A Hybrid Nondominant-Based Genetic Algorithm (NSGA-II) for Multiobjective Optimization to Minimize Vibration Amplitude in the End Milling Process

Aluminium is a noncorrosive, lightweight material used to fabricate parts for the aerospace, automobile, and construction industries. Due to the low-temperature resistance, more heat is generated. At the same time, in machining, tremendous eforts are taken to keep friction and chatter to a minimum and to attain better quality and perfect output, and also more attention is required while selecting the machining process parameters. Spindle speed, rate of feed, radial and axial depth of cut, and radial rake angle of the tool are the parameters utilized to machine aluminium 6063 using the HSS tool on CNC milling to estimate spindle and worktable vibration using a prediction model. In this study, the design of the experiment of the response surface methodology approach is used to create a second-order statistical equation for experimentation with the Design-Expert v12 software. Te performance characteristics are analyzed using the ANOVA method. Te spindle speed achieved the lowest vibration between 2000 and 3000rpm. Next, axial and radial depths were the most vibration-afecting parameter compared to the rate of feed and radial rake angle of the tool. To fnd the best feasible response, the nondominant sorting genetic algorithm II (NSGA II) approach was trained and tested using MATLAB software. Using a Pareto-optimal technique, the optimum worktable vibration ranged from 0.00284 to 0.00165mm/s 2 , whereas the spindle vibration ranged from 0.02404 to 0.01336mm/s 2 . Te predicted values were found to be in an excellent argument when Pareto-optimal solutions are used.


Introduction
In recent years, manufacturers have focused on predicting machining process parameters to minimize machining processes and to meet customer expectations of low cost and high quality.Furthermore, predicting machining parameters that can reduce machine and tool vibrations and produce high-quality materials is crucial in machining.Some discrepancies existed between the simulated and measured cutting forces inside each cycle due to self-excited vibration during the milling process [1].Nowadays, vibration analysis gives a great interest to researchers.Signifcant eforts are needed to monitor and minimize vibrations in cutting tools and machines.To measure and analyze the vibrations, vibration testers, piezoelectric sensors, microelectromechanical sensors, proximity probes, laser Doppler vibrometers, and accelerometers are used.
Te cutting process was performed using a cubic boron nitride (CBN) end mill cutter and AISI-D3 steel material to study the relationship between the wear in the tool and vibration amplitude.Vibration is commonly measured using an accelerometer.Te author [2] found that increased tool wear is caused by an increase in vibration amplitude at high peak frequencies.
Te vibration amplitude signal signifcantly increases when the fank is substantially damaged.Tis results from wearing one fute in an experiment employing accelerometers while milling mild steel workpieces with HSS four-fute end mills to monitor tool wear evolution [3].To increase steel machinability, the infuence of vibration on surface unevenness and tool wear was estimated using an end milling process.Surface unevenness and wear in the tool can be reduced using two-dimensional vibration-assisted microend milling (2-D VAMEM) [4].Sivasakthivel et al. [5] measured the acceleration amplitude using a fast Fourier-transform (FFT) analyzer of Al 6063 material, and the tools used for experimentation were HSS end mill.Te input parameters considered are the helix angle of a cutting tool, spindle speed, feed rate, and the axial and radial depth of cut and found that the tool's helix angle is the most relevant parameter in lowering vibration amplitude.Te research reported by Shen et al. [6] experimented using aluminium alloys to minimize the ultrasonic vibration.Te bottom region of the slot and the vertical sidewall region of the place were studied in a slot-milling machine experiment to decrease surface unevenness.According to the F-test, the most infuential factors are spindle speed and the rate of feed.Surface unevenness and vibration amplitude are minimized considering spindle speed, axial cutting depth, feed, and heating temperature as input parameters are optimized using the analysis of variance (ANOVA) approach with the design of experiment (DoE) of response surface methodology (RSM) and the desired function approach.In the vertical milling center experiment, surface morphology and vibration were assessed using a ball-end-coated carbide insert.According to Amin et al. [7], the temperature and cutting speed have the greatest infuence on the resulting surface quality and maximum vibration amplitude.Experiments were performed using AISI H13 steel with a TiAlN tool to measure the tool's surface roughness, tool wear, and vibration and concluded that wear in the tool and surface roughness depend primarily on tool instability [8].A fnite element model is developed to fnd the vibration on milling machines.Te experimental work was completed, and data were obtained using statistical analysis, which revealed that surface roughness is mostly impacted by the dynamical features of the cutting tool [9].
Milling machine experiments were performed on C45E4 steel; optimization was carried out using a Gaussian process regression model, which analyzes surface roughness while taking into account tool vibration.Te Gaussian process regression (GPR) model was used to forecast surface roughness, and the results indicated that vibration in the tool is also an important factor that impacts surface quality [10].Te researcher [11] designed a mathematical model using the RSM optimization technique to measure tool vibration, surface quality, and tool wear during milling EN31 tools.Te conclusion stated that the feed is the infuencing element impacting surface quality and cutting speed afects the tool.Te vibration analysis experiments were conducted by Sivasakthivel et al. [12] on tool holders and workpiece holders to machine aluminium Al 6351 material considering the rake angle, nose radius, cutting speed, feed rate, and depth of cut.Te regression model was developed, optimization was performed using a genetic algorithm, and the nose radius showed a negative value during sensitivity analysis.Te author [13] stated that tool life, quality of machined products, and functional behavior are primarily afected by vibration.Studies have been conducted using magnetic rheological fuid dampers to suppress vibration using AISI 4340 steel with 45HRC indexable inserts on milling machines.As a consequence, the damper decreases tool vibration while increasing cutting performance.Te cutting resistance and vibration are measured by [14] during the simulation experiment using fnite element modeling.Experiments have shown that improved tool orientation has been found to minimize machining distortion and increase surface quality.
Vibration analysis and Hilbert-Huang transforms (HHTs) were used to monitor the condition of the milling machine and compare the results with FFT.As a result, HHTs revealed the diference in cutting edge due to its high time/frequency resolution, and the cutting fuid increased the vibration damping in the milling machine [15].A 55NiCrMoV6 steel ball-end milling machine experiment measured the cutting resistance and vibration.Surface roughness topography was investigated utilizing the signalto-sound (S/N) ratio and grey rational analysis (GRA) with the surface angle of inclination and tool overhang taken into account.As a result, the angle of inclination and tool overhang have a substantial impact on the produced force and vibration values.Selecting the best tilt angle and tool overhang will decrease surface quality [16].Vibrations that occur when a fat-end cutter arrives at a workpiece during machining can have a signifcant impact on the quality of the fnished part, tool life, and machining efciency.To address this issue, a combination of modeling, simulations, and experiments can be used to understand, predict, and mitigate these vibrations [17].Te experiments were designed using signal processing technology to measure the acceleration signals in spindle vibrations of tools generated during the end mill process.Te protective chamfer, along with a relatively high rake angle, is a successful criterion to minimize the amplitude of vibration under most cutting settings [18].
At various vibration amplitudes and cutting velocities, compression studies were conducted on the surface integrity of the Ti-6Al-4V end mill.Surface roughness values increase as the cutting rate and vibration amplitude increase, resulting in a considerable microvibration texture in rotary ultrasonic elliptical end milling (RUEEM) in the form of burrs divided into machined surfaces [19].Lin et al. [20] generated a regression model to forecast the surface roughness of Al 6061 materials and optimized using artifcial neural networks to measure vibration during end mills based on cutting factors.Estimating prediction accuracy can be ensured by emphasizing the results of the developed prediction model derived from cutting conditions that consider machining stability.A milling experimental study was conducted by a researcher [21] to improve the surface unevenness of AISI 52100 bearing steel and the impact level of the workpiece.Te researcher [22] evaluated the surface roughness in the slot end-mill process with a variable helix angle for aluminium machining.A statistical analysis was conducted to measure surface roughness and proved that variable helix angles can produce much higher surface 2 Advances in Materials Science and Engineering roughness quality than standard helix angle tools.Te experimental investigation of cutting vibration during microend-milling of a straight groove involves studying and measuring the vibrations generated during the machining process.Based on the results, spindle speed is the most critical characteristic that causes cutting vibration [23].
Using the Taguchi technique, cutting factors were studied.Experiments have shown that cutting speed and tooth contact feed were the most inducing factors afecting the surface of the material in both cases.Regression and ANOVA techniques were utilized to anticipate the acceleration amplitude.Based on the results, cutting speed has a signifcant infuence on the vibration amplitudes [24].Te purpose of the study was to improve the machinability of AISI H13 material and identify the ultrasonic vibrations caused in the end milling process.Te results revealed a continual change in mechanical loading caused by the tool's vibratory motion, resulting in a decrease in cutting forces [25].Implementing a stacked generalization ensembles model for milling tool wear state recognition by vibration signal analysis can enhance the efciency and safety of machining processes by enabling proactive maintenance and tool replacement, thereby minimizing downtime and optimizing tool usage [26].
In general, vibration is undesirable phenomenon, resulting in unpleasant motions and dynamic stresses, and also afects machining performance, particularly surface fnish and tool life.A survey of the previously published literature indicates that there are a few publications published on optimizing the parameters impacting vibration characteristics during the milling process of aluminium 6063.Furthermore, none of the studies use experimental analysis to evaluate spindle and worktable vibration under real-world conditions, including the spindle speed, feed rate, axial depth of cut, radial depth of cut, and radial rake angle of the tool.Te RSM, ANOVA statistical analysis, and nondominated sorted genetic algorithm (NSGA-II) heuristic approaches were also performed to optimize the milling parameters.

Development of a Mathematical Model
Because the frst-order model tends to be inadequate for a curved response surface, the second-order model is recommended to approximate a section of the genuine response surface with curvature.All of the terms from the frst-order model, as well as quadratic and cross-product terms, are included in the second-order model.Second-order models depict quadratic surfaces like minimum, maximum, ridge, and saddle.Tus, a second-order linear diference equation is a mathematical equation that represents the connection between consecutive terms in a series.A quadratic polynomial form and second-order model can be represented as follows [5]: where y ″ is the approximated response given in the secondorder equation, β 0 is the regression equation's free term coefcients, x 0 , x 1 , . . .are linear factors, x 11 , x 22 , . . .are the quadratic terms, and x 12 , x 13 , . . .are the interaction terms.Te polynomial coefcients are determined using the multiple regression method.Te coefcients are calculated using the statistical software Design-Expert v12.

Experimental Design
In the present work, the spindle speed, feed rate, axial depth of cut, radial depth of cut, and radial rake of the cutting tool have been considered as the process parameters to measure spindle and worktable vibration.
RSM is a set of mathematical and statistical tools used to analyze situations like the one given using an empirical model and also the most informative method of analysis of the result of a factorial experiment.It is a technique that graphs the results of polynomial regression studies in three dimensions to ofer a sophisticated perspective of correlations between combinations of two predictor variables and an outcome variable.
Te output response vibration on the spindle and worktable may be represented as a function of process parameters such as the spindle speed (N), rate of feed (f z ), axial depth of cut (a p ), radial depth of cut (a e ), and radial rake angle of the tool (c).
Using these parameters, the objective is to get the best possible answer for the machine performance.Similarly, several studies have theoretically anticipated output responses using equation (27): where A is the actual acceleration amplitude response (mm/s 2 ), N is the spindle speed (m/min), f z is the rate of feed (mm/rev), a p is the cutting depth (axial) (mm), a e is the cutting depth (radial) (mm), and c is the rake angle of tool (radial) (degree).Te model parameters are k 1 , k 2 , k 3 , k 4 , k 5 (assessed from experimental data), and the letter c stands for the "error" factor.
Te experiments were carried out utilizing the DoE process of RSM as recommended in [28,29].As given in equation ( 2), the spindle speed, feed rate, radial cut depth, axial depth, and radial rake angle of the tool are the input parameters.
A central composite design (CCD) in statistics is an experimental design that may be used in response surface techniques to create a second-order (quadratic) model of the response variable without requiring a full three-level Advances in Materials Science and Engineering factorial experiment.Te design matrix is selected as per CCD, and levels are coded as −2, −1, 0, 1, and 2. As per the recommendation and suggestion given by the Production Technology Handbook (Hindustan Machine Tools [30], the ranges of cutting values and conditions are selected as presented in Table 1.As indicated in Table 2, 32 experiments were designed to estimate the process parameters.Tese experimental designs were developed using the Design-Expert v12 software, which is a tool that helps us in designing experiments and analyzing the data.To anticipate spindle and worktable vibration, a second-order mathematical equation was devised, whose appropriateness was determined using ANOVA [31,32].Te MATLAB software was used for vibration optimization experiments.Te nondominated sorting genetic algorithm (NSGA-II) forecasts spindle and worktable vibration.

Experimentation
Te milling process was carried out in a CNC vertical machining center (Model-S33).A photographic view of the machined specimen and the HSS tool is shown in Figure 1 and Figure 2. Te test specimen of aluminium (Al 6063) alloy with a dimension of 50 mm × 50 mm × 50 mm was prepared as a cube block to perform experimental work.Aluminium 6063 is a medium-strength alloy with moderate fracture toughness at high strength.Aluminium metal's softness, low melting point, and low electrical resistivity are most likely due to the few electrons available for metallic bonding.As a result, considerable attention should be given during machining.Table 2 shows the chemical properties of Al 6063 (Jindal Aluminium, Bengaluru, India).Te HSS end mill cutter was used for cutting operations under dry conditions.When compared to other tool steels, HSS tools have greater heat and wear resistance, durability, and hardness levels.HSS has a low coefcient of friction and great shock resistance, which helps to minimize chipping and breakage during use.
While conducting the test, two unidirectional piezoelectric accelerometers were fxed in the spindle to achieve spindle vibration and another on the worktable to achieve worktable vibration.So, the measured vibration in each channel was designated as Channel 1 and Channel 2 as shown in Table 3. Te LabVIEW TM software was used in the experiment to act as a data acquisition platform to acquire the vibration.

Prediction Using ANOVA.
ANOVA is a statistical test that compares the mean values of diferent groups to determine the infuence of one or more factors.Te ANOVA test is used in a regression analysis to examine the infuence of independent factors on the dependent variable.Moreover, the ANOVA test helps in determining the signifcance of an experiment's results.

Prediction of Spindle Vibration
Using ANOVA.Te spindle vibration was measured using channel 1, whose ANOVA results are shown in Table 4. Te F-value is 37.13, and the p value is less than 0.05 indicating that the model is adequate (signifcant) and there is a 0.01% magnitude chance of occurring due to noise.Te lack of ft F-value is 3.11, which indicates that the model term is insignifcant and there is an 11.71% chance that the lack of ft F-value is caused by noise.Te predicted R 2 of 0.7191 does not match the adjusted R 2 of 0.9589 as closely as one might expect.Te S/N ratio is measured by adeq precision.A ratio greater than 4 is preferred.Te S/N ratio of 22.034 suggests an adequate signal.
Te regression analysis equation of real machining parameters can be expressed as

(4)
Table 6 shows the adjusted R 2 and predicted R 2 values for spindle and worktable vibration.Te R 2 value and the predicted R 2 value are 0.9589 and 0.7191 for spindle vibration.Te R 2 value and the predicted R 2 value are 0.9711 and 0.8795 for worktable vibration.Te R 2 value is 0.9589 for spindle vibration and 0.9711 for worktable vibration.Te coefcient of determination R 2 indicates that the goodness of ft for the models is nearer to 1. So, the model is signifcant.

Interaction Efect for Spindle Vibration (A SV ).
Figure 3(a) shows the interaction plot and its efects on the spindle speed and the rate of feed on spindle vibration.Due to the wide contact area between the cutting tool and the workpiece and tool movement relative to the workpiece, an increase in the feed is followed by an increase in spindle vibration.However, when the spindle speed is between 2000 and 3000 rpm, the vibration is relatively low, indicating that a medium spindle speed produces better results.Figure 3(b) depicts the interaction plot of spindle speed and the depth of cut (axial) on spindle vibration.Te vibration is low when the spindle speed is between 2000 and 3000 rpm.However, when the depth of cut (axial) is between 0.6 and 0.7 mm, vibration is reduced.Tis might be because the contact area is damped, resulting in minimal vibration.Te interaction impact of spindle speed and radial depth on spindle vibration is depicted in Figure 3(c).As with the depth of cut (axial), the same result occurs.When the radial depth of cut is between 0.7 and 0.8 mm, however, vibration is reduced.Tis is due to the increased difculty of moving the tool into the workpiece.Te interaction impact of the spindle speed and radial rake angle of the tool on spindle vibration is shown in Figure 3(d).
When the radial rake angle is low, the vibration is very low.Tis might be because when the rake angle rises, although the speed should be between 2000 and 3000 rpm, the contact area between the tool and the workpiece grows, causing more chips to develop and increased vibration in the     Advances in Materials Science and Engineering spindle.Te interaction impact of the rate of feed and the axial depth of cut on spindle vibration is revealed in Figure 3(e).Figures 3(a) and 3(b) show that when the rate of feed rises, vibration increases and vibration decreases as the axial depth of cut decreases between 0.6 and 0.7 mm.Te interaction impact of the rate of feed and the radial depth of cut on spindle vibration is revealed in Figure 3(f ).As illustrated in Figures 3(a) and 3(c), the same outcome is duplicated.Figure 3(g) indicates that the rate of feed and the radial rake angle have a less interaction impact.Figure 3(h) depicts a plot of predicted vs. actual responses.

Interaction Efect for Worktable Vibration (A WTV ).
Figure 4(a) depicts the interaction efect of the spindle speed and the rate of feed on worktable vibration.Te rate of feed has less infuence on worktable vibration than the spindle speed.Te vibration is at its lowest between 2000 and 2750 rpm, and then, the vibration rises beyond these limits.As previously stated, the spindle vibrations will be more signifcant at lower and higher speeds, resulting in increased worktable vibrations.Tese spindle vibrations will be transmitted to the worktable, resulting in vibrations on the work surface.Te vibration on the worktable is lessened when the rate of feed is increased.Te worktable vibration is high between 0.06 and 0.1 mm/rev.Tis could be because the spindle dampens most of the vibration frequencies at lower and higher rates of feed, resulting in only minor vibrations in the worktable, whereas at an intermittent rate of feed, the vibrations transferred to the spindle are less and the majority of the vibrations are transferred to the worktable.
Te interaction impact of the spindle speed and axial depth of cut on worktable vibration is depicted in Figure 4(b).Tis case probably occurs due to the spindle vibrating more at lower depths of cut (axial) and higher speeds, and these vibrations are transmitted to the worktable, causing worktable vibrations.Still, there was no discernible efect at higher axial depths.When the spindle speed is low, the change in axial depth has little efect on the vibration of the worktable.Te increase in axial depth reduced worktable vibration, while the spindle speed was low between 2000 and 2750 rpm, with a maximum spindle speed of 3500 rpm.Tis might be due to the fact that even when the axial depth increases, the tool continues to remove material from the workpiece at a consistent rate, even at decreased spindle speeds.Figure 4(c) depicts the interaction infuence of the spindle speed and the radial depth of cutover worktable vibration.Te spindle speed should be between 2000 and 2750 rpm, and the radial depth should be between 0.6 and 0.8 mm to reduce worktable vibration.Tis might be because the spindle vibrates more at lower and greater radial depths, conveying this infuence to the worktable as well.Te vibrations will be damped at an intermediate value of the radial depth, resulting in low worktable vibrations.
Te impact of the spindle speed and rake angle on worktable vibration is shown in Figure 4(d).Te worktable vibration was found to be greater at lower rake angles than at higher rake angles.Worktable vibration gets minimum at a rake angle of 15 °-17 °and a radial depth of 0.70-0.80mm.Figures 4(e)-4(g) provide less signifcant graphs of the worktable vibration efects of the axial depth and rate of feed, radial depth and rate of feed, radial rake angle and rate of feed, and axial depth and radial depth, respectively.More radial depth produces a signifcant increase in worktable vibration at higher rake angles because of more contact between the tool and the workpiece, resulting in increased chatter.Figure 4(h) displays a plot of predicted vs. actual worktable vibration responses.

Multiobjective Optimization
Te objective of the function needs to be a minimum for optimizing spindle and worktable vibration.As a result, the process becomes complex, and in order to fnd the optimal solution, one must consider a multiobjective function [29].To optimize spindle and worktable vibration, the nondominant-based genetic algorithm (NSGA-II) is employed.Nondominant-Based Genetic Algorithm (NSGA-II).NSGA-II is an efective search strategy.It is driven natural selection, which is based on Darwin's theory.Te NSGA-II approach expands the scope of the search procedure.Tis approach intended to identify a collection of optimal solutions known as nondominated solutions, also known as the Pareto set.A nondominated solution is one that delivers an appropriate balance between all objectives without degrading any of them.
NSGA-II was used in this work to achieve multiobjective optimization with parameter constraints.Te constructed regression model is used by a genetic algorithm to anticipate the ideal relationship between cutting parameters.Te output responses are spindle vibration and worktable where A SV and A WTV limitations defne the lower and upper limits that must be fulflled.Among parameter ranges, 1500 ≤ spindle speed of the machine(N) ≤ 3500rpm, 0.04 ≤ rate of feed f z  ≤ 0.12 mm rev , 0.5 ≤ depth of cut(axial) a p   ≤ 0.9mm, 0.5 ≤ depth of cut(radial) a e  ≤ 0.9mm, 12 ≤ radial rake angle of tool(c) ≤ 24 degree. ( To achieve the best results, the following conditions were used: (i) A population size of 100 (ii) A population-type vector (iii) A scaling function rank of function (iv) Stochastic uniform crossover, (v) A Gaussian mutation function (vi) A mutation rate of 0.1 (vii) A scattered mutation function (viii) A crossover rate of 1.0 (ix) A generation size of 1000 Figure 5 displays in coded form the performance of ftness value creation and the best individual variable performances.Figure 6 displays the Pareto-optimal frontier scattered points for their results as a consequence of NSGA-II optimization conducted using MATLAB software.Te input parameter grouping of twenty-one sets of nondominated Pareto-optimal solutions based on the NSGA-II analysis is shown in Table 7. Te spindle vibration ranged from 0.02404 to 0.01336 mm/s 2 , while the worktable vibration ranged from 0.00284 to 0.00165 mm/s 2 .Table 6 shows that all of the NSGA-II generated solutions have an equal high level of agreement.According to the research results, all of the solutions created by NSGA-II are equally good.Te selection of Pareto-optimal solutions helps the manufacturing engineer's and researchers' expectations.

Conclusion
Based on the designed experiments, utilizing central composite design, the spindle vibration, and worktable vibration in terms of acceleration amplitude, the experiment is performed on aluminium 6063 using an HSS tool in a CNC milling machine.Te ANOVA technique was used to examine performance characteristics.Te machining variables and their responses are optimized by NSGA-II.
Spindle speed is an important factor in increasing spindle vibration amplitude.To ensure lesser vibration, the spindle speed should be between 2000 and 2800 rpm.In addition to spindle speed, the axial depth of the cut is an important element that contributes to vibration.Te axial depth of the incision should be between 0.7 and 0.8 mm to guarantee low vibration.Te cut's radial depth should be between 0.7 and 0.8 mm.Te radial rake angle should be between 12 °and 18 °for low vibration, and the feed rate should be between 0.04 and 0.08 mm/rev.
Te smallest worktable vibration was achieved by using spindle speeds ranging from 2000 to 3000 rpm, feed rates ranging from 0.1 to 0.12 mm/rev, axial depth of cut ranging from 0.5 to 0.7 mm, radial depth of cut ranging from 0.6 to 0.8 mm, and radial rake angles ranging from 15 °to 18 °.
MATLAB provided 21 nondominated Pareto-optimal solutions based on multiobjective optimization using the nondominated sorted genetic algorithm (NSGA-II).Te result obtained from NSGA-II is reasonably excellent.Te suggested hybrid approach minimizes the cost and reduction of vibration amplitude prediction for real-world machining conditions.
Te future direction of this research will be use of a carbide-cutting tool to conduct machining operations with respect to tool wear and surface roughness, as well as conducting the experiment based on measurement of the cutting force.

Figure 2 :
Figure 2: Photographic view of the machined specimen and the HSS tool.

Factor 18 Design
ASV) (mm/s2) A : Sp in dl e Sp ee d (N ) (r pm ) B : Fe ed R at e (F z) (m m /r ev ) Depth of Cut (ap) = 0.7 D: Radial Depth of Cut (ae) = 0.7 E: Radial rake Angle (γ) = 18 Design-Expert Sofware Design points above predicted value Design points below predicted value (a) C : A xi al D ep th of C ut (a p) (m m ) Spindle Vibration (ASV) (mm/s2) A : Sp in dl e Sp ee d (N ) (r pm ) : Spindle Speed (N) X2 = C: Axial Depth of Cut (ap) Actual Factors B: Feed Rate (Fz) = 0.08 D: Radial Depth of Cut (ae) = 0.7 E: Radial rake Angle (γ) = : Spindle Speed (N) X2 = D: Radial Depth of Cut (ae) Actual Factors B: Feed Rate (Fz) = 0.08 C: Axial Depth of Cut (ap) = 0.7 E: Radial rake Angle (γ) = 18 Design-Expert Sofware Design points above predicted value Design points below predicted value D : R ad ia l D ep th of C ut (a e) (m m ) : Spindle Speed (N) X2 = E: Radial rake Angle (γ) Actual Factors B: Feed Rate (Fz) = 0.08 C: Axial Depth of Cut (ap) = 0.7 D: Radial Depth of Cut (ae) = 0.7 Design-Expert Sofware Design points above predicted value Design points below predicted value E: R ad ia l ra ke A ng le (γ ) (D eg re e) : Feed Rate (Fz) X2 = C: Axial Depth of Cut (ap) Actual Factors A: Spindle Speed (N) = 2500 D: Radial Depth of Cut (ae) = 0.7 E: Radial rake Angle (γ) = 18 Design-Expert Sofware Design points above predicted value Design points below predicted value C : A xi al D ep th of C u t (a p ) : Feed Rate (Fz) X2 = D: Radial Depth of Cut (ae) Actual Factors A: Spindle Speed (N) = 2500 C: Axial Depth of Cut (ap) = 0.7 E: Radial rake Angle (γ) = 18 Design-Expert Sofware Design points above predicted value Design points below predicted value D : R ad ia l D ep th of C u t (a e) (m m )

18 DesignFigure 4 :
Figure 4: Interaction efect for worktable vibration of (a) N vs. f z over A WTV , (b) N vs. a p over A WTV , (c) N vs. a e over A WTV , (d) N vs. c over A WTV , (e)f z vs. a p over A WTV , (f )f z vs. a e over A WTV , (g)f z vs. c over A WTV , and (h) plot of predicted response vs. actual responses (worktable vibration).

Figure 5 :
Figure 5: Performance of ftness value generation and the best and most consistent variable performances in structured form: (a) distance of individuals, (b) average spread of generations, (c) rank histogram of individuals, and (d) GA optimization tool interface.

Table 1 :
Process parameters and their levels.
Te F-value is 53.04, and p values >0.05 show that the model is adequate (signifcant) and there is a 0.01% chance of

Table 2 :
Chemical properties of Al 6063.

Table 6 :
Adjusted R 2 and predicted R 2 values for spindle and worktable vibration.

Table 8 :
[33,34]ion of the model.MATLAB 2013a software was used for optimization purposes to optimize the vibration.Te input objective function utilizes the MOGA methodology[33,34]: to minimize: spindle vibration A SV : A SV N, f z , a p , a e , c  , to minimize: worktable vibration A : A WTV N, f z , a p , a e , c   subject to constraints: A SV ≤ A SV limit, A WTV ≤ A WTV limit,

Table 8
demonstrates that a regression model developed through RSM of DoE utilizing CCD was validated using a confrmatory test.Vibration of both the spindle speed and the worktable is compared between values predicted by NSGA-II and experimental values.Te percentage of error is determined to be within ±2%, confrming the model's validity.