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## TM 720 - Lecture 10

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**TM 720 - Lecture 10**Short Run SPC and Gage Reproducibility & Repeatability 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 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 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 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 TM 720: Statistical Process Control**DNOM Charts**• The computed variable (xi) is part of a sub-sample of size n • xiis normally distributed • nis held constantfor 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: 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 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 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 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 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 TM 720: Statistical Process Control**Standardized Charts – Attributes**• Standardized zi for Proportion Defective: • Plotted variable is • Control Limits: TM 720: Statistical Process Control**Standardized Charts – Attributes**• Standardized zi for Number Defective: • Plotted variable is • Control Limits: TM 720: Statistical Process Control**Standardized Charts – Attributes**• Standardized zi for Count of Defects: • Plotted variable is • Control Limits: TM 720: Statistical Process Control**Standardized Charts – Attributes**• Standardized zi for Defects per Inspection Unit: • Plotted variable is • Control Limits: 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 - σ2gagei.e., measurement error • Total observed variance in product: σ2total = σ2product + σ2gage (system) TM 720: Statistical Process Control**e.g. Assessing Gage Capability**• Following data were taken by one operator during gage capability study. 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. 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 thetolerance 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 TM 720: Statistical Process Control**Estimating Variance Components of Total Observed Variability**• Estimate total variance: • Compute an estimate of product variance • Since : 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 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.: 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 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 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 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 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! 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. 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 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) 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 TM 720: Statistical Process Control**Repeatability & Reproducibility**Reproducibility(Operators) Repeatability(Measuring Device) 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 TM 720: Statistical Process Control**Comparison of Gage Capability Examples**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! TM 720: Statistical Process Control