## 6 Results and Discussion  
Based on the above, we investigated a total of 982,800 different BPNN models to find the optimum NN model for the prediction of tribological performance of the experimentally tested tool steel grades. We also studied combinations of the use or non-use of the normalization technique and the use of one or two hidden layers.  
The developed ANN models were sorted in a decreasing order according to their RMSE value. Based on this ranking, several reliable ANN architectures were found with a value of RMSE equal  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Training Algorithm & Levenberg-Marquardt Algorithm \\ Normalization & Minmax in the range 0.10-0.90 \\ Number of Hidden Layers & 1; 2 \\ Number of Neurons per Hidden Layer & 1 to 30 by step 1 \\ Control random number generation & Rand (seed, generator) \\  & where generator ranges from 1 to 10 by step 1 \\ Training Goal & 0 \\ Epochs & 250 \\ Cost Function & MSE 1; SSE 2 \\ Transfer Functions & Tansig (T) 3; Logsig (L) 4; Purelin (P) 5 \\ \hline \hline \end{tabular}  
* Mean Square Error. 2 Sum Square Error. 3 Hyperbolic tangent sigmoid transfer function. 4 Log-sigmoid transfer function. 3 Linear transfer function.  
\end{table}
Table 5: Training parameters of BPNN models.  
to zero for all investigated cases (one or two hidden layers, with or without use of normalization technique). Amongst the most reliable mathematical models, it was important to select the simplest as the optimum BPNN model. Specifically, the 8-8-1 BPNN model was proposed as the optimum ANN for the prediction of the wear of the three tool steels. Its notation indicates that it corresponds to the case of one hidden layer with eight neurons (the minimum number of neurons), and without the use of any pre-processed normalization technique (Figure 4). As shown therein, the transfer functions are the log-sigmoid transfer function for the hidden layer and the hyperbolic tangent sigmoid transfer function for the output layer.  
Using this proposed optimum ANN model, preliminary \"safe operation area\" maps of the tribological performance of the three steel grades were produced (Figures 5-7). These maps showed a material property on the \(x\)-axis, namely the bulk hardness, achieved by heat treatment, and an operational parameter during service on the \(y\)-axis, namely the applied pressure. Each map corresponded to one specific rotational speed and one surface state, either without or with the surface treatment of nitrocarburizing. In each figure, in addition to the experimentally tested rotational speed values of 300 and 1050 rpm, the predictions for an intermediate value of 600 rpm were also included.  
Figure 4: Architecture of the optimum 8-8-1 BPNN.  
Figure 5: Maps of tribological performance for steel grade A, tested under three different rotational speeds 300, 600, and 1050 rpm: left column without surface treatment (before nitrocarburizing; right column after nitrocarburizing. The blue area corresponds to desirable uniform wear (Mode I), whilst the red one corresponds to non-desirable material behavior (Modes II, III, IV).  
Figure 6: Maps of tribological performance for steel grade B, tested under three different rotational speeds 300, 600, and 1050 rpm: left column without surface treatment (before nitrocarburizing); right column after nitrocarburizing. The blue area corresponds to desirable uniform wear (Mode I), whilst the red one corresponds to non-desirable material behavior (Modes II, III, IV).