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## Basic Training for Statistical Process Control

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**Basic Training for Statistical Process Control**Process Capability & Measurement System Capability Analysis**Outline**• Process Capability • Natural Tolerance Limits • Histogram and Normal Probability Plot • Process Capability Indices • Cp • Cpk • Cpm & Cpkm • Measurement System Capability • Using Control Charts • Using Factorial Experiment Design (ANOVA) • Hands On Measurement System Capability Study Process Capability Analysis**Process Capability**• Process Capability Analysis (PCA) • Is only done when the process is in a state of Statistical Control • Meaning: NO SPECIAL CAUSES are present • Process does not have to be centered to do PCA • Yield will improve if process is centered, but the value is in knowing what / where to improve the process • PCA is done periodically when the process has been operating in a state of statistical control • Allows for measuring improvement over time • Allows for marketing your competitive edge Process Capability Analysis**Process Capability Analysis is performed when there are NO**special causes of variability present – ie. when the process is in a state of statistical control, as illustrated at this point. Improving Process Capability and Performance Continually Improve the System Characterize Stable Process Capability Head Off Shifts in Location, Spread Time Identify Special Causes - Bad (Remove) Identify Special Causes - Good (Incorporate) Reduce Variability Center the Process LSL 0 USL Process Capability - Timing Process Capability Analysis**Process Capability**• Process Capability is INDEPENDENT of product specifications • Most specifications are set without regard for process capability • However, understanding process capability helps the engineer to set more reasonable specifications • PCA reflects only the Natural Tolerance Limits of the process • PCA is done by examining the process • Histogram • Normal Probability Plot Process Capability Analysis**Natural Tolerance Limits**• The natural tolerance limits assume: • The process is well-modeled by the Normal Distribution • Three sigma is an acceptable proportion of the process to yield • The Upper and Lower Natural Tolerance Limits are derived from: • The process mean () and • The process standard deviation () • Equations: Process Capability Analysis** 1 :68.26% of the total area** 2 :95.46% of the total area 3 :99.73% of the total area -3 or LNTL - + +3 or UNTL -2 +2 The Natural Tolerance Limits cover 99.73% of the process output Natural Tolerance Limits Process Capability Analysis**PCA: Histogram Construction**• Verify rough shape and location of histogram • Symmetric (roughly bell-shaped) • Mean = median = mode • Quickly confirm applicability prior to statistical analysis • Can be very hard to distinguish a Normal Distribution from a t-Distribution • Sometimes even a Normal distribution doesn’t look normal • More data and columns (bins) can make a difference • Verify location of process with respect to Specifications • Quick inspection will show what to do to improve the process Process Capability Analysis**C**u m F r e q C u m F r e q C u m F r e q X X X PCA: Normal Probability Plot • A Normal Plot better clarifies whether the distribution is Normal by a visual inspection for: • Non-random patterns (non-Normal) • Fat Pencil Test (Normal if passes) Process Capability Analysis**PCA: Parameter Estimation**• The Normal Plot mid-point estimates the process mean • The slope of the “best fit” line for the Normal Plot estimates the standard deviation • Choose the 25th and 75th percentile points to calculate the slope • The Histogram mode should be close to the mean • The range/d2 (from Histogram) should be close to the standard deviation • Can also estimate standard deviation by subtracting 50th percentile from the 84th percentile of the Histogram Process Capability Analysis**Process Capability Indices**• Cp: • Measures the potential capability of the current process - if the process were centered within the product specifications • Two-sided Limits: • One-sided Limit: Process Capability Analysis**Cp Relation to Process Fallout**• Recommended Minimum Ratios: (D. C. Montgomery, 2001) • Existing Process 1.25 (1-sided) 1.33 (2-sided) • Existing, Safety / Critical Parameter 1.45 1.50 • New Process 1.45 1.50 • New, Safety / Critical Parameter 1.60 1.67 Process Capability Analysis**Process Capability Indices**• Cpk: • Measures actual capability of current process - at its’ current location with respect to product specifications • Formula: Where: Process Capability Analysis**Process Capability Indices**• Regarding Cp and Cpk: • Both assume that the process is Normally distributed • Both assume that the process is in Statistical Control • When they are equal to each other, the process is perfectly centered • Both are pretty common reporting ratios among vendors and purchasers Process Capability Analysis**LSL**USL Process Capability Indices • Two very different processes can have identical Cpk values, though: • because spread and location interact! Process Capability Analysis**Process Capability Indices**• Cpm: • Measures the current capability of the process - using the process target center point within the product specifications in the calculation • Formula: Where target T is: Process Capability Analysis**Process Capability Indices**• Cpkm: • Similar to Cpm - just more sensitive to departures from the process target center point • Not really in very common use • Formula: Process Capability Analysis**Measurement System Capability**• Examines the relative variability in the product and measurement systems, together • Total variation is the result of • Product variation • Gage variation • Operator variation gaging system variation • Random variation Process Capability Analysis**Measurement System Analysis**• Measurement system can be assessed by • X-bar and R-Charts • Using a single part as the rational subgroup • Is easy to visualize • Requires alternate interpretation of the control charts • Designed Experiments • Using Analysis of Variance • Allows assessment of part x operator interactions • Is statistically complex to compute & analyze Process Capability Analysis**X-Bar & R-Chart Method**• Have each operator measure the same part twice - so the part becomes the rational sample unit • Parts should be representative of those to be measured • Use a sample of 20 - 25 parts • Use a representative set of operators • Either collect data from every operator, or • Randomly select from the set of operators • Collect data under representative conditions • Carefully specify and control the conditions for measurement • Randomly sequence the combination of parts and operators • Preserve the time-order of the collected data & note observations Process Capability Analysis**X-Bar & R-Chart Method**• If each operator measures the same part twice: • Variation between samples is plotted on the X-Chart • Out of control points indicate success in identifying differences between parts • Variation within samples is plotted on the R-Chart • Centerline of R-Chart is the magnitude of the gage variation • Out of control points indicate excessive operator to operator variation (fix with training?) Process Capability Analysis**Out of control points indicate ability to distinguish**between product samples (Good) Out of control points indicate inability of operators to use gaging system (Bad) UCL LCL Sample Number X-Bar Control Chart UCL LCL Sample Number R - Control Chart x R X-Bar & R-Chart Method Process Capability Analysis**X-Bar & R-Chart Method**• Precision to Tolerance Ratio (P/T): • “Rule of Ten”: • The measurement device should be at least ten times more accurate than the smallest measurement • Calculations: and • Interpretation: • Resulting ratio should be 0.10 or smaller if the gage is truly capable Process Capability Analysis**X-Bar & R-Chart Method: R & R**• Repeatability: • Inherent precision of the gage • Reproducibility: • Variability of the gage under differing conditions • Environment • Operator • Time … Process Capability Analysis**X-Bar & R-Chart Method: R & R**• Process is the same as before (20 - 25 parts, …): • But we estimate the Repeatability from the Range Mean computed across all the operators and all parts: • And we estimate the Reproducibility from the Range of variability across all operators for each individual part: Process Capability Analysis**X-Bar & R-Chart Method: R & R**• What to do with the information? • More variation is bad, so… • If the reproducibility variation is larger, improve the operators • If the repeatability variation is larger, improve the gaging instrument • Hands-On Experiment • Micrometer Study Process Capability Analysis**Gage Capability Analysis: ANOVA**• A form of Designed Experiment: • Two-Treatment Full Factorial with Replications • Analysis is done using ANOVA software • Comparisons of variance components is through a series of F-tests (either exceed critical region or look for small p-value) • Can distinguish part X operator interaction • Want to find non-significant operator, interaction terms, and a significant part term - capable system! Process Capability Analysis