Multi-objective optimization of fiber laser welding parameters for 316L stainless steel

Carbon emission modeling

In this study, we developed a carbon emission measurement system, as well as a calculation and conversion program for the conversion of electrical energy consumption into carbon emissions. The electric power of the platform is measured in real time, and the energy consumption of the platform is calculated by the curve integral method. As shown in Figure 2, it is the measured energy consumption of each device. The fiber laser welding platform is composed of multiple systems, and the processing process is accompanied by the generation of multi-source carbon emissions. Specifically, the energy consumption can be categorized into the laser generator, refrigeration, robotic, and auxiliary systems.

Figure 2. Power consumption diagram of laser welding system. (A) The power curve of the entire system in one cycle; (B) Power curve of the laser within one cycle; (C) Power curve of the robot within one cycle; (D) Power curve of the cooler within one cycle; (E) Power curve of the compressed gas device within one cycle; (F) Power curve of the protective gas device within one cycle; (G) Power curve of the auxiliary system within one cycle.

The power-carbon emission conversion for laser welding can be expressed as:

where CEF_elec and CEF_gas are carbon emission coefficients of electric energy and gas, respectively. The carbon emission factor CEF_elec of electric energy consumption is 0.6747 kgCO_2-ep/kWh^[14]. The carbon emission factor CEF_gas of protects the gas consumption is 0.9576 kgCO_2-ep/m³. E_laser, E_robot, E_cool, and E_other represent laser power consumption, robot power consumption, chiller power consumption, and other energy consumption, respectively. E_gas and E_cg are the volume consumption of the protective gas and the volume consumption of the compressed gas, respectively.

Tensile strength modeling

Laser process parameters will directly affect the quality of welded joints. In engineering and testing, T_S is widely used in the evaluation of welding performance, which is calculated as follows:

where F_max is the maximum bearing tension of the welded joint, and A is the cross-sectional area of the stretched sample. In addition, I_W is closely related to T_S and weld quality. The tensile sample was prepared according to the GB/T 2651-2008 standard (Tensile Test Method for Welded Joints). The welded workpiece was sectioned by wire cutting, and the specific sample dimensions are shown in Figure 3.

Figure 3. 316L stainless steel base metal.

Depth-to-width ratio modeling

Figure 4 shows a schematic diagram of the weld morphology in laser welding. As a key visual evaluation indicator, weld morphology is widely used for evaluating and judging welding quality. Compared with mechanical performance testing methods such as tensile testing, judging welding quality through weld morphology is not only more efficient, but also significantly reduces costs. The ideal weld seam shape needs to balance narrow width and large penetration depth. Specifically, when the penetration depth is the same, a narrower weld width helps reduce the range of the heat-affected zone, thereby reducing stress concentration and angular deformation phenomena. In the case of consistent weld width, a greater penetration depth is more advantageous in engineering practice, which is also one of the core advantages of fiber laser welding. Therefore, for weld morphology, the ideal aspect ratio is usually large, and this parameter is often used as an important indicator to measure welding quality in relevant literature. The aspect ratio (I_W) of welds can be used to quantify and compare the morphological characteristics under different welding conditions, which is calculated using

Figure 4. Definition of weld depth and height. TH: Width of weld seam on the surface of the workpiece; BH: depth of weld seam along the vertical direction.

where TH is the width of weld seam on the surface of the workpiece, and BH is depth of weld seam along the vertical direction.

Experimental design

The experimental study focused on 316L austenitic stainless steel, a material widely used in automotive and medical industries due to its superior corrosion resistance and weldability. The workpieces, with dimensions of 100 mm × 80 mm × 3 mm, underwent pre-welding preparation, including surface polishing and cleaning with anhydrous ethanol to remove oxides and contaminants. Post-welding, specimens were sectioned via wire-cutting, metallurgically mounted, and etched (HCl + HNO₃ + H₂O) for microstructural analysis.

All welding trials were conducted on an automated platform comprising an IPG YLR-4000 fiber laser (IPG Photonics, Oxford, MA, USA) integrated with an ABB IRB4400 robotic arm (ABB Robotics, Zürich, Switzerland) for stable path control. Argon shielding was applied at 25 L/min through an inclined nozzle. Three governing parameters - laser power (LP), welding speed (WS), and defocus distance (DA) - were selected because they jointly capture the essential balance between weld quality and E_C. Specifically, laser power dictates the total energy delivered to the weld pool, directly affecting penetration depth, fusion stability, and overall energy consumption. Welding speed controls the interaction time between the beam and the material, influencing heat input per unit length, solidification rate, and consequently T_S and emission levels. Defocus distance modifies the effective spot size and energy density distribution, which determines bead morphology, stability, and the I_W of the weld. By varying these parameters, it is possible to explore a wide design space that reflects the practical trade-off between mechanical performance and sustainability objectives. The experimental design employed optimal Latin hypercube sampling (OLHS) to generate 25 representative parameter sets, and the platform configuration is shown in Figure 5.

Figure 5. Fiber laser welding experimental platform (Photographs taken by the authors). PC: Personal computer.

To efficiently explore the parameter space, an OLHS method was employed^[15-17]. This approach ensured uniform coverage of the design space with minimal experimental runs, reducing costs while maximizing data representativeness. Parameter range settings are shown in Table 1.

A total of 25 experimental runs were conducted [Table 2], with responses measured for E_C, T_S, and I_W.

Stack process description

The proposed stacked regression model integrates heterogeneous base learners (RF, XGBoost, and SVR) with a KRG meta-model to leverage complementary learning patterns and enhance generalization. As illustrated in Figure 6, the model follows a two-stage hierarchical structure.

Figure 6. Stacked regression model structure. RF: Random forest; XGBoost: extreme gradient boosting; SVR: support vector regression; KRG: Kriging.

In this study, a stacking framework was constructed to improve predictive accuracy and robustness for laser welding process modeling. The main role of the base models is to generate preliminary predictions for each target variable, which are then input into the meta-model as new features for further optimization and correction. The base models selected include RF, XGBoost, and SVR^[17,18]. These algorithms were chosen due to their complementary learning strategies: RF, as a bagging-based ensemble of decision trees, is robust to noise and captures nonlinear feature interactions; XGBoost, a boosting-based method, excels in modeling complex dependencies and has demonstrated superior accuracy in structured data^[19-21]; SVR leverages kernel functions to capture nonlinear patterns and maintains strong generalization even under limited samples. The meta-model employs KRG, a Gaussian process regression method, which provides smoother and more generalized outputs by exploiting the spatial correlation among the predictions of the base learners^[22,23]. Unlike a simple concatenation of outputs, the stacking process integrates the complementary strengths of the base models and the meta-model through a hierarchical learning mechanism. By feeding the outputs of the base models as inputs to KRG, the framework achieves higher-level integration, yielding final predictions that are both more accurate and more reliable. This design leverages the diversity of the base models and the interpolation capability of the meta-model, thereby enhancing the robustness and generalization of the predictive system compared with any single-model approach.

Using the experimental equipment mentioned earlier to supplement 200 sets of experimental data, 150 sets were randomly selected as the training set to train the basic model (RF, XGBoost, SVR) and KRG meta model, while the remaining 50 sets were used as independent test sets to evaluate the predictive performance of the model. The division of the training and testing sets is stratified according to parameter distribution, ensuring that the two groups maintain consistent statistical characteristics in terms of LP, WS, DA range and output response distribution to avoid model evaluation bias caused by uneven data distribution.

The implementation of the stack model is divided into the following steps:

Step 1: Base model training: Three base models - RF, XGBoost and SVR - were trained using training dataset 1 and target output 2. The prediction result z_j⁽ⁱ⁾ of the JTH base model f_j for sample X_i is obtained as:

z_j⁽ⁱ⁾: Prediction result of the j-th base model for the i-th sample; X_i: The i-th input sample; j: Index of the base model, taking values of 1, 2, 3; i: Index of the sample.

Step 2: Base model prediction: The base models are used to generate prediction results on both the training and test sets. The prediction results from the training set are then concatenated to form a new feature matrix Z. The new feature matrix is constructed as follows:

Step 3: With Z as the input and Y as the output, the KRG meta-model is trained. The prediction of the meta-model is given by:

where f_KRG is the KRG model.

Step 4: The prediction results of the base model on the test set constitute Y1, and the meta-model predicts Y2 to get the final predicted value of the target variable.

The algorithm flowchart is given in Figure 7.

Figure 7. Algorithm flowchart. RF: Random forest; XGBoost: extreme gradient boosting; SVR: support vector regression; KRG: Kriging; MOPSO: multi-objective particle swarm optimization.

To enable the model to more effectively capture the relationship between input features and output targets, improve training efficiency, and eliminate dimensional discrepancies, the data are normalized using

where x is a data point in the original dataset, x_min and x_max are the minimum and maximum values of the feature, respectively, and x_new is the normalized value. This procedure scales all data to the interval [0, 1], making features comparable and preventing performance issues caused by large differences in data scales in certain machine learning algorithms.

RF: Optimal hyperparameters (n_estimators = 200, max_depth = 8) were determined via grid search to balance model complexity and computational efficiency.

XGBoost: Bayesian optimization determined the optimal hyperparameters - learning rate (0.05), maximum tree depth (5), and L2 regularization (λ = 0.1) - to minimize the validation root mean square error (RMSE).

SVR: Cross-validation selected the radial basis function (RBF) kernel parameters - C = 1 and γ = “scale” - to ensure robust generalization in high-dimensional feature spaces.

KRG: A Matérn kernel (ν = 1.5) was used to model the spatial correlations among the base model predictions:

where r is the Euclidean distance between input points (or distance in the parameter space), σ² is the signal variance controlling the amplitude of the function, and l is the length-scale parameter determining the decay rate of correlation.

Hyperparameters were optimized using 10 restarts of maximum likelihood estimation (MLE) to reduce the risk of convergence to local minima.

Optimization model

The MOPSO algorithm is an extension of the traditional particle swarm optimization (PSO) method, designed to handle multiple conflicting objectives simultaneously. The MOPSO algorithm is an extension of the traditional PSO method, designed to handle multiple conflicting objectives simultaneously^[24,25]. While classical PSO is limited to optimizing a single target function, MOPSO enables the discovery of one or more non-dominated solutions under multi-objective conditions, thus providing a well-distributed Pareto front. Its fundamental principle is to simulate the social behavior and motion dynamics of particle populations in the search space, where each particle represents a candidate solution corresponding to a specific combination of process parameters^[26,27]. The algorithm begins by initializing a population of particles within the design space. Each particle is evaluated using the defined fitness functions, which in this study are E_C, T_S, and I_W. During the iterative process, particles update their velocities and positions based on both their individual best solutions (pBest) and the global best solutions (gBest). At the same time, an external archive is maintained to store the current non-dominated solutions, while leader selection and crowding distance mechanisms are applied to preserve population diversity and guide the swarm toward convergence. Through this iterative mechanism, MOPSO gradually evolves the particle population toward the Pareto front, achieving a balanced optimization among multiple objectives. The overall workflow of the algorithm, including initialization, evaluation, update, and archive management, is illustrated in Figure 7.

Analysis of test results

To further evaluate the relative influence of process parameters on the response variables, analysis of variance (ANOVA) was performed. The sum of squares (SS) for each factor (laser power, welding speed, and defocusing distance) was calculated based on the deviations between the mean response at each factor level and the overall mean response. The mean square (MS) values were obtained by dividing SS by the corresponding degrees of freedom (DF), and the F-values were calculated to determine statistical significance. The contribution ratio of each factor was then derived as:

where SS_factor represents the variation attributed to a given process parameter, and SS_total is the total variation in the model. This method quantitatively reflects the proportion of variability explained by each parameter, with higher ratios indicating greater influence on the response. All statistical analyses were carried out using Minitab 19 software to ensure accuracy and reproducibility.

To systematically evaluate the influence of laser welding parameters on E_C, T_S, and I_W, ANOVA and main effect analysis were conducted^[28,29]. The ANOVA results [Table 3] quantify the contribution of each process parameter (laser power, welding speed, and defocusing distance) to the three response indicators, while the main effect plots [Figure 8] visualize the trends of parameter impacts.

Figure 8. Main effects plot for response indicators. (A) Main effect analysis of E_C; (B) Main effect analysis of T_S; (C) Main effect analysis of I_W. E_C: Carbon emissions; T_S: tensile strength; I_W: weld depth-to-width ratio; LP: laser power; WS: welding speed; DA: defocus distance.

As shown in Table 3, WS exhibits the highest contribution (55.4%) to E_C, followed by laser power (LP, 32.5%) and defocusing distance (DA, 12.1%). This indicates that WS is the most critical factor in controlling energy consumption and E_C. For T_S, LP dominates with a 68.5% contribution, significantly surpassing WS (27.1%) and DA (5.1%), highlighting the decisive role of laser energy input in enhancing joint strength. In contrast, I_w is influenced more evenly by LP (45.2%) and WS (38.3%), with DA contributing moderately (16.5%), suggesting that weld geometry and integrity require a balanced adjustment of energy input and processing speed.

By ANOVA, the main effect of a factor on the response is reflected by the mean difference between its levels. The main effect of each process parameter on the three response indicators is analyzed in Figure 8, where the abscissa indicates the different levels of the parameter. In Figure 8A, E_C from electrical energy consumption increases and decreases monotonically with increasing laser power and welding speed, respectively, while changes in defocusing distance have little effect. In Figure 8B, the I_W shows a significant upward trend with increasing laser power, similar to the influence of welding speed. When defocusing distance varies, the weld I_W first decreases and then increases. In Figure 8C, increasing laser power effectively raises T_S, whereas increasing welding speed significantly decreases it, and defocusing distance has a minor effect.

The combined ANOVA and main effect analysis demonstrate a clear trade-off between process parameters:
1. Increasing WS reduces E_C but compromises T_S and I_W.
2. Elevating LP enhances T_S and I_W at the cost of higher E_C.
3. Adjusting DA offers limited optimization flexibility, primarily affecting I_W.

Performance of the stacking prediction model

A total of 200 datasets were used for training and testing, with five-fold cross-validation performed on the testing set. The parameter settings are summarized in Table 4, and the prediction results are presented in Table 5 and Figure 9.

Figure 9. Five-fold cross-validation average prediction accuracy. (A) Comparison of prediction results between the Stacking model and other common proxy models; (B) Comparison of prediction results among the Stacking model, the base models, and the meta-model. E_C: Carbon emissions; T_S: tensile strength; I_W: weld depth-to-width ratio; ELM: extreme learning machine; PSO-BP: particle swarm optimization - back propagation; RBF: radial basis function; MLP: multi-layer perceptron; XGBoost: extreme gradient boosting; SVR: support vector regression; RF: random forest; KRG: Kriging.

Figure 9 and Table 5 present the results of five-fold cross-validation, which clearly demonstrate the superior performance of the stacked model. The consistently high coefficient of determination (R²) values across folds indicate that the stacked model effectively captures the complex relationships between process parameters and output responses, while the relatively low RMSE highlights its precision in prediction. Moreover, the stacked model integrates multiple base learners - RF, XGBoost, and SVR - with a KRG meta-model, enabling the effective combination of the complementary strengths of each base algorithm^[30,31]. This integration enhances the model’s capacity to manage non-linear relationships and high-dimensional data, thereby improving both predictive accuracy and robustness. Compared with traditional regression models, such as linear regression, and individual base learners, the stacked framework achieves significant improvements in accuracy and generalization. The results of five-fold cross-validation further validate the reliability of the stacked model across different data partitions, reinforcing its superiority over single-model approaches.

To further evaluate the robustness and generalization of the proposed stacked model, a noise robustness test was conducted, as shown in Figure 10. Gaussian noise with amplitudes of ±3% and ±5% was added to the input parameters (laser power, welding speed, and defocus distance). The results show that the model consistently maintained high predictive performance, with all R² values remaining above 0.95 and RMSE exhibiting only minor increases relative to the noise-free baseline. These outcomes demonstrate that the stacked model is not overly sensitive to small input perturbations, confirming its stability and reliability in handling realistic variability in process data while maintaining strong predictive accuracy.

Figure 10. Results of the noise robustness test for the Stacking model. (A) R² value; (B) RMSE value. R²: Coefficient of determination; RMSE: root mean square error.

Model interpretation

LIME explains the individual prediction results of stacked regression models by constructing locally interpretable surrogate models. The specific process is as follows: Firstly, 2,000 Gaussian perturbation samples are generated within the feature space neighborhood of the target sample point to form a local dataset. The disturbance range is determined by adaptive kernel width, ensuring neighborhood coverage and feature dimension adaptation. Subsequently, the trained stacking model is used to predict the E_C, T_S, and weld depth to width ratio (I_W) output values of the perturbed samples, and similarity weights are calculated using an exponential kernel function [w_x = exp(-||x - x′||²/σ²)] (w_x: Similarity weight of the Gaussian perturbed sample relative to the target sample; x: Target sample vector in the feature space; x′: Gaussian perturbed sample vector in the neighborhood of target sample x; σ²: Square of the adaptive kernel widt). Based on this weighted dataset, train a ridge regression model (Regularization coefficient α = 0.02) as a local surrogate model, and the linear coefficient is the feature importance weight. The process is implemented by calling the Lime library (v0.2.0.1) in the Python 3.9 environment, Parameter selection has been verified by Monte Carlo simulation: when the number of perturbed samples N > 3,000, the standard deviation of feature weights converges to within 0.1, meeting the engineering confidence requirements^[32].

To elucidate the decision-making mechanism of the stacked regression model, LIME was employed to quantify the feature-specific contributions to predictions. LIME approximates the model’s behavior locally by constructing linear surrogate models around individual predictions, enabling instance-level explanations. The normalized feature importance values for E_C, T_S, and I_W are summarized in Figure 11.

Figure 11. Average importance. (A) The average importance of E_C; (B) The average importance of T_S; (C) The average importance of I_W. E_C: Carbon emissions; T_S: tensile strength; I_W: weld depth-to-width ratio; LP: laser power; WS: welding speed; DA: defocus distance.

LIME analysis reveals the local impacts of laser welding parameters: WS negatively affects E_C (weight = -0.58 ± 0.08), while LP has a positive link (weight = +0.38 ± 0.06). This aligns with ANOVA results showing WS (55.4%) and LP (32.5%) contributions to E_C. For T_S, LP has a dominant positive weight (+0.69 ± 0.05), matching its 68.5% ANOVA contribution, and WS’s negative impact (-0.31 ± 0.04) fits its 27.1% contribution. I_W is jointly controlled by LP (+0.50 ± 0.07) and WS (-0.41 ± 0.06), reflecting their balanced ANOVA contributions (LP: 45.2%, WS: 38.3%). Defocusing distance (DA) has minimal influence in LIME (weight < ±0.1), consistent with its low ANOVA contributions (E_C: 12.1%, T_S: 5.1%, I_W: 16.5%), confirming the model’s statistical robustness in both global and local interpretations.

In order to gain a more comprehensive understanding of the model’s decision-making process, this study combines LIME and PDP. LIME explains individual predictions through local surrogate models, highlighting feature contributions for specific instances. However, LIME’s focus on single data points limits its ability to capture the overall impact of features across the entire dataset. PDP complements this by visualizing the global average effect of individual features on model predictions, demonstrating how changes in feature values systematically influence model outputs. This combination ensures a thorough and multi-faceted understanding of the model, providing a robust theoretical foundation for optimizing laser welding parameters.

PDP is used to quantify the global marginal effect of a single feature on the model output. When analyzing, select the target feature (such as LP) and evenly divide it into 100 grid points within its value range (1.0-2.0 kW). To balance computational efficiency and statistical reliability of small sample data (n = 25), a conditional distribution sampling strategy is adopted: based on the joint distribution of experimental data, feature dependency relationships are modeled through kernel density estimation (KDE); Extract 100 sets of x_s^(k) samples from the conditional distribution P for each grid point x_c; Calculate marginal effects using [$$ \stackrel\frown{f}_s(x_s^{(k)})\frac{1}{100}\sum_{100}^1f(x_s^{(k)},x_c^{(j)}) $$, $$ \stackrel\frown{f}_s(x_s^{(k)}) $$: Estimated marginal effect of the model output when the target feature x_s​ takes the value of the k-th grid point; x_s^(k): Value of the target feature corresponding to the k-th grid point; x_c^(j): The j-th conditional distribution sampling value of the features except the target feature; j: Index of conditional distribution samples for non-target features; k: Index of grid points for the target feature]. This method reduces the computational complexity to 25% of the traditional full sample method, with an accuracy loss of less than 2%. The final generated PDP curves shown in Figure 12A-I reveal the conflict mechanism between laser power and welding speed. The grid density M = 100 was determined through sensitivity testing (when M > 80, the marginal effect curve MSE < 10^-4); The two-dimensional PDP adopts a 20 × 20 grid to quantify the parameter interaction effect.

Figure 12. PDP analysis for laser welding parameters. (A) Partial dependence of E_C vs. LP; (B) Partial dependence of E_C vs. WS; (C) Partial dependence of E_C vs. DA; (D) Partial dependence of T_S vs. LP; (E) Partial dependence of T_S vs. WS; (F) Partial dependence of T_S vs. DA; (G) Partial dependence of I_W vs. LP; (H) Partial dependence of I_W vs. WS; (I) Partial dependence of I_W vs. DA. PDP: Partial dependence plot; E_C: carbon emissions; LP: laser power; WS: welding speed; DA: defocus distance; T_S: tensile strength; I_W: weld depth-to-width ratio.

PDPs analyze how variations in individual features (while keeping other features constant) affect the predicted output^[33]. For instance, the PDP for E_C illustrates the relationship between welding speed and E_C, clearly showing that as welding speed increases, E_C decreases, aligning with the insights gained from LIME. PDPs provide a more holistic view of the feature-target relationship by aggregating over all data points, thus offering a better understanding of the feature’s impact on model performance. In summary, while LIME gives local, instance-level explanations of the model’s behavior, PDPs provide global insights, aggregating the effects of features on the target variables. Together, they offer complementary perspectives: LIME helps understand individual predictions, while PDPs reveal the overall trends and interactions between process parameters and outputs, thereby enhancing the interpretability of the model.

Analysis of E_C: As shown in Figure 12A and B, when the defocusing distance is maintained at a constant value, the reduction of laser power and the increase of welding speed will significantly reduce E_C. This is because the increased welding speed can effectively reduce system uptime, thereby reducing energy consumption and E_C. The reduction of laser power directly reduces the total input of light energy into heat energy. In addition, as shown in Figure 12B and C, when the laser power is maintained at a constant value, the change of defocusing distance has a small impact on E_C, while the change of welding speed has a greater impact on E_C, which shows that the greater the welding speed, the lower the E_C, which further proves the critical impact of welding speed on the energy efficiency of the system. As shown in Figure 12C, when the welding speed and laser power are kept at a fixed value, the increase in the defocusing distance has a small positive effect on E_C.

Analysis of T_S: As shown in Figure 12D and E, both the increase of laser power and the decrease of welding speed contribute to the improvement of T_S when the defocusing distance is constant. This is because the increase in laser power provides a higher energy input, which is conducive to the formation of the weld pool and deep penetration, thus improving the welding strength. At the same time, the reduction of welding speed extends the time of laser interaction with the material, and further enhances the quality of the weld pool. As shown in Figure 12E, when the laser power is maintained at a fixed value, the increase of welding speed will lead to a significant decrease in T_S, which may be because the welding speed is too high, resulting in too fast cooling of the weld pool, and the complete welding structure is not fully formed. As shown in Figure 12F, when the welding speed and laser power as maintained at a constant value, the DA can further improve the T_S, while the change of defocusing distance has relatively little influence on the T_S.

Analysis of I_W: As shown in Figure 12G and H, when the defocusing distance is maintained at a constant value, the increase of laser power and the decrease of welding speed can significantly improve the I_W. This is because higher laser power can improve the depth and consistency of the weld, while lower welding speed can reduce the instability of the weld pool, thereby improving the I_W of the weld. As shown in Figure 12H, when the laser power and defocusing distance are maintained at constant values, the reduction of welding speed has a particularly significant effect on the improvement of the I_W, which indicates that a slower welding speed can reduce the defects generated in the welding process. As shown in Figure 12I, when the welding speed and laser power remain constant, an increase in defocusing distance significantly reduces the aspect ratio.

In summary, the combination of LIME and PDP provides a comprehensive interpretation of the model, ensuring that the optimization of laser welding parameters is guided by both local and global insights into the relationships between input parameters and output responses. This multi-faceted analysis not only enhances the model’s transparency but also ensures the robustness and reliability of the optimization outcomes.

Optimization model

The fitness function for the MOPSO algorithm is based on three RBF neural network prediction models trained collectively. The hybrid algorithm was implemented in Python with the following parameter settings: number of particle swarms p = 100, number of iterations c = 50, inertia weight w = 0.8, local velocity factor c₁ = 0.1, global velocity factor c₂ = 0.1, number of mesh aliquots m = 10, and external archiving threshold t = 300. This configuration represents the optimal combination of parameters we have derived after analyzing the size of the experimental data, evaluating empirical parameter suggestions found in relevant literature, and conducting multiple iterative tests. Figure 13 shows the Pareto optimal solution set and the trend between individual process parameters (horizontal coordinate) and optimization objectives (vertical coordinate).

Figure 13. Pareto optimal solution set. (A) 3D Pareto solution set; (B) Mapping of Z-axis and X-axis; (C) Mapping of Y-axis and X-axis; (D) Mapping of Z-axis and Y-axis. I_W: Weld depth-to-width ratio; E_C: carbon emissions; T_S: tensile strength.

Optimize model evaluation

To evaluate the performance of the proposed optimization framework, MOPSO was benchmarked against the NSGA-II and the decomposition-based sparrow search algorithm (SSA). Each algorithm was executed 30 times using the Deb–Thiele–Laumanns–Zitzler (DTLZ) benchmark functions DTLZ5 and DTLZ6. The hypervolume (HV) and inverted generational distance (IGD) metrics were recorded for each run, and the results were visualized through box plots, as shown in Figure 13. Larger HV values and smaller IGD values indicate better convergence and diversity of the obtained solution sets.

As illustrated in Figure 14, the MOPSO algorithm consistently outperformed the other two methods in terms of both HV and IGD. This indicates that the solution set generated by MOPSO is closer to the true Pareto front, reflecting superior convergence and distribution characteristics. Consequently, the overall performance of the proposed MOPSO optimization model is demonstrated to be highly effective in balancing convergence and diversity.

Figure 14. HV and IGD box plots. HV: Hypervolume; IGD: inverted generational distance.

Process parameter decision and verification

According to the LIME and PDP analyses above, increasing welding speed can reduce E_C, but T_S and I_W will decrease. Conversely, increasing defocusing distance or laser power can improve T_S and I_W, but E_C will also increase, meaning all three objectives cannot be optimized simultaneously. To balance these objectives while considering the high cost of laser welding tests, eight parameter sets from the Pareto solution interval were selected for final optimization and validation. The test results are shown in Figure 15. The optimized parameters reduced average E_C by 28.98%, increased average T_S by 25.93%, and increased the average I_W by 13.08%. Overall, the improvements in E_C and T_S were more pronounced than those of I_W.

Figure 15. Validation of optimization results for (A) E_C; (B) I_W; and (C) T_S. E_C: Carbon emissions; I_W: weld depth-to-width ratio; T_S: tensile strength.