AN INTEGRATED APPROACH OF GRA COUPLED WITH PRINCIPAL COMPONENT ANALYSIS FOR FRICTION STIR WELDED AM20 MAGNESIUM ALLOY

ABSTRACT


INTRODUCTION
Magnesium alloys possess highly desirable properties that make them an appealing option to replace aluminum and steel in mechanical and structural applications.These properties include exceptional stiffnessto-weight ratio, high damping capacity, the lowest density of all engineering metallic materials, and the ease of recycling [1].As a result, magnesium alloys have become increasingly popular in various practical applications due to their lightweight nature as a replacement for aluminum alloys [2].
In conventional welding processes, magnesium alloy components suffer from several issues, including low strength, hot cracking, alloy segregation, partial melting zones, and porosity in the welded joint.As a result, the mechanical properties and corrosion resistance of the welded joints tend to decrease [3]- [5].To address the aforementioned limitations, the friction stir welding (FSW) process can be employed as a viable option to weld magnesium alloys.
Friction stir welding (FSW) is a solid-state joining process that has become increasingly important in welding technology.Initially developed for welding aluminum alloys, FSW has been extended to a variety of materials such as magnesium, copper, steel, and composites.During FSW, the material being welded is heated and softened by the frictional heat generated between the surface of the plates or components being welded and the contact surface of a specialized rotating tool [6].
This research presents firstly the material and process parameters selection followed by the Taguchi GRA coupled with PCA.Then, we discuss and analyze the ANOVA and find which parameters contribute the most.At the final, we will suggest the optimum settings for better output.

Taguchi Methods
The Taguchi method, also known as the Taguchi design of experiments, is a statistical technique developed by Genichi Taguchi in the 1950s.The method aims to optimize product and process design by identifying the optimal combination of design parameters that minimize the effects of variations in manufacturing and usage conditions [7].
According to Taguchi, quality is defined as minimum loss to society and can be measured by the consistency of performance.This consistency is achieved when the performance is close to the target with minimum variation.To enhance quality, Taguchi introduced a two-step optimization approach: first, identify the factor-level combination that reduces performance variability, and second, adjust the factor levels to bring the performance closer to the target.Taguchi's techniques for quality improvement include the use of orthogonal arrays to design experiments, the calculation of signal-to-noise ratios, and the use of parameter design and tolerance design to optimize product and process design [8].The Taguchi method has been widely applied in various fields, including manufacturing, engineering, and management.Its key concepts and techniques have been shown to be effective in improving product quality, reducing costs, and enhancing customer satisfaction.Some examples of its applications include process optimization in manufacturing systems, machining operations, and drilling of composites [9] [10].

GRA with PCA
Grey relational analysis (GRA) is used for solving interrelationships among the multiple responses.In this approach, a grey relational grade is obtained for analyzing the relational degree of the multiple responses.Grey relational based approach has been used to solve multi-response problems in the Taguchi methods [11].The GRA theory adopts the Grey theory, which is derived from the mixing of clear and unclear information.For example, Black is denoted as vague information, which is considered rudimentary information.In contrast, white signifies completely clear information.However, some information falls in between black and white, referred to as Grey, information that has some things that are clear and unclear or less perfect [12].
However, in real cases, this method doesn't work at best, Hotelling and Pearson developed principal component analysis (PCA), which calculates prioritized weights for each quality response.Kumar

Experimental Work
The experiments use the Taguchi method and specific orthogonal array to maximize the number of process parameters included in the empirical matrix and their levels and minimize the number of experiments.The design of the orthogonal array is influenced by the number of factors and their degrees of freedom (dof) for each factor.Table 1 shows the factors used and their levels.The factors and their levels that were taken into consideration in this study are shown in Table 1.There are three factors that have three levels and one factor with only two levels, so the mixed levels approach will be used in this study.The full factorial of this experiment is 2x3 3 = 54 trials.Based on Taguchi method, this mixed level of experiment can be done by using L18 orthogonal array.The array has 4 columns with mixed degree of freedom based on its level.Table 2 shows the L18 orthogonal array.This design is applied to generate multi-responses approach of weld quality parameters [18].There are eight weld quality parameters, namely ultimate tensile strength (UTS), yield strength (YS), percentage of elongation (% E), compressive stress (CS), bending angle, average hardness at the nugget zone (NZ), thermo mechanical affected zone (TMAZ) and heat affected zone (HAZ) were measured after the experiment and are given in Table 3.The larger the better S/N ratio as computed from Equation (1): The smaller the better S/N ratio as computed by Equation (2): where x is number of replications and yij is measured observation.
Welding process has multiple responses and welding quality sturdily depends upon optimizing all responses simultaneously.Therefore, researchers frequently employ GRA coupled with PCA as weights for optimization of multiple responses simultaneously.These techniques are entirely different from traditional single response optimization.These are effective statistical methods and offer quite successful results in obtaining a combination of parameters for multiple response optimizations [23].Figure 1 depicts the concept of PCA-GRA.
As a result, a larger-the-better criterion has been chosen for these quality characteristics.The normalized results can be expressed as Equation (3).
Thus, the smaller-the-better is used, as represented in Equation ( 4).
where   * () are the generated grey relational values, while    () and    () are the largest and smallest values of   () for  ℎ trial, respectively. = 8 is the number of response variables.The eighteen observations of the experiments are comparability sequence   (),  = 1,2, … ,18,.The best normalized results should be equal to 1. Therefore, for achieving better performance, we expect larger value of normalized results.
Data normalization is followed by calculation of grey relational coefficients (GRC) that explains the relationship between desirable and real experimental normalized results.Expression of GRC   () is determined, as follows in Equation (5).

𝜉 (𝑦
where  0 () = | 0 * () − ()| is deviation sequence, defined as absolute of difference between reference sequence  0 * () and comparability sequence   * ().The identification or distinguishing coefficient (), takes value as   [0, 1], which is generally and in this study were set as 0.5 [24].Grey relational grade (GRG) provides information about correlation strength between the experimental runs, which is computed by weighted mean of respective GRC's for all experimental.GRG value lies between 0 and 1,   [0, 1].Usually, an experimental run with larger GRG is considered the ideal case, which indicates the strength of correlation between corresponding experiments and the ideally normalized value.When equal weights are opted for all quality responses, Equation ( 6) is used for GRG calculation.
In some applied application, weights of quality characteristics are different likewise weights obtained from Principal Component Analysis (PCA).In such cases, Equation ( 6) is modified as Equation ( 7) by applying a weight [25].
where   ( 0 * ,   * ) is GRG for  ℎ experimental run,  is number of quality response,   is weight of  ℎ quality response and ∑    =1 = 1.
PCA is a powerful multivariate statistical technique for multi-objective optimization [26] that reduces the complexity, correlation, vagueness, and dimensions of information by simplifying and combining numerous allied arrays into few uncorrelated arrays and principal components.PCA employs linear permutation for conserving unique information to maximum extent [27].Thus, it converts multi-response optimization to single response optimization without compromising original information [28].It starts by setting a structure of linear combinations arrays of multi-responses.The GRC's computed for response variables is employed to form a matrix, presented as Equation ( 8).
where   () is GRC of each quality responses,  = 1, 2, … , , experiments and  = 1, 2, … , , quality responses.In this research,  = 18 and  = 8.Thereafter, the coefficient correlation matrix can be produced by the following Equation ( 9) where (  (),   ()) is covariance of sequences   () and   ().  () is standard deviation of sequence   () and   () is standard deviation of sequence   ().The eigen values and eigen vectors are computed from   array as per Equation ( 10) Thereafter, eigenvalues (  ) and eigenvectors (  ) of square matrix R are used to determine the uncorrelated principal components (PC's) by using Equation ( 11) where   corresponds to  ℎ principal component.Eigenvalues and principal components are arranged in descending order with respect to explained variance.Therefore, first eigenvalue associated with first PC accounts for largest variance contribution.Eigenvalues corresponding to eigenvectors are presented in the Table 4 for case 1 and Table 5 for case 2.

Implementation of Taguchi GRA with PCA
In the present investigation, Taguchi GRA has been applied for selection of optimal parameter settings.All the eight output responses presented in Table 3 were standardized using Equations (3) or Equation (4).Two optimization scenarios were considered: In Case-1, all quality parameters were assumed to be "higher the better," which means that the purpose of Case-1 was to maximize all quality parameters.In Case-2, UTS, YS, CS, and bending angle were considered "higher the better" while percentage elongation and average hardness values at NZ, TMAZ, and HAZ were considered "lower the better" with the goal of maximizing UTS, YS, CS, and bending angle and minimizing percentage elongation and hardness values at the same time.The choice between Case-1 or Case-2 depends on the user's preference or specific application requirements.Case-1 is preferable when high values of quality parameters are desired, while Case-2 is more suitable when the tensile property needs to be high and the hardness needs to be low.

Case-1: All the Output Responses are taken as "Higher the Better"
For Case-1, all the outputs responses are taken as "higher the better".It is preferable when high values of quality parameters are desired.Hence, for this case, it uses Equations (3) to do normalization and the results are shown in Table 6.After getting the normalized data, GRCs are calculated using Equations ( 5), and the results of the GRCs are obtained in Table 7.
Then, the results of the GRCs will be subjected to PCA analysis, which will later be used to calculate the weighting when calculating the GRG.The component used is the first component that shown in Table 4.The weight is the squared result of the eigenvector PC1, which will later calculate the GRG value using Equations ( 7) and the results can be seen also in Table 7 along with the rank order.From the results of the GRG weighting, it was found that the value of GRG was around from 0 to 1 with results trial no. 9, 1, and 2 sequentially are the three best rank trials.
After getting W-GRG of each trial, then it can be calculated the Average W-GRG of each parameter, including plunging depth (PD), tool rotation speed (RPM), welding speed (WS), and shoulder diameter (SD).From that Average W-GRG, the rank and optimum parameters can be found.Table 8 shows the optimum parameters level (PD1, RPM3, WS3 and SD1) namely plunging depth 0.12 mm, tool rotation speed 1100 rev/min, welding speed 132 mm/min, and shoulder diameter 16 mm.The ANOVA analysis in Table 9 shows the performance of statistical significance and the percentage contribution to each parameter.Unfortunately, in this case, all of the parameters are not significant, with the percentage contribution to each parameter sequentially starting from the largest SD with 22.49%, PD with 14.99%, RPM with 4.14% and WS with 4.08%.Then, the variance contribution of each response can be seen in Table 10.The ANOVA analysis also need to be checked of residual assumption.The assumption that required to be satisfy are normal distribution of residual.In this case, the residual normal distribution test using Anderson-Darling Test in Figure 2 with the result that this residual doesn't satisfy the normal distribution.In this analysis also calculate the variance contribution of response variables for first PC shown at Table 10 which is conclude that UTS is the highest value.For Case-2, some parameters UTS, YS, CS, and bending angle were considered "higher the better" while percentage elongation and average hardness values at NZ, TMAZ, and HAZ were considered "lower the better".This case is more suitable when the tensile property needs to be high and the hardness needs to be low.Hence, for this case, it needs to do normalization by using Equations (3) and the results are shown in Table 11.After getting the normalized data, then GRCs are calculated using Equations (5) and the results of the GRCs are obtained in Table 12.Then, the results of the GRCs will be subjected to PCA analysis which will later be used to calculate the weight when calculating the GRG.The component that used is the first component that shown in Table 5.The weight is the squared result of the eigen vector PC1, which will later calculate the GRG value using Equations ( 7) and the results can be seen also in Table 12 along with the rank order.From the results of the weighted GRG, it was found that the value of GRG was around from 0 to 1 with results trial no. 1, 11, and 2 sequentially are the three best rank trials.After getting W-GRG of each trial, then it can be calculated the Average W-GRG of each parameters including plunging depth (PD), tool rotation speed (RPM), welding speed (WS), and shoulder diameter (SD).From that Average W-GRG, the rank and optimum parameters can be found.Table 13 shows the optimum parameters level (PD1, RPM1, WS2 and SD1) namely plunging depth 0.12 mm, tool rotation speed 600 rev/min, welding speed 98 mm/min, and shoulder diameter 16 mm.The ANOVA analysis in Table 14 shows the performance of statistical significance and the percentage contribution to each parameter.In this case all of the parameters are not significance with the percentage contribution to each parameter sequentially starting from the largest SD with 38.44%, RPM with 24.32%, PD with 4.62% and WS with 2.40%.The variance contribution of each response can be seen in Table 15.The ANOVA analysis also need to be checked of residual assumption.The assumption that required to be satisfy are normal distribution of residual.In this case, the residual normal distribution test using Anderson-Darling Test in Figure 3 with the result that this residual does satisfy the normal distribution.In The study utilized Taguchi orthogonal arrays to generate a design matrix that encompasses the entire parametric space with a limited number of experiments.The experiments were done based on the Taguchi orthogonal array design, which is frequently used to optimize engineering problems [19]-[21].However, the Taguchi method is a single-optimization process and cannot effectively handle the optimization of multiple responses, which is required for several processes[22].