Hybrid Axial Distances in Spherical Region of Central Composite Designs (CCDs)

Response Surface Methodology (RSM) relies heavily on Central Composite Designs (CCDs) for modeling, analysis, and optimization of complex multivariate systems. However, the choice of a suitable axial distance (alpha) continues to pose a major challenge in research. Existing classical spherical and rotatable axial distances most of the time push the star points into extreme operational regions that are not feasible, as the number of factors (k) increases, while practical and cuboidal alphas are heavily affected by variance inflation. Mean–based alternative axial distances (arithmetic, harmonic, and geometric) address boundary constraints but are highly sensitive to extreme values, thereby leading to boundaries that are unconstrained as well as spikes in the scaled prediction variance (SPV). To mitigate this gap, this study proposed a new class of variance–based hybrid axial distances for the CCDs in spherical regions. Variability across classical alphas was accounted for by the proposed variance–based alphas, so as to control bias of extreme values; this was done by directly factoring in dispersion instead of raw averages. The new set of variance–based alphas was constructed using factor spaces ranging from k=2 to k=10 and evaluated against existing axial distances using D–optimality, which is a numeric/alphabetic criterion, as well as Variance Dispersion Graphs (VDGs) and Fraction of Design Space Graphs (FDSGs). The study found that for higher–dimensional design spaces (k≥8), the proposed variance–based alphas were effective in preventing boundary SPV inflation, maintaining stable, low prediction variance across more than 90 percent of the volume of the design space. Furthermore, the well–conditioned information matrices produced competitive D–efficiencies compared to the practical and rotatable alphas. In conclusion, the proposed variance–based alphas provide a mathematically sound, operationally feasible alternative for high–dimensional RSM experiments in the spherical regions.

Hybrid Axial, Distances, Spherical Region

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