**4. Response surface methodology multi-objective optimisation**

In this research, multi-objective optimisation was performed using RSM to determine the optimum process parameters of abrasive waterjet contour cutting of AISI 304L with varied thicknesses using MINITAB 19 software. The following optimisation objectives were stated as follows:

$$f\_1 = \text{Min}\left(\mathbb{R}\_a\right) \tag{8}$$

$$f\_2 = \text{Min}\,(K\text{TA})\tag{9}$$

$$f\_{\clubsuit} = \mathbf{Max}\,\,(\mathbf{MRR})\tag{10}$$

RSM optimisation was performed using the models with the highest determination of coefficients, i.e., R<sup>2</sup> , R<sup>2</sup> adj and R<sup>2</sup> pred. Accordingly, the regression models utilised to minimise surface roughness were Ra3 for 4 and 8 mm and Ra2 for 12 mm. MRR1 and KTA 1 models were used for all material thicknesses.

The Regression models utilised in multi-objective optimisation for varied thicknesses of AISI 304L were expressed by Eqs. (8)-(16).

$$\mathbf{R}\mathbf{a}\_{4\text{mm}} = \mathbf{1.4256} + 0.003222\,\mathbf{X}\_1 - 0.000217\,\mathbf{X}\_2 - 0.001133\,\mathbf{X}\_3 \tag{11}$$

$$\text{KTA}\_{4\text{mm}} = 0.9674 + 0.002414 \,\text{X}\_1 - 0.000235 \,\text{X}\_2 - 0.000932 \,\text{X}\_3 \tag{12}$$

$$\text{MRR}\_{4\text{mm}} = -84.2 + 1.752\,X\_1 + 0.126\,X\_2 + 0.5103\,X\_3 \tag{13}$$

$$\mathbf{Ra}\_{\\$mm} = \mathbf{1.76} + \mathbf{0.008869} \,\mathbf{X}\_1 - \mathbf{0.000577} \,\mathbf{X}\_2 - \mathbf{0.001920} \,\mathbf{X}\_3 \tag{14}$$

$$\text{KTA}\_{8\text{mm}} = 1.386 + 0.006143 \, X\_1 - 000520 \, X\_2 - 0.002039 \, X\_3 \tag{15}$$

$$\text{MRR}\_{8\text{mm}} = -1\text{19} + 2.94\text{1}X\_1 + 0.272\text{3}X\_2 + 0.777X\_3 \tag{16}$$

$$\text{Ra}\_{4\text{mm}} = 2.3854 + 0.004276 \, X\_1 - 0.000446 \, X\_2 - 0.001924 \, X\_3 \tag{17}$$

$$\text{KTA}\_{4\text{mm}} = 1.5981 + 0.006568 \,\text{X}\_1 - 0.000107 \,\text{X}\_2 - 0.002319 \,\text{X}\_3 \tag{18}$$

$$\text{MRR}\_{\\$\text{mm}} = -158.8 + 3.867X\_1 + 0.396X\_2 + 0.917X\_3 \tag{19}$$

In simultaneous optimisation, goals and boundaries must be defined for each process parameter. Targets are based on the experimental data obtained, referring to the set highest value of responses for maximising MRR and lowest value of responses for minimising Ra and KTA. In this optimisation, process parameters and defined objectives were assigned to be equally significant. Therefore, the equal weights

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

(wt. = 1) were assigned in order to achieve an equal importance to the process parameters and objectives. The constraints referring to range and limits of the process parameters are detailed below.

Constraints: ≤*Vf* ≤ 150 mm/min ≤ ma ≤ 500 g/min ≤ *P* ≤ 300 g/min

Limits: KTA4mm ≤1*:*03° ,KTA8mm ≤1*:*68° ,KTA12mm ≤2*:*06° Ra4mm mm1*:*58 μm,Ra8mm mm2*:*45 μm,Ra12mm 2m2*:*46 μm MRR4mm ≥216*:*20 mm3*=* min ,MRR8mm ≥367*:*90 mm3*=* min , MRR12mm ≥472*:*00 mm3*=* min

**Table 17** shows the solutions for multi-objective optimisation performed for 4, 8 and 12 mm thickness of AISI 304L. The solution that provides the value of composite desirability nearest to 1 can be considered as the best solution [40]. **Table 17** reveals that solution 1 is the best for 4, 8 and 12 mm material thicknesses, achieving composite desirability values of 0.748448, 0.780587 and 0.786800, respectively. There are three solutions generated from MINITAB application, providing the settings of input variables, achieved values of responses and composite desirability. Solution 1 provides the optimum settings of input parameters i.e., *Vf* for 4, 8 and 12 mm material thicknesses, at the speeds of 95, 90 and 91 mm/min, respectively. The obtained optimum setting for ma and *P* were found to be the same value for all material thicknesses, at 500 g/min and 200 MPa, respectively. **Table 17** presents the minimum achieved values of KTA and Ra and maximum MRR for 4, 8 and 12 mm material thicknesses, featuring at 0.7990 , 1.283 μm and 297.98 mm<sup>3</sup> /min; 1.068<sup>0</sup> , 1.694 μm and 514.97 mm<sup>3</sup> /min and 1.448<sup>0</sup> , 1.975 μm and 667.07 mm<sup>3</sup> /min, respectively.

An optimisation plot presenting how the variables affected the predicted responses is shown in **Figure 10**, detailing the composite desirability for multiobjective (D) and single-objective optimisation (d). Current variable settings for the input parameters are presented in the figure, alongside with lower and upper limits. **Figure 10** shows a three-sectioned line graph representing the correlation of KTA, Ra and MRR against traverse speed (*X*1), abrasive mass flow rate (*X*2) and waterjet pressure (*X*3).

From the figure, it can be observed that abrasive waterjet contour cutting responses demonstrate a comparable behaviour against input parameters for all material thicknesses. The highest rate of material removal and lowest value of surface roughness and Kerf taper angle were achieved by employing a rate of 150 mm/min speed, 500 g/min abrasive mass flow rate, and 300 MPa of waterjet pressure. Increasing water pressure, alongside high velocity abrasive mass flow rate, produces a greater collision of abrasive particles, generating higher rate of material removal and reducing surface roughness and kerf taper angle [52].

The surface roughness displayed an incrementing value that ranged from 4–13% as the rate of traverse speed increased from 90 to 150 mm/min. As the speed increases per unit of area over time, the kinetic energy containing abrasives gradually decreases, resulting in greater evidences of rough surfaces [52]. Consequently, RSM optimisation



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

**Figure 10.** *Response optimisation plot. (a) t = 4 mm (b)* t *= 8 mm (c)* t *= 12 mm.*

has shown that a lower level of traverse speed can produce a better quality of cut surface. Additionally, surface roughness in this study shows an increasing value ranging 2–5%, as the waterjet pressure increases and the abrasive mass flow rate from 200 to 300 MPa and 300 to 500 g/min, respectively. In this study, it is confirmed that augmenting abrasive flow rate and waterjet pressure, up to a specific range, lowers the value of surface roughness. When higher values of traverse speed are employed, the material removal exhibits an increasing rate that ranges from 16–20%. In addition, increasing rate of material removal was achieved with a range of 5–9%, as the rate of abrasive mass flow and waterjet pressure increased from 200 to 300 MPa and 300 to 500 g/min, respectively. AWJM produces a high level of kinetic energy, driving a higher level of speed and waterjet pressure alongside with abrasive mass flow rate, which in turn generates higher cutting area per unit of time and generates a larger amount of eroded material [53]. Therefore, the rate of material removal is directly proportional to traverse speed, abrasive mass flow rate and waterjet pressure. **Figure 10** shows that kerf taper angle values increase as the rate of traverse speed increases from 90 to 150 mm/min. With continuous reduction in the number of abrasive particles, as the traverse speed increases, the cohesion on metal material decreases, generating a higher tapering angle [52]. The kerf taper angle in this study was reduced by 2–7%, as the abrasive mass flow and waterjet pressure were increased from 200 to 300 MPa and 300 to 500 g/min, respectively. A higher waterjet pressure alongside with abrasive mass flow rate reinforces the collision of abrasive particles on the target material, causing the reduction of kerf taper angle [51].

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