1 Introduction

Al alloy has the characteristics of low density, high strength, corrosion resistance, good formability and low cost. Thus, it is widely used in aerospace, automotive, shipbuilding, plastic molding, and electronics applications [1,2,3,4,5]. In general, thin-walled parts refer to lightweight structural parts composed of various thin plates and stiffeners, and the wall thickness is less than 2.5 mm or the ratio between wall thickness and outline size is less than 1:10 [6]. During machining of thin-walled parts, the thin sections elastically deform occurs due to the low stiffness [7]. This is a common problem in the machining of Al alloy thin-wall components. Surface roughness is also one of the significant indexes to measure the performance of thin-walled parts. The value of surface roughness directly affects the wear resistance, fatigue strength, corrosion resistance, lubrication and friction of thin-walled parts [8, 9]. Thus, machining of Al alloy thin-wall components essentially requires precise setting of machining parameters to obtain the desired process performance i.e. dimensional accuracy and superior quality surface finish [6].

Recently, some researchers studied on the prediction of surface roughness, thin-walled parts deformation, and optimization of the cutting parameters. Wang et al. [10] carried out cutting experiments on AlMn1Cu, and established a surface roughness prediction model based on cutting parameters through partial least squares regression. Hao et al. [3] introduced the tool wear model and the tool forced deformation model into the basic roughness prediction model to improve the roughness model, which is a comprehensive prediction model for the surface roughness of curved thin-walled parts. Khanghah et al. [11] established an empirical model of stainless steel 316 using the RSM and ANOVA. Then, it is associated with simulated annealing to obtain the minimum optimal combination of parameters for the burr characteristics in both up-milling and down-milling processes. Zeng et al. [12] proposed a forward milling process parameters optimization method based on the real-coded self-adaptive genetic algorithm and Grey relational analysis, and validated the effectiveness of the model through additional experiments. Vipindas et al. [13] investigated the influences of surface roughness and top burr on the milling of Ti–6Al–4V with 0.5 mm and 1 mm cemented carbide tools by ANOVA, and reported the significant factors. Neelesh et al. [14] proposed a surface roughness model of high-speed machining Ti–6Al–4V by ANOVA with stepwise backward elimination method, and obtained the optimal combination of process parameters with minimum surface roughness.

In addition, Taguchi method was also applied widely to analyze the significance of cutting parameters on surface roughness during milling and optimized the process parameters [15,16,17]. Mahesh et al. [18] proposed a second-order mathematical models of surface roughness using RSM. Then, the genetic algorithm (GA) was applied to obtain the optimum cutting parameters via MATLAB programming. Kan and Lmalghan et al. [19, 20] constructed a comprehensive prediction model of surface roughness integrated into particle swarm optimization (PSO) algorithm. Dikshit et al. [4] studied the effect of cutting parameters on surface roughness in high-speed milling of Al2014–T6, and established an empirical mathematical model for surface roughness. It reported that the most significant is cutting speed followed by feed rate. Ying et al. [21] established the surface roughness model through BP artificial neural network algorithm with considering material removal rate, and optimized the processing parameters. Sheng et al. [22] established a regression model of cutting force and surface roughness through experimental results, and obtained the optimal milling parameters for thin-walled parts via non-dominated sorting genetic algorithm (NSGA-II) to. Karkalos et al. [23] developed an ANN model for predicting the surface roughness of titanium alloy and indicated that the ANN model was able to predict the surface roughness with a mean error of 0.97%. Gao et al. [24] proposed a deformation control strategy by modifying tool location point for deformation of thin-walled curved surface parts. Wang et al. [25] applied the particle swarm algorithm to establish the machining residual stresses model, and validated the prediction model by two cases. Qiong et al. [26] applied the Finite difference method (FDM) to predict the aerospace thin-walled plates and compared with the finite element method (FEM). Yue et al. [27] proposed a method to calculate the surface error with considering the chip thickness and coupling force. Song et al. [28] applied thin palate theory in developing a time-space discretization method for milling of thin-walled parts stability. The effect of residual stress on the thin-walled parts deformation was investigated with considering the heat treatment process, cutting parameters and machining process [29,30,31,32,33]. Sepp et al. [34] proposed a milling error prediction method for the outer surface of thin-walled parts based on the selection range of tools and process parameters. Wang et al. [35] proposed a methodology of 3D machining allowance modeling and analyzed the machining deformation of aerospace thin-walled parts.

Although many studies have been done in optimization of milling parameters, fewer surveys are performed with consideration of machining deformation and surface quality simultaneously in milling thin-walled plates. Moreover, most of the previous papers are available for identifying the optimal machining parameters by incorporating the soft computing techniques Taguchi method, GA, BP, PSO, NSGA-II and ANN. Therefore, this study explores the feasibility for multi-objective optimization with the RSM-ABC method, which is employed to optimize the Ra-Fd, Ra-Td and TWD simultaneously in Al alloy thin-walled parts machining process. In this paper, experiments were conducted to investigate the influence of cutting parameters on surface roughness and machining deformation simultaneously. Especially for the spindle speed and feed rate with a wider range were selected to increase the cutting efficiency. The models of Ra-Fd, Ra-Td, and TWD were developed using RSM with considering spindle speed (S), feed rate (f), transverse size (ts), and depth of cut (doc). The significance of cutting parameters on surface roughness and machining deformation was analyzed by ANOVA. A comparison of RSM-optimum function and ABC algorithm optimum programming was performed to obtain the best cutting conditions leading to minimum Ra-Fd, Ra-Td and TWD simultaneously. Finally, the optimization cutting parameters was implemented through experiments, which validates the feasibility of the proposed cutting strategy.

2 Experimental Procedure

The high-speed milling operations were conducted on Quick Jet AV1612 machining center that is equipped with HEIDENHAIN CNC system for precise control of machining process, which has a maximum spindle speed of 20,000 rpm and feed rate of 25 m/min. The workpiece selected for the experiment was Al alloy 5083 that has a rectangle block shape of 140 mm × 70 mm × 5 mm. The workpiece was mounted on a special fixture by applying 6 bolts further clamped on the bed of machine tool. The composition of workpiece material is given by Table 1.

Table 1 Chemical composition of workpiece Al alloy 5083
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In this test, a three-edge carbide flat end mill was employed which has a diameter of 12 mm, a helix angle of 40°, a radial rake angle of 13°, and a radial relief angle of 14°. The cutting tool path was programmed by using NX 10 software, and the follow periphery tool path strategy was employed to guarantee the machined surface quality. Figure 1 shows the schematic of the setup milling experiment. The chosen cutting parameters and their ranges are shown in Table 2.

Fig. 1
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Experimental setup

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Table 2 Cutting parameters and their ranges
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In this study, following the investigation method of surface roughness proposed from Refs. [36,37,38]. The surface roughness is investigated along two directions: feed direction and transverse direction. According to scallop height model [36, 37], the values of feed rate and transverse size are the important control factors in the calculation of the surface residual height. In addition, surface roughness along the feed direction and along the transverse direction has different spatial patterns [39]. Thereby, the surface roughness evaluated in Ra pattern was investigated in two directions can achieve optimum surface quality of the milling process with considering control factors.

Experiments were carried out on the constructed test rig. For each experimental run, the thin-walled part deformation, average surface roughness in the feed direction and transverse direction was measured respectively. Surface roughness was measured using an Olympus laser confocal microscope as shown in Fig. 2. The final values of Ra-Fd and Ra-Td under each experiment were calculated by the mean of 5 locations in the feed direction and transverse direction, respectively. The machining deformations of the thin-walled parts were detected using the global status three coordinate measuring machine (CMM) as shown in Fig. 3. In this study, following the measuring method of thin-walled plate deformation proposed from Refs. [40,41,42], the machining deformation values of workpiece are measured along two directions (length direction and width direction) in response to before milling and after milling respectively. The position of the measured points on the top wall is shown in Fig. 4a. Before milling, a measuring point on fixture surface is set as basis point and its deformation is set to zero. Then, the CMM is used to measure the deformation variation by moving along two directions for the unmachined surface and record the data. After machining, the deformation measuring process is repeated on the finished surface, and the measurement points are arranged as the same with the unmachined surface. The deformation of each measured point is obtained by the difference of deformation values between unmachined surface and finished surface. According to Ref. [43], the maximum value of the measured deformation is suggested to develop the response model. Therefore, the obtained maximum deformation value is used in this study. The deformation measuring process for thin-walled plate is shown in Fig. 4.

Fig. 2
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Olympus laser confocal microscope

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Fig. 3
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Global status three-coordinate measuring machine

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Fig. 4
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Deformation measuring process for thin-walled plate a unmachined surface, b finished surface

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3 Design of Experiments

In this paper, the primary objective of the experimental investigation is to analyze the influence of machining parameters on machining deformation and surface quality simultaneously. The selection of machining parameters in milling Al alloy thin-walled plates affects deformation, quality, and productivity of the machined parts [22]. Various parameters influence on surface roughness and machining deformation are investigated according to Refs. [6, 22, 41,42,43]. Among the machining parameters, the following control factors have the significant effect on surface roughness and machining deformation that are selected for evaluation in this paper: spindle speed, feed rate, transverse size, and depth of cut. The levels adopted for the control factors were defined considering tool supplier recommendations. The influences of spindle speed, feed rate, depth of cut, and transverse size were studied in terms of Ra-Fd, Ra-Td, and TWD. In order to test the effect of parameters on the three indicators, the RSM was used. The relationships between the parameters and responses are given as follows:

$$R_{a} - Fd = f\left( {S, f, ts, doc} \right)$$
(1)
$$R_{a} - Td = f\left( {S, f, ts, doc} \right)$$
(2)
$$TWD = f\left( {S, f, ts, doc} \right)$$
(3)

where Ra-Fd, Ra-Td, and TWD are the functions of the spindle speed, feed rate, transverse size, and depth of cut. In RSM, optimal design method is commonly used to construct the curvature of the model, which consisting of categorical factors and constrained regions. In addition, the evaluation of performance of developed model is analyzed by ANOVA, which helps to find out the significance of cutting parameters on the Ra-Fd, Ra-Td, and TWD respectively.

In this study, the analysis was performed using the Design Expert 8 software. All experimental runs were based on optimal design method that contains a total of 25 experimental runs. Four parameters (S, f, ts, doc) were selected as input variables to measure the Ra-Fd, Ra-Td, and TWD values as output results. 25 experiments were carried out and examined at various parametric conditions as list in Table 3.

Table 3 Plan of experiments and their responses
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4 Results and Discussion

In this study, for each experimental run, the Ra-Fd, Ra-Td, and TWD were measured respectively. Table 3 provides the measured Ra-Fd, Ra-Td, and TWD results in response to the various parametric factors. In RSM, the residual analysis was applied to check the adequacy of the proposed Ra-Fd, Ra-Td, and TWD models. In addition, ANOVA was applied to analyze the significance of cutting parameters on the Ra-Fd, Ra-Td, and TWD respectively. In the analysis, the associated p value of less than 0.05 indicates that the cutting parameters and their interactions are statistically significant.

4.1 Analysis of Surface Roughness in Feed Direction

Figure 5 shows two sample results of surface roughness at different parametric factors. It can be noticed that the cutting parametric factor influences the Ra-Fd. The normal plot of residuals and residuals versus run number for the Ra-Fd are shown in Fig. 6. For residual normal plot results, it can be observed that the points are closeness (close) to the straight line. In addition, there is no special pattern and unused structure in residuals versus run number. The above results conclude the proposed model adequacy.

Fig. 5
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Measured surface roughness for different cutting parameters

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Fig. 6
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Residual plot for surface roughness in feed direction

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According to Table 4, ANOVA results show that the spindle speed, feed rate, depth of cut and transverse size are the most significant factors on Ra-Fd. In addition, the most significant interaction factor is the depth of cut-transverse size interaction. Optimal design method containing 25 experimental runs was applied to establish the statistical model. According to the actual variables, the RSM model of surface roughness Ra-Fd is given by:

$$\begin{aligned} R_{a} - Fd & = 2.68319 - 0.00003S - 0.360952doc + 0.088515ts + 0.000511f \\ & \quad + 0.558146doc \cdot ts - 2.49357 \times 10^{ - 8} f^{2} \\ \end{aligned}$$
(4)
Table 4 Modified ANOVA results for surface roughness in feed direction
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The statistical model with \(R^{2}\) and \(R_{adj}^{2}\) are list in Table 4. \(R^{2}\) shows that the error of the regression model as a percentage of the total error (the multivariate correlation coefficient) is 94.72%, and the adjusted error as a percentage of the total error \(R_{adj}^{2}\) (that is, the modified multivariate correlation coefficient) is 92.96%, which is less than \({ }R^{2}\).Also, the values of \(R^{2}\) and \(R_{adj}^{2}\) are both greater than 85%.

The main effect represents direct the influence of cutting parameters on response variable. Figure 7 plots the main effect plot for Ra-Fd. It can be noted that the Ra-Fd correlates negatively with spindle speed. At low spindle speed, the built up edge (BUE) is formed and the chip fracture readily producing the rough surface. As the spindle speed increases, the BUE vanishes and chip fracture decreases, hence the surface roughness decreases [44, 45]. Thereby, a higher spindle speed leads to a smaller Ra-Fd.

Fig. 7
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Main effect plot for surface roughness in feed direction

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The Ra-Fd almost linearly increases as the depth of cut and transverse size increase. This is because the cutting area and the cutting force increase with the increasing of the depth of cut and transverse size. Application of large and intermittent cutting forces during milling of low rigidity components results in regular deflections of thin wall during the machining operation. This causes the material removal in an uneven fashion and poor surface quality [6]. Therefore, a larger depth of cut or transverse size will lead to a larger surface roughness.

In addition, a non-linear increase of Ra-Fd with increasing of feed rate is observed. This is attributed to a higher feed rate leads to a higher cutting force and cutting heat temperature in the machining zone, thus cut chips deposit onto the work surface which results in poor surface quality [6, 45]. Hence, a high feed rate result in a large Ra-Fd. From above results, it is observed that a smaller Ra-Fd can be achieved by increasing the spindle speed or decreasing the feed rate, depth of cut, and transverse size.

4.2 Analysis of Surface Roughness in Transverse Direction

It can be also noticed from Fig. 5 that the cutting parametric factor influences the Ra-Td. Figure 8 shows the normal plot of residuals and residuals versus run number for the Ra-Td. Residual normal plot results shows the points are closeness to the straight line. Furthermore, no special pattern and unused structure is observed in residuals versus run number.

Fig. 8
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Residual plot for surface roughness in transverse direction

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Table 5 presents ANOVA results for the Ra-Td. It is observed that the spindle speed, feed rate, transverse size, and depth of cut are the most significant factors on Ra-Td. In addition, interactions affecting the Ra-Td are: the most significant is spindle speed-feed rate interaction followed by depth of cut-feed rate interaction.

Table 5 Modified ANOVA results for surface roughness in transverse direction
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According to the actual variables, the RSM model of Ra-Td is given by:

$$\begin{aligned} Td & = 3.34602 + 0.000024S - 0.268298doc - 0.102436ts + 0.000402f \\ & \quad - 1.66911 \times 10^{ - 8} Sf + 0.000299doc \cdot f \\ \end{aligned}$$
(5)

The statistical model with \(R^{2}\) and \(R_{adj}^{2}\) are list in Table 5. \(R^{2}\) shows that the error of the regression model as a percentage of the total error is 96.49%, and the adjusted error as a percentage of the total error \(R_{adj}^{2}\) is 95.32%, which is less than \(R^{2}\).Also, the values of \(R^{2}\) and \(R_{adj}^{2}\). are both greater than 85%.

The main effect plot for Ra-Td is shown in Fig. 9. As seen from Fig. 9, the Ra-Td decreases with increasing of the spindle speed, meanwhile, the Ra-Td increases as the feed rate and depth of cut increase. The explanations for these phenomena are already described in Sec. 4.1. For the transverse size, it found that the Ra-Td presents descending trend as transverse size increases. The observed trend is consistent with the pattern reported by other researchers [36, 37]. From above results, we reach the conclusion that a smaller Ra-Td can be obtained by increasing the spindle speed and transverse size or reducing the depth of cut and feed rate.

Fig. 9
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Main effect plot for surface roughness in transverse direction

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4.3 Analysis of Thin-Walled Part Deformation

Figure 10 shows the normal plot of residuals and residuals versus run number for the TWD. Residual normal plot results shows the points are closeness to the straight line. Furthermore, no special pattern and unused structure is observed in residuals versus run number.

Fig. 10
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Residual plot for thin-wall parts deformation

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Table 6 presents ANOVA results for the TWD. As shown in Table 6, the depth of cut is the most significant factors on the TWD. In addition, interactions affecting the TWD are: the most significant is spindle speed- depth of cut interaction followed by spindle speed -feed rate interaction.

Table 6 Modified ANOVA results for thin-walled part deformation
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The RSM model of TWD in terms of actual variables is given by:

$$\begin{aligned} TWD & = 0.04095 + 0.000013S - 0.990185doc + 0.001619ts + 5.70469 \times 10^{ - 6} f \\ & \quad + 0.000034S \cdot doc - 4.42882 \times 10^{ - 10} S \cdot f - 7.41009 \times 10^{ - 10} S^{2} + 1.89589doc^{2} \\ \end{aligned}$$
(6)

The statistical model with \(R^{2}\) and \(R_{adj}^{2}\) are list in Table 6. \(R^{2}\) shows that the error of the regression model as a percentage of the total error is 90.78%, and the adjusted error as a percentage of the total error \(R_{adj}^{2}\) is 86.17%, which is less than \(R^{2}\).Also, the values of \(R^{2}\) and \(R_{adj}^{2}\) are both greater than 85%.

Figure 11 shows the main effect plot for TWD. For the spindle speed, when the spindle speed reaches 11,653 rpm, the TWD increases up to 0.065 mm, after that, it starts to decrease as the spindle speed future increases. This is due to the residual stress of cutting and the cutting force increase with increasing of the spindle speed, which results in a larger deformation of thin-walled parts. However, when the spindle speed exceeds 11,653 rpm, the machining vibration tends to be stable as the cutting force and friction force decrease, hence the TWD decreases with increasing of the spindle speed.

Fig. 11
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Main effect plot for thin-wall parts deformation

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In case of depth of cut, when the depth of cut reaches 0.151 mm, the TWD decreases up to 0.061 mm, and then, it starts to increase as the depth of cut increases rapidly. This is due to the involvement of the cutter in the cutting process becomes larger as the depth of cut increase, which cancels out a significant vibration of the workpiece, thus, the milling process become more stable, and the TWD becomes smaller. While the depth of cut increases, the removal material volume and the tool vibration increases, which leads to the TWD increases rapidly. Furthermore, the TWD is found to be increasing as the feed rate and transverse size increase, which is due to the increasing in feed rate or transverse size makes the cutting force and the temperature between tool and chip thickness become higher, thus a higher feed rate or transverse size will result in a higher TWD.

From above analysis, it can be concluded that the Ra-Fd, Ra-Td are mainly influenced by the spindle speed, feed rate, transverse size and depth of cut, while the TWD is mainly influenced by the depth of cut. Thereby, we reach the conclusion that the depth of cut is the significant effect on the Ra-Fd, Ra-Td, and TWD simultaneously.

5 Results of Multi-objective Optimization

5.1 RSM-Optimum Function

This study aims to minimize the amplitude of deformation of thin-walled part and at same time minimize the surface roughness in feed direction and transverse direction. Moreover, the response parameters (Ra-Fd, Ra-Td, and TWD) are nonlinear multi-objective optimization problem due to the non-linear relationship between surface roughness, thin-wall part deformation and cutting parameters. The objective function is defined as follows:

$${\text{Minisize }}(R_{a} - Fd, R_{a} - Td, TWD)$$
(7)

The constraints on the cutting parameters are as follows:

$$\begin{aligned} & 9000\,{\text{rpm}} \le S \le 15000\,{\text{rpm}} \\ & 3\,{\text{mm}} \le ts \le 7\,{\text{mm}} \\ & 3600\,{\text{mm}}/{\min} \le f \le 9000\,{\text{mm}}/{\min} \\ & 0.1\,{\text{mm}} \le doc \le 0.3\,{\text{mm}} \\ \end{aligned}$$

The optimal machining parameter predicted by RSM optimization functions are spindle speed 14,994 rpm, feed rate 3600 mm/min, transverse size 3.8 mm, and depth of cut 0.1 mm. Meanwhile, the prediction results of Ra-Fd, Ra-Td, and TWD are 4.263 µm, 3.938 µm and 0.049 mm respectively, as list in Table 8.

5.2 ABC Algorithm Optimum

In this study, the ABC was used to find a set of optimal cutting parameters to achieve the optimal surface quality, that is, Ra-Fd, Ra-Td, and TWD simultaneously reach minimum values under a combination of optimal spindle speed, feed rate, depth of cut and transverse size. The ABC is a widely used meta-heuristic tool for multi-objective optimization with the advantages such as few control parameters, easy to implement, simple calculation, and it has a faster rate of convergence. Therefore, ABC algorithm is carried out to solve this multi-objective optimization problem by MATLAB programming. The optimization of cutting parameters by multi-objective ABC is obtained according to Fig. 12 as follows:

Fig. 12
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Multi-objective ABC optimization flow chart

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Step 1 Initialize the parameters, both of the number of the employed bees and onlooker bees are set as 80. The maximum number of iterations are set as 500, and then, generate the food locations.

Step 2 The colonies are equally divided into 3 sub-colonies, and then, perform the select operation independently to each of the sub-objective functions in their individual sub-colonies.

Step 3 This is employed bee phase, where greedy selection is applied to select new food locations by calculating nectar amount for both of the original food locations and their neighborhood.

Step 4 In the onlooker bee phase, greedy selection is used to select new food locations after applying the roulette wheel selection to the results of the step 2.

Step 5 Determine if there is a solution to give up, if there is, then replace it with a randomly produced new location, or go straight to the next step.

Step 6 Memorize the best solution.

Step 7 The ABC continues to iterate until it reaches the maximum iteration number and finally outputs the optimal solution.

The convergence of the Ra-Fd, Ra-Td, and TWD are obtained after iteration as shown in Fig. 13. It can be seen that the Ra-Fd, Ra-Td, and TWD are continuously reduced via the continuous optimization of ABC. After convergence of the ABC, 15 best optimal points are obtained as the Pareto optimal points. The Pareto optimal solutions are listed in Table 7. In the actual machining process, to obtain the best surface quality, it is necessary to consider three responses (Ra-Fd, Ra-Td, and TWD) reach the smallest value stimulatingly. Therefore, the No.1 is considered as the optimal solution.

Fig. 13
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The ABC optimum value of fitness

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Table 7 Multi-objective solutions of ABC
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5.3 Comparison Between RSM And ABC Optimal Results

From above analysis, both RSM-optimum function and ABC algorithm can obtain the best values of Ra-Fd, Ra-Td and TWD respectively. Moreover, the RSM and ABC optimal results are compared as list in Table 8. Compared with RSM results, the values of Ra-Fd, Ra-Td and TWD obtained by ABC algorithm are improved by 2.7%, 2.2% and 16% respectively. Thereby, it can be concluded that the ABC algorithm can obtain a better cutting strategy.

Table 8 Comparison between RSM and ABC optimal results
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Experiment was also conducted to validate the proposed cutting strategy. Comparison between prediction and experimental results are shown in Table 9. It can be seen that the prediction errors of the ABC algorithm optimum for the Ra-Fd, Ra-Td and TWD are 2.3%, 5.7% and 4.6% respectively.

Table 9 Comparison between experimental and ABC optimal results
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6 Conclusions

In this paper, the influence of spindle speed, feed rate, depth of cut and transverse size on Ra-Fd, Ra-Td and TWD were investigated by milling of Al alloy 5083. Especially for the spindle speed and feed rate with a wider range were selected to increase the cutting efficiency. The Ra-Fd, Ra-Td and TWD prediction models were established by RSM, and the influence of cutting parameters on the Ra-Fd, Ra-Td and TWD were analyzed using ANOVA. For optimization process, a comparison of RSM-optimum function and artificial bee colony (ABC) algorithm optimum programming was performed to find the best cutting conditions leading to minimum Ra-Fd, Ra-Td and TWD simultaneously. According to analysis results, the following observations were drawn:

  • The response models are obtained for Ra-Fd, Ra-Td, and TWD in function of control factors. Three models present good data variability explanation with \(R^{2}\) and \({\text{R}}_{adj}^{2}\), which proves the developed models are useful to study the process effects in the results.

  • ANOVA results show that the Ra-Fd, Ra-Td are mainly influenced by the spindle speed, feed rate, transverse size and depth of cut, while the TWD is mainly influenced by the depth of cut. Thereby, it can be concluded that the effect of depth of cut is much more pronounced than the effects of other factors on the Ra-Fd, Ra-Td, and TWD simultaneously.

  • Compared with traditional RSM optimization results, the ABC algorithm can obtain a better cutting strategy, and displays extensive favorable. The ABC algorithm optimum parameters are found to be spindle speed 15,000 rpm, depth of cut 0.1 mm, transverse size 3 mm, feed rate 3601 mm/min.

  • Confirmation experiment was performed to validate the proposed cutting strategy, the experimental results show a good coherence to the calculated results from ABC algorithm, which validates the performance of the proposed cutting strategy.

  • The RSM-ABC methodology used in this paper is detailed and can be applied in other manufacturing processes as a multi-objective modeling and optimization procedure. Moreover, the obtained results can be employed in the milling of Al alloy thin-walled parts, ensuring good surface roughness and low machining deformation.