lass="ff9">−0.09758 0.00600 16.275 0.000
Cutting feed 9.26208 0.30958 29.918 0.000
Spindle speed 0.00009 0.00005 2.017 0.047
S = 0.0881180; R-Sq = 95.69% ; R-Sq(adj) = 95.46%
Table 6. Linear regression model for ln(torque).
ln(Torque) = 1.408 + 0.374 Al2O3 (vol%) 0.254 Gr (vol%)+ 8.933
Cutting feed (mm/rev) + 0.0002 Spindle speed (rpm)
Term Coef SE Coef T p-value
Constant 1.40802 0.114153 12.334 0.000
Al2O3 0.37386 0.015232 24.545 0.000
Gr 0.25409 0.015232 16.682 0.000
Cutting feed 8.93298 0.786467 11.358 0.000
Spindle speed 0.00019 0.000119 1.616 0.110
S = 0.223857; R-Sq = 93.02%; R-Sq(adj) = 92.65%
speed is considered marginal, but still important in this
model with p-value less than 0.05. The final regression
model is:

12
34
ln3.813 0.1370.0976
9.262 0.0001
th
F
XX
XX
 
 (14)
where Fth represents thrust force, X1 represents Al2O3
(vol%), X2 represents Gr (vol%), and X3 represents cut-
ting feed and X4 represents spindle speed.
Similarly for (ln(torque)) model, the final regression
model is:

12
ln Torque1.4080.3740.254+8.9333
X
XX  (15)
where X1 represents Al2O3 (vol%), X2 represents Gr
(vol%), X3 represents cutting feed (mm/rev), and X4 re-
presents spindle speed (rpm).
The signs of the parameters in the model presented in
Tables 5 and 6 were examined. Positive signs mean the
response output (either thrust force or torque) values go
in the same direction as the parameter, and negative signs
imply the opposite.
To test the prediction performance of the ln(thrust
force) and ln(torque) regression models, the absolute re-
lative errors were computed based on experimental and
predicted values. The absolute relative error (ARE) was
computed based on the following equation:

Predicted valueExperimental value
ARE %Experimental value
(16)
Average absolute relative errors (ARE) were 12.59% and
7.10% for ln(thrust force) and ln(torque), respectively.
Although these error levels could be accepted in some
cases, better prediction models may give better predi-
ctability and lower error values. This leads us to use
ANN for prediction purposes instead of MRA because it
performs well and gives a better mapping between inputs
and outputs.
5.2. Artificial Neural Network Results
The ANN was implemented using fully developed feed
forward backpropagation network. The models for cut-
ting forces are identified by using the alumina (Al2O3)
particles contents, graphite (Gr) particles contents, cutting
feeds (f) and spindle speeds (N) as input data and thrust
force and cutting torque as the output data. An 4-10-2
ANN topology was used, which consists of four input
nodes, one hidden layer (10 neurons in the) and two out-
puts (thrust force and torque).
The inputs in ANN nodes must be a numerical value
and fall in the closed interval [0,1]. The input data were
normalized n the range between 0 and 1 using the follow-
ing formula:
Normalized value
input valueminimum value
maximum valueminimum value
(17)
Output values resulting from ANN were also in the
range [0,1] and converted to their equivalent values
based on reverse method of normalization technique.
All of the original 81 machining conditions were ran-
domly divided into three datasets including a training,
validation and testing datasets. The training set contained
57 (70%) data points were used to build the network, 12
data points (15%) were used to measure network gene-
ralization and another 12 points (15%) were used as a
testing set of the neural network. Sigmoid activation fun-
ction was selected to be the transfer function in the
hidden layer and linear function was used between hidden
layer and output layer (Figure 9). After many trials,
learning rate and momentum were experimentally se-
lected to be 0.03 and 0.9, respectively. Levenberg-
Marquardt training algorithm was used to train this ANN.
Training, testing and validation process were terminated
after 106 cycles and further iterations had insignificant
effect on error reduction. The obtained MSE value was
0.00182. Hence, one can conclude that a simple archite-
cture can be used efficiently without loss of prediction
accuracy. Table 7 summarizes ANN training and valida-
tion parameters as well final training error.
However, the main quality indicator of a neural net-
work is its generalization ability, in other words, its abi-
lity to predict accurately the output of unseen data; this
was achieved by testing dataset. Absolute relative errors
Copyright © 2012 SciRes. JMMCE
A. MAYYAS ET AL.
Copyright © 2012 SciRes. JMMCE
1046
Figure 10 shows the comparison between experi-
mental torque and thrust force values and corresponding
ANNs outputs for the total dataset (training, validation
and testing). These figures show that the significant por-
tion of points clusters along the diagonal line, which in
turn is a good indication of performance of training algo-
rithm. The correlation coefficients (R2) between experi-
mental and predicted outputs—values exceed 0.99 for all
training, testing and validating datasets. These values
show the accuracy of prediction ability obtained from
ANN.
Figure 9. Architecture of ANN with 4-10-2 topology.
5.3. Comparison between MRA and ANN
Prediction Models
between experimental and predicted values from ANN
were used to evaluate the performance of the proposed
ANN in prediction technique. The mean absolute relative
errors for the second ANN were: 1.66% for torque and
0.78% for thrust force for testing dataset. These levels of
error are satisfactory and smaller than errors that nor-
mally arise due to experimental variation and instrument-
tation accuracy.
A visual comparison was established between the fitted
and experimental values (Figures 11 and 12) for testing
and validation datasets of ANN, respectively. Both fi-
gures show that the predicted values from ANN appro-
ximate the experimental values much more than the
Figure 10. Comparisons between experimental thrust force values and corresponding ANN outputs (4-10-2 structure).
A. MAYYAS ET AL. 1047
Table 7. Summary of ANN parameters.
Neural network parameters
Network type Feed forward BP (Levenberg-
Marquardt training algorithm)
Network architecture 4-10-2
Number of hidden layer 1
Number of hidden neuron One hidden layer: 10
Transfer function Sigmoid: input hidden layer);
(Linear: hidden layeroutput layer)
Number of training examples 57
Number of testing examples 12
Number of validating examples 12
Learning rate 0.03
Momentum factor 0.9
Number of epochs 106
Mean squared error (MSE) 0.00182
MRA model does. This was also proved by getting smaller
absolute relative errors. Experimental value columns in
Figures 11 and 12 represent the actual values with 10%.
When compared to the experimental values with predi-
cted values, it can be seen that ANN outputs lie in good
prediction ranges compare to MRA outputs.
Now, which prediction method is better and when
should each one be used to predict and optimize the
drilling process in this situation? In the case of develop-
ing empirical relations, MRA model is preferred over
ANN model because it is an explicit model while the
ANN model is a black box. In the other direction, when
data are sparse or not generated from designed experi-
ments, MRA may not be able to produce a better model
than ANN; then the ANN modeling method and its asso-
ciated model may be preferred to the MRA method and
its model if such a model is available.
6. Conclusions
Two modeling techniques were used to predict the thrust
force and torque, namely multiple regression analysis
(MRA) and artificial neural network (ANN). Modeling
the drilling process using MRA and ANN approach pro-
vides a systematic and effective methodology for the
prediction. Both MRA and ANN revealed that rein-
forcement fractions were the important factors that in-
fluence the responses (i.e. thrust force and torque) fol-
lowed by the cutting feed rate. However, spindle speed
seemed insignificant in both models.
Many ANN architectures have been used to model the
collected experimental data. The best neural network
configuration was (4-10-2) which was trained using 57
training examples, tested using 12 examples and vali-
dated using 12 examples.
The results of ANN models showed close matching
(a)
(b)
Figure 11. Experimental vs. predicted values from ANN
and MRA for testing dataset: (a) thrust force; and (b) tor-
que.
(a)
(b)
Figure 12. Experimental vs. predicted values from ANN
and MRA for validation dataset: (a) thrust force; and (b)
torque.
Copyright © 2012 SciRes. JMMCE
A. MAYYAS ET AL.
1048
between the model outputs and the measured outputs.
The mean absolute relative errors were 0.82% for torque
and 2.89% for thrust force models, while MRA model
error values were 7.10% and 12.59%, respectively.
Hence, these models can be used efficiently for predic-
tion potentials for non-experimental patterns which, in
turn, save experimental time and cost. It was shown that
ANN performs well in mapping nonlinear relationships
between inputs and outputs. If both MRA and ANN
models are considered they will provide statistically sat-
isfactory prediction results. ANN methodology consumes
less time and gives higher accuracy. Hence, modeling the
drilling process using ANN is more effective compared
with MRA. The two proposed models are good in mod-
eling and predicting the drilling forces, which in turn can
provide a valuable tool for many similar applications of
modeling methods in engineering design and manufac-
turing. The developed modeling methods in this paper
can aid the prediction, optimization, and improvement of
drilling processes and the selection of cutting parameters
in the case of drilling aluminum-based materials.
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