TM 720: Statistical Process Control TM 720 - Lecture 10 Short Run SPC and Gage Reproducibility & Repeatability 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

TM 720: Statistical Process Control Assignment: Reading: Finish Chapters 7 and 9 Sections 7.4 – 7.8 Sections 9 – 9.2 Assignment: Access Excel Template for Individuals Control Charts: Download Assignment 7 for practice Use the data on the HW7 Excel sheet to do the charting, verify the control limits by hand calculations Solutions for 6 and 7 will post on Thursday Review for Exam II 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

TM 720: Statistical Process Control Review Shewhart Control charts Are for sample data from an approximate Normal distribution Three lines appear on all Shewhart Control Charts UCL, CL, LCL Two charts are used: X-bar for testing for change in location R or s-chart for testing for change in spread We check the charts using 4 Western Electric rules Attributes Control charts Are for Discrete distribution data Use p- and np-charts for tracking defective units Use c- and u-charts for tracking defects on units Use p- and u-charts for variable sample sizes Use np- and c-charts with constant sample sizes 11/7/2018 TM 720: Statistical Process Control

TM 720: Statistical Process Control Short Run SPC Many products are made in smaller quantities than are practical to control with traditional SPC In order to have enough observations for statistical control to work, batches of parts may be grouped together onto a control chart This usually requires a transformation of the variable on the control chart, and a logical grouping of the part numbers (different parts) to be plotted. A single chart or set of charts may cover several different part types 11/7/2018 TM 720: Statistical Process Control

TM 720: Statistical Process Control DNOM Charts Deviation from Nominal Variable computed is the difference between the measured part and the target dimension where: Mi is the measured value of the ith part Tp is the target dimension for all of part number p 11/7/2018 TM 720: Statistical Process Control

TM 720: Statistical Process Control DNOM Charts The computed variable (xi) is part of a sub-sample of size n xi is normally distributed n is held constant for all part numbers in the chart group. Charted variables are x and R, just as in a traditional Shewhart control chart, and control limits are computed as such, too: 11/7/2018 TM 720: Statistical Process Control

TM 720: Statistical Process Control DNOM Charts Usage: A vertical dashed line is used to mark the charts at the point at which the part numbers change from one part type to the next in the group The variation among each of the part types in the group should be similar (hypothesis test!) Often times, the Tp is the nominal target value for the process for that part type Allows the use of the chart when only a single-sided specification is given If no target value is specified, the historical average (x) may be used in its’ place 11/7/2018 TM 720: Statistical Process Control

Standardized Control Charts If the variation among the part types within a logical group are not similar, the variable may be standardized This is similar to the way that we converted from any normally distributed variable to a standard normal distribution: Express the measured variable in terms of how many units of spread it is away from the central location of the distribution 11/7/2018 TM 720: Statistical Process Control

Standardized Charts – x and R Standardized Range: Plotted variable is where: Ri is the range of measure values for the ith sub-sample of this part type j Rj is the average range for this jth part type 11/7/2018 TM 720: Statistical Process Control

Standardized Charts – x and R Standardized x: Plotted variable for the sample is where: Mi is the mean of the original measured values for this sub-sample of the current part type (j) Tj is the target or nominal value for this jth part type 11/7/2018 TM 720: Statistical Process Control

Standardized Charts – x and R Usage: Two options for finding Rj: Prior History Estimate from target σ: Examples: Parts from same machine with similar dimensions Part families – similar part tolerances from similar setups and equipment 11/7/2018 TM 720: Statistical Process Control

Standardized Charts – Attributes Standardized zi for Proportion Defective: Plotted variable is Control Limits: 11/7/2018 TM 720: Statistical Process Control

Standardized Charts – Attributes Standardized zi for Number Defective: Plotted variable is Control Limits: 11/7/2018 TM 720: Statistical Process Control

Standardized Charts – Attributes Standardized zi for Count of Defects: Plotted variable is Control Limits: 11/7/2018 TM 720: Statistical Process Control

Standardized Charts – Attributes Standardized zi for Defects per Inspection Unit: Plotted variable is Control Limits: 11/7/2018 TM 720: Statistical Process Control

Gage Capability Studies TM 720: Statistical Process Control Gage Capability Studies Ensuring an adequate gage and inspection system capability is an important consideration! In any problem involving measurement the observed variability in product due to two sources: Product variability - σ2product Gage variability - σ2gage i.e., measurement error Total observed variance in product: σ2total = σ2product + σ2gage (system) 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

e.g. Assessing Gage Capability TM 720: Statistical Process Control e.g. Assessing Gage Capability Following data were taken by one operator during gage capability study. 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

e.g. Assessing Gage Capability Cont'd TM 720: Statistical Process Control e.g. Assessing Gage Capability Cont'd Estimate standard deviation of measurement error: Dist. of measurement error is usually well approximated by the Normal, therefore Estimate gage capability: That is, individual measurements expected to vary as much as owing to gage error. 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

Precision-to-Tolerance (P/T) Ratio TM 720: Statistical Process Control Precision-to-Tolerance (P/T) Ratio Common practice to compare gage capability with the width of the specifications In gage capability, the specification width is called the tolerance band (not to be confused with natural tolerance limits, NTLs) Specs for above example: 32.5 ± 27.5 Rule of Thumb: P/T  0.1  Adequate gage capability 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

Estimating Variance Components of Total Observed Variability TM 720: Statistical Process Control Estimating Variance Components of Total Observed Variability Estimate total variance: Compute an estimate of product variance Since : 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

Gage Std Dev Can Be Expressed as % of Product Standard Deviation TM 720: Statistical Process Control Gage Std Dev Can Be Expressed as % of Product Standard Deviation Gage standard deviation as percentage of product standard deviation : This is often a more meaningful expression, because it does not depend on the width of the specification limits 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

Using x and R-Charts for a Gage Capability Study TM 720: Statistical Process Control Using x and R-Charts for a Gage Capability Study On x chart for measurements: Expect to see many out-of-control points x chart has different meaning than for process control shows the ability of the gage to discriminate between units (discriminating power of instrument) Why? Because estimate of σx used for control limits is based only on measurement error, i.e.: 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

Using x and R-Charts for a Gage Capability Study TM 720: Statistical Process Control Using x and R-Charts for a Gage Capability Study On R-chart for measurements: R-chart directly shows magnitude of measurement error Values represent differences between measurements made by same operator on same unit using the same instrument Interpretation of chart: In-control: operator has no difficulty making consistent measurements Out-of-control: operator has difficulty making consistent measurements 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

Repeatability & Reproducibility: Gage R & R Study TM 720: Statistical Process Control Repeatability & Reproducibility: Gage R & R Study If more than one operator used in study then measurement (gage) error has two components of variance: σ2total = σ2product + σ2gage σ2reproducibility + σ2repeatability Repeatability: σ2repeatability - Variance due to measuring instrument Reproducibility: σ2reproducibility - Variance due to different operators 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

TM 720: Statistical Process Control Ex. Gage R & R Study 20 parts, 3 operators, each operator measures each part twice Estimate repeatability (measurement error): Use d2 for n = 2 since each range uses 2 repeat measures Operator i xi Ri 1 22.30 1.00 2 22.28 1.25 3 22.10 1.20 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

TM 720: Statistical Process Control Ex. Gage R & R Study Cont'd Estimate reproducibility: Differences in xi  operator bias since all operators measured same parts Use d2 for n = 3 since Rx is from sample of size 3 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

TM 720: Statistical Process Control Ex. Gage R & R Study Cont'd Total Gage variability: Gage standard deviation (measurement error): P/T Ratio: Specs: USL = 60, LSL = 5 Note: Would like P/T < 0.1! 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

Comparison of Gage Capability Examples TM 720: Statistical Process Control Comparison of Gage Capability Examples Gage capability is not as good when we account for both reproducibility and repeatability Train operators to reduce σ2reproducability from 0.1181 Since σ2repeatability = 1.0195 (largest component), direct effort toward finding another inspection device. σ2 repeatability σ2 reproducibility σ2 product P/T Single operator 0.8865 0.0967 Three operators 1.0195 0.1181 1.0263 0.1120 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

Gage Capability Based on Analysis of Variance TM 720: Statistical Process Control Gage Capability Based on Analysis of Variance A gage R & R study is actually a designed experiment Therefore ANOVA can be used to analyze the data from an experiment and to estimate the appropriate components of gage variability Assume there are: a parts b operators each operator measures every part n times 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

The measurements, yijk, are represented by a model TM 720: Statistical Process Control The measurements, yijk, are represented by a model where constant m – overall measurement mean r.v. ti – effect from part differences r.v. bj – effect from operator differences r.v. tbij – joint effect of parts & operator differences r.v. eijk – error from measuring instrument with i = part (i = 1, …, a) j = operator (j = 1, …, b) k = measurement (k = 1, …, n) 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

The Variance Components for the Gage R&R Study Using the Model TM 720: Statistical Process Control The Variance Components for the Gage R&R Study Using the Model The variance of an observation yijk is So: is the variance from parts is the variance from operators is the joint variance from parts & operators is the variance from measuring instrument 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

Repeatability & Reproducibility TM 720: Statistical Process Control Repeatability & Reproducibility Reproducibility (Operators) Repeatability (Measuring Device) 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

Gage R&R – ANOVA Method StatGraphics Output TM 720: Statistical Process Control Gage R&R – ANOVA Method StatGraphics Output ANOVA Table Source Sum Squares Df Mean Square F-Ratio P-Value -------------------------------------------------------- Oper 0.95 2 0.475 Part 957.758 19 50.4083 Oper*Part 128.717 38 3.38728 3.42 0.0000 Residual 59.5 60 0.991667 Total 1146.92 119 Operator variable: Operator Part variable: Part Trial variable: Trial Measurement variable: Measurement 3 operators 20 parts 2 trials Estimated Estimated Percent Sigma Variance of Total ------------------------------------------------ Repeatability 0.995825 0.991667 45.29 Reproducibility 0.0 0.0 0.00 Interaction 1.09444 1.19781 54.71 R & R 1.47969 2.18947 100.00 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

Comparison of Gage Capability Examples TM 720: Statistical Process Control Comparison of Gage Capability Examples σ2repeatability σ2reproducibility σ2gage P/T Single operator 0.8865 0.0967 Three operators (Tabular Method) 1.0195 0.1181 1.0263 0.1120 (ANOVA Method) 0.9958 1.0944 1.4797 0.1614 11/7/2018 TM 720: Statistical Process Control (c) 2002-2006 D.H. Jensen & R.C. Wurl

TM 720: Statistical Process Control Questions & Issues Topics for Exam II: Shewhart Continuous Variable Control Charts X-bar and R; X-bar and S-charts Control Limits from samples or standards using table Western Electric Rules Shewhart-Like Discrete Variable Control Charts P, NP, C, U-charts Defectives vs. Defects; Variable or Constant Sample Sizes Control Charts for Individual Measurements X and Moving Range; Moving Average, EWMA, CUSUM Short Run Statistical Process Control DNOM and Standardized charts (continuous / discrete) Gage Repeatability and Reproducibility Control Chart Method – only! 11/7/2018 TM 720: Statistical Process Control