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Manufacturing Systems Estimation Using Neural Network Models 295

w here /x^ an d - M f c | lzk = exp H (2 )where the norm is Euclidean. No bias terms are needed when Gaussian basis functions are used.

The output layer of the RBF network is l inear and produces a weighted sumof the outputs of the hidden layer, where the sum is calculated by the matrixmultiplication given in Eq. (3).

d2

dL

Wn W 21Wl2 W 22

WlL W 2L

WK1W K2

W KL

Zlz %

ZK

(3)

The strength of the connections between the fcth hidden unit and the Zth outputunit is denoted by weight wu- Term di, where I = 1 , . . . , L , is t he Ith component ofthe network ou tpu t vector for one in p ut /o ut p ut pair . Th e l inear ou tpu t layer function may also include a bias term Aoi- An allowance for nonlinearity in the outputlayer is possible, provided th e transf er function is invertib le. M oody an d D arke n(1989), Broomhead and Lowe (1988), and Hassoun (1995) are popular citations fortheory on Radial Basis Function networks.Training the NetworkTraining of RBF networks is most computationally efficient when a hybrid learningmethod, combining linear supervised learning and linear self-organized learning isused. Supervised learning rules adjust th e netwo rk pa ram eters to move networkoutputs closer to target outputs and self-organized learning rules modify parameters in response to network inputs only. Th e com bination of local re prese ntationand linear learning offers tremendous speed advantages relative to other architecture s such as back prop agation . T he hybrid learning meth od is an exam ple of atraining strategy that decouples learning at the hidden and the output layers, madepossible for RBF networks because of the local receptive field nature of the hiddenunits. Und er the hybrid learning m etho d, recep tive field centers and w idths arefirst dete rm ined using a self-organizing or feedforward tec hniq ue. Th en , a supe rvised feedback procedure that optimizes total error is used to adjust the networkweights and biases that connect the hidden and output layers.