Comparative Prediction and Modelling of Surface Roughness in Milling of AL-7075 Using Regression Analysis and Neural Network

The cutting tool CNC machine has been used to perform the experimental work. Al- 7075 was used to determine the result of the optimization process on the Al parameters. Work piece dimensions of AL was 100 × 150 × 3 mm during the milling process. The significant chemical structure of Al-7075 alloy is listed in Table 1. The value of the surface roughness was calculated with the surface roughness tester. The input parameters [speed of cut (rpm) with the feed rate (mm/rev) and the depth of cut (mm)] were used to estimate the value of the surface roughness (Ra) after machining process through performing orthogonal array L27 experiments. Since this input variables are multi-level parameters with non-linear responses. Therefore, three level tests were considered for each parameter. Table 2 demonstrates the data from experiments for the Al-7075 from orthogonal array L27.

Table 1. Chemical composition of Al-7075 alloy

Table 2. The experimental data for the Al7075 from orthogonal array L27

2.1 Modelling of surface roughness using design of experiments

The design of the experiment considers as a technique used to analyze and model a system response from multi input variables. Therefore, in the current study the full factorial design (FFD) has been used to study the effect of the system input parameters [speed of cut (rpm), the feed rate (mm/rev) and the depth of cut (mm)] with three levels as depicted in Table 3.

Table 3. System process parameters and their levels.

$\begin{aligned} R_{a}=b_{0}+b_{1} * & X_{1}+b_{2} * X_{2}+b_{3} * X_{3}+b_{11} * X_{1}^{2} \\ &+b_{22} * X_{2}^{2}+b_{33} * X_{3}^{2}+b_{12} \\ & * X_{1} * X_{2}+b_{13} * X_{1} * X_{3}+b_{23} \\ & * X_{2} * X_{3}+b_{123} * X_{1} * X_{2} * X_{3} \end{aligned}$                  (1)

where: Ra – represents the output surface roughness Ra (µm); X₁–represent cutting speed (rpm); X₂– feed rate (mm/rev); X3– depth of cut (mm). b₁, b₂, b₃– represent regression coefficients for cutting speed (rpm), feed rate (mm/rev) and depth of cut (mm). b₁₁, b₂₂, b₃₃– represent the squared coefficients of the main factors. b₁₂, b₁₃, b₂₃, b₁₂₃– represents the interactions among coefficients. While the b0 is a free number.

2.1.1 Regression analysis and design matrix experimental results

According to the obtained experimental data, Statistical software release 10.0 has been implemented to analyse the significance of the independent variables (factors) individually (cutting speed (rpm), feed rate (mm/rev) and depth of cut (mm). Beside their interactive coefficients as numerical values on the indicated output the surface roughness Ra (µm). Along these lines, a mathematical model relating the actual domain of independent factors with the experimental results of surface roughness has been developed using the Gauss-Newton approach through, a quadratic polynomial regression model. The fitting model results are depicted in Table 4. seeing that the main parameters with their interaction (b₀, b₁, b₂, b₃, b₁₁, b₂₂, b₃₃, b₁₂, b₁₃, b₂₃, b₁₂₃) which illustrate the high significance with p-values.

Table 4. Regression analysis and significant coefficients according to ANOVA model

While Table 5 appropriately verified the perfection of fitness model by ANOVA results, in which the F-value of (6.439) refers to model significance. Moreover, the model fitting validity was proved in Figure 2.a&b. It is obvious that, the value of R² = 0.987 reverse slight difference between the experimental and predicted response calculations. This indicate that the surface finishing process has been properly described by the proposed quadratic polynomial model with the process parameter that contain one dependent variable and three independent variables.

Table 5. Verified the perfection of fitness model by ANOVA results

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(a)

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(b)

Figure 2. (a) Predicted and actual value for surface roughness; (b) Interaction of cutting speed and feed rate against surface roughness

2.2 Modelling of surface roughness using artificial neural network (ANN)

ANN has great importance due to its ability to solve the nonlinear problems [28]. Besides, ANN is used in prediction of the surface roughness in this study. ANN model with several layers was presented where the data were come from input layer and then were processed in hidden layers. Hyperbolic tangent sigmoid was the depended transfer function of the hidden layer. The hyperbolic tangent sigmoid function which is used in the proposed neural network's hidden layers, is preferred than the traditional sigmoid function because it has steady state at zero, or by other words, this function centers data in a better way for the next layer, as the output's mean is near to zero [29].

On the other hand, the hidden layer's output was fed to output layer which has the linear transfer function. The change in weights was evaluated according to Eq. (2):

$\Delta W_{j i}(n)=\alpha \Delta W_{j i}(n-1)+\eta \delta(n) Y_{i}(n)$         (2)

where, $\Delta w j(n)$ is the weights' changes: i and j =1,2,3…n; $\alpha$ represents momentum coefficient. $\delta j$ represents error; $\eta$ represents learning rate parameter and $\mathrm{Y}(n)$ represents the output at the $n t h$ iteration.

After each iteration, weights would be fed back until the minimum error percentage is obtained, which means good training of the network is achieved. Where best case takes place at regression equalling 1, at this point, the neural network will be stopped. Figure 3 shows the presented ANN model structure: Where, W: weights, and b: bias.

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Figure 3. ANN model structure

Training is the first step in ANN where the inputs (cutting speed, feed rate and depth of cut) were fed to the network in addition to the surface roughness as the desired output. At first, the weights are set in random manner, which will be altered using the Back propagation algorithm to get a satisfactory performance level. Two essential methods of weights initialization exist: zero initialization and random initialization which is used in the proposed ANN model, where this technique prevents the neurons from same features learning of the inputs, this technique breaks the symmetrical results with better accuracy than zero initialization [30]. The parameters of the proposed ANN are:

3 neurons in input layer;

2 hidden layers with 10 neurons;

1 neuron in output layer;

Learning rate is 0.05;

Momentum constant is 0.95.

Weights and biases are created randomly by the MATLAB Toolbox of neural network as follows:

Weights [1,1] (to layer 1 from input 1):

$\left[\begin{array}{ccccc}0.394 & 3.307 & 1.166 ; 1.755 & 1.714 & -1.315 ; \\ -0.777 & -2.321 & 0.896 ; 2.207 & 0.092 & -2.122 ; \\ 2.811 & -0.616 & 1.627 ;-1.829 & -2.126 & -0.909 ; \\ 1.014 & -.0 .616 & 2.842:-5.297 & 0.353 & -0.362 ; \\ -2.262 & -1.385 & -1.951 ;-0.953 & -1.801 & -3.515\end{array}\right]$

Weights [2,1]: $\left[\begin{array}{rlcrr}0.431 & 0.104 & 0.413 & 0.552 & 0.562 \\ -0.357 & -0.740 & 0.634 & 0.527 & 0.293\end{array}\right]$  

bias: b[1] ( bias to layer 1): $\left[\begin{array}{rrrrr}-2.681 ; & -2.435 ; & 1.894 ; & 0.0524 ; & -0.552 ; \\ 0.660 ; & 0.499 ; & 1.098 ; & -3.399 ; & 1.625\end{array}\right]$  

bias: b[2]: [0.656]

The (MSE) Mean Square Error is calculated during the learning process using Eq. (3):

$\mathrm{MSE}=\frac{1}{2} \sum_{i=1}^{n}\left|T_{i}-Y_{i}\right|$             (3)

where $\mathrm{T}_{i}$ and $\mathrm{Y}_{i}$ are the Target and output values of surface roughness respectively. Then, weights which are between hidden and output layers are adjusted and calculated by Eq. (1). For developing the presented ANN model, the network is trained using 27 experiments. The network showed good training and learning to predict the output value of surface roughness for testing and validation.

It is obvious from Figure 4 above that during the training process most of the predicted and experimental values coincide very well on regression line which reaches to R=0.996in training values, it is found that regression is equal to 0.999 and 0.978 for validation and testing respectively, while regression was 0.987 in overall values performance which is a very good result. Generally, many reasons may lead to the existence of some actual values that are not very consistent to the predicted ones. These reasons may be caused by faults in experimental results due to environment, instruments and observations. Besides to the fact that the residuals are not being both negative and positive then the neural network model will not deliver a coincidence between predicted and actual values.

In Figure 5 above, it is clear that the performance of the presented network depended on the amount of the mean square in relation with the increment in number of epochs, where very good training was achieved, and the best validation performance was 0.006 at epoch number 2.

Table 6 shows the percentage of error of the regression models and Artificial Neural network. Each maximum error obtained in the regression models and Artificial Neural network is (0.209) and (0.287), respectively, Table 6 also shows that the regression models and Artificial Neural network minimum errors are (-0.002) and (-0.008), respectively. It can be concluded that the Artificial Neural network model produces more high accuracy results than the regression model based on the R², MSE and errors evaluated.

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Figure 4. Neural network training regression

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Figure 5. ANN Mean square error

Table 6. Comparison between regression model and the neural network prediction