**3. Results and discussion**

### **3.1 Regression models and analysis for surface roughness**

The multi-linear regression coefficients are summarised in **Table 5**, exhibiting the correlation between the input parameters and the output surface roughness for straight-line, inner and outer arc profiles for material thicknesses of 4, 8 and 12 mm. The values of coefficients for all profiles and thicknesses demonstrate a similar trend, showing that constant and variable *X*<sup>1</sup> is positive and variables *X*<sup>2</sup> and *X*<sup>3</sup> are negative. The coefficient indicates the change in the mean response relating in the variation of the specific term, whilst the other term in the model remains constant. The relationship between a term and response is denoted by the sign of the coefficient [44]. The negative correlation coefficient denotes an inverse relationship between variables and responses; and therefore, if it is positive as the coefficient increases, the response mean value also increases. Therefore, an increasing rate of traverse speed (*X*1) results in an incremental value of surface roughness. Moreover, an increasing rate of abrasive mass flow and waterjet pressure indicates/obtains a decreasing value of surface roughness. The values of R2 , R2 adj and R2 pred for 4, 8 and 12 mm ranged from 94.33– 99.08%, 90.94–98.52% and 88.66–96.17%, respectively. This indicates that regression models denote an acceptable confirmation of the relationship between the independent variables and Ra response, which denotes a high significance of the model. Therefore, the multi-linear model is reliable and can be utilised in the optimisation of process parameters. It can be observed that the R2 , R<sup>2</sup> adj and R<sup>2</sup> pred obtained from straight-line, inner and outer arcs profiles have a uniform gap of at least 2%, which is comparable for all material thicknesses. Hence, this minimal gap denotes an insignificant difference between the surface roughness achieved from straight and curvature profiles [36].

The results detailed in **Table 5** show that the highest value of R2 , R<sup>2</sup> adj and R2 pred for 4 and 8 mm material thickness are achieved in Ra3 with the values of 97.26, 94.84 and 92.45%; 98.64, 97.82 and 95.06%; 99.08, 98.52 and 96.17% respectively. Thus, Ra2 achieved the highest percentage of R2 , R2 adj and R<sup>2</sup> pred for 12 mm material thickness


 **5.** *Summary of multi-linear regression coefficients for Ra.*
