An Integrated Method for Multi-Objective Optimization of Multi-Pass Fe50/TiC Laser Cladding on AISI 1045 Steel based on Grey Relational Analysis and Principal Component Analysis

As an essentially multi-input multi-output process, determination of optimal conditions for laser cladding normally requires multi-objective optimization. To understand multi-response coupling, the effects of processing parameters on the morphology quality of multi-pass laser claddings of Fe50/TiC on medium carbon steel AISI 1045 were investigated based on composite central design using response surface methodology. Multiple responses, including clad width, flatness, and non-fusion area, were transformed into a single objective through grey relational analysis, with weights objectively identified by principal component analysis. The correlation between grey relational grade (GRG) and process parameters was established by regression analysis. The results show that the GRG response model has excellent goodness of fit and predictive performance. A validation experiment was conducted at the process condition optimized for maximum GRG. The relative error of the predicted optimal GRG is 4.87% whereas those of interested individual objectives, i.e. clad width, flatness, and non-fusion area, are 5.73%, 2.97%, and 6.73%, respectively, which verifies the accuracy of the established model. The investigation of mechanical properties suggests the hardness of substrate can be improved from 20 HRC to 60 HRC and wear resistance to over 8.14 times better.


Introduction
Laser cladding is a widely used surface engineering technology which applies high density laser energy to the substrate with additive powder to rapidly obtain metallurgical bonding and significantly improve mechanical, physical, and chemical properties of the substrate surface, such as wear, corrosion, and oxidization resistance [1,2]. Due to its low dilution, small distortion, and better surface quality compared with conventional processes, laser cladding technology has been implemented in numerous fields like metallurgical mining, energy transportation, machinery manufacturing, and aerospace [3,4].
Quality assessment of laser claddings normally considers a variety of aspects including morphology, properties, and production attributes, leading to an essentially multi-objective evaluation problem. Some academics considered multiple responses of laser cladding process independently when conducting optimization. The authors of [5][6][7] used regression analysis (RA) to In this work, a novel method integrating GRA with principal component analysis (PCA) is proposed to investigate the multiple responses of multi-pass Fe50/TiC laser cladding on cylindrical substrates composed of AISI 1045 carbon steel. A composite central design (CCD) was developed based on response surface methodology (RSM) to evaluate the effects of significant factors (power ratio, laser power, defocusing distance, and overlapping ratio) on the surface quality of single layer multi-pass claddings with only limited experimental runs. The multiple responses, including clad width, flatness, and non-fusion area, were integrated into a single objective using grey relational analysis with response weights objectively determined by PCA. The values of grey relational grade (GRG) were then found to be an exponential parabola function of process parameters. The established regression model was validated with an experiment of optimized process condition.

Experimental Set-up
The laser cladding system used in the work, as depicted in Figure 2, is comprised of a laser head FDH0273 (Lasermesh, USA) with a focus length of 300 mm, an industrial robot M-710iC/50 (FANUC, Japan), a high power fiber laser YLS-3000 (IPG, Germany) cooled by a TFLW-4000WDR-01-3385 laser chiller (Tongfei, China), a gas-conveyed coaxial powder feeding machine CR-PGF-D-2 (Songxing, China), and a rotational Leadshine worktable. The cladding powder as blown onto the substrate by carrier/shielding argon gas at a pressure of 0.5 MPa, with purity contents listed in Table 3.

Powder and inert gas flow
Deliver lines Powder and inert gas feeding system   Table 3. Purity information about shielding argon gas.

Impurity Name
Before cladding, the substrate was cleaned with acetone to remove surface impurities and dried in a vacuum. The cladding powder was mixed according to the designated mass fraction in a YXQM-2L planetary ball mill machine at a speed of 300 rpm for 2 h, followed by 30 min vacuum drying at a temperature of 120 • C. Cladding powders were then deposited onto the substrate at ambient temperature. After laser cladding, the specimen was sectioned by wire-EDM in the longitudinal direction to prepare metallographic samples of 20 mm×3 mm×5 mm, which were then mounted, ground, and polished for geometrical morphology using a KH-1300 three-dimensional (3D) digital microscopy system (Hirox, Japan) after etching in 4% nital.

Design of Experiment
RSM is a comprehensive optimization method for experiment design and modelling. Compared with orthogonal design, RSM is capable of establishing correlations between targets and influencing factors with high accuracy but lower experiment cost [19,20]. We used central composite design to produce an experimental matrix for four factors with five levels (Specifically, 24 axial points and 6 replicated centre points). The four investigated factors were powder ratio (PR) of TiC, laser power (LP), defocusing distance (DD), and overlapping ratio (OR). Table 4 illustrates the studied factors and their levels. The detailed experimental matrix was obtained according to CCD as illustrated in Table 5. A laser beam of 4 mm in diameter was obtained when focused on the substrate surface. OR is the ratio of overlapped distance (D) between two adjacent passes to the width of single pass (W s ), as demonstrated in Figure 3. The ranges of the four variables were selected based on previous investigation. Other parameters were remained constant: scanning speed = 7 mm/s and gas-powder flow rate = 1100 L/h. The response variables investigated included clad width (W), flatness (F) of multi-pass, and non-fusion areas (A nf ), as illustrated in Figure 4 and defined by Equations (1) and (2) [21], respectively. The experimental results are appended in Table 5.
where A c is the multi-pass clad area, W is the total width of 8 passes, H is the maximum height of the clad layer, and A i nf is the non-fused area between adjacent ith and i + 1th pass.

Multi-Objective Approach
In this section, a new approach integrating GRA with PCA for weight determination is proposed to optimize the multi-objective problem, with the detailed procedure outlined below.

Selection of Signal-to-Noise Ratios
In signal processing, signal-to-noise ratio (SNR) is commonly used to understand the extent to which a desired property is affected by background noise. The higher the SNR, the better the product performance [22,23]. To obtain good quality multi-pass laser claddings, the SNRs of three responses were investigated to reveal the effects of the process parameters. The performance characteristics can be classified into three categories: the Larger the Better (LB), the Smaller the Better (SB), and the Nominal the Better (NB) [23,24]. Specifically, the clad width (W) and flatness (F) are considered as LB, whereas the non-fusion area (A nf ) is set to be SB, as calculated by where n is the total number of experiments and y i is the output value of the ith experimental run.

Normalization of SNR Values
As the three responses have different ranges, normalization of SNR values was conducted to obtain a common scale. In grey relational generation, the normalized responses correspond to the LB criterion, which can be expressed as [25,26]: where y * i (k) are the data after normalization, y i (k) is the response SNRs, and max y i (k) and min y i (k) are the maximum and minimum of the S/N data, respectively.

Calculation of Grey Relational Coefficients
The grey relational coefficient can be computed as follows [27,28]: , which are the ideal sequence (always 1), and the normalized value for the kth response. ψ is a distinguishing coefficient 0 ≤ ψ ≤ 1, where ψ = 0.5 for all quality characteristics. ∆ max and ∆ min are the largest and smallest values of ∆O i (k), respectively.

Determination of Response Weights
Conventionally, the weights for responses in grey relational generation are subjectively selected and usually set impartially. However, for laser cladding, different responses in terms of clad profile and mechanical properties normally require objective weight assignment. PCA, as a dimension reduction technique, is useful for identifying significant features of high-dimensional data and projects them onto a new subspace with equal or fewer dimensions [29]. The contributions of the three responses, i.e., clad width, flatness, and non-fusion area, were determined with PCA using the following procedure [30,31].

Selection of Principal Components
(a) Establish original data sequence for multiple responses by Equation (6).
where m is the number of experiment runs, n is the number of output variables, and y represents the observed response value. For this work, m = 30 and n = 3. (b) Compute the correlation efficient matrix using Equation (7).
where Cov(y i (j), y i (l)) is the covariance of data sequence y i (j) and y i (l), and ηy i (j) and ηy i (l) are the standard deviations of y i (j) and y i (j), respectively.
(d) Select principal components. With contribution rate M k and cumulative contribution rate N k obtained from Equations (9) and (10), the top k principal components are selected when N k ≥80% which accounts for the most significant features.
(e) Calculate principal component loads that reflect the correlation extent between principal component M k and original variable Y j , by:

Calculation of Response Weights
(a) Identify weights for principal components in linear combination using Equation (12): (b) Compute coefficients in the score model by Equation (13): (c) Obtain weights after normalization by:

Calculation of Grey Relational Grade
Grey relational grade (GRG) is the weighted sum of grey relational coefficients, as shown in Equation (15): where ∑ n i=1 ξ * i = 1, and β i is the weight of the ith response identified by PCA. Table 6 illustrates the SNRs of the three responses. The maximum SNR values were obtained at the 26th, 7th, and 28th experimental runs for the three responses (S/N W = 23.7504 dB, S/N F = −0.5476 dB, and S/N A = 16.3631 dB), respectively. Therefore, an optimized condition cannot be achieved for the three variables simultaneously. Table 6. Signal-to-noise ratios (SNRs) of response targets.

Response Weights
The results of principal component analysis are illustrated in Table 7. From Table 7, the eigenvalues for the first and second principal components are greater than one (λ 1 = 1.847, λ 2 = 1.001), which indicates their more significant impacts. The cumulative contribution of the first and second eigenvalues was 94.926%. The load coefficients of two components were thus obtained, as listed in Table 8.  The coefficients for the linear combinations of two principal components are shown in Table 9. With the coefficients, the corresponding combinations of three variables were obtained as depicted in Equation (16):  The score model was then established by Equation (17):

Grey Relational Grade
The results of grey relational grade were computed by applying the weights determined in the previous section, as illustrated in Table 11. The table shows that the largest GRG (0.7388) was obtained from the 28th experimental run, which suggests the optimum process parameter combination. Table 11. Grey relational analysis (GRA) results.

GRG Response Surface Model
To optimize the process parameters for multi-pass laser cladding, it was necessary to establish the correlation between the process parameters and the GRG values. Through significance analysis of GRG values, as depicted in Table 12, a natural logarithm transformation of GRG was well described as a quadratic model of the investigated parameters. By identifying the regression coefficients, the response surface model of GRG was obtained as shown in Equation (18).  Table 13 shows the results of the ANOVA for the response GRG using stepwise regression to eliminate the insignificant factors. The table shows that the p-value of model Prob(F) was less than 0.01% and lack-of-fit was greater than 73.78%, indicating that the correlation model has an existence probability of more than 99.99% and the lack of fit is caused by ambient random noise. Therefore, the selected model presented good fitting accuracy with respect to the studied factors. The value of adequate precision was 17.469, indicative of high model resolution. The coefficients of determination, R 2 , adjusted R 2 , and predicated R 2 were all approximate to one. The difference between the adjusted R 2 and predicted R 2 was less than 0.2. The geographical comparison between the predicted and experimentally obtained GRG values is illustrated in Figure 5. The closeness of data points to the line y = x also indicates their minimal divergence. Thus, the selected model for GRG is capable of explaining the process and has good prediction accuracy.   Figure 6 depicts the distribution of GRG residuals. In the figure, the residuals are relatively uniformly distributed along the straight line, which suggests that the residuals are in normal distribution and not caused by systematic errors. The GRG value, as further suggested by Table 13, is affected by all the factors: PR, LP, DD, and OR. GRG is also affected by the interactions of PR and LP, and LP and DD, and the second order terms of PR, DD ,and OR. The individual effects of each factor on the GRG value are depicted in Figure 7. The GRG is positively proportional to LP, but concave functions of PR, DD, and OR. GRG decreases with increasing weight fraction of TiC and/or overlapping ratio.   Figure 8 illustrates the effects of powder ratio and laser power on GRG. The figure shows that the GRG value decreased as the TiC weight fraction increased and laser power decreased. This occurred because more laser energy was required for melting the increased amount of TiC powder. Less powder was then cladded on the substrate, resulting in narrower cladding layers of equal passes. The addition of more TiC powder improved the viscosity of the melt liquid. Flattening of the melt cladding powder was thus prohibited with a rougher surface obtained. The interaction between PR and LP can be explained by the shadowing effect [32,33]. As more TiC is mixed in the cladding powder, a much larger amount of laser energy is required due to the absorption and shadowing by TiC particles. Less melted powder would, conversely, inhibit the energy circulation within the cladding powder.  The two-dimensional (2D) contour plot and 3D response surface of GRG with respect to laser power and defocusing distance are illustrated in Figure 9. The GRG value also increased with increasing of LP and decreasing absolute value of DD. This occurred because the increased laser energy melted much more cladding powder to produce wider cladding passes. In contrast, departure of the laser focus from the substrate also scattered the energy into a wider spot, which cladded less powder onto the substrate with resultingnarrow passes. The flatness of multi-pass cladding layer was not significantly affected by LP or DD.
(a) 2D contour plot for LP and DD (b) 3D response surface for LP and DD Figure 9. Response of GRG to LP and DD.

Validation of GRG Model
In the previous section, the correlation between GRG values and processing parameters was established as a quadratic regression model. A further experiment was conducted to validate the feasibility and achievability of the optimal conditions. Table 14 shows the optimization criteria and parameter limits for the model optimization. The GRG was set to be a maximum objective and influencing factors ranged between -1 and 1.
The experimental and predicted results for the optimal condition are compared in Table 15. The table shows that the relative errors of the predicted GRG and sub-responses (W, F, and A nf ) were less than 6.8%, which indicates the effectiveness of the established GRG response model. Figure 10 demonstrates that little fluctuation occurred at the top surface of the corresponding cladding and a small amount of non-fused zone was detected for the third and fourth passes. Therefore, multi-pass laser cladding can be optimized to have a maximum GRG of 0.739 with desirable characteristics.

Mechanical Properties
In this section, the mechanical properties of the cladding layer produced under the optimized process conditions were investigated. Figure 11 illustrates the distribution of microhardness alongside the cladding layer, both longitudinally and laterally. Figure 11a shows that the microhardness decreases stepwise with increasing depth of clad layer. It is suggested that the hardest area occurs in the sub-surface zone of clad layer, probably due to the low temperature and large solidification rate, leading to refined grain size and increased hardness [34].
The microhardness in the lateral direction was measured every 0.5 mm at a distance of 0.6 mm outward from the substrate, as depicted in Figure 11b. The microhardness along clad passes fluctuated slightly about 60 HRC, except for two points at the clad edges. In addition, the overlapped area was softer than other areas in the cladding layer due to coarser grain produced due to the remelting effect [35].  Figure 12 shows the microstructure of the optimized clad layer. The microstructure of the clad area was mainly comprised of dendrite and cellular TiC, as demonstrated in Figure 12a, whereas petal shaped cellular grain was found in the overlapped area (Figure 12b). This occurred due to the heating effect of the former pass and remelting of the overlapped area, which facilitated grain growth. The EDS results at points A-D, as shown in Figure 13, suggest element dispersion during laser cladding compared with the chemical composition of cladding powder.
When conducting overlapping cladding, the overlapped area of the solidified pass participates in the following pass cladding. It deflects the energy input into the forming melt pool, thus affecting the thermal gradient distribution and heat transfer of the melt pool. The differences in composition, viscosity, and density between the deposited and solidifying pass also influence the convection of melt material, thereby leading to composition segregation in the cladding layer [36,37].   Figure 14 compares the wear resistance results of the substrate and cladding layer with the optimized process parameters in the test configuration as illustrated in Table 16. The coefficient of friction reduced from 0.72 to 0.38 by adding the cladding powder. The worn surfaces of the substrate and the optimized cladding layer are macroscopically compared in Figure 15. The worn surfaces of both samples had similar profiles with sloughing scratches parallel to the sliding direction. The profile width of the substrate was larger than that of the optimized clad laye, mainly due to the reinforcement addition of hard phase TiC, which has better wear resistance than the α-Fe. 3D profile of worn surfaces, as depicted in Figure 16, suggesting that the substrate was also deeper than that of the optimized clad layer. The wear volume of both samples were computed to be 4573.68×10 −6 mm 3 and 562.13×10 −6 mm 3 , respectively. Therefore, the wear resistance of the substrate can be improved by about 8.14 times by applying the clad layer under the optimal process conditions.

Conclusions
To produce multi-pass laser claddings with good surface quality under multiple evaluation criteria, a serial of experimental runs, proposed by composite central design (CCD), were conducted to establish the regression model of the grey relational grade (GRG) and process parameters. Grey relational analysis (GRA) was applied to combine three process responses-clad width, flatness, and non-fusion area-into one objective. These findings provide useful guidance for process optimization of laser cladding of complex parts with multiple objectives considered. Based on the results, the following conclusions were drawn: • Principal component analysis is capable of objectively deciding the contribution of the three responses to the integrated GRG, i.e., 48.88%, 11.27%, and 39.85%, respectively. The GRG value is mainly affected by the clad width and non-fusion area.

•
The GRG value of the three variables (clad width, flatness, and non-fusion area) is exponentially determined by the second-order model of the process parameters. The GRG response model has excellent goodness of fit and predictive capacity.

•
The GRG value is relevant to all four investigated parameters. The surface quality of multi-pass laser claddings can be improved by increasing laser power, decreasing weight fraction of TiC and overlapping ratio, and/or focusing the laser spot on the surface. • By integrating SNRs with GRA, the multi-response laser cladding process can be transformed into single objective problem and thus viably optimized for individual optimum objectives (maximized clad width and flatness, and minimized non-fusion area).

•
The hardness of AISI 1045 substrate can be improved from 20 to 60 HRC and the wear resistance becomes 8.14 times better (wear volume reduced by about 87.7%) by adding Fe50/TiC powder under the optimal process conditions.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Abbreviations
The following abbreviations are used in this manuscript: