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parameters are modified to the direction of minimizing the difference between the calculated output and targeted output during training.
3.2.2 Formation of decision making pattern
In order to create the decision making pattern, computation of the maximums and minimums of two objective functions within a proper range of the process factors that cover a slightly wider scope than that of the experiments should be initially carried out. Once the maximum and minimum of the objective functions are determined, the scaled values of the objective functions within the range are regularly arranged on a two-dimensional table and every possible combination of two objective functions within the range is ranked with respect to the supervisor’s preference. The final decision making pattern is completed by scaling
the ranking into values between 0 and 1. Figure 3 demonstrates the surface of the NN decision maker where the approximation errors at the sampling points can be ignored.
3.2.3 Structure of NN decision maker and error back-propagation
The NN decision maker has the structure of a multi-layer neural network that consists of an input layer, four hidden layers and an output layer, which can be expressed as 2-6-8-7-5-1. The main advantage of multi-layer neural networks exists in the generalization ability for nonlinear systems. By training the finite number of a discrete data set,a system that responds to continuous inputs can be created respectively. The convergence condition of the least square error is resolved at 0.0001, and after about 20,000 cycles,training is terminated.
4 Results and discussions
The thixoforging experiments were performed under the experimental conditions, and the pistons were obtained as shown in Fig. 4.In order to measure the hardness of the products, the piston was cut vertically. Figure 5 shows the measurement location for the hardness test. Rockwell (B scale) was used for the hardness test. Hardness value, hardness average,and standard deviation are shown in Table and a prediction of values beyond the trained data is possible, which is similar to nonlinear interpolation. Error back propagation, one of the neural network training skills, has been a model for multi-layer neural networks. It does not use a sort of feedback operation but errors are back-propagated to modify the connection factors, such as connection weights and momentums of neural networks during training. The following equations explain the mechanics of error back propagation for the multi-layer neural network.
Here, Wji(t) is a connection weight between neurons,ΔWji(t) is the gradient of Wji(t), t is the number of the training cycle, Oi is an output value of the ith neuron, Oj is the output value of the jth neuron, and dj is the expected output of the jth neuron. is the differential value of the sigmoid function jth neuron at the middle layers, η is the learning rate, and α is the momentum factor.Learning rate parameters play a role in scaling the adjustments to weights, and momentum factors are used in scaling the adjustments from a previous iteration and then
are added to the adjustments in the current iteration. The least square error is used as convergence measure. It is also known as error of delta rule (Eq. 8). In this study, learning rate and momentum factor are set to 0.7 and 0.9, respectively. The convergence condition of the least square error is resolved at 0.0001, and after about 20,000 cycles, training is terminated. 4 Results and discussions The thixoforging experiments were performed under the experimental conditions, and the pistons were obtained asshown in Fig. 4. In order to measure the hardness of the products, the piston was cut vertically. Figure 5 shows the measurement location for the hardness test. Rockwell (B scale) was used
for the hardness test. Hardness value, hardness average, and standard deviation are shown in Table 1.
4.1 Variation analysis of the process factors
The aim of variation analysis is to verify the influence of the process factors, such as the die temperature, the liquid fraction and compression holding time on the goal factors, such as the hardness average and the standard deviation of hardness. Tables 2 and 3 describe the result of variation analysis about the 23 factorial designs which is the former