The statistical analysis was performed using Minitab software package 17. The Taguchi L27 orthogonal statistical designs were followed to optimize the experimental procedure. The processing variables were screened by ANOVA analysis. These factors were modeled by using response surface methodology (RSM). The results were analyzed by multiple regression analysis through least square method. All the terms of model were tested and verified statistically by F-test at probability levels (p<0,05). R-square and Adj-R-square were used to perform the developed the models. After the fitting models, 3D surface plots were applied to evaluate the process parameters. Finally, desirability function based simulated angling algorithm was used to screen the optimum parameter levels. The experimental produce is shown in (Figure 2).

Figure 2:
Experimental procedure.

Response surface method (RSM) is a statistical based technique. RSM method involves three stages. The first stage is the selection of independent variables of major effects on the process through screening studies and the delimitation of the experimental region. The second stage is the mathematic-statistical apply of the obtained experimental data through the fit of a polynomial function and evaluation of the model’s fitness. The third stage is the verification of the necessity and possibility of performing a displacement in direction to the optimal region and selection of optimum values of parameters. The general form second order polynomial equation is given Equation 1:

(1)

According to Equation 1 Y is the predicted reaction or reactions (_^Ra and _^Rz ), _^Xi and _^Xj are variables, β₀ a constant, β_i, β_ii and β_ij of linear, quadratic and the second-order terms, respectively and k is the number of independent parameters (k=4 in this study) and ϵ is the error term.

The traditional experimental design is time consuming and costly. _^L27 orthogonal design was used to overcome this issue. Especially, this method is used by manufacturing process and engineering analysis. These arrays of different combination for variables were applied by the Taguchi approach to analyze the experimental study. While 81 experiments are required to investigate all the effects of the parameters, 27 experiments are sufficient thanks to Taguchi orthogonal array. According to the Taguchi orthogonal array, the minimum number of experiment and coded variables were computed by using Equation 2 and Equation 3.

The Taguchi experiments trial number:

(2)

The terms of L and P are displayed number of levels and number of parameters.

(3)

In terms of _{^{Xi, Xo}} and Δ_x are coded variable, actual variable, center palace, and altered variable. (Table 1) shows the number of parameters and their levels.

Table 1:
CNC experimental design procedure.

In this study, four parameters namely spindle speed, feed rate, depth of cut and tool radius were selected as input parameters while _^Ra and _^Rz were selected as output variables. (Table 2) shows the experimental parameters and their results.

Table 2:
_^L27 experimental design results.

The results were analyzed by conducting Minitab 17 software. Correlation of the experimental roughness value and the estimated values from the regression equation were shown in (Table 3). The experimental data were analyzed by the polynomial models such as linear, 2-way interaction and full quadratic models. The adequacy of models performed indicated that linear, interaction and quadratic models had lower p-value (<0,05). (Table 3) displays the R² and Adjusted-R² values were found low in linear and interaction models. R-square and Adj-R-square values were described with Multiple Linear Regression Analysis (MLR). Quadratic models were calculated to have maximum R² and Adj- R² values. Hence the quadratic models were chosen for _^Ra and _^Rz in this study.

Table 3:
Regression models, R² and Adj-R².

Table 4 and Table 5 indicated variance analysis for _^Ra and _^Rz . In this case f, n, d, r, n², d², fd, fn, nd, nr, fr are significant factors. Contribution (PC %) of each factor on the total variation is computed as percentage. The most effective parameters for _^Ra and _^Rz were interaction between feed rate and spindle speed, their contribution to the model values were 37,16 % and 34,47 %, respectively, on the surface roughness. The coefficient of determination R² for the _^Ra and _^Rz of were 95,91 % and 96,12 % respectively. The Adjusted _^R2 values for _^Ra and _^Rz were computed as 91,15 % and 95,93 % respectively.

Table 4:
ANOVA results for _^Ra.

Table 5:
ANOVA results for _^Rz.

Normal Probability Plot (NPP) was applied to evaluate the data for normality. Residual is the mean difference between the observed value and the predicted or fitted value. If the residuals fall approximately along straight line, they are then normally distributed. In contrast, if the residuals do not fall fairly close to a straight line, they are then not normally distributed (Antony 2014). If the model is correct and if the assumptions are satisfied, the residuals should be structures; in particular, they should be unrelated to any other variable including the predicted response. Figure 3a, Figure 3b, Figure 4a and Figure 4b show that the residuals generally fall on a straight line implying that the errors were disturbed normally, meaning was the experimental data come from a normal population. As a result of the residuals, no unusual structures were apparent.

Figure 3:
(a) Fitted value for _^Ra and (b) NPP of residuals for _^Ra.

Figure 4:
(a) Fitted value for R_z and (b) NPP of residuals for R_z.

Response surface provides the relationship between the factors with each other and the response variable. (Figure 5a) shows the surface plot for _^Ra in CNC machining of wood material with three levels of feed rate and three levels of spindle speed for mid-level hold value of depth of cut with 4 mm and tool radius of 10 mm. The surface roughness increased with decreases the spindle speed and increases the feed rate. (Figure 5b) shows the surface plot for _^Ra in CNC machining of wood material with three levels of depth of cut and three levels of tool radius for mid-level hold value with feed rate of 5 m/min and spindle speed of 15000 rpm. The surface roughness decreased with a decrease of tool radius and depth of cut. (Figure 5c) shows the surface plot for _^Ra in CNC machining of wood material with three levels of feed rate and three levels of tool radius for mid-level hold value with spindle speed of 15000 rpm and 4 mm depth of cut. The surface roughness increased with the increase in tool radius and feed rate. (Figure 5d) shows the surface plot for _^Ra in CNC machining of wood material with three levels of feed rate and three levels of depth of cut for mid-level hold value with spindle speed of 15000 rpm and tool radius of 10 mm. The surface roughness increased with the increase in feed rate and depth of cut. (Figure 5e) shows the surface plot for _^Ra in CNC machining of wood material with three levels of spindle speed and three levels of tool radius for mid-level hold value with 5 m/min of feed rate and depth of cut of 4 mm. The surface roughness increased with the increase in tool radius and decreases with spindle speed. (Figure 5f) shows the surface plot for _^Ra in CNC machining of wood material with three levels of depth of cut and three levels of spindle speed for mid-level hold value with 5 m/min of feed rate and tool radius of 10 mm. The surface roughness increased with the increase in depth of cut and decrease with spindle speed.

Figure 5:
According to 3D surface plots (a, b, c, d, e, f) of different values processing parameters for_^Ra.

(Figure 6a) shows the surface plot for R_z in CNC machining of wood material with three levels of feed rate and three levels of spindle speed for mid-level hold value of depth of cut with 4 mm and tool radius of 10 mm. The surface roughness increased with decreases the spindle speed and increases the feed rate. (Figure 6b) shows the surface plot for ^_Ra in CNC machining of wood material with three levels of depth of cut and three levels of tool radius mid-level hold value with feed rate of 5 m/min and spindle speed of 15000 rpm. The surface roughness decreased with a decrease of tool radius and depth of cut. (Figure 6c) shows the surface plot for ^_Ra CNC machining of wood material with three levels of feed rate and three levels of tool radius for mid-level hold value with 15000 rpm of spindle speed and depth of cut of 4 mm. The surface roughness increased with the increase in tool radius and feed rate. (Figure 6d) shows the surface plot for ^_Ra in CNC machining of wood material with three levels of feed rate and three levels of depth of cut for mid-level hold value with 15000 rpm of spindle speed and tool radius of 10 mm. The surface roughness increased with the increase in feed rate and depth of cut. (Figure 6e) shows the surface plot for ^_Ra in CNC machining of wood material with three levels of spindle speed and three levels of tool radius for mid-level hold value with 5 m/min of feed rate and depth of cut of 4 mm. The surface roughness increased with the increase in tool radius and decreases with spindle speed. (Figure 6f) shows the surface plot for ^_Ra in CNC machining of wood material with three levels of depth of cut and three levels of spindle speed for mid-level hold value with 5 m/min of feed rate and tool radius of 10 mm. The surface roughness increased with the increase in depth of cut and decrease with spindle speed.

Figure 6:
According to 3D surface plots (a, b, c, d, e, f) of different values processing parameters for R_z.

The aim of present work was to find the optimal processing levels leading to minimum values of _^Ra. and _^Rz . Two second order mathematical models were created by using RSM. Moreover, backward elimination method was carried out for determining the non-significant terms in mathematical models. To formulate the optimization problem, proposed models were presented in Equation 4 and Equation 5:

(4)

(5)

These models were solved with two different methods namely, desirability function and simulated angling algorithm. Firstly, the desirability function was applied to determine the initial optimum parameter levels. Then, the results were used the initial point for the simulated angling algorithm. The functions terms were optimized within the specified range. The range of machining conditions was selected and given as follows Equation 6a, Equation 6b, Equation 6c, Equation 6d:

Within ranges of processing parameters,

6(a)

6(b)

6(c)

6(d)

The limitations of cutting variables value of spindle speed, feed rate, depth of cut and tool radius were 12000 rpm, 2 m /min, 2 mm and 8 mm as lower limits while 18000 rpm, 8 m/min, 6 mm and 12 mm were selected as upper limits, respectively.

This is commonly applied engineering problems for the optimization process. The function value is range 0 from 1. In the present work, selected quality characteristic was smaller-the-better for minimizing the surface roughness value. This equation was presented in Equation 7:

(7)

Where T symbolizes the target value of the i th response, _^yi , L symbolizes the acceptable lower limit value, U symbolizes the acceptable upper limit, for this response and W represents the weight. In (Table 6), optimal machining parameters for minimizing _^Ra was found with spindle speed of 17450 rpm, feed rate of 2,9 m/min, tool radius of 8,0 mm and depth of cut of 2,0 mm. optimal machining parameters for minimizing _^Rz was spindle speed of 16990 rpm, feed rate of 2,0 m/min, tool radius of 8,0 mm and depth of cut of 2,0 mm.

Table 6:
Taguchi- DF results for _^Ra and _^Rz.

From the Figure 7 and Figure 8, the optimum values for _^Ra and _^Rz were 3,1287 µm and 11,254 µm, respectively.

Figure 7:
Response optimization plot for _^Ra.

Figure 8:
Response optimization plot for _^Rz.

In this study, simulated angling algorithm was applied to optimize the machining parameters. It is the one of the most useful metaheuristics method for solving the engineering problems due to easy application. It is a widely used method in solving optimization problems. SA is a probabilistic and single solving method based on the annealing process of metallurgy. The algorithm starts with an initial point using random solution, S, then a simple random change is produced to actual solution (S’). Obtained solution was compared with new solution by creating the objective function. The new solution is accepted when the objective f(S’) is smaller than other objective. This solution is recognized with a probability of exp^{-(f(S’)-f(S))/T} . The term of T is a control variable and it is reduced from a relatively to near zero. SA was applied by solving the fitness function formulated in Equation 4 and Equation 5. In (Table 1), the range of processing variables was defined to solve the surface roughness problem. Initial points of SA are given in (Table 7).

Table 7:
Initial points of SA for _^Ra and _^Rz .

Matlab optimization toolbox was used to solve the surface roughness problem. The values of the SA parameters such as annealing function, reannealing interval, temperature function, initial point of temperature, and probability function. Matlab software was used to determine the best optimal results using different simulated angling parameters. The best solution of these parameters was given in (Table 8).

Table 8:
Selection of SA parameters to determine the optimal solution.

Minimum_^Ra value was resulted with spindle speed of 17377 rpm, feed rate of 2,012 m/min, tool radius of 8,00 mm and depth of cut of 2,009 mm using simulated angling algorithm. Minimum_^Rz value was obtained spindle speed of 16980 rpm, feed rate of 2,004 m/min, tool radius of 8,001 mm and depth of cut of 2,003 mm using simulated angling algorithm. The results of Matlab SA toolbox were given in Figure 9a, Figure 9b, Figure 10a, Figure 10b. According to Figure 9a, Figure 10a, the best fitness values for _^Ra and _^Rz were 2,474 µm and 10,674 µm, respectively. In the Figure 10a, Figure 10b, the optimal solutions of _^Ra and _^Rz were performed at the 3106 th and 4712 th iteration, respectively.

Figure 9:
(a) Variation of fitness function and best individuals (b) Converged values of parameters.

Figure 10:
(a) Variation of fitness function and best individuals (b) Converged values of parameters.

The Taguchi orthogonal design based desirability function and desirability function integrated simulated angling algorithm results were given in (Table 9). From the (Table 9), the minimum _^Ra and R_z values were 2,474 µm and 10,674 µm. For this reason, the RSM-DF-GA approach should present an efficient methodology in order to minimize the surface roughness.

Table 9:
Results of experiments and optimum values by Taguchi-RSM-DF and Taguchi-RSM-DF-SA models for _^Ra and _^Rz .

^♠Corresponding author: ender.hazir@istanbul.edu.tr