Response Optimization

Desirability Function (Derringer-Suich)

Maps a response value y onto a dimensionless desirability d ∈ [0, 1] — the building block of multi-response optimisation.

Statusvalidated

Version1.0.0

Minitab equivalentStat > DOE > Response Optimizer (Goals tab)

Description

Three flavours depending on the goal: maximise maps linearly (or with a shape exponent) between an unacceptable lower bound and a fully satisfactory upper bound; minimise is the mirror image; target-is-best is two-sided with peak d=1 exactly at the target. Values outside the acceptable range receive d=0 — no 'soft' penalty. The shape exponents s, t control how demanding the function becomes near the bounds: s>1 makes it concave (more demanding near the upper bound), s<1 convex (more lenient).

Formulas

max

d_{max} (y) = ⎩ ⎨ ⎧ 0 (\frac{y - L}{U - L})^{s} 1 y \leq L L < y < U y \geq U

Larger-the-better with lower bound L (unacceptable) and upper bound U (satisfactory)

min

d_{min} (y) = ⎩ ⎨ ⎧ 1 (\frac{U - y}{U - L})^{s} 0 y \leq L L < y < U y \geq U

Smaller-the-better — mirror image: L is the desired value, U the still-acceptable maximum

target

d_{tgt} (y) = ⎩ ⎨ ⎧ (\frac{y - L}{T - L})^{s} (\frac{U - y}{U - T})^{t} 10 L < y < T T < y < U y = T sonst

Target-is-best: two-sided function with peak exactly at target T. Asymmetric shoulders via shape exponents s and t.

Assumptions

Response y is measurable on a continuous scale with interpretable bounds
Lower bound L < upper bound U; for target additionally L < T < U

Limitations

Values outside [L, U] receive d=0 — no graceful degradation beyond the bounds
Shape exponents change desirability non-linearly; choosing them requires understanding their effect

References

Derringer, G. & Suich, R. (1980). Simultaneous Optimization of Several Response Variables. Journal of Quality Technology 12(4): 214-219
Myers, R.H., Montgomery, D.C., Anderson-Cook, C.M. (2016). Response Surface Methodology, 4th Ed., Wiley — Chapter 6