ENGM 620: Quality Management

1. ENGM 620: Quality Management Session 8 – 30 October 2012 • Process Capability

2. 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

3. 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

4. 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:

5.  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

6. 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:

7. Process Capability Ratio Note • There are many ways we can estimate the capability of our process • If σ is unknown, we can replace it with one of the following estimates: • The sample standard deviation S • R-bar / d2

8. Process Capability Indices • Cpk: • Measures actual capability of current process - at its’ current location with respect to product specifications • Formula: Where:

9. 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

10. LSL USL Process Capability Indices • Two very different processes can have identical Cpk values, though: • because spread and location interact!

11. PCR and an Off-Center Process • CPK = min (CPU, CPL) • Generally, if CP = CPK, then the process is centered at the midpoint of the specifications • If CP≠ CPK, then the process is off-center

12. Probably Different Probably NOT Different Definitely NOT Different Comparison of Variances • The second types of comparison are those that compare the spread of two distributions. To do this: • Compute the ratio of the two variances, and then compare the ratio to one of two known distributions as a check to see if the magnitude of that ratio is sufficiently unlikely for the distribution. • The assumption that the data come from Normal distributions is very important. Assess how normally data are distributed prior to conducting either test. Definitely Different

13. 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:

14. 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:

15. 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

16. 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

17. 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

18. 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?)

19. 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

20. 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

21. X-Bar & R-Chart Method: R & R • Repeatability: • Inherent precision of the gage • Reproducibility: • Variability of the gage under differing conditions • Environment • Operator • Time …

22. 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: