Multi-objective optimization of FSW variables on joint properties of AA5754 aluminum alloy using Taguchi approach and grey relational analysis

The friction stir welding (FSW) process was applied to butt-weld AA5754 plates, and the joints of each weld case were identified by their tensile strength, hardness, and impact toughness. The combined effects of the process variables rotation speed, welding speed, and tilt angle have been evaluated in the present study using Taguchi's design of experiments approach and grey relational analysis (GRA). A multi-objective optimization approach (Taguchi method and GRA) was used to optimize and rank the process variables. The developed models are evaluated for sufficiency, and the most significant terms are found using the ANOVA approach. The results of the ANOVA showed that the rotational speed has a maximum contribution of 55.24%, 59%, and 46.27% in obtaining the optimal values of tensile strength, hardness, and impact toughness, respectively. Finding an optimal parameter that offers the maximum value of all replies requires the use of the GRA and Taguchi Technique by Similarity to Suitable Solution. The two methods provide the same results, and the optimal conditions are a rotational speed of 1000 rpm, a welding speed of 60 mm/min, and a tilt angle of 2.5 degrees. The optimal values for tensile strength, hardness, and impact toughness, respectively, were found to be 136 MPa, 85.25 HV, and 13 J.


Introduction
Aluminum alloys are prominent engineering materials with many industrial uses due to their strength and light weight [1].For instance, the structural parts of aircraft can be made from the 2xxx and 7xxx series, and parts for automobiles can be made from the 5xxx and 6xxx series [2][3].Due to its superior mechanical qualities, friction stir welding is currently utilized extensively in the welding of aluminum alloys [4][5][6].A rotating, non-consumable tool with a pin and shoulder is inserted into the joint line between the plates and moved along the joint line.This procedure produces frictional heat, which welds the junction.The process's maximum temperature increase is lower than the material's melting point.Because of this, the material does not provide any benefits over traditional fusion welding procedures, such as improved mechanical properties, reduced distortion, lower residual stresses, and fewer weld flaws [7].
The profile form and dimensions of the tool pin and tool shoulder of the welding tool have an impact on the rate of heat generation as well as the plastic flow of the workpiece during welding.
Induced forces along the tool axis, heat generation per unit of time, and the flow of weld material on the workpiece are all directly influenced by the pin shape and surface characteristics like threads, tappers, etc. [8,9].Therefore, constrained heating and material flow are the two main functions of the tool, and the tool geometry has a significant role in determining both during the machining process (as well as the traverse rate) [10][11][12][13].As a result, a key parameter that ensures proper joining during welding is the tool's rotation, traverse speed, angle, and shape [14].
The flow of material during welding and the quantity of heat produced are both influenced by the tool's tilting angle.The interaction between the workpiece and tool shoulder, which encourages material flow across the tool, is confirmed by a little rise in the tool's tilt angle.Defective welds happen from a tool tilt angle that is too high because it elevates the pin from the weld's root.Therefore, choosing the right parameter settings is crucial to getting the best output results.To find the ideal parametric range settings, a thorough literature review is conducted.Marathe et al. [15] studies on FSW of AA6061 plates used three different tool profiles, including square, tapered, and cylindrical.At a tool speed of 2750 rpm and a welding speed of 10 mm/min, tapered pins were used to produce the highest possible ultimate tensile strength.Using ANOVA, Deviah et al. [16] discovered that the greatest Vickers microhardness value of 94.8 VHN was achieved when welding joints made of the dissimilar alloys 6.61 and 5083 at tool speeds of 1120 rpm, 70 mm/min, and 20 tool tilt angles.According to Panda et al. [17] hypothesis, welding AA6061 plates using a threaded cylindrical profile tool at 900 rpm and 60 mm/min produced a joint with an enhanced tensile strength of 160.7 MPa.Shah et al. [18] used the Taguchi technique to determine the best parameters for welding AA6061 and AA7075 aluminum alloys, and they concluded that 1000 rpm tool rotation, 110 mm/min transverse speed, and 30 degrees of tool tilting produced the highest tensile strength.To study the effects of process factors on mechanical strength and microstructural changes, Bhojani et al. [19] performed FSW on a hybrid aluminum composite.When welding AA2014-T651 and AA6063 T651, Ranjith et al [20] selected the tool offset, angle of tilt, and pin diameter as input parameters.The maximum tensile strength was 371 MPa using the values of 4° tool tilt angle, 0.5 mm offset towards advancing side, and 6 mm pin diameter.For unlike nonferrous aluminum alloy and titanium alloy, Palani et al. [21] studied the effects of rotation speed, tool pin profiles, and welding speed on tensile strength, tensile elongation, and joint efficiency while utilizing FSW parameters.Following welding speed and the tool pin profiles discovered by ANOVA, rotation speed is the crucial factor in determining the quality of the welded joints.To improve the joint qualities of the tool pin profiles, rotating speed, and welding speed employing GRA for durability and medium grain particles at the nugget zone, Santhanam et al. [22] introduced the friction stir welding alloy AA6063-O in mixed condition.To examine the significance of process parameters for the mechanical characteristics and variability in microstructure of both the standard FSW and mixed FSW joints, variance analysis (ANOVA) is used.Ravikumar and others [23] Optimization of unlike FSW joints between AA6061T-651 and AA7075T-651 aluminum alloys are calculated by ANOVA using grey relational grade from grey analysis and significant parameter contribution.Other parameters include rotation, welding velocity, pin profiles, ultimate tensile strength, and hardness.According to experimental findings, this unique technique is suitable for welding process responses.To optimize the set of parameters for FSW in terms of total fracture energy, crack initiation energy, and crack propagation energy in welds for rotating welding speed, traversing speed, and tool tilt angle, Raweni Abuajila et al. [24] explored the Taguchi design.To evaluate the impact of various parameters, ANOVA was utilized.
In this study, the Taguchi method and Grey relational analysis were used to improve the multiple characteristics of the friction stir-welded AA5754 alloys.Investigations are conducted into how each FSW process parameter affects tensile strength, hardness, and impact toughness.Three response characteristics are combined in GRA to provide a single performance metric known as the grey relational grade (GRG) and the FSW of AA5754 has been demonstrated to have a single optimal conditions.

Materials and methods
In this study, a single pass weld was performed on 5754 aluminum alloy with dimensions of 100 × 100×6 mm.Tables 1 and 2, respectively, contain information on the chemical composition and mechanical characteristics of this alloy.The FSW technique is shown in Fig. 1.The vertical milling machine with a hardened tool attachment was employed to fabricate the joints following the plates were securely clamped using a specialized fixture (Fig. 1a).Pin tool plunging into the AA5754 plates (Fig. 1b).Figures 1c and 1d, respectively, show the travel of a tool along the joint line and its removal once the weld is complete.A hardened H13 steel tool with a threaded pin profile of M6 mm, a height of 5 mm, and a shoulder diameter of 18 mm is utilized for this technique (Fig. 1e).
Following welding, the quality of the welded joint is evaluated using a variety of mechanical tests, including tensile, impact toughness, and hardness tests.The test samples were cut employing a wire cut electrical discharge machine into standard sizes in accordance with the ATSM standard.
The tensile samples have been trimmed perpendicular to the welded connection in accordance with ASTM E8 standard.Vickers' microhardness tester with a 100-g load and a 10-second dwell period was used to determine the microhardness of the FSW specimens.To assess the impact toughness of the weld joint, the Charpy impact test was performed.The V-notch sample for the Charpy test was constructed perpendicular to the welding connection, with the notch located in the middle of the connection.An optical microscope was used to carry out microstructural analysis.Before displaying the surface morphology of all samples, the surfaces were all ground and polished to a mirror-like finish.The samples were etched using Keller's reagent in accordance with ASTM E407 Standard Practice for Micro etching Metals and Alloys.Scanning electron microscopy (SEM) Model Quanta 250FEG (Field Emission Gun) was used to analyze joint fracture morphology.

Taguchi method
The Taguchi approach, which uses an orthogonal array of investigations, greatly reduces the variance for the experiment while allowing for an optimal possible configuration of process control variables.To assess and optimize the welding parameters, the Taguchi methodology, a method for generating process improvements, was employed [25,26].By choosing the main variables that influence the process [27][28][29] and optimizing the methods to get the best results, these enhancements strive to improve the desirable features and reduce flaws.To examine the variation in the output responses, the input parameters are grouped and clustered in the L9 orthogonal array (OA) order [30,31].There are various independent variables that can control the FSW process.In the present investigation, the three process variables of tool rotation speed, welding speed, and tilt angle were conducted.The L9 design of experiment by Taguchi was modified in this work to reduce the number of experiments.Table 3 depicts three variables and three levels of the control variables.
The L9 orthogonal array was generated for three variables and levels, as shown in Table 4, and nine different arranged sets of tests were run applying this array.
The Taguchi technique employs a statistical performance metric known as the signal-to-noise (S/N) ratio to assess the process variables.As an objective function for optimization, the S/N ratio is a logarithmic function of the desired output [32,33].By evaluating the S/N ratio of the measured values, the optimal parameter combinations were identified.Each quality characteristic's S/N ratio can be calculated separately, and regardless of the performance characteristic category, a higher S/N ratio equates to higher quality characteristics.Nominal the best, Smaller the better, and Larger the better are the three categories for the commonly employed standard S/N ratios.In this study larger-the-better evaluation is employed to calculate the S/N ratio, which is defined by: =1 (1)

Weldment Threaded pin Tool Shoulder
where n is the number of evaluations and y is the result of those observations of the i th performance characteristic.Also, the statistical technique analysis of variance (ANOVA) is used to examine the effects of different input parameters on establishing the levels of output responses in the weld joints.The ANOVA test is used to determine the significance of the process variables that affect the mechanical characteristics of weld junctions [34].Additionally, by using the findings of the ANOVA, it is possible to estimate the influence of each parameter on the response.Table 3: Design variables and their levels Table 4: The L9 orthogonal array applied to the evaluations.

Grey relational analysis (GRA)
GRA is the approach utilized when there are multiple attribute situations.With this approach, a grey relational grade (GRG) is used as the basis for the overall assessment of the multiple response procedure.Here, optimization is accomplished by reducing the complex numerous process values to a single GRG, and the best parametric combination that would produce the highest value of GRG is then assessed.To prepare the original data for analysis in the grey relational analysis, data preprocessing is first carried out.Normalization is a transformation carried out on a single data input to scale the data into an appropriate range and spread it uniformly for further analysis.In this research, the observed values are linearly normalized in the range between 0 and 1, also known as the grey relational generating range.to avoid the impact of using various units and to cut down on variability.According to the higheris-better criterion, the normalised compressive strength can be represented as: The frequency can be stated as, according to the lower-the-better criterion: Where   (K) denotes the value following the formation of the grey connection.The minimum value of   () for the kth response is min   (), and the maximum value of   () for the kth response is max   ().
The number of experiments is i = 1, 2, 3, and the number of replies is k = 1, 2, 3.The relationship between the reference sequence and the compatibility sequence is then determined by computing the grey relation coefficient (GRC).Eq.4 can be used to compute the GRC () [23].
Where, ∆  () = ‖  () −   ()‖ =difference between absolute value of x0(k) and xi(k);  is the distinguishing coefficient; 0≤  ≤1 To provide replies equal preference in the present work, the value of  was chosen as 0.5 [35].∆  is the smallest value of ∆  () ∆ and ∆  is the largest value of ∆  ().
The GRG is a gauge of the relevance between two systems or two sequences.The grey relational coefficient for each performance trait is averaged to produce the GRG.The calculated GRG determines the multiple response process's total performance characteristic.The GRG is described as Where ̅  is the mean GRG value at the desired level for the ith parameter, m is the number of process responses.The statistical significance of each component and the percentage contribution of each process parameter to the responses are also determined using the ANOVA approach.

Results and discussions
The best level of each process variable is determined using the S/N ratio method to examine the results, which allows for the correlation of the highest S/N ratio and ensures the greatest value of the tensile strength, hardness, and impact toughness of the welded joints.

Signal-to-noise ratio evaluation
The Taguchi method was used to calculate the S/N ratio and estimate how each element will affect the process' output.One of the fundamental goals of the Taguchi approach is connected to the S/N ratio.It is used to assess quality characteristics and variation from actual values.As a result, the S/N ratio was used to identify the variable level that changes significantly and forecast the blending of the optimal procedure variables [36].Furthermore, these parameters' effects on the mechanical properties of the AA5754 aluminum alloy were observed.S/N ratio at its highest value indicates that the process parameters are at their optimal level.The experimental data for tensile strength, hardness, impact toughness, and the associated S/N ratio, which were determined using Eq. ( 1), are shown in Table 5.According to the largest difference between values (delta statistics), each factor's effects (RS, WS, and TA) were ranked.As shown in Table 6, rotating speed, welding speed, and tilt angle were the welding parameters in decreasing order of significance for the tensile strength of the welded joints.Table 7 demonstrates that, in order of decreasing significance, rotating speed, welding speed, and tilt angle were the welding parameters for the hardness of the welded joints.Table 8 shows that the welding parameters for the impact toughness of the welded joints were rotating speed, welding speed, and tilt angle, in that order of decreasing significance.

Analysis of variance
The analysis of variance (ANOVA) method is used to investigate the significant impacts of controllable factors and their interactions with one another.As a result, it is possible to determine the optimal (optimum) level settings for controllable elements by screening them first and then ranking them using ANOVA [37,38].Tables 9,10, and 11 reveal, respectively, the percentage contribution to tensile strength, hardness, and impact toughness of the tool rotation speed, welding speed, and tilt angle.The contribution percentage represents the relative ability of a component to lessen variation and is a function of the sum of squares (SS) for each significant variable.The significant influence on the process' responses is demonstrated by the high contribution value for this variable.In this investigation, the weld process responses are significantly influenced by the tool rotational speed (RS) and welding speed (WS), both of which are extremely significant variables.According to ANOVA table 9, the rotating speed (RS) is the most important variable, contributing 55.24% to the peak tensile strength of the friction stir welded material.The next two variables that affect tensile strength are welding speed (WS) and tilt angle (TA), which contribute 31.87% and 9.17%, respectively.The tool rotational speed is an extremely significant variable for hardness, accounting for 59 % of the overall variation, as indicated in Table 10.The next contribution to hardness results in 31.8% and 1.74%, respectively, of the welding speed and tilt angle.As shown in Table 11, the tool rotational speed significantly influences impact toughness, accounting for 46.27% of the total variation.The following two variables, which are welding speed and tilt angle, each contribute 40.08% and 10.84% to impact toughness, respectively.where Y is the objective function, RS is the rotational tool speed, WS is the welding speed and TA is the tilt angle.The selected polynomial can be represented as follows for all three variables: Y=0 + 1(RS) + 2(WS) + 3 (TA) + 4 (RS * WS) + 5(RS * TA) + 6 (WS * TA) where 0 is a constant, 1, 2, and 3 the linear term coefficients.4, 5 and 6 the coefficients of the interaction terms.
Develop the first order polynomial equation shown below based on the results of the multiple regression analysis of the design matrix and the response values: Tensile strength (Mpa) = 120.50 The significance of the adjustment for the means equations ( 8), (9), and ( 10) of the tensile strength, hardness, and impact toughness, respectively, is examined using the ANOVA analysis presented in above mentioned tables 9, 10, and 11.The coefficient of determination (R-Sq = 96.3%,92.5%, and 97.2% for tensile strength, hardness, and impact toughness, respectively) has a greater value in the model than the adjusted coefficient of determination (R-Sq(adj) = 95.8%,91.5%, and 96.8% for tensile strength, hardness, and impact toughness, respectively), indicating a strong correlation between the predicted values and the actual results of the experiment.A diagonal line in Fig. 2 depicts the predicted values against the experimental values, showing a uniform random distribution of all the points with a linear correlation.As a result, the model that was developed is deemed adequate and predicts the response with reasonable errors.

Evaluation of experimental results
The main effects plot for tensile strength is displayed in fig.3a.At the extreme levels 1 and 3, the mean tensile strength is low, while at level 2, it is at its highest.While the rotational speed (RS) gets higher to the middle level (1000 rpm) and then drops, the tensile strength of the weld improves.The highest effect is at the intermediate level (60 mm/min) and is also influenced by welding speed (WS), which has a comparable effect.Weld speed and tool rotational speed typically rise as they generate more heat, which has a greater impact on the tensile strength [39,40].The frictional heating and plastic deformation of the material are caused by the rotational speed, which results in mixing and churning of the material around the pin.In addition to the increased friction heating caused by faster tool rotation rates, the material is stirred and mixed more vigorously.
Tensile strength drops as a result of the FSW thermal cycle's peak temperature rising because it produces large, recrystallized grains, exceptional grain growth, and dissolved precipitates [41].
Fig. 3b displays the hardness main effects plot.The weld zone hardness increases as the rotational speed (RS) rises to a midway level (1000 rpm), then declines.The production of coarse grain structure at the weld zone is caused by the increased heat generation at higher tool rotating speeds, which causes more heat to dissipate to the workpiece.This lessens the weld zone's hardness [42,43].The amount of material deformation and frictional heat are reduced as welding speed is increased.According to the general principles of recrystallization, the reduction of the deformation degree prevents dynamitic recrystallization, increasing the recrystallized grain size and raising the hardness of the weld zone as the welding speed increases [44,45].Also, it can be shown in Fig. 3c that the mean impact toughness is lowest at levels 1 and 3 and highest at level 2, respectively.The impact toughness also rises with rotational speed and welding speed.At tool rotational speeds of 800 rpm and 1200 rpm, the impact toughness and tensile strength are lowest, while at 1000 rpm, they are at their highest.Additionally, 60 mm/min weld speed created greater values of impact toughness and tensile strength while the weld speeds (40 and 80 mm/min) have produced lower values.
The tilt angle (TA) has a substantial impact on the material flow.A higher tilt angle increased the frictional force at the tool/workpiece interface, which significantly increased the velocity of the material flow behind the tool.Tool tilt angle influenced the joint tensile and impact toughness, and a substantially parabolic type of variation was seen against the tool tilt angle (the joint strength initially rises, reaches a maximum, and then drops up on increases in tool tilt angle) [46].
Additionally, at greater tool tilt angles, the plunge depth becomes shallower, reducing the amount of base material that the rotating tool shoulder can effectively contact, reducing the amount of plasticized material that can be transferred from the front to the back of the tool, and reducing the strength of the weld joint.Dynamic recrystallization, which took place during the welding process as tilt angle increased due to an increase in total heat generation and the extrusion of base metals into the weld zone, is responsible for the increase in the hardness values in the weld zone [47].The weld zone's relative improvement in hardness with increasing tilt angle is due to the recrystallized grain size that produced the fine grain structure.The input factors are optimized using regression modelling to get the highest tensile strength, hardness, and impact toughness.Figure 4 displays the optimal input values for variables for the highest tensile strength, hardness, and impact toughness.The model predicts that the optimized composite will have a tensile strength of 136 MPa, a hardness of 85 HV, and an impact toughness of 13J, which will be obtained by fabricating it at 1000 rpm rotational speed, 60 mm/min welding speed, and a tilted angle of 2.5 degrees.
Fig. 4 Optimum variables for maximum tensile strength, hardness, and impact toughness

Taguchi approach evaluation of the GRG with variables
Taguchi established an approach determined by an orthogonal array of experiments that significantly reduces experiment variation when process control variables are set to their optimal values.The Taguchi technique applies a statistical measure of performance known as the signalto-noise (S/N) ratio to assess the process variables.Multi-objective optimization problems, however, cannot be optimized using the conventional Taguchi method.The Taguchi approach and Grey relational analysis (GRA) are employed for optimizing multi-objective problems to get around this. Multi-objective optimization problems are reduced to single-objective issues using the grey relational analysis method.The optimal variable combination is then assessed.As a result, the largest grey relational grade (GRG) is acquired [48,49].The data (means) are normalized into a comparable sequence in the range of 0 to 1 pursuant to the GRA approach.Table 12provides the normalized values of each response as determined by Eq. ( 2).These normalized values are employed to determine the grey relational coefficient of each output response applying Eq. ( 4), (Table 13).Subsequently, the grey relational grade for each experiment is determined by averaging the grey relational coefficient using Eq. ( 5) and is also shown in table (13).The result shows that experiment 5 has greater tensile strength, hardness, and impact toughness as it has the largest grey relational grade of 1 within the nine investigations.
Based on the experimental findings, a comprehensive factor analysis of GRG and S/N ratio is carried out.The response for GRG means and S/N ratio means are shown in Table 14.The rotational speed, followed by the welding speed and the tilt angle, was shown to have the greatest influence on the GRG and S/N ratios.
The coefficient of determination (R-Sq) is used to evaluate a model's suitability and serves as an indicator for the model's strength since it includes both significant and non-significant variables, whereas R-Sq(adj) is just related to significant variables.Therefore, R-Sq(adj) value is always less than or equal to R-Sq value.The model developed for GRG has an R-Sq and R-Sq(adj) of 93.48% and 92.55%, respectively.
Table 15.Analysis of Variance for GRG

The optimal value for each variable
Applying the estimated GRG values (Table 13), the mean effect for each level of the parameters was calculated (Table 14).Moreover, Fig. 5 exhibits the main effects of the grey relational grade.Providing the largest GRG value for each factor, the optimal condition is attained at 1000 rpm for rotating speed (RS), 60 mm/min for welding speed (WS), and 2.5 degrees for tilt angle (TA).The depicted optimal welding condition is obviously one of the conditions in the orthogonal experimental design.This state denotes the maximum values for tensile strength, hardness, and impact toughness.exp.#6).As a result of the dynamic recrystallization process, the stir zone in each experiment has roughly equiaxed fine grains [50].Moreover, it has been recognized that dynamic recrystallization occurs during FSW, generating fine, equiaxed grains in the stir zone.It is obvious that when welding speed and tilt angle increase, the size of the stirred zone grains reduces at a constant rotational speed (Fig. 6 a, b) [51].Lower heat input per unit area is responsible for this.
Additionally, as welding speed is increased further and the tilt angle is decreased, the area of the stirred zone is reduced, increasing grain size (Fig. 6 c).This might be explained by the welding tool not having enough time to completely stir the material.Greater heat generation at tool tilt of 1.5 o was the cause of the slightly larger grains in the stir zone.However, when tilt angles increased, the reduction in grain size became more apparent since at 2 o and 2.5 o tool tilt degrees, heat generation dropped as material strain rate increased [52].Additionally, changes in welding speed have greater effects on changes in grain size and precipitation behavior.The joint was constructed utilizing a reduced tool traverse speed, which led to a large heat input because of the lengthy stirring time.
As a result, joints made using lower welding speeds had greater grain sizes than joints made using higher welding speeds.

Conclusion
The Taguchi method and grey relational analysis were used in this investigation to optimize the welding process variable of friction stir welding AAA5754 alloy.According on the present investigation, the following findings can be developed: 1-The optimal procedure variables for the Taguchi technique optimization and GRG are rotating speed at 1000 rpm, welding speed at 60 mm/min, and tilt angle at 2.5 degrees.

2-
The response table of S/N ratios demonstrated that rotational speed, which is ranked 1, has a significant impact on the overall mechanical properties of the weld joint, while welding speed and tilt angle are ranked 2 and 3, respectively.
3-Higher tensile strength, hardness, and impact toughness are demonstrated by the results, which have the largest grey relational grade of 1.
4-A grey relation analysis demonstrated that a rotation speed of 1000 rpm, welding speed of 60 mm/min, and a tilt angle of 2.5 degrees are optimal process variables for producing the highest level of responses.
5-The predicted optimal welding setting produced maximum values for the following properties: tensile strength of 136 Mpa, hardness of 85.25 HV, and impact toughness of 13 J.
6-The study demonstrates that the Taguchi approach and GRA, which use the same set of process parameters to determine the optimal value, are both effective tools for optimizing the multiple objective problem in friction stir welding.

Fig. 1 :
Fig. 1: Photographs of FSW procedure.(a) placing the tool and plates prior plunging, (b) pin tool plunging, (c) tool passing along a joint, (d) pin tool carrying away and (e)FSW tool.

Fig. 2
Fig.2 Relationship between predicted and experimental values for all responses a) tensile

Fig. 3
Fig. 3 Response graph for procedure variables effects on tensile strength (a), hardness (b), and

Fig. 5 Figure 6
Fig. 5 Response graph for procedure variables effects on GRG

Fig. 7
Fig.7represents the tensile fractured surface.The weakest part of a welded joint is indicated by the fracture position.Fracture sites are an indicator of the distribution of strength for a joint without defects[53].The FSW joint, in contrast, consists of dimples of both small and big sizes.A mixedtype fracture with ductile fracture features and grain boundary segregation was visible in the fractography of the joints.Dimples were found that were sparsely populated and devoid of voids, indicating a joint that was free of defects and had a typical fracture.

Table 2
Mechanical properties of AA5754 aluminum alloy.

Table 5 . Experimental results of tensile strength, hardness and impact toughness with their associated
S/N ratios based on ''larger is better''.

Table 9 . Analysis of Variance for tensile strength Table 10. Analysis of Variance for hardness Table 11. Analysis of Variance for impact toughness 3.3 Development of linear and multiple regression model
A mathematical relationship between the independent and dependent variables can be established following the ANOVA analysis to estimate the tensile strength, hardness, and impact toughness of the joints manufactured by the FSW procedure.The linear regression model is written as:

Table 12 .
Estimation of normalized data for each response and deviation coefficients.

Table 13 .
Grey relational coefficient, grey relational grade, and S/N ratios.

Table 14 . The means of GRG and estimated S/N response.
Analysis of variance has been used for examining significant procedure variables and the corresponding percentage contributions to the GRG of FSW joints.The effect of every variable on GRG gets clearly assessed using findings of an ANOVA.The GRG ANOVA analysis is presented in Table15.Table15demonstrates that the variable RS, which has the largest contribution to the overall variability (59.83%), is followed by the variables WS (27.26%) and TA (6.39%).