Issue 
Manufacturing Rev.
Volume 11, 2024



Article Number  14  
Number of page(s)  15  
DOI  https://doi.org/10.1051/mfreview/2024012  
Published online  30 May 2024 
Research article
Prediction models and multiobjective optimization of the single deposited tracks in laser direct metal deposition of 316L stainless steel
^{1}
Faculty of Mechanical Engineering, Le Quy Don Technical University, Hanoi, Vietnam
^{2}
Advanced Technology Center, Le Quy Don Technical University, Hanoi, Vietnam
* email: vtle@lqdtu.edu.vn
Received:
9
January
2024
Accepted:
6
May
2024
Laser direct metal deposition (LDMD) is a metal additive manufacturing process, which uses a laser source to melt metal powder and deposit the molten metal into the part layerbylayer through a nozzle. With suitable process parameters and setting conditions, a component can be fabricated with a full density. In this process, the shape of single tracks is a key indicator, which directly prescribes the quality of the process and the fabricated component. To fabricate a complex component, especially that with thinwall structures with free of defects, controlling the single tracks' geometry and the understanding on the effects of the process parameters are essential. Therefore, this article focuses on studying the effects of process variables on single tracks' attributes in the LDMD process of SS316L and identifying the optimum variables for the deposition of SS316L thin wall structures. The observed results indicated that, among the process parameters (the scanning speed V_{s}, the laser power P_{l}, and the powder feed rate f_{p}), P_{l} exhibits the highest impact contribution to the models of the deposited track width w and the deposited track penetration p with a contribution of 71.83% and 87.68%, respectively. V_{s} exhibits the highest contribution to the models of the deposited track height h a contribution of 49.86%. On the other hand, f_{p} shows an insignificant impact contribution to the w and p models. All the developed models feature a high prediction accuracy with the values of determination coefficients R^{2} of 97.89%, 97.08%, 99.11% for w, h, and p, respectively, indicating that they can be used to prediction w, h, and p with high confidence and precision levels. Moreover, the optimization results achieved by different methods (i.e., GRA, TOPSIS, and PSO+TOPSIS) demonstrated that the PSO and TOPSIS combination can be used to find out the most optimal process parameters (i.e., V_{s} = 6 mm/s, P_{l} = 263.63 W, and f_{p} = 18 g/min) to build thinwalled structures in SS316L by LDMD.
Key words: Laser direct metal deposition / 316L / optimization / GRA / TOPSIS / PSO
© K. Doan Tat et al., Published by EDP Sciences 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Laser direct metal deposition (LDMD) is a metal additive manufacturing (AM) technology that sprays metal powder with a laser source coaxially via a nozzle to melt powder and directly deposit into the workpiece layerbylayer [1–3]. The thickness of layers normally ranges from 0.1 mm to 2 mm. Based on the layer deposition principle, LDMD can build nearnet shape and complex parts without the need of cutting tools and other additional resources as in traditional manufacturing processes (e.g., machining) [4,5]. Nowadays, LDMD is a promising AM technology applied in many fields, for example aeronautics, automobile, molds and dies, repairing of parts, and rapid manufacturing of spare parts [6,7].
In LDMD processes, to create a robust and tightly bonded thick clad layer on the substrate, it is essential to meticulously manage the geometric attributes of an individual deposited track [8–13]. Furthermore, optimizing the processing parameters is important, especially concerning the percentage of porosity and the geometric qualities, to guarantee superior clad performance during the deposition [9–11,14–17]. There are several process parameters that directly influence the quality of single tracks, for example, the laser power P_{l} (W), the powder feeding rate f_{p} (g/min), the scanning speed of the nozzle V_{s} (mm/s), the size of the laser beam spot, the amount of carrying and shielding gas, the powder size, and so on. However, the key process variables identified by previous studies are the laser power P_{l} (W), the powder feeding rate f_{p} (g/min), the scanning speed of the nozzle V_{s} (mm/s) [14,18,19].
Numerous studies have been conducted to explore how the key parameters influence both the geometry of single deposited tracks and the deposited materials' integrity. For example, Errico et al. [20] examined the impact of parameters, including P_{l}, f_{p}, V_{s} and the carrying gas flow rate on microstructures and porosity of SS 304 deposited by LDMD. Ascari et al. [21] conducted an extensive series of analysis to study how process parameters influence geometries, surface roughness, microstructures, tensile strengths, hardness, and productivity in the deposition of bulk materials of SS 316L. Campanelli et al. [22] investigated the effects of P_{l}, f_{p}, V_{s}, and the overlapping distance between successive tracks in assessing the properties of deposited 18 Ni 300 maraging steel. To develop analytical models, which present the relationship between geometry attributes of single deposited tracks (e.g., width, height, penetration, and contact angle) and the process parameters P_{l}, f_{p}, and V_{s}, several authors have adopted the linear model Y = a * X + b with [14,18,19]. The coefficients of the models were determined based on the experimental data with the help of Excel and Matlab software. However, the method to estimate (α, β, and γ) was not detailed in these studies. Most of these works adopted the “trial and error” method so that the determination coefficient of the models (R^{2}) reach a maximum value possible [18].
Concerning the Optimization of the single deposited tracks in the LDMD process, Vaughan et al. [23] proposed a framework for optimizing single tracks considering the laser power, the scanning speed, and the powder feed rate, to obtain a dense bulk components of high strength steels. Zardoshtian et al. [23] adopted the response surface methodology (RSM) to optimize the singletrack geometry of CuCrZr alloy deposited by LDMD. They stated that the estimated optimal parameter, including the scanning speed of 4.5 mm/s, the laser power of 1550 W, and the powder feed rate of 12 g/min enabled producing single track geometries with a good shape, sufficient dilution, and at the maximum rate of deposition. Recently, Javidrad et al. [11] performed a study on the optimization of the process parameter for the LDED process of Inconel 738LC. The authors aimed at minimizing the porosity, while maximizing the production rate and using the RSM method to resolve such a multiobjective optimization problem. Pandey et al. [24] also used the RSM method to optimize the single tracks' geometries when depositing 15Cr5Ni stainless steel. Meanwhile, Du et al. [23] employed the combination of TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) and GRA (GreyRelational Analysis) for optimizing single tracks in laser cladding of 155PH steel.
To the best of the authors' knowledge, until now, very few authors have paid attention to the prediction and multiobjective optimization of the single deposited tracks in LDMD processes. To fill this knowledge gap, this paper aims to develop the prediction models for geometrical attributes of single tracks in LDMD of SS 316L. The models are developed by using experimental data. After that, several multiobjective optimization methods, including GRA, TOPSIS, and PSO (Particle Swarm Optimization) algorithm combined with TOPSIS are considered to identify the optimal process parameters that enable to obtain the expected attributes of single tracks for specific applications. GRA and TOPSIS methods are widely used in different fields to determine the optimal solution among a set of available alternatives [25,26]. On the other hand, PSO is a natureinspired optimization technique used to find all the feasible optimal solutions in the search space [27].
2 Materials and methods
2.1 Materials and equipment
The DLMD system used in this study is comprised of a 0.5 kW fiber laser, a powder feeder system, a deposition nozzle, and a threeaxis CNC mechanism (Fig. 1a). The SS 316L powder was utilized in the experiment. The particle size ranges from 15 to 55 μm with the average size of 25.56 μm and the standard deviation of 9.22 μm (Figs. 1b and 1c) and the chemical composition is given in Tab. 1. SS 316L plates with dimensions of 200 × 100 × 10 mm and its composition indicated in Table 2 were utilized as the substrate. Before the deposition, the metal powder was dried in a vacuum drying oven at 200 °C for 4 h to eliminate moisture. The substrate was polished with sandpaper and then cleaned with acetone and ethanol to remove grease and stains.
Fig. 1 (a) The LDMD system, (b) a SEM image of SS 316L powder, and (c) the powder size distribution. 
Chemical composition of powder and substrate material (wt.%).
Studied variables and their levels.
2.2 Fabrication of samples and data collection
In the LDMD process, there are many process parameters that could influence the geometry and quality of tracks, such as the laser beam spot diameter, the pressure and flowrate of carrying/shielding gas, the particle size distribution, the laser power P_{l}, the powder feeding rate f_{p}, the scanning speed of the nozzle V_{s}, and so on [8,9,11,17,28–31]. However, the laser power, the powder feeding rate, and the scanning speed are identified as the key parameters that most considerably impact on the temperature distribution, the melting pool dimensions, and the geometry of tracks [13,16]. Moreover, in the experiment, the change of these parameters is simpler and more practical than other ones, such as the laser beam spot diameter and the particle size distribution. Therefore, in this research, we consider the scanning speed V_{s}, the laser powder P_{l}, and the powder feeding rate f_{p} for the parametric investigation and optimization, while other parameters were fixed at a constant value. To fabricate the single deposited tracks, the experimental plan was designed according to L9 orthogonal array Taguchi method with three levels for each parameter, as shown in Table 2. As a result, there are nine experimental runs. The nice single tracks were deposited and cleaned, as shown in Figure 2a. All the depositions were performed in a building chamber filled with a shielding gas of argon, where the oxygen level is less than 20 ppm. The rate of shielding and carrying gas was maintained at 6 L/min. The distance from the laser head tip to the substrate's surface was fixed at 15 mm, and the laser beam spot diameter is controlled at 1 mm.
It can be found that, on the top view (Fig. 2a), all the tracks were regular and stable. As a result, we can suppose that all these single tracks have almost constant value of track width and track height, especially in the middle region of each track. Therefore, we decided to analyze the tracks' attributes (including the width w, height h, and penetration depth p) from the middle of their length. This approach enables avoiding the impact of edge effects, and it is also widely used by previous studies in the literature [15,16,32]. Whereby, the crosssections of each single track in the middle regions were extracted, ground, polished and chemically etched to measure the attributes (w, h, and p). The measurement of w, h, and p were performed with the help of an optical microscope (AXIO A2M, Carl Zeiss). The experimental matrix and measured data were presented in Table 3.
The measured data was used to analyze the influence of process variables on the attributes of single deposited tracks and to develop the prediction models. The development of prediction models and the analysis of variance are performed with the aid of Minitab 19 software.
Fig. 2 (a) Deposited single tracks and (b) a crosssection of the deposited single track (6). 
Experimental runs and the collected data.
2.3 Optimization methods
In this study, we aim to find out the optimal process parameters for the building of thinwall structures by the LDMD process. In this case, the width and the penetration of single deposited tracks are preferred to be minimized, while the height of the single deposited track is desired to be maximized. As a result, the multiobjective optimization issue is described as: “Find (V_{s}, P, and f_{p}) to minimize (w and p) while maximize h, subject to 8 ≤ V_{s} (mm/s) ≤ 10, 230 ≤ P (W) ≤ 270, and 8 ≤ f_{p} (g/min) ≤ 12”.
To resolve such a problem, different methods are employed and evaluated to select the best one. As mentioned earlier, the three following methods are examined, including PSO combined with TOPSIS, GRA, and TOPSIS. These methods are briefly described, as follows.
2.3.1 GRA
GRA is one of the methods widely used for the multiobjective optimization in manufacturing processes [33]. This method consists of transforming a multiobjective issue to a monoobjective one. The GRA procedure is composed of the following steps:
(1) Normalizing the kth objective using the following criteria:
if the objective respects the smallestthebest criterion.
if the objective respects the biggestthebest criterion.
where is the value of the kth objective measured at the ith trial, and are the highest and the smallest values of the kth objective measured in the experiment, and is the normalized value of the kth attribute measured at the ith trial. In this study, the objectives (w and p) are normalized according to equation (1) and h is normalized according to equation (2).
(2) Calculating GRCs (greyrelational coefficients) for the normalized data using equation (3):
where and θ_{max} are calculated as equations (4), (5), and (6):
is the reference sequence, for example, , and φ is a coefficient in the range of (0, 1). In this study, φ = 0.5.
(3) Calculating the GRGs (greyrelational grades) by equation (7):
where ω_{j} is the weight of the jth objective with ∑ω_{j} = 1 and 0 < GRG_{j} 1.
Finally, the optimized solution is corresponding to the maximum value of GRG_{j}.
2.3.2 TOPSIS
As mentioned earlier, TOSIS relies on the concept that the optimal alternative is the nearest to the best solution and it is the farthest from the worst solution [34,35]. The main steps of TOPSIS method are expressed, as follows:
(1) Formulating the decision matrix {r_{ij}}_{nxm}, where m is the objective number, n is the trial number in the experiment, and r_{ij} is the value of the jth objective at the ith trial.
(2) Normalizing the decision matrix by equation (8):
(3) Assigning the normalized matrix with the weight of each objective using equation (9):
(4) Determining the best solutions (BS) and the worst solutions (WS) using equations (10) and (11):
(5) Calculating the distance from the BSs and WSs to the feasible solutions using equations (12) and (13), respectively:
(6) Computing the closest index of the BSs using equation (14):
The values of are also in the range of (0, 1), and the optimum solution corresponds to the maximum value of .
In this study, it is remarked that the weight ω_{j }for the jth objective in the TOPSIS and GRA is similar and calculated by the CRITIC method [36]. This approach integrates both the inherent conflict aspect and the level of contrast in determining the weight proportion of each response. The steps and equations associated with this approach are following:
(i) Normalizing the decision matrix {r_{ij}}_{nxm} using equation (15):
where r_{ij} is the normalized value of the ith alternative for jth objective, and are the worst and the best values of the jth objective.
(ii) Calculating the standard deviation of each normalized attribute according to equation (16):
where r_{j} is the average value of the jth normalized objective.
(iii) Constructing the matrix [C_{ij}]_{n}_{xn} with the linear correlation coefficients C_{ij} between the objectives. C_{ij} is computed according to equation (17):
(iv) Calculating the objective information OI_{j} by equation (18):
(v) Identifying the weight ω_{j} or each objective using equation (19):
2.3.3 PSO algorithm
PSO is an optimization algorithm that mimics the collective behavior observed in flocks of birds or schools of fishes. This algorithm involves the initial creation of a population of particles, including their positions (Posi_{i}) and velocities (Velo_{i}) are randomly assigned within allowable ranges. Each particle, specifically its position, represents a potential solution to the problem [28]. In essence, each position can be considered as a vector: Posi_{i} = (Posi_{i,1}, Posi_{i,2}, …, Posi_{i,N}), where ‘i’ signifies the particle index, and ‘N’ denotes the number of design variables, and the particle has a velocity: Velo_{i} = (Velo_{i,1}, Velo_{i,2}, …, Velo_{i,N}). The position of each particle can be seen as a unique point within the solution space. The objective function is used to evaluate each position's performance. During each iteration, the best position for each particle (Posi_{ibest}) and the best position among all particles (Global_{best}) are updated and saved when necessary. Additionally, in each iteration, all the elements of the velocity and position vectors for all particles are updated using equations (20) and (21), respectively:
where i and j present the particle's index and the dimension of each design parameter, respectively. ωs the inertial weight. C_{1} and C_{2} are the coefficients of acceleration. R_{1} and R_{2} are the random numbers induced through a uniform distribution. When the termination criteria are satisfied, the optimum solution is the last best position of all the particles (i.e., the last Global_{best}). The procedure of the PSO algorithm is drawn in Figure 3.
Fig. 3 Flowchart of PSO algorithm. 
3 Results and discussion
3.1 Development of prediction models
In this study, all the prediction models of singletrack attributes (i.e., the width w, the height h, and the penetration p) were developed with the aid of Minitab 19 software. The analysis of variance (ANOVA) for the models was conducted with a confidence level of 95% and a significant level of 5%.
3.1.1 The developed model of w
The prediction model of w is expressed by equation (22), and the ANOVA for the model of w is presented in Table 4. It is shown that the model and all the model terms {V_{s}, P_{l}, and f_{p}} are significant with the Pvalue inferior to 0.05. Among the process variables, the laser power P_{l} shows the highest contribution of 71.83%, followed by the scanning speed V_{s} and the powder feed rate f_{p} with a contribution of 23.85% and 2.20%, respectively. The determination coefficients of the model {Rsq = 97.89%, Rsq (adj) = 96.62%, Rsq(pred) = 92.72%} indicate a high prediction accuracy of the model. Therefore, the developed model of w can be utilized to predict the width of single tracks in the entire space of design with a high reliability level and it can be used as an objective function for the optimization problem.
ANOVA for w.
3.1.2 The developed model of h
The prediction model of h is described by equation (23), and the ANOVA results for this model is given in Table 5. It is indicated that the model and all the model terms {V_{s}, P_{l}, and f_{p}} show the Pvalue inferior to 0.05. Thus, they are significant model terms. In this model, the scanning speed V_{s} features the highest contribution of 49.86%, whereas the laser power P_{l} and the powder feed rate f_{p} show a smaller contribution of 30.85% and 16.38%, respectively. The determination coefficients of the model {Rsq = 97.08%, Rsq (adj) = 95.33%, Rsq(pred) = 90.85%} demonstrate that the model has a high prediction accuracy level. As a result, this model can predict the height of single tracks in the entire space of design with a high reliability level, and it can be used as an objective function for the optimization problem.
ANOVA for h.
3.1.3 The developed model of p
The prediction model of w is expressed by equation (24), and the ANOVA for the model of p is presented in Table 6. It is revealed that the model and all the model terms {V_{s}, P_{l}, and f_{p}} are significant with the Pvalue inferior to 0.05. Among the process variables, the laser power P_{l} shows the highest contribution of 87.68%, followed by the powder feed rate f_{p} and the travel speed V_{s} with a contribution of 6.72% and 4.70%, respectively. The determination coefficients of the model {Rsq = 99.11%, Rsq (adj) = 98.57%, Rsq (pred) = 97.40%} confirm that the developed model of p exhibits a high prediction accuracy. Thereby, this model can be employed to predict the penetration of single tracks in the entire design space, and it can be used for the optimization problem.
ANOVA for p.
3.2 Parametric influence analysis
Figure 4a shows the main effects of the process parameters on the width of single deposited tracks (w). It is shown that w decreases with an increment in the scanning speed V_{s}. A decreasing trend in w is also observed but with a lower rate when the powder feed rate f_{p} increases. On the other hand, w increases with an increase in the laser power P_{l}. Figure 5a shows the interactional effects of P_{l}, V_{s}, and f_{p} on w. It is also revealed that w increases with the increase in P_{l} with all values of V_{s} and f_{p} in the design space. w shows a decreasing trend with the increment in V_{s} for all values of P_{l}, while the interaction effects of (f_{g} and V_{s}) and (f_{g} and P_{l}) on w do not present uniform trends. This observation is in line with the ANOVA results presented in Table 4, where it is indicated that the laser power P_{l} shows the highest contribution of 71.83%, followed by the scanning speed V_{s} with a contribution of 23.85%, while the contribution of f_{p} is only of 2.20%. The effect trend of P_{l} and V_{s} on w observed in this study agrees with the those presented in [18,37]. In the current study and the previous studies [14,18], the w also an increasing trend with an increase in f_{p}. On the other hand, in the study of Sreekanth et al. [37] on the laserDED of Inconel 718, the authors found that w features a decreasing trend with an increase in f_{p}. Such a difference could be explained by the insignificant contribution (only 2.20%) of f_{p} in the developed model of w. Theoretically, an increase in V_{s} leads to decreasing the interaction time of metal powder with the melting pool, resulting in a reduction of the tacking width and the deposited track width w. On the other hand, an increase in P can fuse more powder particles and enlarge the melting pool. Thus, the width track w increases.
Figure 4b shows the direct influence of P_{l}, V_{s}, and f_{p} on h, while Figure 5b exhibits the interaction effects of these parameters on h. It can be seen that h decreases as V_{s} increases from 6 mm/s to 10 mm/s while h increases as P_{l} increases from 200 W to 300 W. Similarly, h exhibits an increasing trend when f_{p} increases from 6 g/min to 18 g/min. These variation trends of h with the process parameter (V_{s}, P_{l}, and f_{p}) variation are also consistent with those observed in previous studies [19,32,38,39]. Due to V_{s} features the highest contribution of 49.86% to the h model (Tab. 5), V_{s} shows negative effects on h (i.e., h decreases when V_{s} increases) for all values of P_{l} and f_{p}. On the other hand, the interaction effects of (f_{g} and V_{s}) and (f_{g} and P_{l}) are not clear. This observation agrees with the ANOVA results given in Table 5, because f_{g} shows a small contribution to the h model (with a contribution of 16.38%). The deposited track height h decreases with an increase in V_{s} can be explained by the reduction of the powder amount per unit length along the deposition direction when V_{s} increases [14,40]. Meanwhile, h increases with an increment in the laser powder P_{l}. This is because when P_{l} increases the energy input increases and the amount of melted powder increases too, resulting in forming a great molting pool. As a result, both the w and h of the deposited track increase.
Figures 4c and 5c present the direct effects and the interactive effects of process variables on the penetration deep p of the deposited track, respectively. It can be observed that p increases for an increment in P_{l} (from 200 W to 300 W), while it is decreased with an augment in V_{s} (from 6 mm/s to 10 mm/s) and f_{p} (from 6 g/min to 18 g/min). In this case, due to the highest contribution to the p model of P_{l} (with 87.68% of contribution), the effect trend of P_{l} on p is the strongest. p increments with an increase in P_{l} for all values of V_{s} and f_{p} in the whole design space. On the other hand, with a small contribution value of f_{p} and V_{s} to the p model (6.72% and 4.70%, respectively), the effect trend of these variables on p is not uniform. The similar observation on the effects of P_{l}, V_{s} and f_{p} on p was reported in previous publications [14,32,38,40,41]. As explained in previous studies, when P_{l} increases, more laser energy input applied to the deposition process increases. As a result, p is deeper.
Fig. 4 Direct influences of parameters on (a) w, (b) h, and (c) p. 
Fig. 5 Interaction influences of parameters on (a) w, (b) h, and (c) p. 
3.3 Optimization results
3.1.1 GRA results
Table 7 presents the results obtained by the GRA method. The normalized objectives were calculated by equations (1) and (2) for {w, p} and h, respectively. The greyrelational coefficients GRC were computed using equations (3)–(6), while the GRGs were achieved using equation (7). It is noted that the weight ω_{j} for each objective (i.e., w, p and h) was estimated from the experimental data by the CRITIC method, and the weight values were employed in both the GRA and TOPSIS methods. In this research, ω_{w} = ω_{p} = 0.26 and ω_{h} = 0.48 Based on the GRGs' value, it is indicated that the run No. 7 with the highest value of GRG (0.703) is considered as the optimal solution. Hence, the optimal parameters estimated by the GRA methods are {V_{s} = 10 mm/s, P_{l} = 200 W, and f_{p} = 18 g/min} that produce a deposited track with {w = 1.386 mm, h = 0.094 mm, and p = 0.042 mm}.
Optimization obtained by GRA.
3.3.2 TOPSIS results
Table 8 shows the results obtained by the TOPSIS method. The normalized data and the weighted normalized data were calculated by equations (8) and (9), respectively. The values of and ere calculated by equations (12)–(14), respectively. Based on the highest value of it is found that the process variables of the run No. 3 (i.e., V_{s} = 6 mm/s, P_{l} = 300 W, and f_{p} = 18 g/min) are considered as the optimal process variables, and they produced the deposited track with {w = 1.745 mm, h = 0.235 mm, and p = 0.116 mm}.
Optimization obtained by TOPSIS.
3.3.3 Results of PSO combined with TOPSIS
As mentioned earlier, the PSO algorithm generates a set of feasible optimal solutions. In this case, the prediction models of w, h, p (Eqs. (22), (23), and (24), respectively) were used as objective functions in PSO algorithms. The parameters of the PSO algorithm were chosen as follows: Both the population size and the repository size are equal to 50. The maximum generation number is 100. The inertia weight is 0.4. The individual confidence and swarm confidence factors are equal to 2. As a result, we obtained the Pareto of 50 feasible optimal solutions, as shown in Figure 6a. It can be found that it is difficult to select one feasible solution manually that satisfies the requirement (i.e., maximizing h while minimizing both w and p), because h increases with an increase in w and p (Figs. 6b and 6c). Therefore, the TOPSIS is utilized to select the most optimal solution among the feasible optimal solutions.
Table 9 shows the results of TOPSIS applied to the 50 feasible optimal solutions. It is shown that the solution No. 14 exhibits the highest value of . As a result, the most optimal solution selected by TOPSIS among the Pareto points is (V_{s} = 6 mm/s, P_{l} = 263.63 W, and f_{p} = 18 g/min), and they produced the deposited track with {w = 1.656 mm, h = 0.208 mm, and p = 0.094 mm}.
Fig. 6 PSO results: (a) Pareto of feasible optimal solutions, (b) w vs. h, and (c) p vs. h. 
Optimization obtained by TOPSIS from the PSO feasible optimal solutions.
3.3.4 Comparison of different optimization results
Table 10 presents the comparison between different optimization results obtained with GRA, TOPSIS, and PSO combined with TOPSIS. Herein, the difference in results between the method A vs. the method B was defined by equation (25), where the Atribute (A) and Atribute (B)s the values of the same attribute obtained by the method A and B, respectively.
It is indicated that the optimal solution obtained by TOPSIS shows the highest value of h, followed by the (PSO + TOPSIS) and GRA, respectively. The h obtained by TOPSIS is higher than that obtained by GRA up to 150%, while the optimal solution generated by (PSO+TOPSIS) shows a higher h value of 122% compared to that of GRA. Similarly, the values of w and p in TOPSIS and (PSO+TOPSIS) are higher than those obtained by GRA.
Compared to the TOPSIS method, the optimal parameters obtained by PSO+TOPSIS method enable producing the single track with the values of w, h, and p lower than those in TOPSIS with a reduction of 5%, 11% and 19%, respectively. As a result, in terms of w and p, the PSO+TOPSIS method provides better results, meanwhile, in terms of h, the TOPSIS is better. Indeed, h is related to the distance upwards the cladding nozzle moves prior to deposing the next layer or the layer height, while the penetration depth p is related to the bonding between the deposited layers through the dilution indicator, d = 100%×p/(p + h). Higher dilutions d lead to porosity and crack generation, which ultimately deteriorates the part's bond quality and mechanical performance [24,32,42]. In the current study, although the TOPSIS method provides a better objective h than the PSO+TOPSIS method (0.235 mm vs. 0.208 mm), it also provide a higher objective p (0.116 mm vs. 0.094 mm), leading to a higher dilution d (33% vs. 31%). Due to this reason, we consider that the PSO+TOPSIS method provides a better solution.
As a conclusion, in this research, compared to GRA and TOPSIS methods, PSO + TOPSIS offers the most appropriate objectives because it gives a solution balancing the objectives in the multiobjective optimization problem − i.e., maximizing h, while minimizing both (w and p).
Finally, the optimal process variables obtained by PSO + TOPSIS (i.e., V_{s} = 6 mm/s, P_{l}_{ }= 264 W, and f_{p} = 18 g/min) were used to build several thinwalled structures as shown in Figure 7. It can be seen that all the asdeposited components have good shape. There are also no major cracks appearing between the asbuilt part and the substrate, as well as between the deposited layers.
Comparison of optimization results obtained by GRA, TOPSIS and (PSO + TOPSIS).
Fig. 7 Thin wall structures deposited with the optimal variables obtained in PSO + TOPSIS (i.e., V_{s} = 6 mm/s, P_{l}_{ }= 264 W, and f_{p} = 18 g/min). 
4 Conclusions
In this research, the influences of process variables (including the laser power P_{l}, the scanning speed of the deposition nozzle V_{s}, and the powder feeding rate f_{p}) on single deposited tracks' attributes in laser direct metal deposition (LDMD) of 316L stainless steel were explored and the optimum process variables for the deposition of 316L stainless steel thinwall structures were identified, considering different multiobjective optimization problems. The main outcomes of this research can be summarized as follows:
The laser powder P_{l} exhibits the highest impact contribution to the models of the deposited track width w and the deposited track penetration p with a contribution of 71.83% and 87.68%, respectively, among the process parameters (the scanning speed V_{s}, the laser power P_{l}, and the powder feed rate f_{p}). The scanning speed V_{s} exhibits the highest contribution to the models of the deposited track height h a contribution of 49.86%. Meanwhile, the powder feeding rate f_{p} shows an insignificant impact contribution to the w and p models.
All the developed models of w, h, and p feature a high prediction accuracy with the values of determination coefficients R^{2} of 97.89%, 97.08%, 99.11%, respectively, which demonstrate that they can be used to prediction w, h, and p with high confidence and precision levels.
Moreover, the optimization results achieved by different methods (i.e., GRA, TOPSIS, and PSO+TOPSIS) demonstrated that the PSO combined with TOPSIS can be used to find out the most optimal process parameters (i.e., V_{s} = 6 mm/s, P_{l} = 263.63 W, and f_{p} = 18 g/min) to build thinwalled structures in SS316L by LDMD.
The asdeposited part built with the optimal process variables shows smooth surfaces and a regular height, and without major cracks, validating their consistency.
Because of a significant role of the single tracks in the LDMD process, this research contributed an insight into the effects of the main process parameters (i.e., the scanning speed V_{s}, the laser power P_{l}, and the powder feed rate f_{p}) on the single tracks and the methods determining the optimal process parameters in LDMD of SS316L, to the academic community and specific applications. However, to provide a comprehensive understanding on the impact of the process parameters on the single tracks, other variables such as the laser beam spot diameter, the pressure and flowrate of carrying/shielding gas, the standoff distance, and the particle size, need to be considered in the future work. The process optimization could also consider other outputs of the single tracks such as surface roughness, wetting angle, porosity, microhardness, etc.
Funding
This research is funded by Vietnam National Foundation for Science and Technology Development under grant number KC4.015/1925.
Conflicts of interest
The authors have no competing interests to declare that are relevant to the content of this article.
Data availability statement
All data generated and analyzed during this study are included in this article.
Author contribution statement
T.K. Doan and V.T. Le are involved in the conceptualization and methodology; T.K. Doan, V.T. Le, and V. N. Duong contributed to the formal analysis and investigation; T.K. Doan and V.T. Le contributed to writing—original draft preparation; all the authors contributed to writing, review, and editing; T.K. Doan acquired the funding.
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Cite this article as: Khoa Doan Tat , Van Thao Le, Nguy Duong Van , Prediction models and multiobjective optimization of the single deposited tracks in laser direct metal deposition of 316L stainless steel, Manufacturing Rev. 11, 14 (2024)
All Tables
All Figures
Fig. 1 (a) The LDMD system, (b) a SEM image of SS 316L powder, and (c) the powder size distribution. 

In the text 
Fig. 2 (a) Deposited single tracks and (b) a crosssection of the deposited single track (6). 

In the text 
Fig. 3 Flowchart of PSO algorithm. 

In the text 
Fig. 4 Direct influences of parameters on (a) w, (b) h, and (c) p. 

In the text 
Fig. 5 Interaction influences of parameters on (a) w, (b) h, and (c) p. 

In the text 
Fig. 6 PSO results: (a) Pareto of feasible optimal solutions, (b) w vs. h, and (c) p vs. h. 

In the text 
Fig. 7 Thin wall structures deposited with the optimal variables obtained in PSO + TOPSIS (i.e., V_{s} = 6 mm/s, P_{l}_{ }= 264 W, and f_{p} = 18 g/min). 

In the text 
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