Portland, OR Process Capability Mythology and Perspective -- the Correct and Incorrect Use of C pk Steve Zagarola 09 Feb 2016

Highlights Glossary / Origins Car and Statistical Analogies Capability Scenarios Performance Conclusions / Recommendations

Glossary Process Capability – What can the process do? Control Chart – Is the process stable / predictable? DOE – (Design of Experiments) What is the full potential? C p – Potential to meet tolerances w/ inherent variation? C pk – Able to meet tolerances accounting for location and inherent variation? C pm – What if the relative COPQ for inherent variation from target? P pk – How is performance considering total variation relative to the tolerances?

Origins 1924 – Chance and Assignable - cause variation – Walter Shewhart 1930s /1980s - Loss Function - Taguchi 1956 – “Capability” - Western Electric 1970s – C p / C pk - Japan 1989 – C pm - L. J. Chan, S. K. Cheng, and F. A. Spiring - JQT P pk – “Performance” – Auto Industry Action Group (AIAG)

5 Capability and Performance Car Analogy

Process Capability Incapable Process

Process Capability Capable Process

To drive thru w/o damage -- 1.Opening wider than car If yes, process is POTENTIALLY CAPABLE (continue to step 2.) If no, process is NOT CAPABLE (get a narrower car or wider opening.) 2.Car sufficiently centered If yes, process is CAPABLE (proceed with process.) If no, process is NOT CAPABLE (center car.) 3.Drive straight If yes, performance successful, car passes thru w/o damage If no, performance unsuccessful Process Capability and Performance

Indices ---

Process Capability and Performance Indices ---

Process Capability and Performance Indices ---

12 Capability and Performance Statistical Analogy

13 Histogram ft. Process Output (Car)

14 1 sigma (σ) ft. = 8 ft. – 6 ft. = 2 ft. mean (μ) Process Output (Car)

15 μ σ 1 σ Process Output (Car) (Analogous Car Width = 6σ) = 2 ft

16 μ σ 1 σ = 2 ft % of outcomes 0 to 12 ft.

17 ft σ 99.73% Histogram to Control Chart

% Histogram to Control Chart ft. 3 σ

% Histogram to Control Chart ft. 3 σ

% Control Chart (drive) ft. 3 σ

Statistically Stable (driving straight, no distractions) ft.

Common Cause Variation Not Statistically Stable (distracted driving) Additional Variation Resulting from Assignable (Special) Cause (σ 2 special cause ) ft.

Total Variation (σ 2 common cause ) + (σ 2 special cause ) Additional Variation Resulting from Assignable (Special) Cause (σ 2 special cause ) Common Cause Variation (σ 2 cc original process ) ft.

Common Cause Variation (σ 2 cc original process ) Additional Variation Resulting from Assignable (Special) Cause (σ 2 special cause ) Total Variation (σ 2 common cause + σ 2 special cause ) ft.

Two Standard Deviations (σ), Two Means (μ) Common Cause Variation (σ 2 cc original process ) σ total σ cc μ overall μ original Total Variation (σ 2 common cause + σ 2 special cause ) (territory traversed) (territory could have traversed)

26 Capability Indices (Car and Driver Ready?)

Cp, Cpk σ cc μ original = 2 ft. 3σ cc ft. R / d 2 - s / c 4 - or cc σ = est. due to common cause variation = cc ^ (undistracted driving)

Cp, Cpk σ cc μ = 2 ft. 3σ cc USLLSL Tolerance = USL - LSL (opening width) (car width)

Cp, Cpk σ cc = 2 ft. 3σ cc USLLSL Tolerance = 12 ft. μ (opening width) (car width)

Cp, Cpk σ cc = 2 ft. 3σ cc USLLSL Tolerance = 12 ft. μ C p = (USL – LSL) / 6σ cc C pu = (USL – μ) / 3σ cc C pl = (μ – LSL) / 3σ cc C pk = min(C pu, C pl) (opening width) (car width)

Cp, Cpk σ cc = 2 ft. 3σ cc USLLSL μ C p = 12/12 = 1 C pu = 6/6 = 1 C pl = 6/6 = 1 C pk = 1 Tolerance = 12 ft. 3 sigma process = 2,700 DPMO (opening width) (car width)

Cp, Cpk σ cc = 1 ft. 3σ cc USLLSL μ Tolerance = 12 ft. C p = (USL – LSL) / 6σ cc C pu = (USL – μ) / 3σ cc C pl = (μ – LSL) / 3σ cc C pk = min(C pu, C pl) (opening width) (car width)

Cp, Cpk σ cc = 1 ft. USLLSL μ C p = 12/6 = 2 C pu = 6/3 = 2 C pl = 6/3 = 2 C pk = 2 Tolerance = 12 ft. 6 sigma process = DPMO 3σ cc (opening width) (car width)

Cp, Cpk σ cc = 1 ft. USLLSL Tolerance = 12 ft. C p = (USL – LSL) / 6σ cc C pu = (USL – μ) / 3σ cc C pl = (μ – LSL) / 3σ cc C pk = min(C pu, C pl) 3σ cc μ (opening width) (car width)

3σ cc Cp, Cpk σ cc = 1 ft. USLLSL μ Tolerance = 12 ft. C p = 12/6 = 2 C pu = 4.5/3 = 1.5 C pl = 7.5/3 = 2.5 C pk = sigma process = 3.4 DPMO 3σ cc (opening width) (car width)

36 Example Capability Scenarios (driving readiness)

37 USLLSL Not Capable, Centered C p = C pk = 0.8

38 USLLSL Capable, Centered C p = C pk = 1.3

39 USLLSL Barely Capable, Centered C p = C pk = 1.0

40 USLLSL Highly Capable, Centered C p = C pk = 2.0

High Potential, Capable, Off-center 41 USLLSL C p = 2.0; C pk = 1.5

High Potential, Not Capable, Off-center 42 USLLSL C p = 2.0; C pk = 0.8

High Potential, Not Capable, Off-center 43 USLLSL C p = 2.0; C pk = -1.0

Not Capable 44 LSL C pk = 0.0 (lower specification only)

Barely Capable 45 LSL C pk = 1.0 (lower specification only)

Highly Capable 46 LSL C pk = 2.0 (lower specification only)

47 Performance

Statistically Stable (driving straight) ft.

Not Statistically Stable (distracted driving) ft.

Two Standard Deviations (σ), Two Means (μ) σ total σ cc μ overall μ original

P pk 51 μ overall σ = est. std. dev. due to total variation total ^ σ total = 3 USLLSL

P pk 52 μ overall P pu = (USL – μ) / 3σ total P pl = (μ – LSL) / 3σ total P pk = min(C pu, C pl ) σ total = 3 USLLSL

P pk 53 σ total = 3 μ overall μ original P pu = (12 – 8) / 9 = 0.4 P pl = (8 – 0) / 9 = 0.9 P pk = USLLSL

P pk 54 σ total = 3 σ cc μ overall μ original P pu = (12 – 8) / 9 = 0.4 P pl = (8 – 0) / 9 = 0.9 P pk = USL σ cc = 2 deg LSL C pu = 3/3 = 1.0 C pl = 3/3 = 1.0 C pk = 1.0

55 σ total = 3 σ cc μ overall μ original P pu = (12 – 8) / 9 = 0.4 P pl = (8 – 0) / 9 = 0.9 P pk = USLLSL σ cc = 2 deg C pu = 3/3 = 1.0 C pl = 3/3 = 1.0 C pk = 1.0 Capable, But unstable

56 Conclusions / Recommendations

1.C p – A first step / doesn’t apply for one-sided specification 2.C pk – OK indicator/ include confidence intervals 3.C pk - Doesn’t guarantee low defect rates 4.Increasing C pk – Value depends on how 5.Stability - Accomplishing capability is meaningless without it 6.P pk – Indicator of performance Ok but can be misleading 7.C pm - Consider its use to maintain focus on true COPQ 8.Priority – Stability / Analysis & reduction of variation / CI Conclusions / Recommendations

ApplicationDOE Control Chart CpCp C pk C pm P pk σ cc σ total Determine Stability Optimize Process Determine Potential Capability Determine Capability Indicate Relative COPQ Indicate Performance to Tolerance When to Use What

WORDS TO THE WISE FROM A FOUR-PART SERIES ENTITLED “THE USE AND ABUSE OF C PK ”, GUNTER -- A CONTRIBUTING TO QUALITY PROGRESS -- STATED (JULY 1989): The greatest abuse of C pk I have seen is that it becomes a kind of mindless effort managers confuse with real statistical process control efforts... in short, rather than fostering never- ending improvement, C pk scorekeeping kills it. IN RESPONSE TO A CRITICAL LETTER GUNTER STATES... Instead of the focus being on what’s necessary for continuous improvement, a new version of the meet- specifications game is being played.

Do first! Process Stability

Use Cp, Cpk when -- New product / process Change in product / process New raw material / people / supplier Process Capability Important Assumptions: In statistical control, normal distribution, subgroups are statistically independent

Use Pp, Ppk when -- Assessing production lot Reporting performance Process Performance Important Cautions: If unstable, P pk > C pk ; estimates of % defective risky; not valid estimate of capability or future performance

Analyze / minimize variation Continual Improvement

Appendix Flemish Process Capability Taguchi Loss Curve Polyester Container Mfg. Illustration of Taguchi Loss Formulas

Flemish Capabilty and Performance

Taguchi Loss Curve

Goal Post Loss σ cc μ = 2 deg 3σ cc USLLSL Tolerance = USL - LSL

TAGUCHI LOSS FUNCTION Loss imparted to society during product use as a result of functional variation and harmful effects Loss (x) = k (x -target ) 2

Goal Post Loss σ cc μ = 2 deg 3σ cc USLLSL Tolerance = USL - LSL

Taguchi Loss Curve σ cc μ = 2 deg 3σ cc USLLSL Tolerance = USL - LSL

Taguchi Loss σ cc μ = 2 deg 3σ cc USLLSL Tolerance = USL - LSL

Taguchi Loss μ 3σ cc USLLSL Tolerance = USL - LSL σ cc = 1 deg

Polyester Container Mfg. Illustration of Taguchi Loss

To injection Desecante regenerati vo Deseca nte operativ o Panel de Control Aire recalentad o Polyester Container Mfg.

Polyester Container Stress Crack Failures

Preform Weight LSLLSL USLUSL Cpk = 1.3 DPMO = 29.7 Cpk = 1.4 DPMO = 9

54.0 gm Cpk = 1.3 Blow Molding Optimized for 54.0 grams LSLLSL 54.4 gm Cpk = 0.6 DPMO = 36,859 DPMO = 40 μ wt.. increases 0.3 gm Stress Crack Resistance (minutes) Weight (gm) To injection

Blow Molding Optimized for 54.3 grams LSLLSL 54.4 gm Cpk = gm Cpk = 1.1 LSLLSL DPMO = 36,859 DPMO = 577 Reset Power & Cam Position Stress Crack Resistance (minutes) Weight (gm) To injection

54.4 gm Cpk = 1.1 LSLLSL 54.0 gm Cpk = 0.44 DPMO = 89,130 DPMO = 577 μ wt.. decreases 0.3 gm Weight (gm) Stress Crack Resistance (minutes) LSLLSL Blow Molding Optimized for 54.3 grams To injection

Formulas Potential Capability (Actual) Capability Taguchi Capability Process Performance cc total R / d 2 - σ = s / c 4 ) - - or cc (σ = estimated standard deviation due to common cause variation = cc ^ ( σ = estimated standard deviation due to common + special cause variation total ^ T = target ) ( ) cc Alternative formulas