## 3 Surrogate Modeling and Multi-Objective Optimization
### Surrogate Modeling: Kriging  
Kriging, a geo-statistical interpolation technique, considers both the distance between known data points and the variability within those points when predicting values for unknown areas [51; 52]. The Kriging method and its implementation using the MATLAB version: R2018b Kriging toolbox in this study are detailed in [53]. The two surrogate models, previously described, were constructed using experimental data (reported in Table 4) and the aid of the Kriging MATLAB toolbox [53]. To establish a relationship between roughness and process parameters, the roughness measured at the center of the product was used. The roughness data at the injection point exhibited significant fluctuations, making it impossible to establish a meaningful relationship between roughness and the process parameters. This was most likely due to the formation of wavelike flow marks at the injection point (see [31; 54]). To assess the reliability and generalization ability of the Kriging models, we applied Leave-One-Out Cross-Validation (LOOCV). In this method, each data point is removed once, and the model is trained on the remaining data to predict the excluded point. The prediction errors from all iterations were then used to calculate the Root Mean Square Error (RMSE), providing a quantitative measure of model accuracy. The resulting RMSEs were 0.299 mm\({}^{3}\), which is low compared to the measured shrinkage range (1.97-6.45 mm\({}^{3}\)), and 0.135 mm, which is slightly higher but reasonable compared to the observed roughness range (0.395-0.993 mm), confirming that the model offers reasonable predictive performance and avoids overfitting. To gain an insight into these relationships, the two surrogate models are plotted in Figures 9 and 10 as functions of the process parameters. The plots in Figure 9 affirm the established patterns regarding the effects of process parameters on volumetric shrinkage. Figure 9 illustrates that increasing mold and melt temperatures leads to higher volumetric shrinkage, consistent with the observations in Figure 9c. Higher melt temperatures can increase shrinkage because the polymer expands more when heated and shrinks more significantly upon cooling. In addition, higher temperatures may lead to more polymer chain orientation during flow, which can cause uneven shrinkage. On the other hand, higher mold temperatures allow for the slower cooling of the material, reducing internal stresses and warpage but increasing shrinkage. It is also observed in Figure 9a that as the parking time rises, volumetric shrinkage tends to decrease. This is because longer packing times ensure that enough material is added to offset shrinkage during cooling. Figure 9b indicates that an increase in both packing pressure and injection pressure results in decreased volumetric shrinkage. This is because higher packing and injection pressures force more material into the cavity during the packing phase, compensating for material shrinkage during cooling. Additionally, Figure 9d demonstrates that volumetric shrinkage declines with longer cooling times. Insufficient cooling time can lead to parts that are not fully solidified when ejected, causing warpage and higher shrinkage as the part cools outside the mold. Figure 9d also indicates that shrinkage is higher at higher speeds, as well as lower speeds. This is because higher injection speeds may lead to uneven shrinkage due to increased shear heating and faster cooling in certain regions, and lower injection speeds may prevent full cavity filling, leading to higher shrinkage or defects like voids. These observations are summarized in Table 5.  
Figure 9: Variation in the volumetric shrinkage plotted for the various process parameters. The color gradient from yellow to blue represents high to low shrinkage values, respectively. (**a**) Shrinkage versus mold temperature and packing time. (**b**) Shrinkage versus injection pressure and packing pressure. (**c**) Shrinkage versus melt temperature and shrinkage versus mold temperature.