Process Capability in the Six Sigma Environment

GE Research & Development Center Process Capability in the Six Sigma Environment C.L. Stanard 2001CRD119, July 2001 Class 1 Technical Information Series

Copyright 2001 General Electric Company. All rights reserved.

Corporate Research and Development Technical Report Abstract Page Title Process Capability in the Six Sigma Environment Author(s) C.L. Stanard Phone (518)387-4131 8833-4131 Component Information Systems Laboratory Report Number 2001CRD119 Date July 2001 Number of Pages 13 Class 1 Key Words Traditional and Six Sigma Introduction to Process Capability Indexes, Stability and Non- Normality This document explores issues the relationship between traditional and Six Sigma process capability analysis and indices for normal, non-normal, and non-standard data. Examples and general remedies for non-standard situations are presented. Manuscript received July 13, 2001

Process Capability in the Six Sigma Environment Chris Stanard, ASA Albany Conference, March 1999

Outline Introduction to Process Capability Indexes (PCIs) Traditional Six Sigma Stability and Non-Normality Examples

Intro to Process Capability Indices General Idea: Compare What a Process Should Do with What It Actually Does General Form of PCIs = specification interval/process spread General Definitions for traditional Process Capability Indexes Cp: inherent process capability k: position of process in relation to specification mean Cpk: position of a 6sigma process in relation to specification mean Cpl: position of of 6sigma process in relation to lower specification limit Cpu: position of 6sigma process in relation to upper specification limit Issues: These each assume normality of the individual parts or processes being measured (not just for averages), stable processes, adequate gauge, no censoring, no correlation, no truncation, and no mixture of distributions. More on Definitions in Handout, Including Six Sigma Z-score definitions Adapted from Kotz and Lovelace, 1998

Six Sigma Capability Index: Zst DPUlt = Defects per unit Long Term DPMOlt = (Total DPUlt x 10^6) / Total Opportunities Zst=Corresponding Value from Normal Table with adjustment for short term to long term measurement (A Z-Shift adjustment of 1.5 is often used. Critical To Quality Characteristic CTQ Zst Index Often Used For Multiple CTQs Finally, another issue relevant to all Process Capability Indexes: The estimate sqrt(variance) of standard deviation is a biased estimator.

Assessing Stability Issues If Process Not Stable, Get It Stable! Assignable vs. Common (Random) Cause Process Centering (particularly Impacts Cpk) Short Term vs. Long Term (including std. deviation estimation) Some Metrics To Consider: Q1/Q3, Q3-Q1, (Q3-Q1)/Q2, X 0.95 -X 0.05, Variance, Skewness, Kurtosis, Mode(s), and appropriate confidence intervals Consider Order Statistics vs. Distributions Incidence of Outliers Time Series Effects The Interpretation of Process Capability Confidence Interval Width

Options If the Process Is Not Normal Transform Data Fit Appropriate Distribution Use Distribution Families (e.g. Pearson, Johnson, etc.) Use Nonparametrics Use Quantiles to estimate X 0.99865 = mean + 3sigma, X 0.50 =mean, X 0.00135 =mean-3sigma Cp=(USL-LSL)/(X 0.99865 - X 0.00135 ) Cpk=min((USL- X 0.50 )/(X 0.99865 - X 0.50 ), Cp=(X 0.50 -LSL)/(X 0.50 - X 0.00135 )) Use of Subgroup Averages Traditionally Used, But There Are Problems

Process Some Processes That Do Not Fit the Normal Distribution Stock Prices, Financial Data Time Between Failures Lamp Life Fatigue Life Chemical Reactions Mechanical Properties of Material Number of Manufacturing Defects Distribution Lognormal Gamma Weibull, Lognormal Weibull, Lognormal, Extreme Value Logistic Extreme Value Poisson

More Characteristics That Do Not Follow the Normal Distribution Diameter, Roundness, Mold Dimensions, Customer Waiting Time,Taper, Flatness, Surface Finish, Concentricity, Eccentricity, Perpendicularity, Angularity, Roundness, Warpage, Straightness, Squareness, Weld Strength, Bond Strength, Tensile Strength, Casting Hardness, Particle Contamination, Hole Location, Shrinkage, Dynamic Imbalance, Insertion Depth, Parallelism, Inductance Source for listing of many of these: Measuring Process Capability by Davis Bothe, McGraw Hill, 1997

GE Silicones Example: Analysis of Tear Data Data Data & Distribution Distribution Fit Fit.206.206 Overlay Chart Overlay Chart Frequency Comparison Frequency Comparison Normal Distribution Gives Zst = 3.35 & DPMO = 31,700 Crystal Crystal Ball Ball Simulation Simulation Forecast: Tear B Forecast Normal Forecast: Tear B Forecast Normal 50,000 Trials Frequency Chart 132 Outliers 50,000 Trials Frequency Chart 132 Outliers.025 1234.025 1234.154.154.103.103 Normal Distribution Mean Normal = 104.71 Distribution Std Mean Dev = 104.71 18.62 Std Dev = 18.62.019.019.012.012 925.5 925.5 617 617.051.051 Input Data Input Data 80.00 100.00 120.00 140.00 160.00 80.00 100.00 120.00 140.00 160.00 One Sided LSL: 70 PPI.006 308.5.006 308.5 0 0 50.00 77.50 105.00 132.50 160.00 50.00 Certainty 77.50 is 96.83% 105.00 from 70.00 to +Infinity 132.50 160.00 Certainty is 96.83% from 70.00 to +Infinity.206.206.154.154.103.103 Overlay Chart Overlay Chart Frequency Comparison Frequency Comparison Extreme Value Distribution Gives Zst = 7.01 & DPMO = 0.018 Forecast: Tear B Forecast Extreme Value Forecast: Tear B Forecast Extreme Value 50,000 Trials Frequency Chart 838 Outliers 50,000 Trials Frequency Chart 838 Outliers.027 1331.027 1331 Extreme Value Distribution.020 998.2 Mode Extreme = 98.79 Value Distribution Scale Mode = 9.99 = 98.79 Scale = 9.99.020.013 998.2 665.5.013 665.5.051.051 Input Data Input Data 80.00 100.00 120.00 140.00 160.00 80.00 100.00 120.00 140.00 160.00 One Sided LSL: 70 PPI.007 332.7.007 332.7 0 0 70.00 87.50 105.00 122.50 140.00 70.00 87.50 105.00 122.50 140.00 You Already Had A Six Sigma Process And Didn t Know It!!

Issues with Transformations: Weibull Example 0.5 + StDev 0.0-0.5-5 -4-3 -2-1 0 1 2 3 4 5 Lam bda Νο!! Surprise!! Box-Cox Transform for normality don t work this time!! Sometimes You Can t Force the Normality Assumption!! Yes!!

Capability Analysis of A Transfer Function Variable: Response Process Capability Analysis for Response Anderson-Darling Normality Test A-Squared: P-Value: Mean StDev Variance Skewness Kurtosis N 61.757 0 1127.50 174.45 30432.7-2.75433 12.4033 1000 USL Target LSL Mean Sample N StDev (ST) StDev (LT) Process Data Process Benchmarks Actual (LT) Potential (ST) 1000.00 1127.50 Sigma (Z.Bench) 1000 144.780 PPM 174.450 0.73 0.73 232427 233365 LSL ST LT Minimum 1st Quartile Median 3rd Quartile Maximum 1116.68 167.12 1173.21-341.39 1084.03 1183.47 1234.72 1368.72 95% Conf idence Interv al f or Mu 1138.33 95% Conf idence Interv al f or Sigma 182.45 95% Conf idence Interv al f or Median 1189.74 Potential (ST) Capability Cp CPU CPL 0.29 Cpk 0.29 Cpm -500 0 500 1000 1500 Overall (LT) Capability Observed Performance Expected ST Performance Expected LT Performance Pp PPM < LSL 158000.00 PPM < LSL 189251.46 PPM < LSL 232427.18 PPU PPL 0.24 158000.00 189251.46 232427.18 Ppk 0.24 O verall(lt ) C apability O bserved Perform ance E xpected ST Perform ance E xpected LT Perform ance Pp PPU PPL Ppk 0.24 0.24 PPM < LS L 158000.00 158000.00 PPM < LS L 189251.46 189251.46 PPM < LS L 232427.18 232427.18

Applying Tranformation Variable: Transform ed( Process Capability Analysis for Response+342 A nderson-d arling N orm ality Test A -S quared: P-Value: M ean StD ev Variance S kew ness K urtosis N Minim um 1st Q uartile M edian 3rd Q uartile M axim um 4.91E +15 16.520 0 5.03E +15 1.88E +15-7.2E -01-1.4E -01 1000 0 3.92E +15 5.47E +15 6.44E +15 9.64E +15 95% C onfidence Interv alfo r M u 5.15E +15 95% C onfidence Interv alfo r S igm a USL USL Target LSL LSL Mean Mean Sample N StDev (ST) StDev (ST) StDev (LT) StDev (LT) Cp CPU CPL Cpk Cpm Process Data 1342 4.3527E+15 1470 7.5909E+15 1000 145 2.8108E+15 174 2.8618E+15 Potential (ST) Capability 0.38 0.38 Box-Cox Transformation, With Lambda = 5 LSL 0 2.00E+15 4.00E+15 6.00E+15 8.00E+15 1.00E+16 1.20E+16 1.40E+16 1.60E+16 StDev 140 135 130 Box-Cox Plot for Response+342 ST LT 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Lambda Last Iteration Info Lambda StDev Low 4.944 128.564 Est 5 128.562 Up 95% C onfidence Interv alfo r M e d ian 5.29E +15 5.58E +15 Pp PPU PPL Ppk Overall (LT) Capability Observed Performance Expected ST Performance Expected LT Performance PPM < LSL 158000.00 158000.00 PPM < LSL 124656.41 124656.41 PPM < LSL 0.38 0.38 128923.07 128923.07 Pp PPU PPL Ppk Overall (LT) Capability Observed Performance Expected ST Performance Expected LT Performance PPM < LSL 158000.00 158000.00 PPM < LSL 124656.41 124656.41 PPM < LSL 0.38 0.38 128923.07 128923.07

Results for the Equation y = x^2: Forecast: Test To Square 2,000 Trials Frequency Chart 8 Outliers.025 49 Forecast: Response 2,000 Trials Frequency Chart 51 Outliers.177 354.018 36.75.133 265.5.012 24.5.089 177.006 12.25.044 88.5-6.00-3.00 0.00 3.00 6.00 0 0.00 5.00 10.00 15.00 20.00 0 Input x Normal Response y Not Normal!!

C.L. Stanard Process Capability in the Six Sigma Environment 2001CRD119 July 2001