wide-bandwidth noise processes, identifying the parameter process for eacht is very hard since the driving noise changes very rapidly. An alternative approach is used, and an auxiliary control problem is introduced to overcome the difficulties. By means of weak convergence methods and comparison control techniques, nearly optimal controls are obtained.This research was supported in part by the National Science Foundation under Grant DMS-9022139 and DMS-9224372.     

Dynamic programming for fuzzy systems with fuzzy environment

A dynamic programming functional equation for a multi-stage decision problem with fuzzy dynamics and environment is formulated and solved by a process of fuzzy interpolation. This is an extension of the Bellman-Zadeh model.