# Optimization of Process Parameters for Laser Cutting Process of Stainless Steel 304: A Comparative Analysis and Estimation with Taguchi Method and Response Surface Methodology

Authors: Vi Nguyen; Faisal Altarazi; Thanh Tran
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The research presents an analysis and comparison of the Taguchi Design and Response Surface Methodology (RSM) in optimizing the laser cutting machine parameters on dimension accuracy for stainless steel products. The paper studies effects of input factors such as cutting speed, nitrogen pressure, power, and frequency on the quality cutting of stainless steel (304) specimens. In this paper, two objectives are examined: targeting laser-cut edge to perpendicular 90 degrees and maximizing the cutting accuracy. The paper proposes a simple formula to optimize both targets by minimizing one new function definition, i.e., dimensional error. An L9 orthogonal array of Taguchi Methodology is adopted to minimize the number of experiments and shorten analysis time to achieve the optimal parameters. These results are compared with RSM. Box–Behnken Design (BBD) type of RSM requires more experiments than the Taguchi approach. RSM regression models as the quadratic functions of the control factors are developed to minimize the dimensional error of cutting products. Then, Analysis of Variance (ANOVA) and graphs will be analyzed to determine the influences of variables on the responses. Both Taguchi’s method and RSM found that the most influential factor on dimension accuracy is cutting speed, followed by laser power. While Taguchi provides good graphic visualization for quickly predicting the optimum condition, it cannot examine the interaction effects as RSM due to the lack of data. Besides, RSM reveals the percentage contribution of factors on dimensional error. Cutting speed has a maximum contribution, i.e., 39% of the total. The interaction of cutting speed and power contributes 16% of the total. In this study, RSM can predict optimum conditions more accurately than Taguchi. There are misleading results from the Taguchi method compared with RSM. However, the difference between these objective values is insignificant. The validation experiments show that the Taguchi method can be a practical approach for optimization problems. It can help reduce cost and time and achieve the desired optimal outputs. With cutting problems requiring high precision, the RSM method is highly recommended for identifying optimal parameter settings and interaction effects. With problems that their experimental runs consume high cost and time, Taguchi can be a suitable method for screening the significant variables. Although Taguchi and RSM are used widely for optimization problems in many fields, choosing the right methodology for various objectives is still a concern with different arguments and needs further research. Therefore, this study could be an adequate reference for parameter optimization problems in various fields.


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## Body

## 1. Introduction

Laser cutting is one of the most widespread methods to shape and separate a metal workpiece into desired parts. Laser material processing provides solutions for precise and complex geometry. Also, it can perform various cutting from hard materials to soften ones. However, this technology requests users’ professional knowledge to apply it effectively. Otherwise, it will cost waste of energy and raw materials. In general, with each type of laser cutting machine and well-defined applications, manufacturers will provide a comprehensive database to set up process parameters. However, a new customized target or geometry usually requires individual parameters optimization. And optimizing a multitude of factors for laser cutting machines might require time-consuming and intensive costs, especially if possible interactions of different process parameters are considered.Many types of laser systems, including oxygen and nitrogen, are currently in existence. The nitrogen laser cutting system provides many benefits, including a high output quality and processing power. Selecting the most suitable parameter settings becomes one of the most practically essential tasks to improve product quality and productivity simultaneously in laser cutting. The effect of laser control factors on the material should be treated as an essential issue in order to increase product quality and productivity [1]. Pradhan and Biswas research examined the impacts of the power factor, including discharge current and applied voltage, on the surface roughness of metallic inputs [2]. In the cutting process, the use of higher cutting speeds causes an increase in the cutting temperature zone, which leads to faster cutting tool wear, then affects dimensional accuracy, surface roughness, and tool life [3, 4]. Kotadiya and Pandya indicated that the laser cutting quality depends on the laser power, gas pressure, cutting speed, beam diameter, beam incident angle, stand-off distance, pulse frequency, and focus positions [5]. Many experiments have been conducted to investigate the effect of process parameters on the quality of cutting surfaces. Different researchers have used different methods to optimize the cutting parameters [6, 7]. However, according to Gvozdev and Golyshev, a reliable method for forecasting the quality of cutting surface and choosing the optimum initial parameters for workpiece sheets has not been developed yet [8]. Thus, it usually takes a lot of time to characterize the optimum conditions that allow for a high-quality cutting surface.Design of experiment (DOE) is a systematic method that is applied to many optimization problems. DOE allows achieving great information from the limited executing experiments. Response Surface Method (RSM) and Taguchi design are two DOE approaches that use widely and prove their effective applications in many fields [9–13]. Venkata Rao and Murthy employed the Central Composite Design (CCD) with 18 experiments to develop statistic models like RSM, Artificial neural networks (ANN), and support vector machine (SVM). These models were to investigate the effects of cutting parameters on surface roughness and workpiece vibration [14]. Tosun and Ozler investigated the effect of cutting parameters on tool life and the workpiece surface roughness by hot-turning optimization. They applied the Taguchi methodology to learn the parameter influences and for multiple performance optimizations [15]. Huehnlein et al. used design of experiments (DOE) and the response surface methodology (RSM) to study the optimization of laser cutting of thin Al2O3 ceramic layers [16]. They studied the process parameters of the laser cutting machine including laser power, velocity, distance from the nozzle to the surface, the pressure of gas, and position to the focus. The paper demonstrated the potential of DOE approach in process optimization. Genna et al. studied the influences of material type, workpiece thickness, cutting speed, and gas pressure on the kerf quality in laser cutting. Based on DOE, they developed a systematic factorial design to test the varying control factors and gave recommendations for optimal cutting conditions for St37-2 low-carbon steel, AlMg3 aluminum alloy, and AISI 304 stainless steel [17].Compared to RSM, Taguchi design helps in reducing significantly the number of tests which will save time and cost for users [18, 19]. Taguchi’s method enables analyzing many process parameters and their effects on multiresponses [20]. Taguchi design is consistent and recognizes control and noise factors. However, its effectiveness and accuracy predictability need to be examined for particular quality problems. Although Taguchi and RSM are used widely for optimization problems in many fields, choosing the right methodology for various objectives is still a concern with different arguments and needs further research.Stainless steel contributes as a widely used material in many fields such as food production, chemical, textile, health, pharmaceutical, nuclear, etc. However, it is a challenging machinable material due to its strength and hardening processing rate (Figure1(b)). Stainless steel (304) is chosen to optimize the laser cutting process in this paper because of its wide applications in production and consumption. The qualities of outputs in terms of dimensional accuracy are evaluation criteria. This research aims to examine the influences of parameter settings on the dimensional error in a laser cutting process. Dimensional error is defined as a combined function with two objectives to minimize, i.e., the inclination of the cut edge and the deviation of the cutting dimension with the target value. The parameters include power supply, nitrogen pressure, laser beam frequency, and cutting speed. The stainless-steel specimens are cut by the 2D laser cutting machine TruLaser 2030. This paper researches two methodologies, i.e., Taguchi and RSM. The authors first use two approaches for finding optimal parameter settings for the laser machines for an objective, minimizing dimensional error. It investigates the relationship of factors, interactions, and responses. Then the research compares the results of two approaches for further applications. Through the optimization problem in the laser cutting process, the paper gives a guideline for applying the suitable methodology in different circumstances.Figure 1
(a) Specimen design according to ASTM D638, all dimensions in mm; (b) SS304 chemical composition and mechanical properties.
(a)(b)


## 2. Materials and Methods

### 2.1. Specimen and Testing Methods

The laser cutting machine (Trulaser 2030) at the laboratory of Vietnamese German University (VGU) was used to carry out experiments (Figure2). A 3 mm thick stainless steel-304 (SS-304) was kept constant throughout the runs. Due to SS- 304 having lower carbon which minimizes carbide precipitation, the research selects it as workpiece material. Specimen is drawn in detailed dimension to input to the laser cutting machine like Figure 1(a). Figure 1(b) shows the information of the chemical composition and mechanical properties of the SS-304 3 mm thickness used in this paper.Figure 2
Illustration for the cutting process.Four control factors were taken for the evaluation process. The ranges of chosen independent variables, which influence the response, and dimensional accuracy, are given in Table1.The Cutting Procedure. First of all, the 3D model of the specimen is designed with the required specifications (Figure 1(a)) in CAD software. Then this CAD model will be converted into a DXF file and input into the laser cutting machine (Trulaser 2030). The cutting material is a 3 mm thick stainless steel-304 (SS-304) sheet. Next, the machine parameter factors are adjusted based on the design matrix table (Tables 2 and 3). The control factors must be input carefully for each different experiment. Then the machine cut the metal sheet into specimen shape with the setup control factors. After the cutting process, the specimens are taken out from the SS-304 sheet frame. This process must be implemented slowly and carefully to avoid damaging the side surface of the output (the contact surface between the cut specimens and the leftover metal sheet). Finally, the measurement process is carried on afterward.Inspection of Specimen. The testing equipment supported for this inspection is a digital caliper which can release a sensitive measuring value (Figure 3). The Mitutoyo caliper is prepared and calibrated precisely before the measurement procedure. The laser cutting angle effect on the cut specimens was measured by measuring the gauge width of the upper and lower surfaces of the samples.Table 1
Main factors and their value ranges.
Control factorUnitMin valueMax value(A) PowerWatt19002000(B) Nitrogen pressureMPa1525(C) FrequencyHz1900020000(D) Cutting speedm/min2.55.5In this dimensional checking, if the top and the bottom gage width have the same value, the laser-cut edge is perpendicular (90 degrees), or the kerf taper angle is 0 degrees. In addition, the dimensional error also is a principal for quality checking. The specification value of the gage width is 6 mm. The deviation of measured width for the top and bottom surfaces of the gage to the target value (6 mm) should be as small as possible.Table 2
Box–behnken design (RSM) and response.
RunOrderPower (A)Nitrogen pressure (B)Frequency (C)Cutting speed (D)KtKbDIFF|Kb − Kt|Y|Kb − 6| + |6 − Kt|WattMPaHzm/minmmmmmmmm11950252000045.955.990.040.062195020200005.56.16.20.10.33195020190002.56.036.0300.0641950251900045.975.9700.0651950201950046.046.050.010.096200020195002.55.986.020.040.0471900251950046.036.070.040.181900151950046.026.030.010.059195015195002.55.9960.010.01101900201900045.966.080.120.12112000202000045.976.020.050.0512195025195005.56.26.30.10.513190020195002.55.996.010.020.0214195020200002.55.996.020.030.0315195015195005.566.10.10.11619502019500466.020.020.02172000151950045.965.980.020.0618190020195005.56.36.300.61919501519000466.10.10.120195020190005.56.156.20.050.3521200020195005.56.016.0100.02221900202000046.016.120.110.13232000201900046.056.080.030.132419502019500466.010.010.0125195025195002.55.965.980.020.06261950152000045.9960.010.01272000251950045.996.020.030.03Table 3
Taguchi experiment results.
NoABCDKtKbYS/N1190015190002.55.955.960.0920.91512190020195003.255.965.980.0624.43731900252000045.865.910.2312.765441950151950045.955.970.0821.93825195020200002.55.916.000.0920.91516195025190003.255.975.980.0526.02067200015200003.255.955.970.0821.938282000201900045.936.060.1317.72119200025195002.55.895.990.1218.4164Figure 3
Measuring needle, width measurement.In this paper, an objective function is developed to minimize dimensional error. Dimensional error is calculated based on the difference between top and underneath gauge diameters and their deviation from the nominal value of 6 mm.(1)Y=Kt−6+Kb−6,where Kt represents the top gage width; Kb represents the bottom gage width.From Figure4 and the objective function, we can see that a minimum value of Y can only be obtained (target value = 0) when the workpiece edges are cut accurately 90 degrees, and the cutting dimensions of the top and bottom gage are both equal to the nominal value, i.e., 6 mm.Figure 4
Section view of the gage section.


### 2.2. Research Methodologies

RSM is a collection of advanced design of experiments (DOE) techniques that are practical and helpful for modeling the relationship between the process factors and the output [11]. RSM is often used to refine models, especially with suspect curvature in their response surface. Rotational Central Composite design (RCCD) and Box–Behnken Design (BBD) are two common types of RSM used for experimentation. BBD is a proficient and powerful design. It is normally used when performing nonsequential experiments. RCCD requires more experiments than BBD. In addition, RCCD may include extreme settings that can be beyond the safe operating limits. But BBD can ensure all factors are in the region of interest. BBD requires three levels per factor. The treatment combinations are at the midpoints of the edges of the experimental space and at the center [10]. BBD has the advantage of requiring fewer experiments than RCCD. Figure 5 shows an example of a Box–Behnken Design for three factors. In this study, RSM in a type BBD is conducted with two goals. The first target is to find the optimal settings to minimize the dimensional error. The second one is to investigate the relationship between factors and responses to understand the system. Figure 6 shows the steps of applying RSM in this paper. Preliminary research was conducted to identify the significant parameters and their range on dimensional accuracy. They are laser power, gas pressure, nitrogen frequency, and cutting speed. Then, RCCD and BBD are compared to select a suitable RSM design. With four control factors, BBD generates 27 experiments corresponding to factors’ values in Table 2. The laser machine is set up based on the design matrix table. Then observed and measured output is recorded for all 27 runs. Based on the collected data, RSM builds a suitable regression model which reflects the relationship between factors variance and the shift of the response. ANOVA table is studied to check the adequacy of the model and analyze the effect of control parameters and their interactions on the dimensional error. Many researchers used it as a powerful and effective tool to analyze the influence of the experimental parameters on the performance characteristics [21, 22].Figure 5
A Box–Behnken design for three factors.Figure 6
Seven steps of RSM.Besides RSM, the research also adopts the Taguchi approach to solve the problem. Taguchi design uses an Orthogonal Array (OA) which requires a limited number of runs. It is a fractional factorial design which is balanced matrix weighted factor levels equally. This paper adopted anL9 orthogonal array for three-level four factors analysis shown in Table 4.Table 4
L9 orthogonal array for 4 factors and 3 levels design (Taguchi).
No. of experimentABCDNo. of experimentABCD111111190015190002.5212222190020195003.25313333190025200004421234195015195004522315195020200002.5623126195025190003.25731327200015200003.25832138200020190004933219200025195002.5The Taguchi approach recommends a signal-to-noise (S/N) ratio for quality optimization. Maximizing the value of S/N of a process means minimizing the effects of the noise factors and variability. The S/N ratio is applied for continuous characteristics and can be divided into three types:Nominal-the-best:(2)SN=10∗logy¯sy2.Smaller-the-better:(3)SN=−10∗log101n∑i=1nYi2.Larger-the-better:(4)SN=−10∗log101n∑i=1n1Yi2,where y is response, y¯ is the average of response value, n is the number of observations, and sy2 is variance of y.In the Taguchi approach, the S/N ratio represents quality characteristics. The higher values of the S/N present better design factor settings to make the system more robust [19]. In this research, a lower-dimensional error is desirable for high cutting quality. Therefore, the type of S/N analysis, smaller-is-better ((3), will be applied.


## 2.1. Specimen and Testing Methods

The laser cutting machine (Trulaser 2030) at the laboratory of Vietnamese German University (VGU) was used to carry out experiments (Figure2). A 3 mm thick stainless steel-304 (SS-304) was kept constant throughout the runs. Due to SS- 304 having lower carbon which minimizes carbide precipitation, the research selects it as workpiece material. Specimen is drawn in detailed dimension to input to the laser cutting machine like Figure 1(a). Figure 1(b) shows the information of the chemical composition and mechanical properties of the SS-304 3 mm thickness used in this paper.Figure 2
Illustration for the cutting process.Four control factors were taken for the evaluation process. The ranges of chosen independent variables, which influence the response, and dimensional accuracy, are given in Table1.The Cutting Procedure. First of all, the 3D model of the specimen is designed with the required specifications (Figure 1(a)) in CAD software. Then this CAD model will be converted into a DXF file and input into the laser cutting machine (Trulaser 2030). The cutting material is a 3 mm thick stainless steel-304 (SS-304) sheet. Next, the machine parameter factors are adjusted based on the design matrix table (Tables 2 and 3). The control factors must be input carefully for each different experiment. Then the machine cut the metal sheet into specimen shape with the setup control factors. After the cutting process, the specimens are taken out from the SS-304 sheet frame. This process must be implemented slowly and carefully to avoid damaging the side surface of the output (the contact surface between the cut specimens and the leftover metal sheet). Finally, the measurement process is carried on afterward.Inspection of Specimen. The testing equipment supported for this inspection is a digital caliper which can release a sensitive measuring value (Figure 3). The Mitutoyo caliper is prepared and calibrated precisely before the measurement procedure. The laser cutting angle effect on the cut specimens was measured by measuring the gauge width of the upper and lower surfaces of the samples.Table 1
Main factors and their value ranges.
Control factorUnitMin valueMax value(A) PowerWatt19002000(B) Nitrogen pressureMPa1525(C) FrequencyHz1900020000(D) Cutting speedm/min2.55.5In this dimensional checking, if the top and the bottom gage width have the same value, the laser-cut edge is perpendicular (90 degrees), or the kerf taper angle is 0 degrees. In addition, the dimensional error also is a principal for quality checking. The specification value of the gage width is 6 mm. The deviation of measured width for the top and bottom surfaces of the gage to the target value (6 mm) should be as small as possible.Table 2
Box–behnken design (RSM) and response.
RunOrderPower (A)Nitrogen pressure (B)Frequency (C)Cutting speed (D)KtKbDIFF|Kb − Kt|Y|Kb − 6| + |6 − Kt|WattMPaHzm/minmmmmmmmm11950252000045.955.990.040.062195020200005.56.16.20.10.33195020190002.56.036.0300.0641950251900045.975.9700.0651950201950046.046.050.010.096200020195002.55.986.020.040.0471900251950046.036.070.040.181900151950046.026.030.010.059195015195002.55.9960.010.01101900201900045.966.080.120.12112000202000045.976.020.050.0512195025195005.56.26.30.10.513190020195002.55.996.010.020.0214195020200002.55.996.020.030.0315195015195005.566.10.10.11619502019500466.020.020.02172000151950045.965.980.020.0618190020195005.56.36.300.61919501519000466.10.10.120195020190005.56.156.20.050.3521200020195005.56.016.0100.02221900202000046.016.120.110.13232000201900046.056.080.030.132419502019500466.010.010.0125195025195002.55.965.980.020.06261950152000045.9960.010.01272000251950045.996.020.030.03Table 3
Taguchi experiment results.
NoABCDKtKbYS/N1190015190002.55.955.960.0920.91512190020195003.255.965.980.0624.43731900252000045.865.910.2312.765441950151950045.955.970.0821.93825195020200002.55.916.000.0920.91516195025190003.255.975.980.0526.02067200015200003.255.955.970.0821.938282000201900045.936.060.1317.72119200025195002.55.895.990.1218.4164Figure 3
Measuring needle, width measurement.In this paper, an objective function is developed to minimize dimensional error. Dimensional error is calculated based on the difference between top and underneath gauge diameters and their deviation from the nominal value of 6 mm.(1)Y=Kt−6+Kb−6,where Kt represents the top gage width; Kb represents the bottom gage width.From Figure4 and the objective function, we can see that a minimum value of Y can only be obtained (target value = 0) when the workpiece edges are cut accurately 90 degrees, and the cutting dimensions of the top and bottom gage are both equal to the nominal value, i.e., 6 mm.Figure 4
Section view of the gage section.


## 2.2. Research Methodologies

RSM is a collection of advanced design of experiments (DOE) techniques that are practical and helpful for modeling the relationship between the process factors and the output [11]. RSM is often used to refine models, especially with suspect curvature in their response surface. Rotational Central Composite design (RCCD) and Box–Behnken Design (BBD) are two common types of RSM used for experimentation. BBD is a proficient and powerful design. It is normally used when performing nonsequential experiments. RCCD requires more experiments than BBD. In addition, RCCD may include extreme settings that can be beyond the safe operating limits. But BBD can ensure all factors are in the region of interest. BBD requires three levels per factor. The treatment combinations are at the midpoints of the edges of the experimental space and at the center [10]. BBD has the advantage of requiring fewer experiments than RCCD. Figure 5 shows an example of a Box–Behnken Design for three factors. In this study, RSM in a type BBD is conducted with two goals. The first target is to find the optimal settings to minimize the dimensional error. The second one is to investigate the relationship between factors and responses to understand the system. Figure 6 shows the steps of applying RSM in this paper. Preliminary research was conducted to identify the significant parameters and their range on dimensional accuracy. They are laser power, gas pressure, nitrogen frequency, and cutting speed. Then, RCCD and BBD are compared to select a suitable RSM design. With four control factors, BBD generates 27 experiments corresponding to factors’ values in Table 2. The laser machine is set up based on the design matrix table. Then observed and measured output is recorded for all 27 runs. Based on the collected data, RSM builds a suitable regression model which reflects the relationship between factors variance and the shift of the response. ANOVA table is studied to check the adequacy of the model and analyze the effect of control parameters and their interactions on the dimensional error. Many researchers used it as a powerful and effective tool to analyze the influence of the experimental parameters on the performance characteristics [21, 22].Figure 5
A Box–Behnken design for three factors.Figure 6
Seven steps of RSM.Besides RSM, the research also adopts the Taguchi approach to solve the problem. Taguchi design uses an Orthogonal Array (OA) which requires a limited number of runs. It is a fractional factorial design which is balanced matrix weighted factor levels equally. This paper adopted anL9 orthogonal array for three-level four factors analysis shown in Table 4.Table 4
L9 orthogonal array for 4 factors and 3 levels design (Taguchi).
No. of experimentABCDNo. of experimentABCD111111190015190002.5212222190020195003.25313333190025200004421234195015195004522315195020200002.5623126195025190003.25731327200015200003.25832138200020190004933219200025195002.5The Taguchi approach recommends a signal-to-noise (S/N) ratio for quality optimization. Maximizing the value of S/N of a process means minimizing the effects of the noise factors and variability. The S/N ratio is applied for continuous characteristics and can be divided into three types:Nominal-the-best:(2)SN=10∗logy¯sy2.Smaller-the-better:(3)SN=−10∗log101n∑i=1nYi2.Larger-the-better:(4)SN=−10∗log101n∑i=1n1Yi2,where y is response, y¯ is the average of response value, n is the number of observations, and sy2 is variance of y.In the Taguchi approach, the S/N ratio represents quality characteristics. The higher values of the S/N present better design factor settings to make the system more robust [19]. In this research, a lower-dimensional error is desirable for high cutting quality. Therefore, the type of S/N analysis, smaller-is-better ((3), will be applied.


## 3. Experimental Results and Analysis

### 3.1. Response Surface Methodology

The cutting process was repeated continuously for 27 specimens with different setting control factors. The difference between top and bottom gage width (DIFF) and the dimensional error (Y) is measured as shown in Table 2.In the experiment results, the bottom gage width of specimens can be higher than the top one due to the cut edge inclination. The accuracy of the cut edge has effects on the tolerances of cutting parts. Therefore, the inclination of the cut edge should be minimized. The difference (DIFF) between the bottom gage width and the top gage width can explain the gradient of the cut. DIFF equal to 0 means that the laser-cut edges are perpendicular.However, the DIFF function is only for predicting the cutting inclination. In some cases, the laser-cut edge is perpendicular (DIFF = 0), but the dimensions of widths are not as desired specification value (6 mm). So objective valueY is calculated as in (1). Y considers both targets, i.e., perpendicular cutting and dimensional accuracy in the function model.The quadratic models were popularly used and verified their accuracy in evaluating the quantitative relationship between laser processing parameters and cutting output [23, 24]. In this paper, a full quadratic function is chosen to build the regression model. A regression equation in uncoded units below shows a correlation between the response function with the parameter variables.(5)yx=b0+∑i=1lbixi+∑i=1l−1∑j=i+1lbijxixj+∑i=1lbiixi2+ε,where b’s are the coefficients of the regression model; xi, xj are the different control factors; l is the total number of control factors; y is the response; and ε is the error.In this research, Minitab software is used to generate Analysis of Variance (ANOVA) tables and regression models. ANOVA tables are used to examine the effect of factors on the response. ANOVA quantitatively examines the sources of variation and relative contributions of each control factor and interaction through the sum of squares technique.Below is the ANOVA table and regression model for the objective function.Regression model:(6)Y=37−0.0117A−0.05555B−0.00318C+3.47D+0.0000B∗B+0.000000C∗C+0.0519D∗D−0.000080A∗B−0.0000001A∗C−0.002000A∗D+0.000009−0.000007C∗D.From the ANOVA analysis in Table5, to identify the significant terms for dimensional error statistical analysis, we use the p value. It is a probability to test the null hypothesis about correlation. The null hypothesis states that there is no correlation between a term (control factors or their interactions) and the dependent variable.Table 5
ANOVA table.
SourceDFAdj SSAdj MSF-valuep valueModel140.4997660.0356985.440.003Linear40.2905500.07263811.070.001A10.0396750.0396756.050.030B10.0192000.0192002.930.113C10.0048000.0048000.730.409D10.2268750.22687534.59<0.001Square40.0828410.0207103.160.055A∗A10.0031150.0031150.470.504B∗B10.0000230.0000230.000.954C∗C10.0049340.0049340.750.403D∗D10.0725930.07259311.070.0062-way interaction60.1263750.0210633.210.040A∗B10.0016000.0016000.240.630A∗C10.0020250.0020250.310.589A∗D10.0900000.09000013.720.003B∗C10.0020250.0020250.310.589B∗D10.0306250.0306254.670.052C∗D10.0001000.0001000.020.904Error120.0787080.006559Lack-of-fit100.0749080.0074913.940.219Pure error20.0038000.001900Total260.578474A significance level of 0.05, which indicates 5% risk type 1, is chosen. Or we can say 5% risk of error when we conclude the model explains the variation in the response. However, it is not true in reality. If the probability is less than or equal to 0.05, we can conclude that there is an association between the changes in the corresponding factor and the variation in the response.For main factors, we can see that power (A) and cutting speed (D) are statistically significant to the output, their p value lesser than 0.05. Interaction of AD has a strong influence on the response. Or we can say that the relationship between power and the dimensional error depends on predictor D, cutting speed. Other terms with a p value larger than 0.05 indicate that there is not enough evidence to conclude the statistically significant association between these terms and the output. These analyses are also presented clearly in the Pareto chart (Figure 7).Figure 7
Pareto chart of RSM.To get more information, the percentage of contribution (PCR) of each factor can simply be calculated by dividing the sum of squares (Adj SS) of each term by the total Adj SS.(7)PCRi=Sum of  squaresAdj SSof each termThe total sum of squaresAdj SS totalx100.Substituting the Adj SS value of each term from the ANOVA table to (7), we can see that the cutting speed (D) has the maximum influence on Y and has a percentage contribution of 39%.(8)PCRD=Sum of squaresAdj SSof DAdj SS totalx100=0.2268750.578474x100=39%.The PCR of other parameters such as power (A), nitrogen pressure (B), and frequency (C) is 7%, 3%, and 1%, respectively. The interaction of A and D has a strong impact on the variance of the output, with a contribution of 16%. PCR of other factors and interactions also can be calculated in the same way for effect analysis and consideration.From the ANOVA table, a lack-of-fit value is determined to predict the variation and the fitness of the model. The probability of Lack of Fit (p = 0.219 > 0.05) indicates that the regression model is sufficient. The model equation, as expressed, is suitable for describing the response. The R-square value of the regression model is 86.39%. It means that the model is explained by 86.39% of the variance in response.Figure8 depicts the main effects on Y when we consider only the influence of independent variables on the response. When laser power (A) increases, it will decrease the value of dimensional error (Y). At the same time, increasing nitrogen pressure (B) will also increase the value of Y. The relation of frequency (C) and cutting speed (D) to Y follow curve functions where the minimal value of Y will correspond to the lowest point on the curve.Figure 8
Main effects plot forY.However, the variance of dimensional error depends not only on the main factors but also on their interactions. Figure9 examines the associating parameters on the response. The more difference between gradients of respective factors is, the stronger the interaction is. We can see that AD has a strong interaction. The following is BD, then AC, and BC. They have weak influences. The interactions of AB and CD almost do not impact the shift of the response.Figure 9
Interaction plot forY.The contour plots (Figure10) and surface plots (Figure 11) examine the relationship between a fitted response and two factors. The contour plot of Y versus D, A shows that the optimal region for Y is strongly impacted by the interaction between A and D. In general, when we increase the cutting speed, we should also increase power to reduce dimensional error. However, at low cutting speed, excessive power can result in bad cutting quality in the high burr or the wide kef width on the corresponding specimens. At the low cutting speed combined with a high level of laser power, the Y value is increased significantly. On the other side, high cutting speed (D = 5.5 m/min) and insufficient power also reduce the dimensional accuracy of output. Even some parts of the specimens cannot be cut under this condition. By observation, the underside may have dross remaining on due to not cutting penetration. Balancing the cutting speed and the power is strongly impacting the cutting quality.Figure 10
Contour plots.
(a)(b)Figure 11
3D surfaces plots.The contour plot ofY versus D and B (hold A = 1900 W, and C = 19500 MPa) shows that the optimal region for Y corresponds to the value of D lower than 3.6. From 15 to 25 MPa is a suitable range of nitrogen pressure for cutting stainless steel. And the cutting dimensional error is not much impacted when varying nitrogen pressure in this range. The effect of control factors and their interactions on the response are visualized clearer on the 3D surface plots (Figure 11).Figure12 presents residual versus fits, the histogram plot, and the constant variance test. From the histogram, we can see the normal distribution of residuals. In the normal probability plot, residuals fall on a straight line error. From the residuals versus fits graph, it is obvious that the residuals do not follow any pattern. They are randomly distributed.Figure 12
Residual plots forY.Using the response optimizer function in Minitab, an optimal parameter set to minimizeY is found (Figure 13). The optimum yield was achieved at the power of 1900 W, nitrogen pressure of 25 MPa, frequency of 19252.5 Hz, and Cutting Speed of 2.5 m/min.Figure 13
Optimal condition.The model predicts that the optimum value ofy is −0.0861. In reality, the value of dimensional error is always nonnegative. There are always errors or residuals between the fitted value and the observed one. Therefore, the value of y calculated from the regression model can be less than 0. The actual optimal value will equal the optimal predicted value plus error. The plot of residuals versus fits for this regression model is shown in Figure 12.


### 3.2. Taguchi Method

Below is theL9 (34) Taguchi design table and the corresponding measured data of top and bottom gage widths. Y is calculated by equation (1). And S/N ratio is obtained by applying equation (3).For example,(9)Y1=Kt−6+Kb−6=5..95−6+5.96−6=0.09,S/N1=−10∗log101n∑i=1nYi2=−10∗log100.092=20.9151.Other values ofY and S/N in Table 3 are calculated similarly to the example.Response Tables6 and 7 are for SN ratio and means. At each level of each factor, the averages of the response values and the SN ratio are calculated. For example, factor A has three levels and three measurements at each level.(10)YA=19000¯=0.09+0.06+0.233=0.12667,SNA=19000¯=20.9151+24.4370+12.7543=19.37.Table 6
Response table for signal-to-noise ratios.
LevelABCD119.3721.6021.5520.08222.9621.0221.6024.13319.3619.0718.5417.47Delta3.602.533.066.66Rank2431Table 7
Response table for means.
LevelABCD10.126670.083330.090000.1000020.073330.093330.086670.0633330.110000.133330.133330.14667Delta0.053330.050000.046670.08333Rank2341Delta value is calculated by subtracting the lowest average from the highest one. Based on the delta value, the rank of the factor effect is defined. From the response table for means, the factor that has the biggest delta value, i.e., has the largest effect onY (ranked 1) is D. The second rank is A, followed by C and B.The same analysis for the SN ratios response table,D is ranked 1, followed by A, B, and C.Figures14 and 15 indicate the change in means and SN ratio when a given parameter varies from one level to another level. These trends match the results found in RSM. Based on Figure 15, the optimal conditions can be predicted. Optimal factors levels that minimize dimensional error or maximize SN ratio are 1950 W power, 15 MPa nitrogen pressure, 19500 Hz frequency, 3.25 m/min of cutting speed.Figure 14
Main effects plot for mean.Figure 15
Main effects plot for SN ratios.The optimum conditions obtained by using RSM and Taguchi are not similar (Table8). Confirmation experiments with predicted optimal parameter sets were carried out for quality verification. Results show that the parameter settings anticipated by RSM obtained a smaller value of dimensional error (Yopt = 0.01) than the Taguchi optimal condition (Yopt = 0.03). The misleading results may be due to a small number of data in the Taguchi method that cannot examine the interaction effects.Table 8
Comparison of predicted optimal conditions of RSM and Taguchi method.
Control factorUnitRSM optimal setTaguchi optimal set(A) PowerWatt19001950(B) Nitrogen pressureMPa2515(C) FrequencyHz1925319500(D) Cutting speedm/min2.53.25Results of validation experiments Yopt (mm)0.010.03


## 3.1. Response Surface Methodology

The cutting process was repeated continuously for 27 specimens with different setting control factors. The difference between top and bottom gage width (DIFF) and the dimensional error (Y) is measured as shown in Table 2.In the experiment results, the bottom gage width of specimens can be higher than the top one due to the cut edge inclination. The accuracy of the cut edge has effects on the tolerances of cutting parts. Therefore, the inclination of the cut edge should be minimized. The difference (DIFF) between the bottom gage width and the top gage width can explain the gradient of the cut. DIFF equal to 0 means that the laser-cut edges are perpendicular.However, the DIFF function is only for predicting the cutting inclination. In some cases, the laser-cut edge is perpendicular (DIFF = 0), but the dimensions of widths are not as desired specification value (6 mm). So objective valueY is calculated as in (1). Y considers both targets, i.e., perpendicular cutting and dimensional accuracy in the function model.The quadratic models were popularly used and verified their accuracy in evaluating the quantitative relationship between laser processing parameters and cutting output [23, 24]. In this paper, a full quadratic function is chosen to build the regression model. A regression equation in uncoded units below shows a correlation between the response function with the parameter variables.(5)yx=b0+∑i=1lbixi+∑i=1l−1∑j=i+1lbijxixj+∑i=1lbiixi2+ε,where b’s are the coefficients of the regression model; xi, xj are the different control factors; l is the total number of control factors; y is the response; and ε is the error.In this research, Minitab software is used to generate Analysis of Variance (ANOVA) tables and regression models. ANOVA tables are used to examine the effect of factors on the response. ANOVA quantitatively examines the sources of variation and relative contributions of each control factor and interaction through the sum of squares technique.Below is the ANOVA table and regression model for the objective function.Regression model:(6)Y=37−0.0117A−0.05555B−0.00318C+3.47D+0.0000B∗B+0.000000C∗C+0.0519D∗D−0.000080A∗B−0.0000001A∗C−0.002000A∗D+0.000009−0.000007C∗D.From the ANOVA analysis in Table5, to identify the significant terms for dimensional error statistical analysis, we use the p value. It is a probability to test the null hypothesis about correlation. The null hypothesis states that there is no correlation between a term (control factors or their interactions) and the dependent variable.Table 5
ANOVA table.
SourceDFAdj SSAdj MSF-valuep valueModel140.4997660.0356985.440.003Linear40.2905500.07263811.070.001A10.0396750.0396756.050.030B10.0192000.0192002.930.113C10.0048000.0048000.730.409D10.2268750.22687534.59<0.001Square40.0828410.0207103.160.055A∗A10.0031150.0031150.470.504B∗B10.0000230.0000230.000.954C∗C10.0049340.0049340.750.403D∗D10.0725930.07259311.070.0062-way interaction60.1263750.0210633.210.040A∗B10.0016000.0016000.240.630A∗C10.0020250.0020250.310.589A∗D10.0900000.09000013.720.003B∗C10.0020250.0020250.310.589B∗D10.0306250.0306254.670.052C∗D10.0001000.0001000.020.904Error120.0787080.006559Lack-of-fit100.0749080.0074913.940.219Pure error20.0038000.001900Total260.578474A significance level of 0.05, which indicates 5% risk type 1, is chosen. Or we can say 5% risk of error when we conclude the model explains the variation in the response. However, it is not true in reality. If the probability is less than or equal to 0.05, we can conclude that there is an association between the changes in the corresponding factor and the variation in the response.For main factors, we can see that power (A) and cutting speed (D) are statistically significant to the output, their p value lesser than 0.05. Interaction of AD has a strong influence on the response. Or we can say that the relationship between power and the dimensional error depends on predictor D, cutting speed. Other terms with a p value larger than 0.05 indicate that there is not enough evidence to conclude the statistically significant association between these terms and the output. These analyses are also presented clearly in the Pareto chart (Figure 7).Figure 7
Pareto chart of RSM.To get more information, the percentage of contribution (PCR) of each factor can simply be calculated by dividing the sum of squares (Adj SS) of each term by the total Adj SS.(7)PCRi=Sum of  squaresAdj SSof each termThe total sum of squaresAdj SS totalx100.Substituting the Adj SS value of each term from the ANOVA table to (7), we can see that the cutting speed (D) has the maximum influence on Y and has a percentage contribution of 39%.(8)PCRD=Sum of squaresAdj SSof DAdj SS totalx100=0.2268750.578474x100=39%.The PCR of other parameters such as power (A), nitrogen pressure (B), and frequency (C) is 7%, 3%, and 1%, respectively. The interaction of A and D has a strong impact on the variance of the output, with a contribution of 16%. PCR of other factors and interactions also can be calculated in the same way for effect analysis and consideration.From the ANOVA table, a lack-of-fit value is determined to predict the variation and the fitness of the model. The probability of Lack of Fit (p = 0.219 > 0.05) indicates that the regression model is sufficient. The model equation, as expressed, is suitable for describing the response. The R-square value of the regression model is 86.39%. It means that the model is explained by 86.39% of the variance in response.Figure8 depicts the main effects on Y when we consider only the influence of independent variables on the response. When laser power (A) increases, it will decrease the value of dimensional error (Y). At the same time, increasing nitrogen pressure (B) will also increase the value of Y. The relation of frequency (C) and cutting speed (D) to Y follow curve functions where the minimal value of Y will correspond to the lowest point on the curve.Figure 8
Main effects plot forY.However, the variance of dimensional error depends not only on the main factors but also on their interactions. Figure9 examines the associating parameters on the response. The more difference between gradients of respective factors is, the stronger the interaction is. We can see that AD has a strong interaction. The following is BD, then AC, and BC. They have weak influences. The interactions of AB and CD almost do not impact the shift of the response.Figure 9
Interaction plot forY.The contour plots (Figure10) and surface plots (Figure 11) examine the relationship between a fitted response and two factors. The contour plot of Y versus D, A shows that the optimal region for Y is strongly impacted by the interaction between A and D. In general, when we increase the cutting speed, we should also increase power to reduce dimensional error. However, at low cutting speed, excessive power can result in bad cutting quality in the high burr or the wide kef width on the corresponding specimens. At the low cutting speed combined with a high level of laser power, the Y value is increased significantly. On the other side, high cutting speed (D = 5.5 m/min) and insufficient power also reduce the dimensional accuracy of output. Even some parts of the specimens cannot be cut under this condition. By observation, the underside may have dross remaining on due to not cutting penetration. Balancing the cutting speed and the power is strongly impacting the cutting quality.Figure 10
Contour plots.
(a)(b)Figure 11
3D surfaces plots.The contour plot ofY versus D and B (hold A = 1900 W, and C = 19500 MPa) shows that the optimal region for Y corresponds to the value of D lower than 3.6. From 15 to 25 MPa is a suitable range of nitrogen pressure for cutting stainless steel. And the cutting dimensional error is not much impacted when varying nitrogen pressure in this range. The effect of control factors and their interactions on the response are visualized clearer on the 3D surface plots (Figure 11).Figure12 presents residual versus fits, the histogram plot, and the constant variance test. From the histogram, we can see the normal distribution of residuals. In the normal probability plot, residuals fall on a straight line error. From the residuals versus fits graph, it is obvious that the residuals do not follow any pattern. They are randomly distributed.Figure 12
Residual plots forY.Using the response optimizer function in Minitab, an optimal parameter set to minimizeY is found (Figure 13). The optimum yield was achieved at the power of 1900 W, nitrogen pressure of 25 MPa, frequency of 19252.5 Hz, and Cutting Speed of 2.5 m/min.Figure 13
Optimal condition.The model predicts that the optimum value ofy is −0.0861. In reality, the value of dimensional error is always nonnegative. There are always errors or residuals between the fitted value and the observed one. Therefore, the value of y calculated from the regression model can be less than 0. The actual optimal value will equal the optimal predicted value plus error. The plot of residuals versus fits for this regression model is shown in Figure 12.


## 3.2. Taguchi Method

Below is theL9 (34) Taguchi design table and the corresponding measured data of top and bottom gage widths. Y is calculated by equation (1). And S/N ratio is obtained by applying equation (3).For example,(9)Y1=Kt−6+Kb−6=5..95−6+5.96−6=0.09,S/N1=−10∗log101n∑i=1nYi2=−10∗log100.092=20.9151.Other values ofY and S/N in Table 3 are calculated similarly to the example.Response Tables6 and 7 are for SN ratio and means. At each level of each factor, the averages of the response values and the SN ratio are calculated. For example, factor A has three levels and three measurements at each level.(10)YA=19000¯=0.09+0.06+0.233=0.12667,SNA=19000¯=20.9151+24.4370+12.7543=19.37.Table 6
Response table for signal-to-noise ratios.
LevelABCD119.3721.6021.5520.08222.9621.0221.6024.13319.3619.0718.5417.47Delta3.602.533.066.66Rank2431Table 7
Response table for means.
LevelABCD10.126670.083330.090000.1000020.073330.093330.086670.0633330.110000.133330.133330.14667Delta0.053330.050000.046670.08333Rank2341Delta value is calculated by subtracting the lowest average from the highest one. Based on the delta value, the rank of the factor effect is defined. From the response table for means, the factor that has the biggest delta value, i.e., has the largest effect onY (ranked 1) is D. The second rank is A, followed by C and B.The same analysis for the SN ratios response table,D is ranked 1, followed by A, B, and C.Figures14 and 15 indicate the change in means and SN ratio when a given parameter varies from one level to another level. These trends match the results found in RSM. Based on Figure 15, the optimal conditions can be predicted. Optimal factors levels that minimize dimensional error or maximize SN ratio are 1950 W power, 15 MPa nitrogen pressure, 19500 Hz frequency, 3.25 m/min of cutting speed.Figure 14
Main effects plot for mean.Figure 15
Main effects plot for SN ratios.The optimum conditions obtained by using RSM and Taguchi are not similar (Table8). Confirmation experiments with predicted optimal parameter sets were carried out for quality verification. Results show that the parameter settings anticipated by RSM obtained a smaller value of dimensional error (Yopt = 0.01) than the Taguchi optimal condition (Yopt = 0.03). The misleading results may be due to a small number of data in the Taguchi method that cannot examine the interaction effects.Table 8
Comparison of predicted optimal conditions of RSM and Taguchi method.
Control factorUnitRSM optimal setTaguchi optimal set(A) PowerWatt19001950(B) Nitrogen pressureMPa2515(C) FrequencyHz1925319500(D) Cutting speedm/min2.53.25Results of validation experiments Yopt (mm)0.010.03


## 4. Conclusions

The paper contributes an extended study about Taguchi and RSM methodologies and their application in optimization problems. In this paper, the influence of four laser control factors (power, pressure, frequency, and cutting speed) on the accuracy of dimension quality was studied and optimized by applying the RSM and Taguchi techniques. This paper proposed a simple formula for a combined function named “dimensional error” to optimize two objectives, i.e., targeting laser-cut edge to perpendicular 90 degrees and maximizing the cutting gage width accuracy. RSM Box–Behnken experimental design is used. The experimental results are examined through a quadratic regression equation and ANOVA table. The effects of main parameters and their interactions on responses are analyzed. Taguchi method requires fewer experimental trials than RSM. OA design allows simultaneously optimizing numerous factors and yields quantitative information. Taguchi method maximizes the SN ratio to identify design parameter settings that reduce the variance in the process and obtain the desired output.The paper proves the effectiveness of both methods in screening significant variables that influence the quality output. While the Taguchi presents a cost-effective approach, the RSM method explains better the interaction effects and the percentage distributions of factors on the variance of responses. Both methodologies found that the most influential parameter on the dimensional error is laser cutting speed, followed by the laser power. Nitrogen pressure and frequency have insignificant effects compared to cutting speed and power factors. RSM proved that the association of cutting speed and laser power has a strong influence on the dimensional error. In RSM, the ANOVA table helps users understand the significant degree of the main factors and their interaction with the output quality. From the ANOVA table, the contribution of the cutting speed was the highest, at 39%, followed by power (A) (7% contribution). Nitrogen pressure (B) and frequency (C) have a weak impact on the response with 3% and 1% contributions, respectively. The interaction of AD has a strong influence on the dimensional error with a 16% contribution. Surface and contour plots visualize the degree changing the value of output created by the varies in the entire range of parameters, whereas in the Taguchi design, the effect of main factors is identified based on the variance in the average value of responses and SN ratio at their different levels. Interactions are aliased with the main effects. Therefore, the interaction effects are not analyzed clearly as by RSM.In this case study, the RSM Box–Behnken Design gave more accurate predictions than the Taguchi technique. There is a difference in Taguchi and RSM optimum conditions. The validation experiments show that RSM defines more precisely the optimal parameter set. However, Taguchi’s design requires smaller experimental trials that can save time and cost. In addition, the anticipated parameter set from Taguchi obtains the output very close to the optimum solution. Taguchi method can also speculate pretty accurately the relations between independent and dependent variables. Therefore, with problems that their experimental runs consume high cost and time, Taguchi can be a suitable method. With high precision cutting problems, the RSM method is recommended for identifying optimal parameter settings. This study also could be a beneficial reference for parameter optimization problems in various fields.



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*Source: 6677586-2022-05-21.xml*