## 4 Concluding Remarks  
In this study, a framework for optimizing injection molding process parameters to minimize surface roughness and volumetric shrinkage in molded products was presented. The procedure involves constructing two surrogate models using the Kriging methodology to describe the relationships between process parameters and the two target outputs: surface roughness and volumetric shrinkage. These models were developed based on data obtained from actual injection molding experiments. Several plots of the surrogate models as functions of the process parameters were generated to gain insights into these relationships. The observations from these plots reveal the following:  
* Volumetric shrinkage generally decreases with increasing injection pressure, packing pressure, packing time, and cooling time. * Surface roughness generally decreases with increasing mold temperature, packing time, injection pressure, and melt temperature. These trends validate known relationships in the field regarding how injection molding parameters influence product quality. A two-objective optimization problem was then formulated and solved using the pattern search algorithm, accounting for the interdependence between surface roughness and volumetric shrinkage. This process generated a Pareto front that highlights the trade-off between the two competing objectives and led to the identification of three optimal parameter sets:  
* Point 1: Minimizes volumetric shrinkage (1.9314 mm\({}^{3}\)) but results in the highest surface roughness (0.55956 \(\upmu\)m). * Point 2: Achieves the lowest surface roughness (0.20557 \(\upmu\)m) but results in the highest volumetric shrinkage (3.9286 mm\({}^{3}\)). * Point 3: Offers the best compromise with a volumetric shrinkage of 2.2348 mm\({}^{3}\) and a surface roughness of 0.28246 \(\upmu\)m.  
Through an analysis of the Pareto front, the optimal balance between conflicting objectives was determined. The dilemma of achieving both suitable surface roughness and minimal volumetric shrinkage, which are conflicting objectives, poses a significant challenge for the injection molding sector. The authors assert that the optimization framework proposed in this study can be readily implemented within the industry. This framework provides plastic injection molding engineers and managers with a potent decision-making tool, empowering them to identify the optimal compromise among process parameters to address both objectives effectively. Additionally, improvements to surface roughness can be achieved by adjusting the process parameters, potentially providing a cost-effective alternative to the more expensive approach of enhancing product roughness by improving mold surface roughness. Conceptualization, S.M.S.M. ; methodology, S.M.S.M. and H.M.O. ; validation, S.M.S.M. ; formal analysis, S.M.S.M. and H.M.O. ; investigation, S.M.S.M. and H.Z.K. ; data curation, \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{**Points**} & **Volumetric** & **Surface** & **Measured** & **Packing** & **Packing** & **Coiling** & **Injection** & **Melt** & **Mold** \\  & **Shrinking** & **Ra** & **Pressure** & **Pressure** & **Time** & **Time** & **Speed** & **Temp.** & **Temp.** \\  & **(mm3)** & **(mm)** & **(bar)** & **(bar)** & **(s)** & **(s)** & **(mm/s)** & **(\({}^{\circ}\)C)** & **(\({}^{\circ}\)C)** \\ \hline P1 & 1.9314 & 0.55956 & 200 & 15 & 56.531 & 800 & 400 & 30 & 9 \\ P2 & 2.2348 & 0.28246 & 250 & 15 & 45.844 & 800 & 400 & 30 & 9 \\ P3 & 3.9286 & 0.20557 & 250 & 31.062 & 35.984 & 800 & 127.03 & 30 & 9 \\ \hline \end{tabular}
\end{table} 
Table 7: The main distinguished points on the Pareto front. S.M.S.M. and H.Z.K. ; writing--original draft preparation, S.M.S.M., H.Z.K. and H.M.O. ; writing--review and editing, S.M.S.M., H.Z.K. and H.M.O. ; visualization, S.M.S.M. and H.M.O. ; supervision, S.M.S.M. ; project administration, S.M.S.M. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable.