Response Surface Methodology with Desirability Function Optimization

Linear regression assumes a linear relationship between factors and responses, giving a simple model but often inadequate fit near optima where curvature is significant. RSM typically uses second-order polynomial models that capture curvature and factor interactions, providing more accurate predictions in regions of interest and enabling reliable optimization.

Common designs include central composite designs (CCDs), which balance efficiency and model quality for second-order models, and Box-Behnken designs, which avoid extreme corners. Use CCD if you need to explore a wide range; Box-Behnken if you want fewer extreme conditions. The choice depends on number of factors, available budget, and region of interest.

For k factors, a full second-order model has (k+1)(k+2)/2 parameters. A central composite design requires approximately 2^k + 2k + 1 runs. For 3 factors, that is roughly 20 experiments. Fewer is possible with model simplification; more is better for robustness and error estimation. Plan for at least 10-20% replicates to estimate pure error.

RSM is designed for continuous factors. For categorical factors, stratify your experiments: run separate RSM studies for each category level, or use mixed-level designs that combine continuous and categorical factors. Alternatively, use coded dummy variables to incorporate categories into the model, though interpretation becomes more complex.

For each response, specify: (1) target value (if applicable), (2) lower and upper specification limits, and (3) preference shape (linear, convex, or concave depending on whether values near the target are preferred). Harrington and Derringer-Suich functions offer standard shapes. Consult with stakeholders to ensure targets align with business or engineering requirements; sensitivity analysis can reveal how robust the optimum is to changes in limits.