## 3 Surrogate Modeling and Multi-Objective Optimization
### Surrogate Modeling: Kriging
These observations are summarized in Table 6.  
Injection molding process parameters, such as packing pressure, mold temperature, cooling time, injection speed, injection pressure, melt temperature, and packing time, play a crucial role not only in determining volumetric shrinkage and surface roughness but also in influencing the overall functional properties of the final product, including mechanical strength, esthetic quality, and dimensional stability. For instance, packing pressure helps compensate for material shrinkage during the cooling phase. When applied adequately, it reduces the formation of sink marks and voids, thereby enhancing the product's dimensional stability and part density [18; 56; 60]. Packing time, which refers to the duration that the pressure is maintained for after the mold is filled, impacts the extent of post-fill compaction. Sufficient packing time contributes to consistent dimensions, whereas excessively long durations may increase cycle time without significant improvements in product quality [56; 60]. Melt temperature influences the viscosity of the polymer melt, affecting both mold cavity filling and the level of internal stress retained after solidification. While higher melt \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** \\ **Roughness** \\ \end{tabular} & **Figure** &
\begin{tabular}{c} **Previously** \\ **Reported** \\ \end{tabular} \\ \hline Injection pressure & Decreased roughness & Figure 10b & [34] \\ Packing pressure & Decreased roughness & Figure 10b & [29] \\ Packing time & Decreased roughness & Figure 10a & - \\ Cooling time & Minimal effect & Figure 10d & - \\  & Roughness dips as injection speed & & \\ Injection speed & increases and then increases again & Figure 10d & - \\  & for higher speeds & & \\ Melt temperature & Decreased roughness & Figure 10c & [29; 34] \\ Mold temperature & Decreased roughness & Figure 10a & [29; 59]. \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The effects of the injection molding process parameters on product roughness. temperatures improve flow characteristics and help prevent short shots, they may also degrade the polymer if thermal thresholds are exceeded, ultimately compromising the mechanical strength of the finished part [61]. Mold temperature significantly affects the functional properties of injection-molded components. Elevated mold temperatures can promote polymer crystallization in semi-crystalline materials such as polypropylene (PP) and polyamide (PA), thereby enhancing tensile and flexural moduli along with tensile yield strength [62; 63]. Cooling time is critical for dimensional accuracy. Longer cooling durations promote uniform solidification, reducing internal stresses and minimizing warpage and shrinkage [64; 65]. Injection pressure also has a notable impact on product functionality. Higher injection pressures improve the ability of molten polymers to fill thin sections and complex geometries while also enhancing dimensional accuracy and reducing defects such as warpage [26]. Lastly, injection speed and melt temperature influence the strength of weld lines. Increasing both parameters has been shown to improve weld line strength, thereby reinforcing the structural integrity of the molded part [66]. It is important to note that semi-crystalline thermoplastics such as HDPE, polypropylene (PP), nylon, and polybutylene terephthalate (PBT) undergo a phenomenon known as supercooling. This occurs when the polymer melt is cooled below its solidification temperature without immediately crystallizing. Supercooling typically arises during rapid cooling in the injection molding process or when there are insufficient nucleation sites to initiate crystallization. This phenomenon influences material shrinkage, as well as mechanical properties and optical clarity.