Surface quality improvement of AZ31 Mg alloy by a combination of modified Taguchi method and simple multi objective optimization procedure

Abstract

The motive of this study is to evaluate the machinability properties of AZ31 Mg alloy in the face milling operations. Depth of cut, rate of feed and speed of cut are three input variables in the face milling process, while surface roughness (Ra) and tool nose radius deviation (dR) are performance characteristics. For three input variables each with three levels, to decrease the number of experiments, an L₉ orthogonal array (OA) of Taguchi was employed which enables an efficient approach for conducting a full-factorial design of experiments. Selection of optimal parameters for the face milling process to achieve the required surface quality is accomplished through the utilization of a modified Taguchi approach and a novel multi objective optimization procedure. The work establishes empirical relationships that correlate the performance characteristics (i.e. Ra and dR) with three input variables. Corrections to the empirical relations were specified to obtain the expected range of Ra and dR. Ra and dR estimates were validated with existing test data.

Keywords:

• AZ31 Mg alloy
• depth of cut
• face milling
• rate of feed
• speed of cut
• surface roughness
• Taguchi method
• tool nose radius

Reviewing Editor:

• Professor Swadesh Kumar Singh, Gokaraju Rangaraju Institute of Engineering and Technology, India

Subjects:

• Mechanical Engineering
• Manufacturing Engineering
• Mathematics & Statistics for Engineers

1. Introduction

The combination of lightweight, high strength, good machinability, excellent damping and thermal conductivity properties make Mg alloys attractive for wide collection of applications in electric vehicle, biomedical, aerospace, electronic industries that prioritize weight reduction, performance optimization and energy efficiency (Muni Tanuja et al., 2022; Tan & Ramakrishna, 2021; Mordike & Ebert, 2001). While the hexagonally close packed (HCP) crystalline arrangement of Mg alloys limits their formability and also active slip systems, which limits their suitability for certain structural applications (Sivam et al., 2018; Brandão et al., 2016). Currently, the predominant research in Mg alloys is primarily focused to plastic processing technologies (such as forging, rolling, drawing, extrusion and pressing) and various heat treatment methods (including annealing, quenching, solid state dissolution and aging) (Jiang et al., 2015; Sivam et al., 2018; Wang et al., 2015; Jiang et al., 2016). In a study by Zakaria et al. (2022) to compensate the lack of cooling in dry machining, a new submerged convective cooling (SCC) approach was used to perform the turning process of AZ31 Mg alloy, which increased the cutting performance.

An analysis is conducted to evaluate the effects of cryogenic and dry machining conditions on high speed milling of Mg alloy in the regime of high cutting speeds, since Mg alloy chips are highly flammable, it essential to find effective ways to reduce the temperature effect in the cutting areas. The effects of these cutting conditions on the milling process are being investigated to improve the safety and performance of Mg alloy machining (Jouini et al., 2023; Ghosh et al., 2007). In their study, Lu et al. (2016) examined the effect of cutting variables (i.e. depth of cut, rate of feed) on microstructure, surface roughness and work hardening. As speed of cut increases, depth of stress layer, roughness and hardening decrease. On the contrary, it was noticed that with higher the depth of cut and rate of feed increase, these parameters show an upward trend. Speed of cut, axial and depth of cut in radial direction, and feed per tooth are important to achieve increased machining stability and excellent surface finish, and the recommended a tool with a 20⁰helix angle for finish milling process (Zagórski et al., 2022). By refining the microstructures with unique elements, it may be possible to improve the mechanical properties, heat treating the material and thermo-mechanical treatment as well. By designing a die for severe plastic deformation (SPD), improved homogeneity of deformation can be observed in the pressed sample (Anantha et al., 2023a, 2023b; Moradpour et al., 2018). Anantha et al. (2023a, 2023b) implemented a genetic algorithm (GA) multi-criterion optimization tool to find the optimal die geometry (Groove width = 3 mm, Groove angle = 50° and Coefficient of friction = 0.22).

The S/N ratio transformation introduced by Taguchi (Ross, 1989; Konduri et al., 2017) offers a way to account for variability in repeated tests by consolidating the output responses into a single value. Without implementing the S/N ratio transformation, several optimal solutions have been achieved in various applications. These include minimizing damage to composites caused by drilling (Singaravelu et al., 2012), ensuring collision-free satellite separation processes (Koneru et al., 2022), increasing the performance of heat pipes (Ramya et al., 2018; Koneru et al., 2023), optimizing the design of planetary gears (Miladinovic et al., 2021), improving manufacturing processes (Satyanarayana et al., 2018; Rajyalakshmi & Boggarapu, 2018; Rajyalakshmi & Rao, 2019; Satyanarayana et al., 2019; Dharmendra et al., 2019; Dharmendra et al., 2020; Satyanarayana et al., 2021; Buddi et al., 2018), optimizing painting processes using robot (Danthala et al., 2021), increasing the performance of a biodiesel engine using additives (Sanjeevannavar et al., 2022) and stir casting technique (Saritha et al., 2019).

Santhakumar and Iqbal (2022) conducted interesting studies on the effect of milling parameters (speed of cut, depth of cut and rate of feed) on surface quality and tool nose radius in Mg alloys (see Figure 1). They implemented the transformation of S/N ratio to the measured Ra and dR choosing the smaller-the-better as standard. They performed a grey relational Taguchi analysis to adjust the cutting parameters of dry face milling for achieving lower Ra and dR. In fact, it has been recommended to apply the transformation of S/N ratio to the repeated test response to take care for the experimental variance, while most researchers have applied this transformation to the mean value of the repeated test data.

Figure 1. Experimental setup for dry face milling of AZ31 Mg alloy (Santhakumar & Iqbal, 2022): (A) Coolant nozzle (OFF); (B) Collet; (C) Workpiece.

Encouraged by the research of the aforementioned researchers, the problem is re-examined by applying a modified Taguchi method and a reliable multi objective optimization procedure for selecting a group of input process parameters (viz. depth of cut, rate of feed and speed of cut) to achieve higher Ra and lower dR in face milling of AZ31 Mg alloy.

2. Analysis

Dry face milling is accomplished on AZ31 Mg alloy by means of a cutting tool of 12 mm $\phi$ TiCN-coated carbide grade inserts. The dimensions of the work-piece are $200\mathit{\text{ mm}}\times 200\mathit{\text{ mm}}\times 200\mathit{\text{ mm}}.$ The experimentation setup involved utilizing a 3-axis VMC (Vertical Machining Center) equipped with a 7.5 kW motor and a spindle speed maintained below 8000 rpm. Speed of cut (A), rate of feed (B) and depth of cut (C) are milling parameters. The AZ31 Mg alloy’s chemical composition (Santhakumar & Iqbal, 2022) in Table 1 confirms with others (Hsiang & Lin, 2019; Karthik et al., 2021; Goyal et al., 2022; Ömer, 2019).

Table 1. AZ31’s chemical composition.

Chemical	Composition (wt. %)
	Hsiang et al. [37]	Karthik et al. [38]	Goyal et al. [39]	Omer [40]	Santhakumar and Iqbal [36]
Al	2.5–3.5	3.42	3.4	3.23	3.08
Zn	0.6–1.4	1.15	1.2	0.84	0.76
Mn	>0.20	0.38	0.20	0.21	0.15
Ca	<0.04	–	0.04	–	0.001
Si	<0.10	0.025	0.1	0.014	0.01
Ni	<0.005	–	0.005	0.00042	0.002
Cu	<0.05	0.025	0.05	0.0014	0.001
Fe	<0.005	0.0041	0.005	0.0032	0.005
Other	<0.30	–	–	–	0.0001
Mg	Balance	94.99	95.00	95.7	95.99

To perform dry milling experiments, each input parameter in Table 2 is assigned with three levels. Taguchi’ L₉ orthogonal array (OA) is considered for conducting fewer experiments and set the input parameters for each test run. For $n_l=3$ levels to each input parameter, the number of input parameters ($n_p$) according to the Taguchi’s L₉ OA can be found from (Ross, 1989) (1) $$N{}_{\mathit{\text{Tag}}}=1+n_p\left(n_l-1\right)\Rightarrow n_p=\frac{\left(N_{\mathit{\text{Tag}}}-1\right)}{\left(n_l-1\right)}=\frac{\left(9-1\right)}{\left(3-1\right)}=\frac{8}{2}=4$$(1)

Table 2. Different levels of dry milling processing variables and measured performance characteristics of AZ31 Mg alloy.

(a) Dry milling parameters with their levels
Dry milling parameters	Designation	Level: 1	Level: 2	Level: 3
Speed of cut (mm/min)	A	40	60	80
Rate of feed(mm/tooth)	B	0.05	0.1	0.15
Depth of cut(mm)	C	0.3	0.5	0.7
Fictitious	D	$d_1$	$d_2$	$d_3$
(b) Output characteristics
Experimental Run	Dry milling parameters at different levels				Performance characteristics[36]
	A	B	C	D	Surface roughness, ‘Ra’ ($\mu m$)	Tool nose radius deviation, ‘dR’(mm)
1	1	1	1	1	0.8303	0.92
2	1	2	2	2	1.6697	0.95
3	1	3	3	3	2.1580	1.01
4	2	1	2	3	0.8037	0.90
5	2	2	3	1	1.8521	0.97
6	2	3	1	2	2.2594	1.08
7	3	1	3	2	0.6630	0.89
8	3	2	1	3	0.9178	0.98
9	3	3	2	1	2.1036	1.12

For $N_{\mathit{\text{Tag}}}=$9 tests, 4 input parameters can be accommodated. In this case, only three input parameters (i.e. A, B and C) were set and hence, a dummy (fictitious) parameter is introduced i.e. D in Table 2 to assess the error in estimates for output responses. This aspect was thoroughly discussed by Miladinovic et al. (2021). Measured data of dry milling process for the nine sets of input parameters are tabulated in Table 2.

An analysis of variance (ANOVA) was carried using the measured data of performance characteristics presented in Table 2 and Table 3 summarizes the results of the ANOVA. Results depicted in Table 3 clears that, rate of feed (B) has the maximum contribution to ‘Ra’ and ‘dR’ of 85.1% and 85.6%, respectively. Speed of cut (A) and depth of cut (C) have contributions of 8% and 2.5% to Ra, while 4.2% and 5% contribution to dR. The sum of the contributions of all the process variables (A, B, C and D) to ‘Ra’ and ‘dR’ is 100%. The fictitious parameter (D) has contribution of 4.3% to Ra, while 5.2% to dR. Majority in manufacturing problems depend on experimentation to observe the phenomenon prior to mathematical modeling. Some unknown parameters may influence the process resulting scatter in the performance characteristics in repeated tests. Though the Taguchi’s L₉ OA is applicable to four parameters with three levels, the present problem specifies only three parameters with three levels. Considering the unknown fourth parameter (fictitious) as D, ANOVA is performed and arrived the %Contribution. The parameter D can be treated as the unknown influencing parameter (or error in measurements if any) of the performance characteristics.

Table 3. ANOVA results indicating the significance of milling process parameters.

Milling process factors	First mean	Second mean	Third mean	SOS (sum of squares)	%Contribution
‘Ra’: The grand mean = 1.4731$\mu m$
A	1.5527	1.6384	1.2281	0.28099	8.0
B	0.7657	1.4798	2.1737	2.97390	85.1
C	1.3358	1.5257	1.5577	0.08629	2.5
D	1.5953	1.5307	1.2932	0.15190	4.3
‘dR’: The grand mean = 0.98 mm
A	0.96	0.9833	0.9967	0.00207	4.2
B	0.9033	0.9667	1.07	0.04247	85.6
C	0.9933	0.99	0.9567	0.00247	5.0
D	1.0033	0.9733	0.9633	0.00260	5.2

The utilization of the modified Taguchi method (Ramya et al., 2018; Koneru et al., 2023; Miladinovic et al., 2021; Satyanarayana et al., 2018) enables the determination of the range of estimates in mechanical properties (such as Surface roughness [Ra], Tool nose radius [dR]) within a given range. By considering the specified process parameters as input variables, namely Speed of cut (A), Rate of feed (B) and Depth of cut (C), this approach facilitates process design by providing insights into the expected scatter in repeated experiments. By considering $\phi$ as a measure of performance (Ra or dR) and employing the additive law (Ross, 1989), Equation (2)(2) $$\widehat{\phi }={\overline{\phi }}_g+\sum _{i=1}^{n_p}\left({\overline{\phi }}_{\mathit{\text{il}}}-{\overline{\phi }}_g\right)=\sum _{i=1}^{n_p}{\overline{\phi }}_{\mathit{\text{il}}}-\left(n_p-1\right){\overline{\phi }}_g$$(2) allows for the prediction of $\stackrel{⌢}{\phi }$ based on the levels of the input variables. (2) $$\widehat{\phi }={\overline{\phi }}_g+\sum _{i=1}^{n_p}\left({\overline{\phi }}_{\mathit{\text{il}}}-{\overline{\phi }}_g\right)=\sum _{i=1}^{n_p}{\overline{\phi }}_{\mathit{\text{il}}}-\left(n_p-1\right){\overline{\phi }}_g$$(2)

${\overline{\phi }}_{\mathit{\text{il}}}$ can be obtained from ANOVA Table 3. Subscript ‘$i$’ with $i=1,2,3,4$ stands for the milling process parameters (A, B, C and D), and ‘$l$’ with $l=1,2,3$ stands for the 1-mean, 2-mean, 3-mean of the performance indicator (Ra or dR) to the levels of (A, B, C and D). ${\overline{\phi }}_g$ is the grand mean of $\phi$ for all 9 test runs. Considering Ra as the performance indicator, ${\overline{\phi }}_g$ = 1.4731$\mu m$ and ${\overline{\phi }}_{22}$=1.4798 in ANOVA Table 3, which corresponds to the 2-mean of Ra for the level-2 of the milling process parameter, B. Here, the estimates of Ra and dR from Table 4 are utilized for comparing with the experimental results obtained from the 9 test runs conducted using Taguchi’s L₉ OA (orthogonal array). $n_p$= 3 and $n_p$= 4 in Equation (2)(2) $$\widehat{\phi }={\overline{\phi }}_g+\sum _{i=1}^{n_p}\left({\overline{\phi }}_{\mathit{\text{il}}}-{\overline{\phi }}_g\right)=\sum _{i=1}^{n_p}{\overline{\phi }}_{\mathit{\text{il}}}-\left(n_p-1\right){\overline{\phi }}_g$$(2) provide the estimates of the performance characteristics by excluding and including the contribution of the parameter, D. It is clearly visible about the discrepancy in the estimates in Table 4 for $n_p$=3, whereas in case of $n_p$=4, the estimates are identical to the test data. The corrections to the estimates for $n_p$=3 are determined by calculating the minimum and maximum deviation of mean values related to the dummy parameter (D) from the grand mean. The estimated range of the performance characteristics can be obtained after application of these corrections to the estimates. Applying corrections −0.1799 µm and −0.01667 mm to the estimates of Ra and dR, one can obtain lower bound values. The upper-bound values of Ra and dR are attained by employing corrections of 0.1222 µm and 0.0233 mm, respectively, to the corresponding estimates.

Table 4. Estimates of Ra and dR.

Test Run	Milling process parameters				Test [36]	Estimate Equation (2)(2) $$\widehat{\phi }={\overline{\phi }}_g+\sum _{i=1}^{n_p}\left({\overline{\phi }}_{\mathit{\text{il}}}-{\overline{\phi }}_g\right)=\sum _{i=1}^{n_p}{\overline{\phi }}_{\mathit{\text{il}}}-\left(n_p-1\right){\overline{\phi }}_g$$(2)			Estimated range
	A	B	C	D		<table_head>$n_p=3$ </table_head>	R.E (%)	<table_head>$n_p=4$ </table_head>	Minimum	Maximum
Ra ($\mu m$)
1	1	1	1	1	0.8303	0.7080	14.7	0.8303	0.5281	0.8303
2	1	2	2	2	1.6697	1.6121	3.40	1.6697	1.4322	1.7343
3	1	3	3	3	2.158	2.3379	−8.30	2.1580	2.1580	2.4602
4	2	1	2	3	0.8037	0.9836	−22.4	0.8037	0.8037	1.1059
5	2	2	3	1	1.8521	1.7298	6.60	1.8521	1.5499	1.8521
6	2	3	1	2	2.2594	2.2018	2.50	2.2594	2.0219	2.3240
7	3	1	3	2	0.663	0.6054	8.70	0.6630	0.4255	0.7276
8	3	2	1	3	0.9178	1.0977	−19.6	0.9178	0.9178	1.22
9	3	3	2	1	2.1036	1.9813	5.8	2.1036	1.8014	2.1036
dR (mm)
1	1	1	1	1	0.92	0.8967	2.5	0.92	0.88	0.92
2	1	2	2	2	0.95	0.9567	−0.7	0.95	0.94	0.98
3	1	3	3	3	1.01	1.0267	−1.7	1.01	1.01	1.05
4	2	1	2	3	0.9	0.9167	−1.9	0.90	0.90	0.94
5	2	2	3	1	0.97	0.9467	2.4	0.97	0.93	0.97
6	2	3	1	2	1.08	1.0867	−0.6	1.08	1.07	1.11
7	3	1	3	2	0.89	0.8967	−0.8	0.89	0.88	0.92
8	3	2	1	3	0.98	0.9967	−1.7	0.98	0.98	1.02
9	3	3	2	1	1.12	1.0967	2.1	1.12	1.08	1.12

To illustrate the simplicity of the present analysis, the estimates of Ra in Test Run-1 of Table 4 are presented below. The levels of (A, B, C and D) in Test Run-1 are (1, 1, 1, 1) and the mean values of Ra from ANOVA Table 3 are: ${\overline{\phi }}_{11}=$1.5527$\mu m;$ 0.7657$\mu m;$ ${\overline{\phi }}_{31}=$1.3358$\mu m;$ and ${\overline{\phi }}_{41}=$1.5953$\mu m.$ The grand mean, ${\overline{\phi }}_g$ = 1.4731$\mu m.$

Using Equation (2)(2) $$\widehat{\phi }={\overline{\phi }}_g+\sum _{i=1}^{n_p}\left({\overline{\phi }}_{\mathit{\text{il}}}-{\overline{\phi }}_g\right)=\sum _{i=1}^{n_p}{\overline{\phi }}_{\mathit{\text{il}}}-\left(n_p-1\right){\overline{\phi }}_g$$(2) , estimate of Ra with exclusion of D ($n_p=3$) in Test Run-1: $${\overline{\phi }}_{21}=\widehat{\phi }={\overline{\phi }}_{11}+{\overline{\phi }}_{21}+{\overline{\phi }}_{31}-2\times {\overline{\phi }}_g=1.5527+0.7657+1.3385-2\times 1.4731=0.7080\mu m$$

Using Equation (2)(2) $$\widehat{\phi }={\overline{\phi }}_g+\sum _{i=1}^{n_p}\left({\overline{\phi }}_{\mathit{\text{il}}}-{\overline{\phi }}_g\right)=\sum _{i=1}^{n_p}{\overline{\phi }}_{\mathit{\text{il}}}-\left(n_p-1\right){\overline{\phi }}_g$$(2) , estimate of Ra with inclusion of D ($n_p$=4) in Test Run-1: $$\widehat{\phi }={\overline{\phi }}_{11}+{\overline{\phi }}_{21}+{\overline{\phi }}_{31}+{\overline{\phi }}_{41}-3\times {\overline{\phi }}_g=1.5527+0.7657+1.3385+1.5953-3\times 1.4731=0.8303\mu m$$

The corrections to the estimates of Ra for $n_p$=3 are the minimum and maximum deviation of mean values related to the dummy parameter (D) from the grand mean: $$\mathrm{\text{min}}\left\{{\overline{\phi }}_{41},{\overline{\phi }}_{42},{\overline{\phi }}_{43}\right\}-{\overline{\phi }}_g=\mathrm{\text{min}}\left\{1.5953,\text{1}.5307,\text{1}.2932\right\}-1.4731=1.2932-1.4731=-0.1799\mu m$$ $$\mathrm{\text{max}}\left\{{\overline{\phi }}_{41},{\overline{\phi }}_{42},{\overline{\phi }}_{43}\right\}-{\overline{\phi }}_g=\mathrm{\text{max}}\left\{1.5953,\text{1}.5307,\text{1}.2932\right\}-1.4731=1.5953-1.4731=0.1222\mu m$$

Applying these corrections to the estimate of Ra with exclusion of D, the minimum and maximum estimates of Ra obtained for Test Run-1 are: 0.5281 (=0.7080 − 0.1799) and 0.8302 (=0.7080 + 0.1222).

For the specified levels of input parameters (A, B and C) in Taguchi L₉ OA, the performance characteristics (Ra and dR) are evaluated using additive law (2). Appropriate corrections are applied to obtain the lower and upper limits of Ra and dR. Test data (Santhakumar & Iqbal, 2022) in Table 4 fall within the estimated range. It is possible to estimate the performance characteristics for all combinations of the three levels and three milling process parameters using the additive law (2).

For the combination of three levels of three input parameters, 27 (=3³) sets were created. These are $(((A_i,B_j,C_k),k=1\mathrm{}\mathit{\text{ to}}3),j=1\mathit{\text{ to}}3),i=1\mathit{\text{ to}}3)$ in which subscripts to parameters (A, B, C) are the levels. They represent the full factorial design of experiments and allotted the sequence numbers from 1 to 27. The set of input parameters to the sequence numbers 1, 5, 9, 11, 15, 16, 21, 22 and 26 represent the test Runs 1–9 in Taguchi’s L₉ OA. Using Equation (2)(2) $$\widehat{\phi }={\overline{\phi }}_g+\sum _{i=1}^{n_p}\left({\overline{\phi }}_{\mathit{\text{il}}}-{\overline{\phi }}_g\right)=\sum _{i=1}^{n_p}{\overline{\phi }}_{\mathit{\text{il}}}-\left(n_p-1\right){\overline{\phi }}_g$$(2) , Ra and dR values were calculated for the framed 27 sets of input parameters. The performance characteristics (Ra and dR) for the above 27 sets of parameters including the test data of Taguchi’s L₉ OA in Table 4 are shown in Figures 2 and 3. Test data fall between the lower and upper limits.

Figure 2. Comparison between test results (Santhakumar & Iqbal, 2022) and Ra estimates.

Figure 3. Comparison between test results (Santhakumar & Iqbal, 2022) and dR estimates.

Transforming the three levels of the process parameters to −1, 0 and 1, it is possible to represent mean values (see ANOVA Table 3) of the performance indicator (Ra or dR) in a quadratic polynomial form and use them in the additive law (2) to obtain empirical relationship for Ra and dR in terms of input variables (A, B, C). The transformed process parameters are ${\xi }_1=\frac{1}{20}(A-60);$ ${\xi }_2=20B-2;$ and ${\xi }_3=5C-2.5.$ Using the mean values of Ra and dR in ANOVA Table 3, the following empirical relationships are developed. (3) $$\mathit{\text{Ra}}=1.6978-0.16227{\xi }_1-0.248{\xi }_1^2+0.704{\xi }_2-0.0102{\xi }_2^2+0.110933{\xi }_3-0.0789{\xi }_3^2$$(3) (4) $$\mathit{\text{dR}}=0.98+0.018333{\xi }_1-0.005{\xi }_1^2+0.083333{\xi }_2-0.02{\xi }_2^2-0.01833{\xi }_3-0.015{\xi }_3^2$$(4)

To illustrate the effectiveness of Equations (3)(3) $$\mathit{\text{Ra}}=1.6978-0.16227{\xi }_1-0.248{\xi }_1^2+0.704{\xi }_2-0.0102{\xi }_2^2+0.110933{\xi }_3-0.0789{\xi }_3^2$$(3) and (4)(4) $$\mathit{\text{dR}}=0.98+0.018333{\xi }_1-0.005{\xi }_1^2+0.083333{\xi }_2-0.02{\xi }_2^2-0.01833{\xi }_3-0.015{\xi }_3^2$$(4) , the three levels of Test Run-1 having a set of parameters (A = 40, B = 0.05, C = 0.3) are considered. The transformed parameters for this set of parameters are (${\xi }_1=-1,{\xi }_2=-1,{\xi }_3=-1$). Using them in Equations (3)(3) $$\mathit{\text{Ra}}=1.6978-0.16227{\xi }_1-0.248{\xi }_1^2+0.704{\xi }_2-0.0102{\xi }_2^2+0.110933{\xi }_3-0.0789{\xi }_3^2$$(3) and (4)(4) $$\mathit{\text{dR}}=0.98+0.018333{\xi }_1-0.005{\xi }_1^2+0.083333{\xi }_2-0.02{\xi }_2^2-0.01833{\xi }_3-0.015{\xi }_3^2$$(4) result $$\begin{array}{c}\mathit{\text{Ra}}=1.6978-0.16227\times (-1)-0.248\times {(-1)}^2+0.704\times (-1)-0.0102\times {(-1)}^2\\ +0.110933\times (-1)-0.0789\times {(-1)}^2\\ =1.6978+0.16227-0.248-0.704-0.0102-0.110933-0.0789\approx 0.7080\mu m\end{array}$$

Applying corrections −0.1799 and 0.1222 µm, lower and upper bound values of Ra obtained are 0.5281 µm and 0.8303 µm, which are in good agreement with those in Table 4. $$\begin{array}{c}\mathit{\text{dR}}=0.98+0.018333\times (-1)-0.005\times {(-1)}^2+0.083333\times (-1)-0.02\times {(-1)}^2\\ -0.01833\times (-1)-0.015\times {(-1)}^2\\ =0.98-0.018333-0.005-0.083333-0.02+0.01833-0.015\approx 0.8967\mathit{\text{ mm}}\end{array}$$

Applying corrections −0.01667 and 0.0233 mm, lower and upper bound values of dR obtained are 0.88 and 0.92 mm, which are in good agreement with those in Table 4.

The above illustration confirms the validity of the developed empirical relationships (3) and (4). For Ra and dR. Applying corrections −0.1799 µm and −0.01667 mm to Ra and dR for Equations (3)(3) $$\mathit{\text{Ra}}=1.6978-0.16227{\xi }_1-0.248{\xi }_1^2+0.704{\xi }_2-0.0102{\xi }_2^2+0.110933{\xi }_3-0.0789{\xi }_3^2$$(3) and (4)(4) $$\mathit{\text{dR}}=0.98+0.018333{\xi }_1-0.005{\xi }_1^2+0.083333{\xi }_2-0.02{\xi }_2^2-0.01833{\xi }_3-0.015{\xi }_3^2$$(4) , lower bound values of Ra and dR can be obtained. Upper-bound values of Ra and dR are achieved by employing corrections 0.1222 µm and 0.0233 mm to Ra and dR for Equations (3)(3) $$\mathit{\text{Ra}}=1.6978-0.16227{\xi }_1-0.248{\xi }_1^2+0.704{\xi }_2-0.0102{\xi }_2^2+0.110933{\xi }_3-0.0789{\xi }_3^2$$(3) and (4)(4) $$\mathit{\text{dR}}=0.98+0.018333{\xi }_1-0.005{\xi }_1^2+0.083333{\xi }_2-0.02{\xi }_2^2-0.01833{\xi }_3-0.015{\xi }_3^2$$(4) . Results from Equations (3)(3) $$\mathit{\text{Ra}}=1.6978-0.16227{\xi }_1-0.248{\xi }_1^2+0.704{\xi }_2-0.0102{\xi }_2^2+0.110933{\xi }_3-0.0789{\xi }_3^2$$(3) and (4)(4) $$\mathit{\text{dR}}=0.98+0.018333{\xi }_1-0.005{\xi }_1^2+0.083333{\xi }_2-0.02{\xi }_2^2-0.01833{\xi }_3-0.015{\xi }_3^2$$(4) are closer to those from additive law (Equation (2)(2) $$\widehat{\phi }={\overline{\phi }}_g+\sum _{i=1}^{n_p}\left({\overline{\phi }}_{\mathit{\text{il}}}-{\overline{\phi }}_g\right)=\sum _{i=1}^{n_p}{\overline{\phi }}_{\mathit{\text{il}}}-\left(n_p-1\right){\overline{\phi }}_g$$(2) ).

3. Results and discussion

By examining ANOVA Table 3, it is possible to determine a set of milling process parameters $A_3{B}_1{C}_1$ (with subscripts indicating the respective levels of parameters) that result in the minimum Ra value and for achieving the minimum dR, the parameters $A_1{B}_1{C}_3$ are necessary.

The sets of parameters for achieving the minimum Ra and minimum dR values are distinct from each other. In order to obtain one common set of milling parameters to minimize both Ra and dR, the current multi objective functions (Ra and dR) are converted into one single objective function, as presented in previous works (Koneru et al., 2022; Ramya et al., 2018; Koneru et al., 2023). In ANOVA results (Table 3), the values of Ra and dR were normalized with the respective maximum values, ${(\mathit{\text{Ra}})}_{\mathrm{\text{max}}}$= 2.5459 µm; and ${(\mathit{\text{dR}})}_{\mathrm{\text{max}}}$= 1.1233 mm. Defining ${\zeta }_1\left(\equiv \frac{\mathit{\text{Ra}}}{{(\mathit{\text{Ra}})}_{\mathrm{\text{max}}}}\right)$ and ${\zeta }_2\left(\equiv \frac{\mathit{\text{dR}}}{{(\mathit{\text{dR}})}_{\mathrm{\text{max}}}}\right),$ By minimizing the functions ${\zeta }_1$ and ${\zeta }_2,$ one can achieve the minimum values for Ra and dR. By specifying positive weighing factors as ${\omega }_1$ and ${\omega }_2$ (that satisfy: ${\omega }_1+{\omega }_2=1$) as explained in the work (Koneru et al., 2022; Ramya et al., 2018; Koneru et al., 2023), one single objective function is formulated to minimize both Ra and dR can be written as, (5) $$\zeta ={\omega }_1{\zeta }_1+{\omega }_2{\zeta }_2$$(5)

By minimizing the single objective function ($\zeta$), the optimal milling parameters can be determined, provides minimum Ra and dR values. The minimum $\zeta$ Ra and dR with $A_3{B}_1{C}_1$ and $A_1{B}_1{C}_3$ is produced by minimizing for ${\omega }_1$=1 ($\Rightarrow {\omega }_2$ =0). In ANOVA Table 3, these two cases are displayed with bolded numerals. To achieve common optimal process conditions, two weighing factors (${\omega }_1$ and ${\omega }_2$) are taken into account.

In ANOVA Table 3, the mean values of Ra and dR are normalized by ${(\mathit{\text{Ra}})}_{\mathrm{\text{max}}}$= 2.5459 m and ${(\mathit{\text{dR}})}_{\mathrm{\text{max}}}$ = 1.1233 mm, respectively. The mean values of ${\zeta }_1$ and ${\zeta }_2$ are calculated. Equation (5)(5) $$\zeta ={\omega }_1{\zeta }_1+{\omega }_2{\zeta }_2$$(5) results in a single objective function $(\zeta ),$ when mean values of ${\zeta }_1$ and ${\zeta }_2$ are multiplied with equalized weighing factors (and ${\omega }_2$). A set of optimal parameters was determined as $A_3{B}_1{C}_1,$ from the minimum mean values of $\zeta$ to obtain the minimum Ra and dR (see Table 5). Table 6 lists milling parameters for particular conditions with estimates of Ra and dR.

Table 5. Mean values of the single objective function $(\zeta )$ in Equation (6) with different sets of weighing factors (${\omega }_1,$ ${\omega }_2$).

Milling parameters	First mean	Second mean	Third mean	Optimal solution
${(\mathit{\text{Ra}})}_{\mathrm{\text{min}}}:$ ${\omega }_1=1\mathit{\text{and}}{\omega }_2=0$
A	0.6099	0.6435	0.4824	$A_3{B}_1{C}_1$
B	0.3007	0.5813	0.8538
C	0.5247	0.5993	0.6118
${(\mathit{\text{dR}})}_{\mathrm{\text{min}}}:$ ${\omega }_1=0\mathit{\text{and}}{\omega }_2=1$
A	0.8546	0.8754	0.8873	$A_1{B}_1{C}_3$
B	0.8042	0.8606	0.9526
C	0.8843	0.8813	0.8517
${(\mathit{\text{Ra}})}_{\mathrm{\text{min}}}$ and ${(\mathit{\text{dR}})}_{\mathrm{\text{min}}}:$ ${\omega }_1=\frac{1}{2}\mathit{\text{and}}{\omega }_2=\frac{1}{2}$
A	0.7322	0.7595	0.6848	$A_3{B}_1{C}_1$
B	0.5524	0.7209	0.9032
C	0.7045	0.7403	0.7317

Table 6. Milling parameters and the predictions of Ra and dR under particular conditions.

Milling parameters				Surface roughness, Ra (µm)	Tool nose radius deviation, dR (mm)
Optimal set	Speed of cut, A (mm/min)	Feed rate, B (mm/tooth)	Depth of cut, C (mm)
Single objective optimization - ${(\mathit{\text{Ra}})}_{\mathrm{\text{min}}}$
$A_3{B}_1{C}_1$	80	0.05	0.3	0.2036–0.5058	0.9167–0.9567
Single objective optimization -${(\mathit{\text{dR}})}_{\mathrm{\text{min}}}$
$A_1{B}_1{C}_3$	40	0.05	0.7	0.075–1.0522	0.8433–0.8833
Multi-objective optimization - ${(\mathit{\text{Ra}})}_{\mathrm{\text{min}}}$ and ${(\mathit{\text{dR}})}_{\mathrm{\text{min}}}$
$A_3{B}_1{C}_1$	80	0.05	0.3	0.2036–0.5058	0.9167–0.9567

Figure 4 depicts the change in Ra and dR with the speed of cut (A) for the specified rate of feed, $B_1=0.05\mathit{\text{ mm}}/\mathit{\text{tooth}}$ and depth of cut, $C_1=0.7\mathit{\text{ mm}}.$ Surface roughness (Ra) shows increasing-decreasing trend, whereas the tool nose radius deviation (dR) shows increasing trend with the speed of cut (A). Figure 5 represents the variation of Ra and dR with rate of feed (B) for the specified speed of cut, $A_3=80\mathit{\text{ mm}}/\mathrm{\text{min}}$ and depth of cut, $C_1=0.7\mathit{\text{ mm}}.$ Ra and dR increase with the rate of feed (B). Figure 6 indicate the variations of Ra and dR with depth of cut (C) for the specified speed of cut, $A_3=80\mathit{\text{ mm}}/\mathrm{\text{min}}$ and rate of feed, $B_1=0.05\mathit{\text{ mm}}/\mathit{\text{tooth}}.$ R ${\omega }_1$a shows increasing trend, whereas dR shows increasing-decreasing trend with the depth of cut (C). The curves in Figures 4–6 are generated from the developed empirical relations (3) and (4) and applying the corrections for the lower and upper bounds.

Figure 4. (a) Variation of Ra with the speed of cut (A) for a given rate of feed, $B_1=0.05\mathit{\text{ mm}}/\mathit{\text{tooth}}$ and depth of cut, $C_1=0.7\mathit{\text{ mm}}.$ (b) Variation of dR with the speed of cut (A) for a given rate of feed, $B_1=0.05\mathit{\text{ mm}}/\mathit{\text{tooth}}$ and depth of cut, $C_1=0.7\mathit{\text{ mm}}.$

Figure 5. (a) Variation of Ra with the rate of feed (B) for the a given speed of cut, $A_3=80\mathit{\text{ mm}}/\mathrm{\text{min}}$ and depth of cut, $C_1=0.7\mathit{\text{ mm}}.$ (b) Variation of dR with the rate of feed (B) for the specified speed of cut, $A_3=80\mathit{\text{ mm}}/\mathrm{\text{min}}$ and depth of cut, $C_1=0.7\mathit{\text{ mm}}.$

Figure 6. (a) Variation of Ra with the depth of cut (C) for the specified speed of cut, $A_3=80\mathit{\text{ mm}}/\mathrm{\text{min}}$ and rate of feed, $B_1=0.05\mathit{\text{ mm}}/\mathit{\text{tooth}}.$ (b) Variation of dR with the depth of cut (C) for the specified speed of cut, $A_3=80\mathit{\text{ mm}}/\mathrm{\text{min}}$ and rate of feed, $B_1=0.05\mathit{\text{ mm}}/\mathit{\text{tooth}}.$

Industries expect simple, reliable and easy-to-implement procedures to solve optimization problems. Several researchers (Satyanarayana et al., 2019) are using many algorithms like grey relational analysis (GRA) without assessing the capabilities in existing theories. Application of the S/N ratio transformation (Kurra & Regalla, 2015; Wankhede et al., 2023) may not provide any improvement over the optimal solutions for the performance characteristics of a single test run. Here, a modified Taguchi method is followed and found the range of the performance characteristics to the specified set of process variables and a simple multi-objective optimization technique. Empirical relations for Ra and dR are developed and validated with test data. Statistical analysis in this article is performed on Excel sheets without using Minitab/Design Expert.

4. Conclusions

In this article, the machinability properties of the AZ31 Mg alloy were examined during the face milling process. Three levels were allocated to each of input parameters: depth of cut, rate of feed and speed of cut and the Taguchi’s L₉ OA was considered. To achieve the desired minimal ‘Ra’ and ‘dR’ in the workpiece, the study used a reliable and efficient multi-objective optimization technique. This article concludes the following points:

• ANOVA results confirm that rate of feed (B) has maximum influence on ‘Ra’ with a contribution of 85.1% and on tool nose radius deviation (dR) with a contribution of 85.6%.

• The optimal milling process variables to achieve a minimum surface roughness (Ra) are a speed of cut of 80 mm/min, a rate of feed of 0.05 mm/tooth and a depth of cut of 0.3 mm.

• A speed of cut of 40 mm/min, a rate of feed of 0.05 mm/tooth and a depth of cut of 0.7 mm are the optimal milling process variables to achieve the minimum ‘dR’.

• While an optimal set of parameters to achieve a minimum ‘Ra’ and ‘dR’ is a speed of cut of 80 mm/min, a rate of feed of 0.05 mm/tooth and a depth of cut of 0.3 mm.

Author contributions

Conceptualization: M.T.A., N.R.B. and S.K.; Methodology: N.R.B. and M.T.A.; Data Collection: P.P.T.P. and S.K.; Analysis and Results interpretation: N.R.B. and M.T.A.; Validation of Results: T.B. and S.K.; Preparation of Draft Manuscript: M.T.A., N.R.B., P.P.T.P.; Manuscript Review and Editing: K.I.U. and R.D.; visualization: T.B., R.D., M.T.A. and N.R.B.; supervision: T.B. and K.I.U. The final version of the manuscript was reviewed and approved by all authors.

Disclosure statement

The authors disclosed no potential conflicts of interest.

Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article.

Additional information

Funding

No funding was received for this work.

Notes on contributors

Muni Tanuja Anantha

Muni Tanuja Anantha M.Tech (Ph.D), Assistant Professor, Anurag University, Hyderabad, Sheet Metal Processing.

Sireesha Koneru

Sireesha Koneru M.Tech (Ph.D), Assistant Professor, KLEF deemed to be university, Guntur, Design.

Parameshwaran Pillai Thiruvambalam Pillai

Parameshwaran Pillai Thiruvambalam pillai PhD (Manufacturing Engineering), University College of Engineering, BIT Campus, Tiruchirappalli, Fracture Mechanics, Composite Materials.

Nageswara Rao Boggarapu

Nageswara Rao Boggarapu Ph.D, Professor, KLEF deemed to be university, Theoretical and Applied Mechanics, Non-Destructive Testing.

Tanya Buddi

Tanya Buddi Ph.D, Associate Professor, GRIET, Hyderabad, Wood based Bio-Composites, Nano Composites, Sheet Metal Forming.

Kseniia Iurevna Usanova

Kseniia Iurevna Usanova M.Tech, Peter the Great St.Petersburg Polytechnic University: Sankt-Peterburg, Renewable Energy Resources.

Rajesh Deorari

Rajesh Deorari Ph.D, Uttaranchal University, Dehradun, Face milling, Powder Metallurgy.