The widespread use and demand for plastic products worldwide have caused manufacturers to covet high productivity and product quality. Most plastic products are produced using the injection molding technique. This technique is characterized by long cooling times, which affect the production cycle and product quality. According to the literature, cooling during the injection molding process can be significantly affected by the design of the cooling channels. This study is, therefore focused on multi-factor design optimization of circular cross-section conformal cooling channels for multiple responses. The Taguchi design-of-experiments approach was adopted in this study. The key variables of conformal cooling channels that were studied involved diameters, depths, and pitches. Solidworks® was used for 3D design and numerical simulation to determine the cooling time, volumetric shrinkage, warpage, and sink marks. Multi-response optimization was then conducted using the Taguchi-Grey Relational Analysis technique. Results show that the optimal cooling channel design has a minimum; diameter of 8 mm, a depth of 12 mm, and a pitch of 16 mm. Additionally, Analysis of Variance (ANOVA) revealed that the diameter is the significant cooling channel design parameter contributing to all the responses concurrently, with the most significant percentage of 80.26%. Comparing the conformal and straight cooling channel designs, superior performance was noted for the former against the latter, with the optimal design recording an improvement of 29.35%, 5.99%, 19.77%, and 38.85% in the cooling time, volumetric shrinkage, warpage, and depth of sink marks, respectively.