Response Optimization

Composite Desirability D (Geometric Mean)

Blends several individual desirabilities d_i into a single composite D ∈ [0, 1] via the weighted geometric mean.

Statusvalidated

Version1.0.0

Description

D = (∏ d_i^{w_i})^{1/Σw_i}. Key property: the moment any d_i = 0, D collapses to 0 — no individual response is allowed to leave its acceptable range, otherwise the solution is invalid (hard veto). Weights w_i ≥ 0 bias the geometric mean toward higher-priority responses.

Formulas

D = (i = 1 \prod m d_{i}^{w_{i}})^{1/ \sum_{i} w_{i}}

Weighted geometric mean — log-linearised: Σ w_i · ln(d_i) / Σ w_i, then exponentiated

Veto

\exists i : d_{i} = 0 ⟹ D = 0

Zeros are not 'averaged out' — a single d_i = 0 forces D = 0

Assumptions

Each d_i ∈ [0, 1] (e.g. from an individual response's desirability function)
Weights w_i ≥ 0; at least one w_i > 0

Limitations

The hard veto can be unintentionally strict when d_i values are near zero but not exactly zero
The geometric mean is symmetric in the d_i (beyond weighting) — it prefers balanced solutions over extreme ones

References

Derringer, G. & Suich, R. (1980). Simultaneous Optimization of Several Response Variables. Journal of Quality Technology 12(4): 214-219
Harrington, E.C. (1965). The desirability function. Industrial Quality Control 21(10): 494-498