**2. Methodology**

In this study, three major steps were employed, consisting of abrasive waterjet contour cutting experiments, regression modelling and optimisation. The experiment, modelling and optimisation procedures are presented in **Figure 1**. The experiment was conducted using the Taguchi L9 orthogonal array to analyse the impacts of input parameters, i.e., traverse speed, abrasive mass flow rate and waterjet pressure. Desirability analysis using response surface methodology is employed for the experimental results of material AISI 304L. In this desirability analysis, multi-responses are considered. It establishes the optimum set of the selected process parameters on the performance characteristics.

A regression model was developed using the machining process parameters from the experimental execution to extract mathematical models. A linear stepwise regression analysis was performed to predict the surface roughness, material removal rate

**Figure 1.**

*Multi-objective optimisation process flow chart.*

and kerf taper angle value. The reliability of the models generated was assessed based on coefficient of determination (R2 , R<sup>2</sup> adj & R<sup>2</sup> pred). However, supposing that regression models are not within the acceptable range or do not provide preferable values of coefficients of determination set by the decision-maker, it is anticipated that these models will not provide precise prediction. Therefore, the selected parameter setting conflicts with the response variables, denoting the necessity for modification of independent variables or experimental design [16].

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

Referring to **Figure 1**, after achieving the fittest models, a multi-objective optimisation was performed by using response surface methodology with the objectives of maximising material removal, whilst minimising surface roughness and kerf taper angle. The number of solutions and iterations (i = 1 to n) may vary, depending on the machining process requirements to establish the best alternative or solution. Hence, if the composite desirability is not within the tolerable array, several iterations repeating the response surface optimisation were executed. Subsequently, if these repetitions reached the maximum number of iterations and the composite desirability is not attaining adequate values, modifying the design of experiments and the corresponding independent variables or its values is necessary [16]. Moreover, in some cases, other soft computing techniques should be considered [17].

### **2.1 Material and experimental design**

In this work, the material machined in the experiments was AISI 304L with varied thicknesses of 4, 8 and 12 mm. The assigned material thicknesses with differing uniform gaps were used to gain a better yield of variations in AWJM cutting behaviour. Stainless steel, such as AISI 304L, is widely used in fabrication industries, where it is recognised for its high strength and corrosion and heat resistance. This results from its high alloying content of Cr and Ni [18]. The chemical and mechanical composition of this material is detailed in **Table 1**.

The setup consisted of an OMAX MAXIEM 1515 abrasive waterjet machine, possessing a direct drive pump and dynamic cutting head with maximum pressure of 413.7 MPa and cutting area of 2235 mm length and 1727 mm width. The cutting head is comprised of a mixing chamber for abrasive and waterjet, along with a nozzle diameter of 0.56 mm and a jet impact angle of 90°. An abrasive garnet with a mesh size of #80 was utilised for abrasive waterjet cutting experiments. The unit is inclusive of IntelliMax software, where the experiment setup conditions were uploaded and entered. The cutting head can move in the Z-axis over a distance of 305 mm, with a maximum traverse speed of 12,700 mm/min. Standoff distance was designated to 1.5 mm in agreement with recommended range for abrasive waterjet machining in previous works [20, 21]. The AWJM setup and process parameters are demonstrated in **Figure 2**.


### **Table 1.** *Chemical and mechanical composition of AISI 304L [19].*

Upon completion of the experiments, the roughness of the machined surfaces was quantified by a surface roughness tester (TR200 model). **Figure 2** presents the cut surface captured by LEICA M80, which indicates the measurement area for the roughness. The kerf top and bottom width were measured using a LEICA M80 optical microscope model. Moreover, rate of material removal and kerf taper angle were calculated using Eqs. (1) and (2), respectively [11]. The roughness of the cut surface determined according to the ISO/TC 44 N 1770 standard, (μm); *Wt* is width of the cut surface at the jet inlet, (mm]; *Wb* is the width of the cut surface at the jet outlet, (mm); u is the angularity or perpendicular deviation, (mm); α°- inclination angle of the cut surface, (°); MRR is the Material Removal Rate, (mm<sup>3</sup> /min); *t* is the thickness of the material (mm) [22].

$$MRR = ht\left(\frac{W\_t + W\_b}{2}\right) V\_f \tag{1}$$

$$KTA = Area \left(\frac{W\_t + W\_b}{2ht}\right) \tag{2}$$

The input parameters considered in abrasive waterjet contour cutting in this experiment included traverse rate (*Vf* ), abrasive flow rate (ma) and water pressure (*P*), as these parameters have been demonstrated in previous studies as having significant impacts in AWJM applications [10, 12, 23, 24]. Surface integrity, kerf geometries

**Figure 2.** *AWJM setup and process parameters.*

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

and low material removal rate evidence has been reported in machining of AISI 304L, requiring further improvement [4, 25]. Furthermore, taper angles formed in AWJM demonstrate different inclinations as contour curvature radius differs [26]. Hence, quality and productivity are an intensified demand in various manufacturing fields and are significant performance indicators for machining processes. Therefore, in this study, material removal rates (MRR), surface roughness (Ra) and kerf taper angle (KTA) have been chosen as process parameter characteristics for abrasive waterjet contour cutting investigations, due to their influence against the selected input parameters. The levels of the considered independent variables, responses and coding assignment have been detailed in **Tables 2** and **3**.

Abrasive waterjet cutting was executed for three different profiles, representing straight-line, inner arcs and outer arcs, as part of the completed twelve profiles, as demonstrated in **Figure 2**. The abovementioned profiles were selected to confirm a broad array of complicated machining profiling applications. The levels of profiles employed showed occurrences of surface roughness, low machining rate and inaccuracies of cut geometries in regard to previous works [27, 28], recommending further studies, predominantly for difficult-to-cut materials, such as AISI 304L (**Figure 3**).

The design of experimentation (DOE) was carried out using the Taguchi approach in MINITAB 19 software. The Taguchi method is useful in determining the best combination of factors under desired experimental conditions, reducing the large number of experiments which would be required in traditional experiments as the number of process parameter increases [29, 30].

In Taguchi's approach, selection of the appropriate orthogonal array depends on aspects such as: the number of input and response factors along with the interactions that are of key significance; number of levels of data for input factors; and required resolution of experiment and limitations cited on cost and performance [29, 31]. With this specific advantage, this method is suitable in conducting experiments with an appropriate number of tests to determine the optimal combination and significance of the selected factors [32]. The relevant variation in thicknesses dictates different material responses. Therefore, Taguchi L9 orthogonal array was executed for three


### **Table 2.**

*Levels of input process parameters.*


**Table 3.** *Output parameters for varied profiles.*

### **Figure 3.**

*Abrasive waterjet contour cutting profiles.*


### **Table 4.**

*Taguchi* L9 *orthogonal array.*

levels of material thicknesses (t), i.e., 4, 8 and 12 mm, as presented in **Table 4**. The AWJM performances were analysed accordingly by the applied material thickness.

### **2.2 Modelling and multi-objective optimisation**

A mathematical model was developed to associate the input process parameters to the response's characteristics. To achieve this, a linear regression was employed to develop models for the prediction of responses. The empirical model for the prediction of the responses in regard to controlling parameters was established by linear regression analysis. Regression analysis was then applied to obtain the interactions between independent and dependent variables [33]. Multi-linear regression involves regression analysis of dependent and independent variables exhibiting a linear relationship [34]. It stipulates the relationship between two or more variables and a response variable by fitting a linear equation to examine data. The value of the independent variable x or process parameter is correlated with a value of the dependent variable, y, which is the output parameter. In general, this analysis is applied to investigate the degree of relationship between multiple variables fitted by a straight line [33].

In general, regression model is expressed by Eq. (3) [33].

$$
\lambda y = \infty + \beta\_1 \mathbb{1}\_1 + e \tag{3}
$$

$$\text{Where}\\
\text{in}: \mathcal{e} = \mathcal{y}\_1 - \hat{\mathcal{y}}\_1 \tag{4}$$

where, *y* = dependent variable, α = constant, *x*<sup>1</sup> = Independent variable, *β*1= coefficient of independent variable*x*1, e = error, *<sup>y</sup>*1= regression line values and *<sup>y</sup>*b<sup>1</sup> <sup>=</sup> actual observation.

If this involves more than one variables, then it is categorised as multi-regression as shown in Eq. (5) [33].

$$\mathbf{y} = \mathbf{x} + \beta\_1 \mathbf{x}\_1 + \beta\_2 \mathbf{x}\_2 + \beta\_3 \mathbf{x}\_3 + \dots \dots \dots \beta\_n \mathbf{x}\_n + \mathbf{e}\_n \tag{5}$$

A multi-linear regression analysis can be employed *to fit a predictive model to an observed data set of values of output and input variables*. The obtained results of surface roughness, material removal rate and kerf taper angle were expressed in terms of the input parameters such as traverse speed (*X*1) abrasive mass flow rate (*X*2) and waterjet pressure (*X*3).

*The predicted values are functional for optimising the parameters by providing an adequate comprehension of the significant parameters.* The percentage of error between the experimental data and acquired predicted values has been calculated based on Eq. (6) [33]. The relative percentage of error was acceptable at <20% [35].

$$Error = \frac{1}{n} \sum\_{n}^{1} \left[ \frac{Response\_{(experiment)} - Response\_{(predicted)}}{Response\_{(experiment)}} \right] \text{(\text{\textquotedblleft}degree\text{\textquotedblright})} \tag{6}$$

The performance of the established regression model was assessed by statistical approaches to confirm the goodness-of-fit of the model and the impact of the predicted variables. Following this, the significance and effectiveness of the developed models were validated by *analysis of variance*. *Analysis of variance (ANOVA) is a statistical method that facilitates the evaluation of comparative influences for each control parameter* [36, 37]. The significance of input parameters including traverse speed, abrasive mass flow rate and waterjet pressure were investigated using *p-* values and determination of coefficient (R<sup>2</sup> ). In this work, a confidence interval of 95% (*p* < 0.05) has been applied that is in alignment with previous works [29, 38, 39]. A 95% confidence interval means that there is only a 5% chance of being the wrong estimation; therefore, the influence of each process parameter or other interactions on the responses is considered insignificant if their *p*-values were estimated at more than 0.05 [37].

The determination of coefficient (R2 , R<sup>2</sup> adj and R<sup>2</sup> pred) refers to the percentage variation of responses ranging from 0–100%. These indicators determine the adequacy of the model against obtained experimental data and predicted observation. This R<sup>2</sup> , R2 adj and R2 pred value of ≥80%, proved a better model fits of the obtained data [35].

Response surface methodology (RSM) can be utilised for multi-objective optimisation. This multi-desirability is based on multi-response optimisation using an objective function D(X), denoted as desirability function [40]. This method translates each response (*yi* ) into a desirability function (di), differing in the array of 0 ≤ di ≤ 1,

where desirability function =0 indicates an undesirable response and desirability function =1 represents a fully desired response [41]. The objective function D is specified by Eq. (7) [40].

$$D = (d\_1 X d\_2 X \dots \dots d\_n)^{\mathbb{V}\_n} = \left(\prod\_{i=1}^n d\_i\right)^{\mathbb{V}\_n} \tag{7}$$

The effectiveness of multi-objective optimisation is anticipated based on the method used for establishing priority weights for each response characteristics [42]. Generally, equal importance is set for selected responses; hence, weights may differ depending on the machining process requirements in order to establish the most suitable solution [43].

A simultaneous optimisation process was employed to determine the levels of resulting to the maximum overall desirability. The responses namely Ra, MRR and KTA were optimised concurrently to assess the set of input process parameters with the objectives of maximising MRR and minimising Ra and KTA.
