quality-engineering

name	quality-engineering
description	Quality engineering — SPC, control charts (X-bar/R/S/IMR), process capability (Cp/Cpk/Ppk), Six Sigma DMAIC, MSA Gauge R&R, DOE, APQP, control plan, PFMEA.
metadata	{"priority":7,"promptSignals":{"phrases":["quality","SPC","control chart","Cpk","Six Sigma","gauge R&R","DMAIC","APQP","process capability","MSA"],"minScore":3}}

Quality Engineering — Complete Skill

Statistical Process Control (SPC)

Control Chart Types

Chart	Data Type	Sample Size	Use When
X-bar & R	Variables	n = 2-10	Most common manufacturing
X-bar & S	Variables	n ≥ 10	Better estimate of σ
Individuals & MR (IMR)	Variables	n = 1	Slow processes, auto industry
p-chart	Attribute (proportion)	Variable n	% defective
np-chart	Attribute (count)	Fixed n	Number defective
c-chart	Attribute (count)	Fixed area	Defects per unit (Poisson)
u-chart	Attribute (count)	Variable area	Defects per unit, variable sample

X-bar & R Chart

Control limits: UCL_X̄ = X̄̄ + A₂ × R̄ LCL_X̄ = X̄̄ - A₂ × R̄ UCL_R = D₄ × R̄ LCL_R = D₃ × R̄ (= 0 for n ≤ 6)

Control chart constants (A₂, D₃, D₄): n=2: A₂=1.880, D₃=0, D₄=3.267 n=3: A₂=1.023, D₃=0, D₄=2.574 n=4: A₂=0.729, D₃=0, D₄=2.282 n=5: A₂=0.577, D₃=0, D₄=2.114 n=10: A₂=0.308, D₃=0.223, D₄=1.777

Estimated σ = R̄/d₂ (d₂: n=2→1.128, n=5→2.326, n=10→3.078)

IMR Chart (Individuals)

X chart: UCL = X̄ + 2.66 × MR̄; LCL = X̄ - 2.66 × MR̄ MR chart: UCL = 3.267 × MR̄; LCL = 0 σ̂ = MR̄/1.128

Western Electric Rules (Out-of-Control Signals)

Rule 1: 1 point beyond ±3σ Rule 2: 2 of 3 consecutive points beyond ±2σ (same side) Rule 3: 4 of 5 consecutive points beyond ±1σ (same side) Rule 4: 8 consecutive points on same side of centerline Rule 5: 6 consecutive points trending in one direction Nelson rules add: 14 points alternating up/down

Process Capability

Cp and Cpk

Cp = (USL - LSL) / (6σ) — potential capability (centered process) Cpk = min[(USL - X̄)/(3σ), (X̄ - LSL)/(3σ)] — actual capability

σ from control chart: σ̂ = R̄/d₂ (short-term, within-subgroup) Ppk = min[(USL - X̄)/(3s), (X̄ - LSL)/(3s)] — long-term with s = sample std dev

Capability targets: Cpk ≥ 1.33: capable (industry minimum for general) Cpk ≥ 1.67: very capable (safety-critical, Automotive AIAG) Cpk ≥ 2.0: excellent (Six Sigma: DPMO = 3.4 at 1.5σ shift) Cpk < 1.0: incapable — process produces nonconforming parts

Cpm (Taguchi) = Cp / √(1 + ((μ-T)/σ)²) T = target value; penalizes deviation from nominal

DPMO from Cpk: DPMO = 2 × 10⁶ × Φ(-3 × Cpk) × (1 + 0.5σ shift for long-term) Cpk=1.33 → ~64 DPPM; Cpk=1.67 → ~0.6 DPPM

Process Performance (Pp, Ppk)

Pp = (USL-LSL)/(6s_total) — uses total s, includes between-subgroup variation When Cpk >> Ppk: large between-subgroup variation (setup, shift changes) When Cpk ≈ Ppk: stable, well-controlled process

Measurement System Analysis (MSA)

Gauge R&R (AIAG MSA Manual)

%GRR = (σ_gauge / σ_total) × 100% σ_gauge² = σ_repeatability² + σ_reproducibility² σ_reproducibility² = σ_appraiser² + σ_appraiser×part²

Acceptance criteria: %GRR < 10%: excellent measurement system 10% ≤ %GRR < 30%: conditional (may be acceptable based on use) %GRR ≥ 30%: unacceptable — improve gauge

NDC (Number of Distinct Categories) = 1.41 × (σ_part / σ_gauge) NDC ≥ 5 required (can distinguish 5 levels of part variation)

ANOVA method (preferred): Partitions: Part, Appraiser, Part×Appraiser interaction, Repeatability EV (Equipment Variation) = 5.15 × σ_repeatability AV (Appraiser Variation) = 5.15 × σ_reproducibility

Linearity and Bias

Bias = (measured average - reference value) — systematic error Linearity: bias varies across measurement range → check bias at 5+ reference values Acceptable bias: within ±10% of tolerance (AIAG)

Attribute Gauge R&R

Kappa statistic: κ = (P₀ - P_e)/(1 - P_e) κ ≥ 0.90: excellent agreement; κ < 0.70: unacceptable

Six Sigma — DMAIC

Define

Project charter: problem statement, goal (SMART), scope, team, timeline SIPOC: Suppliers → Inputs → Process → Outputs → Customers VOC (Voice of Customer) → CTQ (Critical to Quality) translation CTQ tree: need → driver → CTQ metric with LSL/USL

Measure

Current process baseline: defect rate, DPMO, σ-level Data collection plan: operational definitions, sample plan (30+ parts min) σ-level = NORMSINV(1-DPMO/10⁶) + 1.5

Analyze

Root cause tools:

• 5-Why: trace cause chains to root
• Fishbone (Ishikawa): 6M — Man, Machine, Material, Method, Measurement, Mother Nature
• Pareto chart: 80/20 — vital few vs. trivial many
• Hypothesis testing: t-test (compare means), F-test (compare variances), chi-square (attribute data)
• Regression: Y = f(x₁, x₂, ...) — identify significant X's

Improve

DOE (Design of Experiments):

• Full factorial: 2^k design (k factors, 2 levels each)
• Fractional factorial: 2^(k-p) — confounding, resolution III/IV/V
• Response Surface: Box-Behnken, Central Composite Design (CCD)
• ANOVA to identify significant factors; regression equation for Y

Taguchi L9, L18: robust design — minimize sensitivity to noise S/N ratio = -10 log(MSD); larger-the-better/smaller-the-better/nominal-the-best

Control

Control plan: control characteristic, specification, measurement system, sample size/frequency, reaction plan SPC charts on key X's and Y's PFMEA update: reassess RPN after controls implemented Mistake-proofing (Poka-Yoke): prevent, detect, react — prevent is highest level Control plan handoff to production

PFMEA (Process FMEA)

RPN Scoring

RPN = Severity (S) × Occurrence (O) × Detection (D), 1-10 each High RPN or high S: priority for action regardless of O and D

AIAG AP (Action Priority, 2019 AIAG-VDA): High (H): act immediately; Medium (M): consider action; Low (L): review at discretion AP based on S first, then O, then D — severity 9-10 with any high O is always High AP

Severity scale (1-10): 10: safety — no warning; 9: safety — with warning; 8: loss of primary function; ...; 1: no effect

Occurrence scale (1-10): 10: ≥1 per 2 items; 9: 1/8; ...; 3: 1/15,000; 2: 1/150,000; 1: <1/million

Detection scale (1-10): 10: no detection; 9: hard to detect; ...; 2: almost certain to detect; 1: guaranteed detection (error-proof)

APQP (Advanced Product Quality Planning)

5 Phases (AIAG)

1. Plan and Define — VOC, design goals, quality plan
2. Product Design and Development — DFMEA, DVP&R, design review
3. Process Design and Development — Process flow, PFMEA, control plan, MSA plan
4. Product and Process Validation — PPAP elements, production trial run, MSA
5. Feedback, Assessment, Corrective Action — SPC, lessons learned

PPAP (Production Part Approval Process)

Submission levels 1-5 (Level 3 = full submission most common) 18 elements including: PFMEA, control plan, MSA, process capability (Cpk≥1.67 for initial)

Output

Provide: control chart type selection, UCL/LCL values, Cp and Cpk [vs. 1.33 target], %GRR [vs. 10% threshold], NDC [vs. 5 minimum], Six Sigma level and DPMO, PFMEA RPN and AP rating, control plan recommendations.