## 3 Surrogate Modeling and Multi-Objective Optimization
### Surrogate Modeling: Kriging
(**d**) Shrinkage versus injection speed and cooling time. Figure 10 offers valuable insight into the correlation between surface roughness and injection molding process parameters. Figure 10a suggests that increasing mold temperature leads to a decrease in roughness, a trend also evident in Figure 10c. This behavior  
Figure 10: Variation in surface roughness (Ra) for the various process parameters (The color gradient from yellow to blue represents high to low roughness values, respectively): (**a**) roughness versus mold temperature and packing time; (**b**) roughness versus injection pressure and packing pressure; (**c**) roughness versus melt temperature and mold temperature; and (**d**) roughness versus injection speed and cooling time. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Injection Molding Parameters** &
\begin{tabular}{c} **Observed Effects of Increasing the** \\ **Process Parameter Value on** \\ **Volumetric Shrinkage** \\ \end{tabular} & **Figure** & **Reference** \\ \hline Injection pressure & Decreases Volumetric shrinkage & Figure 9b & [55] \\ Packing pressure & Decreases Volumetric shrinkage & Figure 9b & [18; 55; 56] \\ Packing time & Decreases Volumetric shrinkage & Figure 9a & [18; 56; 57] \\ Cooling time & Decreases Volumetric shrinkage & Figure 9d & [18; 58] \\ Injection speed & Increases Volumetric shrinkage & Figure 9d & - \\ Melt temperature & Increases Volumetric shrinkage & Figure 9c & [18; 55] \\ Mold temperature & Increases Volumetric shrinkage & Figure 9a,c & [55; 57] \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The effects of the injection molding process parameters on the volumetric shrinkage of the product. has been reported in [29; 59]. This behavior is most likely attributed to the fact that higher mold temperatures allow the molten polymer to flow smoothly and conform closely to the smooth mold surface, thereby improving surface finish. This figure also shows that roughness decreases with increasing packing time. Longer packing time may help maintain the material in a molten state for a longer period, allowing for better surface replication, as well as reducing voids, sink marks, and surface defects caused by shrinkage, improving surface smoothness. Figure 10b illustrates that increasing the injection pressure leads to a reduction in roughness, a finding consistent with the results reported in [34]. Higher injection pressure improves mold surface replication by forcing the molten polymer into fine details of the mold and thus reducing voids and sink marks, leading to smoother surfaces. This figure also shows that for low injection pressures, as the packing pressure is increased, roughness decreases. This trend is reported in [29]. This trend is, however, reversed for higher packing pressure. It is observed in Figure 10c that as the melt temperature increases, the roughness decreases, which is a behavior that is also reported in [29; 34]. Higher melt temperatures decrease the polymer's viscosity, allowing for the better flow and replication of mold surface details and thus reducing surface roughness. Lastly, Figure 10d reveals that cooling time has minimal impact on surface roughness, corroborating the findings reported in [33]. However, the injection speed shows an interesting behavior (not seen in the literature) where the roughness dips as the injection speed increases and then increases again for higher speeds. This increase could be attributed to the flow becoming turbulent at higher speeds, leading to surface defects like flow marks and jetting. On the other hand, low speeds may result in incomplete mold filling, resulting in an increase in surface roughness.