#### 4.1. Analysis of Experimentation Using the FIS

This section is included in order to demonstrate that the FIS is able to model the behavior of the response variables more efficiently than by using RSM. Data shown in

Table 1 and

Table 2 are employed in order to develop a FIS which can be then employed to obtain the technological tables following the procedure previously mentioned in

Section 3. As can be seen in Reference [

1], a method for obtaining the technological tables from a conventional design of experiments along with multiple linear regression techniques was proposed, where technological tables were obtained for TiB

_{2}, which is a sintered ceramic material and in Reference [

5] technological tables were obtained for B

_{4}C, SiSiC and WC-Co. However, as was previously mentioned, if the regression is not able to adequately predict the behavior of a response variable, the technological tables obtained from these models will not be accurate. In this section, the proposed methodology in this present study is applied for the case of the EDM of Inconel

^{®} 600. However, it should be mentioned that this methodology could be applied for other kinds of materials.

Figure 1 shows the membership functions that were used to fuzzify the inputs. As can be observed, triangular functions were selected for the inputs. On the other hand, the present study assumes that it is possible to linearly vary the parameters in the EDM equipment in order to be able to select the values obtained from the technological tables which are determined to be optimal ones. If this is not possible, the FIS would have to be used on the possible values of these independent variables. As

Table 2 shows, the design of experiments does not continuously vary the values of the independent variables; thus, it is possible that the optimal values are not selected if only these values are considered. In addition, it may be that there are levels vacant when establishing the levels of roughness, which is the dependent variable that was selected as

${\mathrm{output}}_{1}$ since, as explained above, it is one of the most widely used parameters for characterizing surface quality and, therefore, its determination is of great importance and interest in industry.

In this present study, the FIS was obtained using Matlab™2019b. Therefore, from

Table 2, it is possible to directly obtain the set of rules that make up the FIS. As previously mentioned, a Sugeno FIS was employed by using the Fuzzy Logic Toolbox™ of Matlab

^{TM}2019b [

40]. Mamdani systems are more intuitive and the rules are easier to understand, making them more suitable for expert systems, developed from human knowledge [

40,

42,

43]. On the other hand, the defuzzification process for a Sugeno system is more computationally efficient compared to that of a Mamdani system [

40,

42,

43].

Figure 4 shows the employed FIS which was developed from the rules shown in

Table 3. This table shows the rules implemented in the fuzzy system, in symbolic format, codified from the outputs. For each output value, a FIS was developed. In this way, it is possible to model the behavior of Ra, MRR, and EW for each of the manufacturing strategies.

The codification shown in

Table 3, which was obtained from

Table 2, is “current intensity, pulse time, and duty cycle”:

$\u201cI\left(i\right)Ti\left(j\right)dc\left(k\right),\mathrm{output}\left(1=and,2=or\right):weight\u201d$. In this case,

$weight=1$, so that each rule has the same effect relative to others [

40], where the numbering 1, 2, 3, and 4 is employed for the inputs in order to select the levels of the variables. As can be observed in

Table 1, these variables have four levels. For example, the levels for the intensity are given by {2 A, 4 A, 6 A, and 8 A}. Therefore, these values are coded as {1, 2, 3, and 4} in

Table 3. The same procedure is applied for both pulse time and duty cycle. In the case of the output, there are 64 values which are obtained from the DOE with the different input conditions. That is, for the case of Ra, for instance,

$111,1\left(1\right):1$ corresponds to the following:

That is,

where the input values

$\u201cI\left(i\right),\text{}Ti\left(j\right),\text{}dc\left(k\right)\u201d$ and the outputs

$m{f}_{1}\dots m{f}_{n}$ are selected from

Table 1.

The FIS was generated directly from experimental data. Therefore, as shown later, the precision of the obtained results is much higher than that obtained using RSM.

Figure 5,

Figure 6,

Figure 7 and

Figure 8 are included to compare the response surfaces obtained with the proposed methodology using the FIS and those obtained from the RSM, as done in Reference [

2], where the experimental data were fitted by using a second degree polynomial, which is shown by Equation (6).

As can be observed in

Figure 7 and

Figure 8, the results obtained with the FIS are close to those obtained with the regression as a consequence of Ra and MRR being fitted adequately by a quadratic polynomial, as can be seen from the coefficients of determination of the fit and from the RMSE and mean absolute error (MAE) statistics, which are shown in Equation (7) and in

Table 4. However, as

Table 4 shows, this is not the case for the electrode wear (EW), which is shown in

Figure 5; hence, it is possible to conclude that the FIS is more accurate than RSM. Therefore, it is able to predict more adequately the values of the response, within the range of study, than the RSM.

Figure 6,

Figure 7 and

Figure 8 show a comparison between the EW, Ra, and MRR results obtained with the RSM and with the FIS. As can be observed in

Figure 7 and

Figure 8, differences are not significant as a consequence of the fact that experimental Ra and MRR results are well fitted by a second-order polynomial, such as that shown in Equation (6). However, this is not the case for electrode wear, as shown in

Figure 5 and

Figure 6. As

Table 4 shows, the polynomial model is not accurate and, in this case, the differences between the FIS and the regression model are significant. Therefore, data provided by the FIS are more accurate than those obtained by using the RSM, and the technological tables are more accurate if the FIS is used instead of the regression model.

Figure 9 shows the response surfaces for both Ra and MRR obtained with the proposed methodology using the FIS for the case of positive polarity. Equation (7) shows the statistical parameters that were used to determine the precision of the models used for modeling the dependent variables, that is, Ra, MRR, and EW. As can be observed in

Table 4, the FIS accuracy is higher than that provided by the RSM. Data shown in

Table 4 were obtained by using Matlab™2019b.

As can be observed in

Table 4, the fuzzy inference system fits all the data perfectly, which is logical since the FIS was built according to the procedure shown in the previous section. However, this is not the case with the RSM which, despite using all the DOE points for the determination of the models, is not able to adequately adjust the electrode wear surface response. Therefore, the values predicted by the regression have lower accuracy than those predicted by the FIS. In this case, the polynomial models for the case of both roughness and material removal rate are acceptable. Nevertheless, the precision is lower than that of the FIS. In any case, in other types of experimentation in which there is less precision in the least squares adjustments, the employment of the FIS becomes more important since it adjusts to all the points of the model.

In Torres et al. [

2], the model with the highest value of

$adjusted{R}^{2}$ was selected. However, in this present study, the model with all the regression coefficients is used because these models have higher

${R}^{2}$ values than those shown in Reference [

2] and, with the aim of considering all the effects in the models such as the models shown in Reference [

2], some of the independent variables could be eliminated.

Figure 10b shows that it is possible to analyze the experimental results in a similar way to that done with regression models. It is shown that the most important effects are the current intensity and the pulse time, followed to a lesser extent by the duty cycle. In addition, by using the FIS, the values obtained are more precise, as can be seen in

Table 4. As can be observed, the differences between the values predicted by the regression model and those predicted by the FIS are significant. Specifically, in the case of positive polarity, the regression model does not adequately predict the behavior of electrode wear, as can be seen in

Table 4. Therefore, the results provided by the regression model when predicting electrode wear are not accurate. In this case, the FIS is shown to have significant advantages over the regression model. Specifically, it is shown that, with increasing intensity, there is less wear on the electrode, which is logical because, as seen in

Figure 11a,b, if the intensity decreases, so does the removal of material, while the surface roughness assumes smaller values, with the wear of the electrode in these cases being greater, which is in good agreement with experimental values. Finally,

Figure 12 shows the interaction effects plot. As can be observed, the most significant interactions are those related to the current intensity and the pulse time. On the other hand, it is observed that the differences between both the FIS and the regression are significant, as a consequence of the fact that the regression model is not able to adequately predict the behavior of the electrode wear. In addition,

Table 4 shows that the FIS is able to predict the behavior of the response variables more adequately than the regression, which is logical as a consequence of the methodology employed for defining the FIS. Hence, the fit is perfect in the case of the FIS, and this is not so in the case of the regression model. Therefore, the technological tables with values provided by the FIS are more accurate than those provided by conventional methods.

Figure 11 shows that the current intensity is the variable that has the greatest impact on both Ra and MRR, which is logical since, within the values considered in the present study, a higher intensity reflects higher material removal and worse surface roughness. On the other hand, it can be observed in

Figure 11b that the pulse time affects the material removal rate to only a slight extent and that, approximately for values of the pulse time within the range

$50\mathsf{\mu}\mathrm{s}Ti\left(\mathsf{\mu}\mathrm{s}\right)75\mathsf{\mu}\mathrm{s}$, the material removal rate stands at its maximum value, being constant when the current intensity and the duty cycle are at their average values.

Figure 12${b}_{\left(3x3\right)}$ shows that it is possible to analyze the interaction effects between factors by using the FIS in a similar way to conventional analysis of factorial 2

^{k} experiments along with regression models. These factors are represented in an array

$\left(3files\times 3columns\right)$. The results were generated by analyzing the variation of one factor between its maximum and minimum levels, when all the other factors were held at their average level. For example, in

Figure 12 ${b}_{\left(1,2\right)}$, it is shown that, when the current intensity is held at its lowest level, the electrode wear values are lower with increasing pulse time, when the duty cycle is at its average level of 0.45%. Moreover, if the current intensity is held at its highest level, the electrode wear values are lower than those obtained when the current intensity is held at its lower level. On the other hand, in the case of duty cycle, which is represented in

Figure 12 ${b}_{\left(1,3\right)}$, it is shown that the electrode wear remains approximately constant versus the duty cycle when the pulse time is held at a constant value of 62.5 µs, showing that the electrode wear values are independent of either higher or lower values of intensity. A similar analysis could be done with all the interaction effects.

Figure 13 shows the interaction plots effect, using the FIS, for the three independent variables under study in the case of positive polarity when Ra and MRR are considered as response variables.

Figure 14 and

Figure 15 show the main effects plot and the interaction effects plot for the case of negative polarity, using the FIS. As can be observed, a similar behavior to that of positive polarity is obtained. The same comments regarding the precision of the models are applicable in the negative polarity case.

As demonstrated in this section, the response surfaces generated with the FIS have greater precision than those obtained with the RSM; thus, the technological tables are determined according to the methodology described in the previous Section. It should be mentioned that it was considered necessary to develop the previous analysis in order to show the higher accuracy of the FIS model to predict the surface roughness, the material removal rate, and the wear of the electrode.