Desirability Optimization

What is the Derringer-Suich desirability function?

The Derringer-Suich desirability function is a method for multi-response optimization that transforms each response into a dimensionless desirability score (0 to 1), then combines them into an overall desirability using the geometric mean. A desirability of 1 represents an ideal value; 0 means the response is outside acceptable limits.

Three goal types define how each response is transformed: Maximize (d increases from Lower to Upper bound), Minimize (d decreases from Lower to Upper bound), and Target (d peaks at the target value and decreases toward both bounds). Shape parameters s₁ and s₂ control the curvature: s=1 gives linear desirability, s<1 gives convex (easier to satisfy), s>1 gives concave (harder to satisfy).

The overall desirability D is the geometric mean of all individual desirabilities: D = (d₁ × d₂ × ... × dₖ)^(1/k). This ensures that if any single response is completely unacceptable (dᵢ = 0), the overall desirability is also 0, preventing trade-offs that sacrifice one critical quality attribute.

Formula: Maximize: d = ((y - L) / (U - L))^s for L ≤ y ≤ U Minimize: d = ((U - y) / (U - L))^s for L ≤ y ≤ U Target: d = ((y - L) / (T - L))^s1 for L ≤ y ≤ T d = ((U - y) / (U - T))^s2 for T ≤ y ≤ U Overall D = (d₁ × d₂ × ... × dₖ)^(1/k)

Example Calculation

Two responses: Yield (maximize, L=70, U=95, observed=88, s=1) and Cost (minimize, L=10, U=50, observed=25, s=1). d_yield = (88-70)/(95-70) = 0.72. d_cost = (50-25)/(50-10) = 0.625. Overall D = (0.72 × 0.625)^(1/2) = √0.45 = 0.671. A reasonable but improvable combined desirability.

When to Use This Calculator

• When optimizing a process with multiple conflicting quality responses that must be balanced simultaneously (e.g., maximize strength while minimizing cost)
• During product formulation when each ingredient affects multiple responses differently and trade-offs must be systematically evaluated
• After RSM analysis to combine multiple response surface models into a single optimization objective
• When comparing alternative process settings — overall desirability provides a single score that accounts for all quality requirements
• During design review to quantify how well a proposed solution meets all specification requirements relative to other alternatives

Common Mistakes to Avoid

• Setting bounds too tight so that no feasible solution achieves D > 0 — start with realistic bounds based on what the process has historically achieved, then tighten gradually
• Using equal shape parameters (s=1) for all responses when some are much more critical — increase s for critical responses to penalize deviations from target more severely
• Ignoring that overall D = 0 when any individual d = 0 — if one response is completely unacceptable, overall desirability collapses regardless of how good other responses are; ensure all bounds are achievable
• Optimizing desirability without considering factor interactions — desirability scores are computed from observed values; for prediction, combine desirability with response surface models
• Treating desirability as an absolute metric — D = 0.7 is not inherently 'good' or 'bad'; it depends on the tightness of bounds and shape parameters; use D for comparison between alternatives, not as a standalone acceptance criterion

How to Interpret Results

• Overall D = 0 — at least one response is completely outside acceptable limits; the solution is infeasible as-is
• Overall D = 0.01-0.3 — most responses are far from their ideal values; significant process improvement is needed
• Overall D = 0.3-0.6 — a compromise solution exists but with substantial trade-offs; investigate which response is pulling D down
• Overall D = 0.6-0.8 — a good solution that meets most requirements well; fine-tuning may improve individual responses
• Overall D > 0.8 — excellent; all responses are close to their ideal values simultaneously; this solution is a strong candidate for implementation

Related Standards & References

• Derringer & Suich (1980), Journal of Quality Technology — The original paper defining the desirability functions and geometric-mean combination implemented here
• Harrington (1965), Industrial Quality Control — The earlier desirability concept that Derringer and Suich generalized with shape parameters
• NIST/SEMATECH e-Handbook §5.5.3 — Multiple response optimization guidance using the desirability approach
• Montgomery, Design and Analysis of Experiments — Standard textbook treatment of multi-response optimization with desirability functions after RSM

Frequently Asked Questions

How do I set the shape parameters (s1, s2)?

Shape s=1 gives linear desirability (equal importance across the range). s<1 (e.g., 0.5) makes it easier to achieve high desirability — use when you're fairly tolerant of deviation. s>1 (e.g., 2 or 3) makes it harder to achieve high desirability — use for critical responses where being close to the target matters greatly. Most practitioners start with s=1 and adjust based on engineering judgment.

Why use the geometric mean instead of the arithmetic mean?

The geometric mean ensures that any single response with d=0 (completely unacceptable) drives the overall desirability to 0 — you cannot compensate for a failed response by excelling at others. The arithmetic mean would allow such trade-offs, potentially hiding a critical quality failure. This 'veto power' property makes the geometric mean the correct choice for multi-response optimization.