A fuzzy set based solution method for multiobjective optimal design problem of mechanical and structural systems using functional-link net

The principle of solving multiobjective optimization problems with fuzzy sets theory is studied. Membership function is the key to introduce the fuzzy sets theory to multiobjective optimization. However, it is difficult to determine membership functions in engineering applications. On the basis of rapid quadratic optimization in the learning of weights, simplification in hardware as well as in computational procedures of functional-link net, discrete membership functions are used as sample training data. When the network converges, the continuous membership functions implemented with the network. Membership functions based on functional-link net have been used in multiobjective optimization. An example is given to illustrate the method.     

Solving fuzzy equations using evolutionary algorithms and neural nets

 In this paper we use evolutionary algorithms and neural nets to solve fuzzy equations. In Part I we: (1) first introduce our three solution methods for solving the fuzzy linear equation AˉXˉ + Bˉ= Cˉ; for Xˉ and (2) then survey the results for the fuzzy quadratic equations, fuzzy differential equations, fuzzy difference equations, fuzzy partial differential equations, systems of fuzzy linear equations, and fuzzy integral equations; and (3) apply an evolutionary algorithm to construct one of the solution types for the fuzzy eigenvalue problem. In Part II we: (1) first discuss how to design and train a neural net to solve AˉXˉ + Bˉ= Cˉ for Xˉ and (2) then survey the results for systems of fuzzy linear equations and the fuzzy quadratic.     

An application of fuzzy set theory to the external performance evaluation of distribution centers in logistics

 The purpose of this paper is to propose an algorithm for external performance evaluation in the area of logistics from retailers' viewpoint under fuzzy environment. The fundamental concepts we have adopted include the factor analysis, eigenvector method, fuzzy Delphi method, fuzzy set theory and multi-criteria decision-making method. We use factor analysis to condense twenty external performance sub-criteria into six criteria to construct the hierarchical structure of external performance evaluation of distribution centers. The fuzzy Delphi method is integrated with the eigenvector method to form a set of pooled weights of the extracted criteria. The concepts of triangular fuzzy number and linguistic variables are used to assess the preference ratings of linguistic variable, ‘importance’ and ‘appropriateness’. Through the hierarchy integration, we obtain the final scores of distribution centers' performance. Then we use a revised Chang and Chen's ranking method to rank the final scores of distribution centers for choosing the best distribution center in the area of logistic management.     

A neural network and fuzzy rule base hybrid for pattern classification

 This paper presents a novel hybrid of the two complimentary technologies of soft computing viz. neural networks and fuzzy logic to design a fuzzy rule based pattern classifier for problems with higher dimensional feature spaces. The neural network component of the hybrid, which acts as a pre-processor, is designed to take care of the all-important issue of feature selection. To circumvent the disadvantages of the popular back propagation algorithm to train the neural network, a meta-heuristic viz. threshold accepting (TA) has been used instead. Then, a fuzzy rule based classifier takes over the classification task with a reduced feature set. A combinatorial optimisation problem is formulated to minimise the number of rules in the classifier while guaranteeing high classification power. A modified threshold accepting algorithm proposed elsewhere by the authors (Ravi V, Zimmermann H.-J. (2000) Eur J Oper Res 123: 16–28) has been employed to solve this optimization problem. The proposed methodology has been demonstrated for (1) the wine classification problem having 13 features and (2) the Wisconsin breast cancer determination problem having 9 features. On the basis of these examples the results seem to be very interesting, as there is no reduction in the classification power in either of the problems, despite the fact that some of the original features have been completely eliminated from the study. On the contrary, the chosen features in both the problems yielded 100% classification power in some cases.     

On the Initialisation of Sammon’s Nonlinear Mapping

The initialisation of a neural network implementation of Sammon’s mapping, either randomly or based on the principal components (PCs) of the sample covariance matrix, is experimentally investigated. When PCs are employed, fewer experiments are needed and the network configuration can be set precisely without trial-and-error experimentation. Tested on five real-world databases, it is shown that very few PCs are required to achieve a shorter training period, lower mapping error and higher classification accuracy, compared with those based on random initialisation. Received: 20 April 1999, Received in revised form: 08 July 1999, Accepted: 05 August 1999     

Approximation capabilities of multilayer fuzzy neural networks on the set of fuzzy-valued functions

This paper first introduces a piecewise linear interpolation method for fuzzy-valued functions. Based on this, we present a concrete approximation procedure to show the capability of four-layer regular fuzzy neural networks to perform approximation on the set of all d_p continuous fuzzy-valued functions. This approach can also be used to approximate d_∞ continuous fuzzy-valued functions. An example is given to illustrate the approximation procedure.