## 3 Results
### Residual Analysis  
Figure 4 displays the residuals plot for Assembly I, where a larger dataset is available. The residuals plotted against the fitted values exhibit a random scatter around the zero line without any discernible patterns or systematic structures. This randomness indicates that the assumptions of constant variance of residuals and linearity are satisfied. The consistent spread of residuals across all levels of fitted values suggests that the model adequately captures the underlying relationship between the predictors and the response variable. Figure 5 presents the normal Q-Q plot for the residuals. This plot reveals that most of the residuals align closely along the reference line, indicating that they are approximately normally distributed. This alignment suggests that the normality assumption of the residuals is reasonably satisfied, which is crucial for the validity of parametric tests within the regression framework. Minor deviations observed at the tails of the distribution indicate slight departures from normality, which are common in practical data analysis. These minor deviations are unlikely to significantly impact the results due to the robustness of statistical tests against mild violations of normality. Overall, the approximate normality of the residuals confirms the appropriateness of the linear regression model and supports the validity of the hypothesis tests and confidence intervals computed in the analysis. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Predictor** & **Sum of Squares** & **df** & **B** & _p_**-Value** \\ \hline Assembly I & & & & \\ \hline Locating scheme & 13.08 & 2 & 3088.95 & 0 \\ Sequence & 0.30 & 2 & 70.37 & 0 \\ Main weld path (F) & 0.01 & 1 & 1.60 & 2.17 \(\times\) 10\({}^{-1}\) \\ Secondary weld path (G) & 0.01 & 1 & 0.21 & 5.52 \(\times\) 10\({}^{-1}\) \\ Locating scheme and sequence & 0.25 & 4 & 29.65 & 3.66 \(\times\) 10\({}^{-9}\) \\ \hline Assembly II & & & & \\ \hline Locating scheme & 15.46 & 1 & 182.74 & 0.0054 \\ Sequence & 3.02 & 1 & 35.75 & 0.0268 \\ Main weld path (F) & 3.03 & 1 & 35.89 & 0.0267 \\ Secondary weld path (G) & 0.003 & 1 & 0.04 & 0.8637 \\ Tertiary weld path (H) & 0.008 & 1 & 0.01 & 0.7804 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: ANOVA results for welding displacement. Figure 4: Residual-fitted plot for reference Assembly I. Furthermore, the plots of the main effects and the significant interactions were generated to visualize the influence of each factor on welding displacement. Figure 6 illustrates the main effects of the locating scheme, Figure 6a, welding sequences, Figure 6b, weld path directions of the two dominant paths, Figure 6c,d, and the interaction of the most significant factors. For Assembly I, the significant factors were the locating scheme and the welding sequences, Figure 6e,f, respectively. Figure 5: Quantile–quantile plot for reference Assembly I.  
Figure 6: Cont. For Assembly II, the main effects were the locating scheme, visualized in Figure 7a, followed up by the weld sequence, in Figure 7b, and the weld paths F and G, in Figure 7c,d, respectively. The main interactions were between the locating scheme and the welding sequence visualized in Figure 7e and the interaction between the sequences and the weld path F's direction, Figure 7f. In both assemblies, the locating scheme, representing the fixturing points and the orientation of those points, had a dominant influence on the displacements after welding and springback. The second main factor for both assemblies was the welding sequence, where changing the order of applying the weld segments was shown to have a significant effect on the displacement after springback. Finally, the weld path direction, where the forward or backward pass of the weld is determined, in both assemblies was shown to be relatively less significant for the displacements after welding and springback. Figure 8a visualizes the displacements of the assembly welded in the N-O locating scheme and the optimal welding sequence and weld path direction.