1 Introduction

Inconel is a refractory superalloy that contains a large amount of nickel and chromium. It is used in many technological applications because of its mechanical performance at relatively high temperatures up to 650 °C. Because of the low thermal conductivity, machining of this material is characterized by rapid wear of the cutting tools, poor surface quality, high cutting forces, and high vibrations [1]. For these reasons, Inconel is classified as a difficult material to machine. This is why this refractory alloy has aroused a great deal of interest in the scientific community and particularly in the field of machining [2]. A lot of research has been done on the machining of this material in order to find the optimal conditions that guarantee easy machining, high surface quality, minimum energy consumption, and maximum productivity at low cost [3, 4]. Devillez et al. [5] performed surface integrity tests when machining Inconel 718 with and without lubrication using a carbide metal tool. The results found show that dry machining with a coated carbide tool leads to potentially acceptable surface quality with residual stresses and microhardness values of the same order as those obtained under wet conditions using the optimized (Vc) value. Cantero et al. [6] analyzed the mechanisms of tool wear (cemented carbide, ceramic, and CBN) when turning Inconel 718. Park et al. [7] evaluated cutting tool wear in different lubrication environments when machining Inconel 718. Rahman et al. [8] examined the effect of cutting conditions on the machinability of Inconel 718. Wear (Vb), workpiece roughness (Ra), and cutting force (Fz) were used as performance indicators for tool life. Mahesh et al. [9] presented a detailed literature review of heat generation during machining of Inconel 718 and its influence on various output machining parameters. The study was interested in the multiple possibilities to reduce the cutting temperature. Behera et al. [10] investigated the dry machining performance of Inconel 825 (difficult to cut) using uncoated carbide and MT-CVD TiCN-Al2O3-coated carbide inserts. The results reveal that the coated tool results in a 39.79% reduction in tangential cutting force at (Vc = 121 m/min). Deshpande et al. [11] developed surface roughness prediction models considering cutting parameters, cutting force, sound, and vibration during turning of Inconel 718 with cryogenic and non-treated. Tebassi et al. [12] performed a modeling study of (Ra) and (MRR) when turning Inconel 718 (35 HRC) with the ceramic tool (Al2O3 + SiC Whisker). Thirumalai et al. [13] applied the method (TOPSIS) to optimize the cutting parameters during turning of Inconel 718. Deshpande et al. [14] used the (ANN) method to predict roughness (Ra) using cutting parameters, cutting force, sound, and vibration when turning Inconel 718. The authors confirm that the developed (ANN) models predict (Ra) with more than 98% precision.

Recently, in (2022), several other researches have been performed on Inconel machining, among them are the following: (1) Tan et al. [15] performed a turning study on the surface integrity of Inconel 718 machined by two carbide and ceramic inserts. The effects of cutting parameters on surface roughness, residual stresses, and micro-hardness were considered. (2) Alsoruji et al. [16] proposed the Taguchi analysis coupled with the (GRA) method for obtaining an optimal cutting regime that satisfies high (MRR), lower taper angle, and minimum roughness, in laser beam drilling of Inconel 718. (3) Xu et al. [17] presented an experimental and numerical investigation during the machining of Inconel 718 with worn tools. Chip formation, cutting force variation, tool wear and thermal distribution, and microstructure were examined. (4) Sivalingam et al. [18] exploited two methods (MCDM), namely (ARAS) and (CODAS) in order to optimize the operating parameters when turning Inconel 718 alloy in dry cutting environments and (ASCF). (5) Zahoor et al. [19] compared three optimization methods (PSO, GA, and DF) to minimize (Ra) when milling Inconel 718. The results found reveal that the optimization approach (PSO) is more efficient than the methods (GA and DF).

Optimization in the machining process is considered to be a very important operation that allows the proper choice of cutting conditions in order to improve productivity, ensure the quality of the final product, and reduce the manufacturing cost. The optimization methods applied in machining are numerous [20, 21] and their effectiveness varies from one method to another. The (DF) approach is a multi-objective optimization method, which has been widely used by many researchers in many fields, particularly in machining optimization [22]. Frifita et al. [23], Tebassi et al. [24], Parida and Maity [25], Kuppan et al. [26], Kar et al. [27], Manohar et al. [28], and Świercz et al. [29], have used it successfully in machining Inconel. On the other hand, the (GRA) method is among the most widely used optimization ones in several fields of science and particularly in the field of mechanical manufacturing. This method has proved its effectiveness in determining the optimal operating conditions in machining in order to overcome the problem of multi-response optimization [30]. Karsh and Singh [31] and Sanghvi et al. [32] applied the (GRA) method while machining 625 and 825 Inconel to solve the multi-objective optimization problems of cutting conditions in (EDM). Vikram et al. [33] evaluated the process parameters with (GRA) analysis when machining low machinability materials under dry and wet conditions. Sahu et al. [34] used the (GRA) method for process parameter optimization in electrostatic discharge machining of titanium alloy (Ti6Al4V) and 316L stainless steel. Kant and Dhami [35] performed a multi-response optimization of the parameters using the (GRA) method during abrasive waterjet machining of EN31 steel. Moharana and Patro [36] realized a multi-objective optimization of the machining parameters of EN-8 carbon steel in (EDM) using the (GRA) method. Also, Pradhan [37] and Hanif et al. [38] applied the (RSM and GRA) methods during (EDM) machining of AISI D2 steel to optimize the operating factors. Also, Chaudhari et al. [39] exploited the two approaches (RSM and GRA) coupled with the (PCA) method to find the optimal parameters during (EDM) machining of pure titanium. In order to find the optimal parameters when milling a glass fiber-reinforced polymer, Yaser and Shunmugesh [40] successfully employed the approaches (GRA and DF).

A literature review shows that very few studies have been conducted on the machining of Inconel 718 by a PVD-coated carbide insert, taking into account the tool vibration parameter. Also, the analysis of the literature shows that very little research has been carried out on the machining of Inconel 718 by introducing four input factors and five output technological parameters simultaneously, which constitutes the originality of this work. Moreover, in this machining context, almost no study has considered a comparison between two optimization methods, namely the desirability function (DF) based on regression models and the method (GRA) derived from methods multiple criteria decision (MCDM) according to several desired cases.

The main objective of this article is to develop mathematical prediction models for the output parameters (Vtng, Ra, Fz, Pc, and MRR) using the approach (RSM) and analysis (ANOVA) in order to quantify the influence of the input factors (Vc, f, ap, and r) on the outputs when turning Inconel 718 according to a Taguchi orthogonal array L27 (3^4). Moreover, the (DF) and (GRA) approaches were used and compared in order to perform a multi-objective optimization according to two case studies that can be encountered in the industry. The proposed work concerns all mechanical manufacturing companies, as it provides the information necessary for the analysis of product quality.

2 Experimental procedure

2.1 Couple tool/material to be machined

The material used in this study is the refractory alloy (Inconel 718). It is widely used in the marine, nuclear, space technology, rocket engine, nuclear reactors, petrochemical, submarine, and power plant sectors. The specimen used is a cylindrical bar with a diameter of 70 mm and a length of 350 mm. The chemical composition of Inconel 718 is given in Table 1. The turning tests were performed on a conventional lathe (TOS TRENCIN) model SN40C (Pm = 6.6 kW) following the standard (ISO 3685). The lathe is equipped with a variable speed drive, model ABB series ACS355, with a speed sensor allowing the control of the speed of rotation of the spindle. The cutting inserts used are metal carbide with a (PVD) coating of grade (GC1105) from the firm SANDVIK. The (PVD) coating consists of a thin layer of TiAlN offering high tenacity, excellent adhesion, and even draft wear [41]. The inserts are mounted on a SANDVIK tool holder with the ISO designation PSBNR2525K12 (Fig. 1).

Table 1 Chemical composition of Inconel 718 (%)
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Fig. 1
figure 1

Machine, tool, and material used

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2.2 Measuring equipment

Figure 2 exposes the equipment used for the measurement of the different output parameters. The measurement of roughness (Ra) was performed by a roughness meter (2D) of the MITUTOYO model SJ-210. It consists of a 5-μm diamond tip (feeler) that moves axially over a distance of 4 mm on the machined surface. Measurements were repeated three times by flipping the workpiece through a 120° angle and the average of the measurements was taken. The acquisition of the roughness profiles was performed with a roughness meter software (SJ- Communication-Tool) (Fig. 2a). In order to measure the vibrations of the tool in the tangential direction (Vtng), a vibrometer type VM-6360 accompanied by an accelerometer was used. The acquisition of the measurements was performed using the program (data collection system) (Fig. 2b). To measure the tangential cutting force (Fz) generated during machining, a dynamometer (KISTLER 9257 B) was used (Fig. 2c). The power consumed during machining (Pc) was calculated based on the tangential force (Fz) measured according to Eq. (1). The material removal rate (MRR) was chosen as an index of productivity; it was calculated according to Eq. (2).

Fig. 2
figure 2

Equipment used for experimental measurements

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$${{\varvec{P}}}_{{\varvec{c}}}=\frac{{\varvec{F}}{\varvec{z}}.{{\varvec{V}}}_{{\varvec{C}}}}{60}, \, \left[{\varvec{W}}\right]$$
(1)
$${\varvec{M}}{\varvec{R}}{\varvec{R}}={\varvec{V}}{\varvec{c}}.{\varvec{f}}.{\varvec{a}}{\varvec{p}}\boldsymbol{ }\left[{{\varvec{c}}{\varvec{m}}}^{3}/{\varvec{m}}{\varvec{i}}{\varvec{n}}\right]$$
(2)

2.3 Experimental design

The standard Taguchi L27 (3^4) orthogonal table was adopted as the experimental design. Four input factors were chosen: the first one was assigned to the tool nose radius (r), the second to the cutting speed (Vc), the third to the feed rate per revolution (f), and the last to the depth of cut (ap). Each factor varied in three levels, and the range of variation is shown in Table 2. The choice of cutting conditions is made according to the recommendations of the cutting tool manufacturer (SANDVIK). The design of the experiment’s matrix, adopted according to Taguchi L27 design, gives the different combinations of input factors as it is shown in Table 3.

Table 2 Levels of entry factors
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3 Results and discussion

Table 3 presents the experimental values of the output parameters as a function of the four main cutting factors (r, Vc, f, and ap). These values were obtained as a result of the different combinations of the elements of the cutting regime according to Taguchi orthogonal array L27 during turnings of Inconel 718. After the analysis of the obtained values, it can be seen that the roughness (Ra) varies from (0.296 to 0.929 µm), the tool vibration (Vtng) varies from (1.334 to 3.728 m/s2), the tangential force (Fz) varies from (49.640 to 217.050 N), while the power consumption (Pc) varies from (27.900 to 181.755 W), and for the material removal rate (MRR), it varies from (0.240 to 2.240 cm3/min).

Table 3 Experimental results of performance parameters
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3.1 Analysis of variance (ANOVA)

In order to make the statistical study and determine the contribution of the main factors and the interactions, ANOVA was exploited. The results from the ANOVA for the four output parameters (Ra, Vtng, Fz, and Pc) were obtained for a significance level (α = 0.05), which means a confidence level of 95%. In the ANOVA tables, the values of the degrees of freedom (df) are reported; the sequential sum of squares (SS Seq) was used to quantify the variation in response data explained by each term. Column 3 shows the values of the contributions of the input factors on the output parameters (cont%), followed by the sum of adjusted squares (Adj SS), and then the mean squares (Adj MS). Fisher’s coefficient (F value) and probability value (P value) indicate the level of significance of the factors for each of the output parameters. If P > 0.05, the factor is statistically insignificant on the measured response. On the other hand, if P < 0.05, the factor is statistically significant.

3.1.1 ANOVA for (Ra) and (Vtng)

The results of the ANOVA of (Ra) are indicated in Table 4. The main factors are significant because P < 0.05. The feed rate (f) has a great influence on the criterion (Ra) with (23.39%) of cont%, followed by (Vc) with (16.09%) of cont%, then comes the factors (r and ap) with (3.7% and 0.02%) of cont% respectively. The square term (Vc × Vc) is significant with (3.61%) of cont%. The interactions (r × Vc), (r × f), (Vc × f), and (Vc × ap) are also significant with (3.81%, 19.42%, 4.13%, and 4.97%) of cont% respectively. These results are in perfect agreement with those [24, 42,43,44]. Table 5 shows the ANOVA of the tangential vibration (Vtng). It can be seen that all three factors (r, Vc, and ap) are significant (P < 0.05). Radius (r) and speed (Vc) have the greatest influence with (32.41%) and (31.36%) cont% respectively, then followed by (ap) with (8.77%) cont%. Concerning the square terms (r × r) and (Vc × Vc), they are also significant with (6.23% and 9.41%) of cont% respectively. The rest of the terms do not have a significant influence on (Vtng). Almost similar results were found by Meddour et al. [45], when following the vibration in turning by taking into consideration the cutting conditions and tool nose radius.

Table 4 ANOVA for (Ra)
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Table 5 ANOVA for (Vtng)
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3.1.2 ANOVA for (Fz) and (Pc)

Table 6 presents the results of the ANOVA for effort (Fz). It can be seen that the main factors are significant, as (P < 0.05). The principal contribution (64.11%) is attributed to the factor (ap), it is followed by (f) with (21.18%) of cont%, then follows the factors (Vc and r) with (3.86% and 3.46%) of cont% respectively. The square terms (Vc × Vc), (f × f), and (ap × ap) are significant (cont% < 1.85%). Many researchers claim that the chip cross section (ap × f) is the main factor affecting the stress (Fz) [44, 46, 47]. The interactions (Vc × f) and (Vc × ap) are also significant with cont% of (2.13% and 1.08%) respectively. Concerning the results of (Pc) (Table 7), it is clear that the factors (ap) and (Vc) have the greatest influence with (38.29%) and (36.99%) cont% respectively. These results are very logical, since Eq. (1) indicates that speed (Vc) and effort (Fz) are the main factors controlling (Pc) [48, 49]. The other main factors (f and r) are significant with (8.59% and 2.41%) of cont%. The square term (Vc × Vc) and the interaction (Vc × f) are also significant with cont% of (0.62% and 9.22%).

Table 6 ANOVA for (Fz)
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Table 7 ANOVA for (Pc)
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3.1.3 Main effects plot and contributions

Figure 3 shows the plot of the main effects of the input factors (r, Vc, f, and ap) on the parameters (Ra, Vtng, Fz, and Pc). It is clearly seen in Fig. 3a, that the factors (f, Vc, and r) strongly affect (Ra) respectively, as they have high slopes. In contrast, the factor (ap) does not have a significant effect on (Ra). These results confirm the ANOVA results in Table 4. Several researches confirm that (ap) has little influence on (Ra) during machining [23, 50]. For vibration (Vtng), the plots in Fig. 3b show that the factors (Vc and r) have a significant effect on (Vtng) because they have high slopes. While (ap) comes third, while the feed rate (f) does not have a significant effect on (Vtng). The slope analysis of the main effects of (Fz) Fig. 3c, shows that (ap) strongly affects (Fz), followed by the feed rate (f), as they have the largest slope. Regarding the parameter (Pc) shown in Fig. 3d, we notice that (ap, Vc, and f) respectively strongly affect (Pc), while (r) has a smaller effect on (Pc). Figure 4 summarizes the different contributions of the main factors on the output parameters.

Fig. 3
figure 3

Main effects plot for (Ra, Vtng, Fz, and Pc)

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Fig. 4
figure 4

Cont% of the main factors according to the output parameters

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3.2 Modeling of technological performance parameters

The relationship between input factors (r, Vc, f, and ap) and output parameters (Ra, Vtng, Fz, and Pc) was modeled by quadratic regression equations. Insignificant terms were not excluded from the equations in order to keep the (R2) high. The different mathematical models found and their (R2) are given by Eqs. (3), (4), (5) and (6). These models will be used for the prediction of the output parameters, in the range of variation of the cutting conditions and also for the optimization studies.

$$\begin{aligned}{\varvec{R}}{\varvec{a}}\boldsymbol{ }\left[{\varvec{\mu}}\mathbf{m}\right]&=-0.320+0.509\boldsymbol{ }{\varvec{r}}+0.0099\boldsymbol{ }{\varvec{V}}{\varvec{c}}+8.90\boldsymbol{ }{\varvec{f}}-2.92\boldsymbol{ }{\varvec{a}}{\varvec{p}}\\&-0.149\boldsymbol{ }{{\varvec{r}}}^{2}-0.00012\boldsymbol{ }{{\varvec{V}}{\varvec{c}}}^{2}+19.9\boldsymbol{ }{{\varvec{f}}}^{2}-0.08\boldsymbol{ }{{\varvec{a}}{\varvec{p}}}^{2}\\&+0.0043\boldsymbol{ }{\varvec{r}}\times {\varvec{V}}{\varvec{c}}-4.901\boldsymbol{ }{\varvec{r}}\times {\varvec{f}}+0.748\boldsymbol{ }{\varvec{r}}\times {\varvec{a}}{\varvec{p}}\\&-0.1044\boldsymbol{ }{\varvec{V}}{\varvec{c}}\times {\varvec{f}}+0.0324\boldsymbol{ }{\varvec{V}}{\varvec{c}}\times {\varvec{a}}{\varvec{p}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad{{\varvec{R}}}^{2}=92.02\boldsymbol{\%}\end{aligned}$$
(3)
$$\begin{aligned}{\varvec{V}}{\varvec{t}}{\varvec{n}}{\varvec{g}}\boldsymbol{ }\left[\mathbf{m}/{\mathbf{s}}^{2}\right]&=-0.96+7.75\boldsymbol{ }{\varvec{r}}-0.1228\boldsymbol{ }{\varvec{V}}{\varvec{c}}+14.2\boldsymbol{ }{\varvec{f}}+5.08\boldsymbol{ }{\varvec{a}}{\varvec{p}}\\&-2.122\boldsymbol{ }{{\varvec{r}}}^{2}+0.001043\boldsymbol{ }{{\varvec{V}}{\varvec{c}}}^{2}-51.9\boldsymbol{ }{{\varvec{f}}}^{2}-2.4\boldsymbol{ }{{\varvec{a}}{\varvec{p}}}^{2}\\&-0.01525\boldsymbol{ }{\varvec{r}}\times {\varvec{V}}{\varvec{c}}-\boldsymbol{ }3.99\boldsymbol{ }{\varvec{r}}\times {\varvec{f}}-1.50\boldsymbol{ }{\varvec{r}}\times {\varvec{a}}{\varvec{p}}+0.109\boldsymbol{ }{\varvec{V}}{\varvec{c}}\\&\times {\varvec{f}}+0.0085\boldsymbol{ }{\varvec{V}}{\varvec{c}}\times {\varvec{a}}{\varvec{p}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {{\varvec{R}}}^{2}=92.96\boldsymbol{\%}\end{aligned}$$
(4)
$$\begin{aligned}{\varvec{F}}{\varvec{z}}\boldsymbol{ }\left[{\varvec{N}}\right]&=140.6+123.8\boldsymbol{ }{\varvec{r}}-\boldsymbol{ }5.37\boldsymbol{ }{\varvec{V}}{\varvec{c}}-2049\boldsymbol{ }{\varvec{f}}+589\boldsymbol{ }{\varvec{a}}{\varvec{p}}-29.1\boldsymbol{ }{{\varvec{r}}}^{2}\\&+0.03263\boldsymbol{ }{{\varvec{V}}{\varvec{c}}}^{2}+5461\boldsymbol{ }{{\varvec{f}}}^{2}+861\boldsymbol{ }{{\varvec{a}}{\varvec{p}}}^{2}-0.214\boldsymbol{ }{\varvec{r}}\times {\varvec{V}}{\varvec{c}}\\&-5\boldsymbol{ }{\varvec{r}}\times {\varvec{f}}-84.0\boldsymbol{ }{\varvec{r}}\times {\varvec{a}}{\varvec{p}}+24.78\boldsymbol{ }{\varvec{V}}{\varvec{c}}\times {\varvec{f}}-5.77\boldsymbol{ }{\varvec{V}}{\varvec{c}}\times {\varvec{a}}{\varvec{p}}\quad \quad \quad \quad {{\varvec{R}}}^{2}=98.32\boldsymbol{\%}\end{aligned}$$
(5)
$$\begin{aligned}{\varvec{P}}{\varvec{c}}\boldsymbol{ }\left[{\varvec{W}}\right]&=128.2+68.3\boldsymbol{ }{\varvec{r}}-4.764\boldsymbol{ }{\varvec{V}}{\varvec{c}}-1921\boldsymbol{ }{\varvec{f}}+282\boldsymbol{ }{\varvec{a}}{\varvec{p}}-17.5\boldsymbol{ }{{\varvec{r}}}^{2}\\&+0.01809\boldsymbol{ }{{\varvec{V}}{\varvec{c}}}^{2}+3324\boldsymbol{ }{{\varvec{f}}}^{2}+336\boldsymbol{ }{{\varvec{a}}{\varvec{p}}}^{2}+0.203\boldsymbol{ }{\varvec{r}}\times {\varvec{V}}{\varvec{c}}\\&+17\boldsymbol{ }{\varvec{r}}\times {\varvec{f}}-88.9\boldsymbol{ }{\varvec{r}}\times {\varvec{a}}{\varvec{p}}+31.25\boldsymbol{ }{\varvec{V}}{\varvec{c}}\times {\varvec{f}}+2.87\boldsymbol{ }{\varvec{V}}{\varvec{c}}\times {\varvec{a}}{\varvec{p}} \quad \quad \quad \quad {{\varvec{R}}}^{2}=98.45\boldsymbol{\%}\end{aligned}$$
(6)

3.2.1 Experimental and predicted values and the probability diagram

Figure 5a–d compare the experimental and predicted values by the models found (Eqs. (3)–(6)). It can be seen, that the difference between the values is minimal and which implies a good correlation because all (R2) varies between [92.02 and 98.45%]. Figure 6a–d present the normal probability plot for the parameters (Ra, Vtng, Fz, and Pc). We note that the regression model is statistically significant, as the majority of the residual values are almost aligned on a straight line, which led to the conclusion that the associated errors were normally distributed.

Fig. 5
figure 5

Comparison between experimental and predicted values for (Ra, Vtng, Fz, and Pc)

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Fig. 6
figure 6

Normal probability plot for (Ra, Vtng, Fz, and Pc)

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3.2.2 3D response surface

  1. (a)

    3D response surface for (Ra) and (Vtng)

The 3D graphs of Fig. 7a–c allow to evaluate the simultaneous influence of (f × r) (f × Vc), and (f × ap) on (Ra). We notice on Fig. 7a, that for a minimal value of (f), the radius (r) has almost no effect on (Ra). An increase in (f) leads to an increase in (Ra). But it should be noted that for a large feed, the increase in (r) allows a clear improvement in (Ra). It can be seen from Fig. 7b that for a minimum value of (f), the factor (Vc) has a very slight effect on the evolution of (Ra). On the other hand, for a maximum value of (f), an increase of (Vc) leads to a reduction of the criterion (Ra). Figure 6c shows that the minimum value of (Ra) is obtained with minimum (f) and maximum (ap). The 3D plots in Fig. 8a–c permit to evaluate the simultaneous influence of (Vc × r), (Vc × f) and (Vc × ap) on the vibration (Vtng). Figure 8a indicates that the factor (Vc) and (r) possess almost the same effect on (Vtng). It can be observed from Fig. 8(b) that (Vc) has a more pronounced influence on (Vtng) compared to that of (f); minimum values of the parameter (Vtng) are achieved for high values of (Vc). This is probably due to the reduction of the force (Fz). The minimum value of (Vtng) is obtained with (Vc) maximum and (ap) minimum Fig. 8c. These different 3D curves, prove the inexistence of interaction effect between the different input parameters considered on the variation of (Vtng). This was already found by the ANOVA analysis.

  1. (b)

    3D response surface for (Fz) and (Pc)

Fig. 7
figure 7

3D response surface for (Ra)

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Fig. 8
figure 8

3D response surface for (Vtng)

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The 3D plots in Fig. 9a–c permit to evaluate the simultaneous influence of (ap × r), (ap × Vc), and (ap × f) on (Fz). As found by ANOVA, these curves show that (ap) has the dominant effect, followed by (f) and (Vc), while the effect of (r) remains almost negligible. Figure 9a demonstrates that the force (Fz) is minimal when (ap) and (r) are at their minimum values. Figure 9b indicates that when (ap) is at the lowest value and (Vc) at the highest one, the parameter (Fz) is at its minimum. According to Fig. 9c, the minimum force is obtained with low values of (ap) and (f), i.e., for a small section of the removed chip. The 3D graphs of Fig. 10a–c, allow to evaluate the simultaneous effect of (Vc × f), (Vc × ap), and (ap × r) on (Pc). Figure 10a indicates an interaction between the effect of (Vc) and that of (f); indeed, for a small value of (Vc), the effect of (f) is negligible. Whereas for a maximum value of (Vc), an increase in (f) leads to a considerable change in (Pc). From Fig. 10b, the lowest value of (Pc) is reached when (ap) and (Vc) are at their low level. Figure 10c illustrates that the effect of (r) is very small before that of (ap); when (r) and (ap) are at their minimum value, the parameter (Pc) has the lowest value.

Fig. 9
figure 9

3D response surface for (Fz)

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Fig. 10
figure 10

3D response surface for (Pc)

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4 Multi-objective optimization of cutting conditions

Optimization is an important step in the machining process, which selects the most appropriate cutting conditions to obtain the required values for a single or multiple variables, which usually has a direct economic impact such as machine time or total machining cost [51]. In this study, two multi-objective optimization methods, namely, the desirability function (DF) approach and gray relational analysis (GRA) were used. The goal is to find an optimal cutting regime that satisfies two cases, namely, (Ra-min, Vtng-min, and MRR-max) and (Ra-min, Vtng-min, Fz-min, Pc-min, and MRR-max).

5 Optimization by the maximization method of (DF)

The (DF) approach is a simple optimization technique that is easy to use and to program in statistical software [22, 52]. Our objective is to improve surface quality, reduce energy consumption, increase machining stability, and increase productivity simultaneously. The desirability function (DF) is defined by Eq. (7) [25, 28]:

$${\varvec{D}}{\varvec{F}}={\left(\prod\limits_{{\varvec{i}}=1}^{{\varvec{n}}}{{\varvec{d}}}_{{\varvec{i}}}^{{\varvec{w}}{\varvec{i}}}\right)}^{\frac{1}{\sum_{{\varvec{j}}={\varvec{i}}}^{{\varvec{n}}}{\varvec{W}}{\varvec{i}}}}$$
(7)

di: Individual desirability within [0, 1], defined by the target output, as indicated by Eqs. (8) and (9).

n: Number of responses in the measure;

wi: Corresponding weights.

Target to maximize:

$$\left\{\begin{array}{c}0\\ di\\ 1\end{array}=\left[\frac{{\varvec{Y}}{\varvec{i}}-{{\varvec{m}}{\varvec{i}}{\varvec{n}}}_{{\varvec{i}}}}{{{\varvec{m}}{\varvec{a}}{\varvec{x}}}_{{\varvec{i}}}-{{\varvec{m}}{\varvec{i}}{\varvec{n}}}_{{\varvec{i}}}}\right]\right.\boldsymbol{ }\begin{array}{c}{\varvec{S}}{\varvec{i}}\\ {\varvec{S}}{\varvec{i}}\\ {\varvec{S}}{\varvec{i}}\end{array}\boldsymbol{ }\boldsymbol{ }\boldsymbol{ }\begin{array}{c}{\boldsymbol{\Upsilon}}_{{\varvec{i}}}\le {{\varvec{m}}{\varvec{i}}{\varvec{n}}}_{{\varvec{i}}}\\ {{\varvec{m}}{\varvec{i}}{\varvec{n}}}_{{\varvec{i}}}\le {{\varvec{Y}}}_{{\varvec{i}}}\le {{\varvec{m}}{\varvec{a}}{\varvec{x}}}_{{\varvec{i}}}\\ {\boldsymbol{\Upsilon}}_{{\varvec{i}}}\ge {{\varvec{m}}{\varvec{a}}{\varvec{x}}}_{{\varvec{i}}}\end{array}$$
(8)

Target to minimize:

$$\left\{\begin{array}{c}1\\ di\\ 0\end{array}=\left[\frac{{{\varvec{m}}{\varvec{a}}{\varvec{x}}}_{{\varvec{i}}}-{{\varvec{Y}}}_{{\varvec{i}}}}{{{\varvec{m}}{\varvec{a}}{\varvec{x}}}_{{\varvec{i}}}-{{\varvec{m}}{\varvec{i}}{\varvec{n}}}_{{\varvec{i}}}}\right]\right.\boldsymbol{ }\begin{array}{c}{\varvec{S}}{\varvec{i}}\\ {\varvec{S}}{\varvec{i}}\\ {\varvec{S}}{\varvec{i}}\end{array}\boldsymbol{ }\boldsymbol{ }\boldsymbol{ }\begin{array}{c}{\boldsymbol{\Upsilon}}_{{\varvec{i}}}\le {{\varvec{m}}{\varvec{i}}{\varvec{n}}}_{{\varvec{i}}}\\ {{\varvec{m}}{\varvec{i}}{\varvec{n}}}_{{\varvec{i}}}\le {{\varvec{Y}}}_{{\varvec{i}}}\le {{\varvec{m}}{\varvec{a}}{\varvec{x}}}_{{\varvec{i}}}\\ {\boldsymbol{\Upsilon}}_{{\varvec{i}}}\ge {{\varvec{m}}{\varvec{a}}{\varvec{x}}}_{{\varvec{i}}}\end{array}$$
(9)

Table 8 summarizes the desired objectives and the ranges of the cutting parameters defined for the optimization, as well as the range of variation of the considered technological parameters. Table 9 and Fig. 11 illustrate the optimization results obtained from the desirability function (DF) for the two considered cases. The optimization study of the 1st case (Ra-min, Vtng-min, and MRR-max) led to the regime corresponding to (r = 0.8 mm, Vc = 70 m/min, f = 0.108 mm/rev, and ap = 0.3 mm), which resulted in (Ra = 0.297 µm, Vtng = 2.087 m/s2, and MRR = 2.101cm3/min). While for the 2nd case (Ra-min, Vtng-min, Fz-min, Pc-min, and MRR-max), where all five output parameters are considered simultaneously, the optimal regime obtained is (r = 0.8 mm, Vc = 69.192 m/min, f = 0.088 mm/rev, and ap = 0.197 mm) and which resulted in the different optimized outputs (Ra = 0.280 µm, Vtng = 1.559 m/s2, Fz = 63.769 N, Pc = 81.660 W, and MRR = 0.977 cm3/min). These results indicate that taking into account (Fz) and (Pc), results in a decrease of (53.4%) in (MRR), caused by a reduction in the chip cross section (ap × f). On the other hand, the value of (Ra) remains almost unchanged, while a slight decrease in (Vtng) is observed.

Table 8 Constraints for the two optimization cases
Full size table
Table 9 Results for the two optimization cases (DF)
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Fig. 11
figure 11

Optimization diagram for the two cases

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5.1 Optimization by method (GRA)

The (GRA) method is a technique for solving difficult optimization problems by transforming the multi-objective problem into a single objective [30, 34]. The goal of the (GRA) method is to determine the optimal combination of turning parameters, which allows us to minimize (Ra, Vtng, Fz, and Pc) and maximize the (MRR), following the two cases mentioned above. The steps of the (GRA) method are presented in Fig. 12.

Fig. 12
figure 12

Steps involved in (GRA) method

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  1. (a)

    Normalization

This step consists in normalizing the experimental results of the output parameters, according to the goal of the optimization, in order to make all the responses of the same magnitude in the interval [0–1]. For the minimization of (Ra, Vtng, Fz, and Pc), the normalization was performed according to Eq. (10), and for the maximization of (MRR) by Eq. (11). The results of the normalization are reported in Table 10.

$${{\varvec{x}}}_{{\varvec{i}}}\left({\varvec{k}}\right)=\frac{{\varvec{m}}{\varvec{a}}{\varvec{x}}\left({{\varvec{x}}}_{{\varvec{i}}}^{0}\left({\varvec{k}}\right)\right)-{{\varvec{x}}}_{{\varvec{i}}}^{0}\left({\varvec{k}}\right)}{{\varvec{m}}{\varvec{a}}{\varvec{x}}\left({{\varvec{x}}}_{{\varvec{i}}}^{0}\left({\varvec{k}}\right)\right)-{\varvec{m}}{\varvec{i}}{\varvec{n}}\left({{\varvec{x}}}_{{\varvec{i}}}^{0}\left({\varvec{k}}\right)\right)}$$
(10)
$${{\varvec{x}}}_{{\varvec{i}}}\left({\varvec{k}}\right)=\frac{{{\varvec{x}}}_{{\varvec{i}}}^{0}\left({\varvec{k}}\right)-{\varvec{m}}{\varvec{i}}{\varvec{n}}\left({{\varvec{x}}}_{{\varvec{i}}}^{0}\left({\varvec{k}}\right)\right)}{{\varvec{m}}{\varvec{a}}{\varvec{x}}\left({{\varvec{x}}}_{{\varvec{i}}}^{0}\left({\varvec{k}}\right)\right)-{\varvec{m}}{\varvec{i}}{\varvec{n}}\left({{\varvec{x}}}_{{\varvec{i}}}^{0}\left({\varvec{k}}\right)\right)}$$
(11)

where

Table 10 Normalized experimental results \({{\varvec{x}}}_{{\varvec{i}}}\left({\varvec{k}}\right)\)
Full size table

\({{\varvec{x}}}_{{\varvec{i}}}\left({\varvec{k}}\right)\): Normalized value

\({\varvec{m}}{\varvec{a}}{\varvec{x}}\left({{\varvec{x}}}_{{\varvec{i}}}^{0}\left({\varvec{k}}\right)\right)\): Maximum value of the (kth) response \({{\varvec{x}}}_{{\varvec{i}}}^{0}\left({\varvec{k}}\right)\)

\({\varvec{m}}{\varvec{i}}{\varvec{n}}\left({{\varvec{x}}}_{{\varvec{i}}}^{0}\left({\varvec{k}}\right)\right)\): Minimum value of the (kth) response \({{\varvec{x}}}_{{\varvec{i}}}^{0}\left({\varvec{k}}\right)\)

The value of \({{\varvec{x}}}_{{\varvec{i}}}\left({\varvec{k}}\right)=1\) of the normalized results indicates the best performance characteristic.

  1. (b)

    Calculation of gray relational coefficients (GRC)

The GRC coefficient represents the correlation between the ideal and experimental results. The mathematical formula for GRC (ξ(k)) is given as follows:

$${{\varvec{G}}{\varvec{R}}{\varvec{C}}}_{{\varvec{i}}}=\frac{{\Delta }_{{\varvec{m}}{\varvec{i}}{\varvec{n}}}+{\varvec{\Psi}}{\Delta }_{{\varvec{m}}{\varvec{a}}{\varvec{x}}}}{{\Delta }_{0{\varvec{i}}}+{\varvec{\Psi}}{\Delta }_{{\varvec{m}}{\varvec{a}}{\varvec{x}}}}$$
(12)

where

Δ0i (k): Difference in absolute value between the ideal value \({x}_{0}^{k}\) (k) and \({x}_{i}^{k}\) (k)

Δ: The smallest value of Δ0i (k)

Δ: The Greatest value of Δ0i (k)

Ψ: is the distinction coefficient (Ψ ϵ [0, 1]). In our case the value of Ψ is 0.5.

$${\Delta }_{0{\varvec{i}}}\left({\varvec{k}}\right)=\Vert {{\varvec{x}}}_{0}\left({\varvec{k}}\right)-{{\varvec{x}}}_{{\varvec{i}}}\left({\varvec{k}}\right)\Vert$$
(13)
$${\Delta }_{{\varvec{m}}{\varvec{i}}{\varvec{n}}}=\underset{\forall {\varvec{j}}\in {\varvec{i}}}{{\varvec{min}}}\underset{\forall {\varvec{k}}}{{\varvec{min}}}\Vert {{\varvec{x}}}_{0}\left({\varvec{k}}\right)-{{\varvec{x}}}_{{\varvec{i}}}\left({\varvec{k}}\right)\Vert$$
(14)
$${\Delta }_{{\varvec{m}}{\varvec{a}}{\varvec{x}}}=\underset{\forall {\varvec{j}}\in {\varvec{i}}}{{\varvec{max}}}\underset{\forall {\varvec{k}}}{{\varvec{max}}}\Vert {{\varvec{x}}}_{0}\left({\varvec{k}}\right)-{{\varvec{x}}}_{{\varvec{i}}}\left({\varvec{k}}\right)\Vert$$
(15)

The values of the (GRC) are presented in the Table 11.

  1. (iii)

    Calculation of gray relational grade (GRG)

Table 11 Gray relational grade coefficients (GRCi (k)) and gray relational grade (GRGi)
Full size table

The (GRG) indicates the relationship between the series, it is calculated by the Eq. (16).

$${{\varvec{G}}{\varvec{R}}{\varvec{G}}}_{{\varvec{i}}}=\frac{1}{{\varvec{n}}}\sum_{{\varvec{k}}=1}^{{\varvec{n}}}{{\varvec{G}}{\varvec{R}}{\varvec{C}}}_{{\varvec{i}}}$$
(16)

n: is the number of performance characteristics (in our case n = 5)

i: 1 to m, where m is the number of tests

After the GRG calculation, the optimal level combination is selected on the basis of the gray relational grade (GRG) table. The largest GRG value that is near the ideal normalized value is the optimal combination. Consequently, the optimal level of the process parameters is the level with the largest GRG value. Table 11.

The level corresponding to the maximum of the average values of the “grey relational quality” is the optimal level of the parameters. The optimal combination of the parameters of the Inconel 718 turning process is deduced from Fig. 13.

Fig. 13
figure 13

Gray relational grade (GRG) main effects plot

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From the main effects plot of (GRG) in Fig. 13, the optimal combination for the 1st case (Ra, Vtng, and MRR), the parameters are (r1, Vc3, f2, and ap2). For the 2nd case (Ra, Vtng, Fz, Pc, and MRR), the optimal combination is (r1, Vc3, f1, and ap1). We note that the two optimal regimes found are not included in the L27 experimental design along with the corresponding output values. For this purpose, Eq. (17) was used to calculate the predicted output values [53, 54]. The results found for the two optimization cases are reported in Table 12.

$${\varvec{P}}{\varvec{r}}={{\varvec{A}}}_{{\varvec{G}}}+\sum\limits_{{\varvec{j}}=1}^{{\varvec{n}}}\left[\left({{\varvec{A}}}_{{\varvec{i}}{\varvec{j}}}\right)-{{\varvec{A}}}_{{\varvec{G}}}\right]$$
(17)

where

Table 12 Results for the two optimization cases using the (GRA) method
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Pr: Predicted response value.

AG: General average for each of the output parameters.

Ai, j: Average of the response at the optimal level.

n: Number of machining parameters. In this case, n = 4 (r, Vc, f, and ap).

Table 12 illustrates the results obtained by the method (GRA) for the two optimization cases. The 1st case (Ra-min, Vtng-min, and MRR-max) led to the regime corresponding to (r = 0.8 mm, Vc = 70 m/min, f = 0.12 mm/rev, and ap = 0.2 mm), which gives three outputs (Ra = 0.318 µm, Vtng = 1.740 m/s2, and MRR = 1.680 cm3/min). While for the 2nd case (Ra-min, Vtng-min, Fz-min, Pc-min, and MRR-max), where all output parameters are considered simultaneously, the regime obtained is (r = 0.8 mm, Vc = 70 m/min, f = 0.08 mm/rev, and ap = 0.1 mm), which gives the following responses: (Ra = 0.355 µm, Vtng = 1.277 m/s2, Fz = 31.683 N, Pc = 74.095 W, and MRR = 0.56 cm3/min). The results found show that the consideration of the outputs (Fz) and (Pc) in the optimization causes a decrease of the (MRR) of (66.66%). On the other hand, the value of (Ra) undergoes a slight increase of (11.63%) and the value of (Vtng) enjoys a decrease of (26.60%). This optimization case also showed very low values for both (Fz and Pc), especially the value of (Fz).

5.2 Comparisons between the two optimization methods (DF) and (GRA)

Table 13 exposes the results of the optimal cutting regimes as well as the values of the outputs for the two optimization cases obtained by the two methods (DF) and (GRA). The comparison of the optimal regimes obtained for the 1st case shows that the factors (r and Vc) are the same, but (f) increased by (0.108 to 0.12) mm/rev, and (ap) decreased by (0.3 to 0.2) mm, for (DF) and (GRA) respectively. This led to low roughness (Ra) and maximum (MRR) for (DF) method; on the other side, the tool vibration (Vtng) is lower for (GRA) method. The comparison for the 2nd optimization case indicates that the (DF) method favors the minimization of (Ra) by (21.12%) for a better surface quality as it increased by (26.79%) compared to (GRA). Similarly, (DF) improves (MRR) by (114.10%), for maximum production, as it decreased by (53.29%) compared to (GRA). In contrast, the (GRA) method favors the minimization of the parameters (Vtng, Fz, and Pc) compared to (DF) by (18.09%, 50.32%, and 9.26%) respectively.

Table 13 Results obtained
Full size table

6 Conclusion

The experimental, modeling, and optimization study in finishing turning of Inconel 718 by a coated metal carbide tool (GC1105) led to the following conclusions:

  1. 1.

    The ANOVA of (Ra), indicates that the main factors are significant. The feed rate (f) has the greatest influence with a cont% of (23.39%), followed by (Vc) and the radius (r) with cont% of (16.09% and 3.7%) respectively, lastly, the factor (ap) with a cont% of (0.02%). The terms (Vc × Vc), (r × Vc), (r × f), (Vc × f), and (Vc × ap) are also significant.

  2. 2.

    The ANOVA of (Vtng) reveals that the main terms (Vc, r, and ap) are significant. The radius (r) has the greatest influence with a cont% of (32.41%), (Vc) with a cont% of (31.36%), the depth (ap) with a cont% of (8.77%), and lastly, the factor (f) with a cont% of (1.28%). The square terms (r × r) and (Vc × Vc) are also significant with cont% of (6.23% and 9.41%) respectively.

  3. 3.

    The ANOVA of (Fz) proves that the main factors are significant, and that the factor (ap) has the highest cont% with (64.11%), followed by the factor (f) with a cont% of (21.18%), then comes (Vc) and (r) with cont% of (3.86% and 3.46%) respectively. Note also that all the square terms are significant, as well as the two interactions (Vc × f) and (Vc × ap).

  4. 4.

    The ANOVA of (Pc) indicates that all four main factors are significant and that (ap) is the most important factor affecting (Pc) with (38.29%) cont%. On the other side, the factors (Vc, f, and r) have cont% of (36.99%, 8.59%, and 2.41%) respectively. The interaction (Vc × f) is significant with (9.22%) cont%.

  5. 5.

    The mathematical models of the output parameters (Ra, Vtng, Fz, and Pc) proposed in this study were found to be accurate, all (R2) are high, (R2Ra = 92.02%), (R2Vtng = 92.96%), (R2Fz = 98.32%), and (R2Pc = 98.45%). They can be used for the best predictions of the responses as well as for optimization.

  6. 6.

    Optimization using the (DF) approach was applied on the basis of the mathematical models found, following two desired objectives. The optimal cutting regimes are as follows:

    1. (a)

      1st case: For minimization of (Ra and Vtng) and maximization of (MRR):

      (r = 0.8 mm, Vc = 70 m/min, f = 0.108 mm/rev, ap = 0.3 mm), corresponding to the optimized output values, (Ra = 0.297 μm, Vtng = 2.087 m/s2, and MRR = 2.268 cm3/min).

    2. (b)

      2nd case: For minimization of (Ra, Vtng, Fz, and Pc) and maximization of (MRR):

      (r = 0.8 mm, Vc = 69.192 m/min, f = 0.088 mm/rev, ap = 0.197 mm), which correspond to the optimized of output values, Ra = 0.280 μm, Vtng = 1.559 m/s2, Fz= 63.769 N, Pc= 81.660 W, and MRR= 1.199 cm3/min.

  7. 7.

    The optimization using the (GRA) approach was applied following the same desired objectives as the (DF) optimization method. The optimal cutting regimes are as follows:

    1. (a)

      1st case: For minimization of (Ra and Vtng) and maximization of (MRR):

      (r = 0.8 mm, Vc = 70 m/min, f = 0.12 mm/rev ap = 0.2 mm), which correspond to the optimized of outputs values, Ra = 0.318 μm, Vtng = 1.740 m/s2, and MRR= 1.680 cm3/min.

    2. (b)

      2nd case: For minimization of (Ra, Vtng, Fz, and Pc) and maximization of (MRR):

      (r = 0.8 mm, Vc = 70 m/min, f = 0.08 mm/rev, ap = 0.1 mm), which correspond to the optimized of output values, Ra= 0.355 μm, Vtng = 1.277 m/s2, Fz= 31.683 N, Pc= 74.095 W, and MRR= 0.56 cm3/min.

  8. 8.

    The comparison of the optimal regimes obtained by the two methods (DF and GRA) for the 2nd optimization case indicates that the method (DF) favors the minimization of (Ra) by (21.12%), for a better surface quality, as well as the maximization of (MRR) with an increase of (114.10%). On the other hand, the (GRA) method favors the minimization of the parameters (Vtng, Fz, and Pc) by (18.09%, 50.32%, and 9.26%) respectively.

The present work is by no means exhaustive. It can be extended to the study of the performance gap of tool materials such as ceramics and CBN when machining Inconel 718. Further investigations can be carried out to evaluate the influence of the cutting conditions on the machining technology parameters, and particularly the cutting temperature and the wear of the cutting tools under (MQL) environment. Finally, the application of other modeling and optimization methods (such as SVM, ANN, FL, GA, PSO, and PCA) would lead to better control of the performance parameters.