| name | robust-design |
| description | Robust design (Taguchi method) — parameter design, tolerance design, signal-to-noise ratio (S/N: nominal-is-best, smaller-is-better, larger-is-better), orthogonal arrays (L9, L18, L27), control vs. noise factors, outer array experiment, ANOVA for factor significance, response surface methodology (RSM), Six Sigma DMAIC, Cpk/Cp process capability, Monte Carlo tolerance analysis, and DFSS (Design for Six Sigma) methodology. |
Robust Design (Taguchi Method) — Complete Skill
Robust Design Philosophy
Taguchi's Quality Loss Function
Quality loss function:
L(y) = k × (y - T)² [k = quality loss coefficient; y = response; T = target value]
k = Δ / δ² [Δ = financial loss at tolerance limit δ; L(T±δ) = Δ]
Expected loss:
E[L] = k × (σ² + (μ - T)²) [includes variance + bias²; reduce both for minimum loss]
Taguchi's strategy: move μ → T (nominal) by adjusting control factors; reduce σ by exploiting non-linearity (parameter design); then tighten tolerances only where needed (tolerance design)
Traditional vs. Taguchi:
Traditional: reduce σ by tighter tolerances (costly)
Taguchi: find operating point where response is insensitive to noise (cheaper)
Signal-to-Noise (S/N) Ratios
Three S/N Types
Nominal-is-best (NTB): target response T; minimize variance while keeping mean on target
S/N_NTB = 10 × log₁₀(μ²/σ²) [higher = better; maximize]
Use for: dimensions, temperatures, pressures where target value exists
Smaller-is-better (SIB): response should be as small as possible (zero ideal)
S/N_SIB = -10 × log₁₀(Σy²/n) [mean square response; higher S/N = smaller response]
Use for: noise level, surface roughness, defect rate, vibration amplitude
Larger-is-better (LIB): response should be as large as possible
S/N_LIB = -10 × log₁₀(Σ(1/y²)/n) [mean of reciprocal squares; higher S/N = larger response]
Use for: strength, hardness, life, efficiency
Conversion from S/N to variance:
S/N_NTB = 10 × log₁₀(μ²/σ²); rearrange: σ/μ = 10^(-S/N_NTB/20) → coefficient of variation
Orthogonal Arrays (OA)
Design Experiment with OA
L9 (3⁴) Orthogonal Array:
9 experiments; 4 factors; each at 3 levels; all main effects estimable
| Exp | A | B | C | D |
|---|
| 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 2 | 2 | 2 |
| 3 | 1 | 3 | 3 | 3 |
| 4 | 2 | 1 | 2 | 3 |
| 5 | 2 | 2 | 3 | 1 |
| 6 | 2 | 3 | 1 | 2 |
| 7 | 3 | 1 | 3 | 2 |
| 8 | 3 | 2 | 1 | 3 |
| 9 | 3 | 3 | 2 | 1 |
Balance: each level of each factor appears exactly 3 times; orthogonal pairs → main effects estimated independently
L18 (2¹ × 3⁷): 18 experiments; 1 two-level factor + up to 7 three-level factors
L27 (3¹³): 27 experiments; up to 13 three-level factors (includes 2-factor interactions in some columns)
Standard OA selection:
k factors at 2 levels: use L4 (3 factors), L8 (7 factors), L16 (15 factors)
k factors at 3 levels: use L9 (4 factors), L27 (13 factors)
Mixed: L18, L36
Outer Array (Noise Factor Experiment)
Inner array: control factors (engineer-controllable)
Outer array: noise factors (uncontrollable in production: temperature variation, material lot, humidity)
Each row of inner array × each row of outer array = run a combination; compute S/N from outer array replications
Example:
Inner: L9 with 4 control factors (9 experiments)
Outer: L4 with 2 noise factors (4 conditions per inner experiment)
Total runs: 9 × 4 = 36 (manageable)
Computes: for each of 9 inner experiments → 4 outer results → S/N ratio → select control factor levels maximizing S/N
Analysis Procedure
Main Effects and ANOVA
Main effect plot:
For each factor: average S/N at each level (level 1, 2, 3)
Plot level vs. average S/N → visual identification of important factors and optimal levels
Optimal: select level giving highest S/N for each factor
ANOVA for factor significance:
SS_factor = n_levels × Σ (level_mean - grand_mean)² [sum of squares]
SS_error = SS_total - Σ SS_factors [residual]
F_factor = (SS_factor/df_factor) / (SS_error/df_error) [F-ratio; compare to F-critical table]
p < 0.05: factor significant; p > 0.10: factor can be pooled into error
Percent contribution:
ρ_factor = SS_factor / SS_total × 100% [relative importance; top factors to control]
Confirmation experiment:
After selecting optimal levels → run confirmation experiment → verify predicted S/N matches actual
If Δ_S/N > 2 dB: interaction effects may be significant → consider L27 or RSM
Prediction and Confidence Interval
Predicted optimal S/N:
S/N_predicted = T̄ + Σᵢ (A_best - T̄) + (B_best - T̄) + ... [T̄ = grand mean; Aᵢ_best = best level mean for factor i]
Confidence interval on prediction:
CI = ±√(F_α × V_error × (1/n_eff)) [n_eff = N / (1 + total DOF of factors); F_α from table]
Response Surface Methodology (RSM)
Central Composite Design (CCD)
For quantitative factors: 2-level factorial + star points + center points:
k factors: 2^k factorial + 2k star points (α from center) + n_c center points
Total: 2^k + 2k + n_c experiments
Second-order polynomial model:
y = β₀ + Σβᵢxᵢ + Σβᵢᵢxᵢ² + ΣΣβᵢⱼxᵢxⱼ + ε [curvature + interactions; OLS estimation]
Fit via least squares; check R² and lack-of-fit
Optimization:
∂y/∂xᵢ = 0 → stationary point; classify (ridge analysis, canonical form)
Maximum, minimum, or saddle point (depending on sign of eigenvalues of Hessian)
Box-Behnken Design:
Alternative to CCD; all factorial points at edge midpoints; no corners; rotatable
Fewer high-leverage extreme points; good for physical process limits
Six Sigma and Process Capability
Cp and Cpk
Process capability index Cp:
Cp = (USL - LSL) / (6σ) [process width vs. spec width; Cp ≥ 1.33 for capable process]
Off-center process Cpk:
Cpk = min[(USL - μ)/(3σ), (μ - LSL)/(3σ)]
Cpk ≥ 1.33: capable; 1.0 ≤ Cpk < 1.33: marginal; Cpk < 1.0: not capable
Six Sigma target:
σ_process = (USL - LSL)/12 → Cp = 2.0; with 1.5σ mean shift → Cpk = 1.5; 3.4 PPM defects
Relationship to S/N:
High S/N_NTB = low variance → high Cpk (if mean also centered at target)
DMAIC Process
Define: project charter; customer requirements (CTQ: critical to quality); define defect
Measure: collect baseline data; compute Cpk; identify measurement system variation (gauge R&R)
Analyze: identify root causes (FMEA, fishbone, regression); Pareto of defect sources
Improve: Taguchi parameter design (L9/L18/RSM) → optimize CTQ; validate
Control: control plan; SPC charts (X̄-R, IMR); mistake-proofing (poka-yoke)
Monte Carlo Tolerance Analysis
Simulate variation propagation:
For each design parameter xᵢ: sample from distribution (Normal: μᵢ, σᵢ = tol_i/3)
Compute response y = f(x₁, ..., xₙ) → repeat N = 10,000–100,000 times
Result: distribution of y → Cp, Cpk, PPM defect rate
RSS (Root Sum of Squares):
σ_y ≈ √(Σ (∂y/∂xᵢ × σᵢ)²) [linearized; sensitivity × tolerance RSS; valid for linear functions]
Sensitivity: S_i = ∂y/∂xᵢ; tolerance budget: allocate so each (S_i × σᵢ)² contributes equally
Standards and References
| Standard | Scope |
|---|
| Taguchi, Chowdhury & Wu "Taguchi's Quality Engineering Handbook" (2004) | Definitive Taguchi reference |
| AIAG APQP | Automotive product quality planning; PPAP |
| AIAG MSA | Measurement system analysis (gauge R&R) |
| ASQ Certified Quality Engineer Body of Knowledge | Six Sigma tools |
| ISO 21747 | Statistical methods for process capability |
| NIST/SEMATECH e-Handbook of Statistical Methods | DOE and RSM reference |
Output
Provide: problem definition (response y; target T; noise factors; control factors with levels), S/N type selected (NTB/SIB/LIB; justification), OA selected (L9/L18/L27/CCD; number of experiments; control factor assignments), experiment results (S/N for each experiment; mean effect plot per factor), ANOVA table (SS, df, MS, F, p% contribution per factor; factors pooled or not), optimal factor levels (table: factor → optimal level → expected S/N [dB]; vs. initial S/N [dB]; improvement [dB]), confirmation experiment (predicted S/N [dB]; actual S/N [dB]; within CI?), Cp/Cpk before and after optimization (σ_before [units]; σ_after [units]; Cpk_before; Cpk_after), tolerance analysis (key sensitivity terms S_i × σᵢ [units]; RSS σ_y [units]; PPM defect before/after), and applicable reference (Taguchi Handbook, AIAG APQP, ISO 21747).
