*Multi-objective Optimisation in Abrasive Waterjet Contour Cutting of AISI 304L DOI: http://dx.doi.org/10.5772/intechopen.106817*

with the values of 99.08%, 98.52% and 96.17% accordingly. Therefore, the most fitted and predominant models were Ra3 for both 4 and 8 mm, and Ra2 for 12 mm material thickness. The predicted Ra values of regression models applied for straight-line, inner and outer arcs profiles of three levels of material thicknesses are detailed in **Tables 6**-**8**. The percentage error obtained for 4, 8 and 12 mm AISI 304L thicknesses ranged from �4.22 to 3.44%, 3.30 to 6.71% and � 5.75 to 2.49%, respectively. The errors determined for Ra between the predicted value and experimental results are less than 20%, denoting that these models are reliable for predicting Ra values.

**Figure 4** presents the residual plot for Ra, consisting of normal probability plot, residual versus fits, histogram for residuals and residuals versus experimental values for the most fitted regression models for 4, 8 and 12 mm, at Ra3, Ra3 and Ra2, respectively. Similarly, the normal probability plots for all the material thicknesses demonstrated a close fit to a line in a normal probability graph. The points forming an


### **Table 6.**

*Predicted Ra values of regression models for* t *= 4 mm.*


**Table 7.**

*Predicted Ra values of regression models for* t *= 8 mm.*

*Multi-objective Optimisation in Abrasive Waterjet Contour Cutting of AISI 304L DOI: http://dx.doi.org/10.5772/intechopen.106817*


### **Table 8.**

*Predicted Ra values of regression models for* t *= 12 mm.*

approximately straight-line and falling along the fitted line denotes that the data is normally distributed and there is a good relation between measured and estimated response values [45]. In general, the residuals versus fits and observation graph for each material thickness display that the points are distributed randomly and near both sides of 0, with no distinguished pattern denoting a minimal deviation within residuals and estimated values. This graph plots the difference between the experimental data as predicted on the y-axis and the fitted or predicted values on the x-axis, to validate the assumption that the residuals have constant variance [46].

**Figure 4** also exhibits the histogram graph for Ra, illustrating the distribution or frequency of the residuals for all observations. The data shows the frequency of Ra for 4, 8 and 12 mm material thicknesses to range from �0.02 to 0.03, �0.05 to 0.05 and � 0.02 to 0.02, respectively. The histogram presents distribution of the surface roughness obtained from varying material thicknesses. **Figure 4** histogram of residuals denotes that the residuals are normally distributed. These results reveal a minimal interval of inequalities of the experimental data, indicating that the Ra models meet their assumptions and are well fitted for the accuracy of prediction [46]. The effects of process parameters were established by ANOVA, where surface roughness results are given in **Tables A1**-**A3** in the Appendix section.

### **Figure 4.**

*Residual plots for surface roughness. (a) Ra3 (μm) for* t *= 4 mm (b) Ra3 (μm) for* t *= 8 mm (c) Ra2 (μm) for* t *= 12 mm.*

**Figure 5.**

*Percentage contribution of variables for surface roughness. (a) Ra3 (μm) for* t *= 4 mm (b) Ra3 (μm) for* t *= 8 mm (c) Ra2 (μm) for* t *= 12 mm.*

The impacts of the parameters for all profiles across the three levels of material thicknesses demonstrated a similar trend, denoting traverse speed and waterjet pressure to be significant factors for acquiring p-Values lower than 0.05, as detailed in **Tables A1**-**A3**. Accordingly, this work has established that abrasive mass flow rate is an insignificant input parameter for obtaining p-Values >0.05, ranging from 0.002 to 0.067. **Figure 5** represents the percentage contribution of variables for Ra of the most fitted regression models for 4, 8 and 12 mm material thickness. Overall, traverse speed features as the most influencing parameter, followed by waterjet pressure and abrasive mass flow rate. It can be observed here that the influence of traverse speed decreases, ranging from 69.39 to 58.85%, as the material thickness increases. In AWJM, an increasing traverse speed reduces the number of abrasive particles, leading to higher occurrences of surface roughness [47]. **Figure 5** shows that as the material thickness increases, the percentage contribution of waterjet pressure and abrasive mass flow rate also increases, ranging from 24.09 to 33.1% and 3.77 to 5.31%, respectively. The increasing value of waterjet pressure denotes higher energy, reinforcing a larger amount of abrasive particles obtaining lower surface roughness [48]. Further, an increasing rate of abrasive mass flow breaks down abrasive particles into smaller sizes, resulting in more sharp edges that reduce surface roughness [15]. The percentage errors obtained were less than 20%, indicating acceptable reliability of the models, as described in Eq. (6).

### **3.2 Regression model and analysis for material removal rate**

**Table 9** displays multi-linear regression coefficients of models developed for material removal rate against input parameters i.e., traverse speed (*X*1), abrasive mass flow rate (*X*2) and waterjet pressure (*X*3) for 4, 8 and 12 mm material thicknesses of AISI 304L. Regardless of material thickness and cutting profile category, the input parameter coefficients acquired a positive sign whilst the constant coefficients had a negative sign. The sign of the coefficient denotes the trend of relationship between variables and response [44]. As a result, an increasing rate of traverse speed, abrasive mass flow rate and waterjet pressure, generates a higher rate of material removal. Overall, the coefficient of determination R<sup>2</sup> ranged from 97.79 to 97.92%, with R<sup>2</sup> adj ranging from 96.46 to 96.67% and R2 pred ranging from 92.53 to 94.35%, confirming that all generated regression models were significant. The models were established to be sufficient for accurate forecasting of material removal rate within the assigned levels of input parameters for AWJM of straight and arcs profiles. Furthermore, **Table 9** demonstrated that MRR1 (straight-line), MRR2 (inner *Multi-objective Optimisation in Abrasive Waterjet Contour Cutting of AISI 304L DOI: http://dx.doi.org/10.5772/intechopen.106817*


### **Table 9.**

*Summary of linear regression coefficients for MRR.*

arcs) and MRR3 (outer arcs) attained a uniform gap of at least 2% for R2 , R<sup>2</sup> adj and R2 pred values. This nominal disparity of the coefficient of determination indicates that AWJM performance for straight and curvature profiles are not significantly different from one another [36]. The results detailed in **Table 9** confirm that the highest values of R2 , R<sup>2</sup> adj and R<sup>2</sup> pred for all material thicknesses was attained in MRR1 (straight-line profile) with values of 97.92, 96.67 and 94.35%; 98.86, 98.18 and 95.73%; 98.70, 97.92 and 95.19% respectively. This statistical measurement evaluates the relationship between the model and response variables, indicating that a value nearest to 100% denotes a more reliable model [49]. Therefore, MRR1 regression models are considered as the most fitted model for 4, 8 and 12 mm material thicknesses.

**Tables 10**-**12** present the predicted MRR values using the generated regression models of 4, 8 and 12 mm thickness of AISI 304L for three varied contour profiles. The percentage error acquired for 4, 8 and 12 mm AISI 304L thicknesses ranged from �5.35 to 5.15%, �6.59 to 4.77% and � 5.05 to 6.62%, respectively. The errors determined for Ra between the predicted value and experimental results were less than 20%, indicating models to be well fitted for predicting MRR values.

Plots of all residuals of the best material removal rate (MRR1) for all material thicknesses are represented in **Figure 6**. Overall, the normal probability plots for all the material thicknesses illustrate that the adjacency of the points are linear indicating there is no deviation from the assumptions, because they are normally and independently distributed [46]. Residuals versus fits and observation for MRR1 of straightline, inner and outer arc profiles confirm that there is no skewness or outlier pattern, revealing that individual deviated assumptions have no conflicts or contradictions. **Figure 6** also presents the histogram graph for MRR1, obtaining frequency ranging from �10 to 15 for 4 mm, �15 to 15 for 8 mm and � 18 to 20 for 12 mm material thicknesses. These results signify that the distribution or frequency of residuals for all observations fell in minimal interval or inequalities of the experimental data, justifying the adequacy of the suggested MRR1 models [46].

According to the results presented in **Tables A4**-**A6** in the Appendix section, detailing ANOVA for material removal rate, the effects of the input parameters for


### **Table 10.**

*Predicted MRR values of regression model for* t *= 4 mm.*


### **Table 11.**

*Predicted MRR values of regression model for* t *= 8 mm.*

straight and arc profiles at 4, 8 and 12 mm AISI 304L thicknesses display comparable results. Further, the results reveal that traverse speed and waterjet pressure are statistically and physically significant factors for obtaining *p-*Values<0.05. Hence, the abrasive mass flow rate features as a low impacting input parameter for obtaining *p-*Values greater than the acceptable value of 0.05, ranging from 0.002 to 0.751.

The percentage contribution of variables for the most fitted regression models MRR for 4, 8 and 12 mm material thicknesses are illustrated in **Figure 6**. In general, traverse speed is indicated as the most impacting variable, followed by waterjet pressure and abrasive mass flow rate, with a percent contribution ranging from 71.14– 78.94%, 12.11–24.09% and 2.65–9.03% respectively for all profiles and material thicknesses. It is apparent here that the percentage contribution of traverse speed increases

*Multi-objective Optimisation in Abrasive Waterjet Contour Cutting of AISI 304L DOI: http://dx.doi.org/10.5772/intechopen.106817*


### **Table 12.**

*Predicted MRR values of regression model for* t *= 12 mm.*

### **Figure 6.**

*Residual plots for material removal rate. (a) MRR 1 (mm<sup>3</sup> /min) for* t *= 4 mm (b) MRR 1 (mm<sup>3</sup> /min) for* t *= 8 mm (c) MRR (mm<sup>3</sup> /min) for* t *= 12 mm.*

in range from 71.4 to 77.55% as the material thickness increases. An increasing traverse speed reinforces the contact time of the waterjet with the abrasive on the material, producing a higher volume rate of material to the machine [9]. Contrastingly, the percentage contribution of waterjet pressure and abrasive mass flow rate decreased as the material thickness and traverse speed increased, ranging from 22.42–12.11% and 4.35–9.03%, respectively. The increasing traverse speed and depth or thickness of the material to cut, results in a more prolonged machining process, which gradually leads to subsiding kinetic energy and loss of large of abrasive particles, resulting in reduced effectiveness of abrasive mass flow rate and waterjet pressure during the erosion process (**Figure 7**) [9, 47].

### **3.3 Regression model and analysis for kerf taper angle**

The summary of the multi-linear regression coefficients for kerf taper angle of straight-line, inner and outer arc profiles using 4, 8 and 12 mm material thicknesses are detailed in **Table 13**. The results provide a similar trend, showing the constant sign

**Figure 7.**

*Percentage contribution of variables for material removal rate. (a) MRR 1 (mm<sup>3</sup> /min) for* t *= 4 mm (b) MRR 1 (mm<sup>3</sup> /min) for* t *= 8 mm (c) MRR (mm<sup>3</sup> /min) for* t *= 12 mm.*

as positive, with variables *X*1, *X*<sup>2</sup> and *X*<sup>3</sup> as negative for all profiles and thicknesses. If the coefficient sign is negative, as the variable increases, the response decreases, whereas if the coefficient is positive, the relationship between variables and responses is directly proportional [44]. Therefore, an increasing rate of traverse speed (X1) results in an increasing angle of the kerf taper. Thus, an increasing rate of abrasive mass flow and waterjet pressure reduces the value of kerf taper angle. The values of R2 , R2 adj and R2 pred for 4, 8 and 12 mm ranged from 94.74–99.37%, 91.59–98.99% and 80.11–97.66%, respectively. This confirms that regression models are reliable in representing correlation between variables and responses and can be used in the optimisation of process parameters.

The coefficient of determination (R2 , R<sup>2</sup> adj and R<sup>2</sup> pred) obtained from straightline, inner and outer arc profiles for all material thicknesses had a similar and consistent gap of at least 2%. The AWJM provides comparable behaviour in processing both straight and curvature profiles [36]. The highest values of R2 , R<sup>2</sup> adj and R<sup>2</sup> pred for 4 and 8 mm material thicknesses were attained in KTA1 with values of 97.56, 96.09 and 90.57%; 98.02, 96.82 and 92.01%; 99.37, 98.99 and 97.66%, respectively. These are the most fitted model, to be utilised in the optimisation of the process parameters of this study.

The predicted KTA values using the regression models applied for straight-line, inner and outer arc profiles of the three levels of material thicknesses are detailed in **Tables 14**-**16**. The percentage error obtained for 4, 8 and 12 mm AISI 304L thicknesses ranged between �2.55 to 1.72%, �2.67 to 3.74% and � 3.14 to 2.43%, respectively. The errors calculated for KTA between the predicted value and experimental results were less than the acceptable maximum limit of 20%, indicating the reliability of the models in predicting KTA values.

**Figure 8** illustrates the residual plot for KTA including normal probability plot, residual versus fits, histogram for residuals and residuals versus experimental values. The results showed that the most fitted regression model is achieved from KTA1 for all material thicknesses. Correspondingly, the normal probability plots for all material thicknesses present a near fit to a line in a normal probability graph. The points constructing an approximate straight-line and plotted along the fitted line signifies that the data is normally distributed and there is a good relation between experimental data and predicted values [45]. Predominantly, the residuals versus fits and observation graph for each material thickness exhibit that the points are plotted randomly and near both sides of 0 with no identified pattern denoting a minimal deviation within residuals and estimated values. **Figure 8** also presents the histogram graph for KTA


*Multi-objective Optimisation in Abrasive Waterjet Contour Cutting of AISI 304L DOI: http://dx.doi.org/10.5772/intechopen.106817*

> **Table 13.** *Summaryof*

 *linear regression coefficients*

 *for KTA.*


### **Table 14.**

*Predicted KTA values of regression model for* t *= 4 mm.*


### **Table 15.**

*Predicted KTA values of regression model for* t *= 8 mm.*

illustrating the distribution or frequency of the residuals for all observations. The results show that the frequency of KTA for 4, 8 and 12 mm material thicknesses range from �0.002 to 0.015, �0.05 to 0.05 for 8 mm and � 0.02 to 0.03, respectively. These graphs reveal a minimal interval or inequalities of the experimental data indicating that the KTA regression models are highly fitted to concrete prediction [46].

**Tables A7**-**A9** in the Appendix section detail the results of ANOVA, where it can be observed that the impacts of parameters for all profiles and three levels of material thicknesses demonstrate a similar trend, denoting traverse speed and waterjet pressure to be significant factors for acquiring *p-V*alues lower than 0.05. Thus, the abrasive mass flow rate was found insignificant for achieving *p-*Values >0.05, ranging from 0.002 to 0.245 for all profiles and material thicknesses.

*Multi-objective Optimisation in Abrasive Waterjet Contour Cutting of AISI 304L DOI: http://dx.doi.org/10.5772/intechopen.106817*


### **Table 16.**

*Predicted KTA values of regression model for t = 12 mm.*

### **Figure 8.**

*Residual plots for kerf taper angle. (a) KTA1 (°) for* t *= 4 mm (b) KTA1 (°) for* t *= 8 mm (c) KTA1 (°) for* t *= 12 mm.*

**Figure 9.**

*Percentage contribution of variables for kerf taper angle. (a)* t *= 4 mm (b)* t *= 8 mm (c)* t *= 12 mm.*

**Figure 9** exhibits the percentage contribution of variables for KTA for the most fitted regression models for 4, 8 and 12 mm material thickness. Traverse speed was the most influencing parameter, followed by waterjet pressure and abrasive mass flow rate, in agreement with previous studies [14, 37]. The obtained results have shown that the influence of traverse speed decreases in range from 64.21 to 53.33% as the material thickness increases. An increasing value traverse speed results in the loss of a large number of abrasive particles, continuously dropping as the material thickness also increases, leading to a higher angle of kerf taper [50]. **Figure 9** shows increases of material thickness, the percentage contribution of waterjet pressure and abrasive mass flow rate, ranging from 26.60 to 33.40% and 6.75 to 12.65%, respectively. This increasing value of waterjet pressure resulted in higher energy, generating a larger amount of abrasive particles that result in a lower kerf taper [51]. Moreover, a rising rate of abrasive mass flow breaks down abrasive particles into a smaller scale, generating more sharp points that results in reduction of kerf taper angle [51].
