response surface methodology (rsm)

**Response Surface Methodology (RSM)** is an advanced DOE technique that **models the relationship** between process inputs (factors) and outputs (responses) as a mathematical surface, enabling **optimization** — finding the factor settings that maximize, minimize, or target a specific response value. **Why RSM?** - Factorial designs (2-level) identify which factors are important and provide linear models — but real processes rarely have purely linear responses. - RSM uses **3+ levels** per factor to fit **quadratic (second-order) models** that capture curvature, minima, maxima, and saddle points in the response landscape. - Once the response surface is modeled, the **optimal operating point** can be found mathematically. **The RSM Model** A second-order RSM model for $k$ factors: $$y = \beta_0 + \sum_{i=1}^{k}\beta_i x_i + \sum_{i=1}^{k}\beta_{ii}x_i^2 + \sum_{i