Chapter 4 Control Charts for Measurements with Subgrouping

Chapter 4 Control Charts for Measurements with Subgrouping (for One Variable)

4. 9 Determining the Point of a Parameter Change •

4. 10 Acceptance Sampling and Acceptance Control Chart • Acceptance sampling is not a process control technique. • Acceptance sampling plan specifies the sample size that is to be used and the decision criteria that are to be employed in determining whether a lot or shipment should be rejected. • “You cannot inspect quality into a product” Harold F. Dodge • Studies show that only 80% of non-confirming units are detected during 100% final inspection. • Acceptance sampling should be used only temporarily. • Problems with acceptance sampling plans include the fact that the producer’s risk and the consumer’s risk can both be unacceptably high.

4. 10. 1 Acceptance Control Chart • Acceptance control limits are determined from the specification limits (Far from 3 range) • APL (Acceptable Process Level): the process level that yields product quality to be accepted 100(1 - )% • : Probability of rejecting an APL • RPL (Rejectable Process Level): the process level that yields product quality to be rejected 100(1 - )% • : Probability of accepting an RPL • p 1: acceptable % of units falling outside the spec. • p 2: rejectable % of units falling outside the spec.

4. 10. 1 Acceptance Control Chart • (4. 4) (4. 5) (4. 6) (4. 7)

4. 10. 1. 1 Acceptance Chart Example •

Table 4. 2 Data in Subgroups Obtained at Regular Intervals Subgroup 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 X 1 72 56 55 44 97 83 47 88 57 13 26 46 49 71 71 67 55 49 72 61 X 2 84 87 73 80 26 89 66 50 47 10 39 27 62 63 58 69 63 51 80 74 X 3 79 33 22 54 48 91 53 84 41 30 52 63 78 82 69 70 72 55 61 62 X 4 49 42 60 74 58 62 58 69 46 32 48 34 87 55 70 94 49 76 59 57 X-bar 71. 00 54. 50 52. 50 63. 00 57. 25 81. 25 56. 00 72. 75 47. 75 21. 25 42. 50 69. 00 67. 75 67. 00 75. 00 59. 75 57. 75 68. 00 63. 50 R 35 54 51 36 71 29 19 38 16 22 26 36 38 27 13 27 21 17 S 15. 47 23. 64 21. 70 16. 85 29. 68 13. 28 8. 04 17. 23 6. 70 11. 35 11. 53 15. 76 16. 87 11. 53 6. 06 12. 73 9. 98 12. 42 9. 83 7. 33

4. 11 Modified Limits • If the specification limits were at k , the limits would be widened by (k-3)

4. 12 Difference Control Charts • The general idea is to separate process instability caused by uncontrollable factors from process instability due to assignable causes. • This is accomplished by taking samples from current production and also samples from what is referred to as a reference lot. • The reference lot consists of units that are produced under controlled process conditions except for possibly being influenced by uncontrollable factors. • Since both samples are equally influenced by uncontrollable factors, any sizable differences between sample means should reflect process instability due to controllable factors.

4. 12 Difference Control Charts •

4. 12 R-chart of Difference •

4. 12 Paired Difference Control Charts •

4. 13 Other Charts •

4. 14 Average Run Length (ARL) • If the parameters were known, the expected length of time before a point plots outside the control limits could be obtained as the reciprocal of the probability of a single point falling outside the limits when each point is plotted individually. • The expected value is called the Average Run Length (ARL). • It is desirable for the in-control ARL to be reasonably large. • The parameter-change ARL should be small.

4. 14 Average Run Length (ARL) •

4. 1 Weakness of the ARL Measure • The run length distribution is quite skewed so that the ARL will not be the typical run length • The standard deviation of the run length is quite large

4. 15 Determining the Subgroup Size • By convenience: 4 or 5 • Economic design of control charts • Using graphs (such as given by Dockendorf, 1992) • The larger the subgroup, the more power a control chart will have for detecting parameter changes. • Survey showed most respondents used subgroup size of about 6.

4. 15. 1 Unequal Subgroup Sizes • May caused by missing data • Minitab (with 2 columns) Variable Sample Size (VSS) • Smaller sample sized is used if the sample mean falls within “warning limits” (2 -sigma limits) • Larger sample sized is used if the sample mean falls between warning limits (2 -sigma limits) and control limits. • Superior in detecting small parameter changes

4. 16 Out-of-Control Action Plans (OCAPs) A flow chart with • Activator: out-of-control signal (limits + run rules) • Checkpoint: potential assignable causes • Terminator: action taken to resolve the condition

4. 17 Assumptions for Control Charts •

4. 17. 1 Normality • For R-, S-, and S 2 -charts, the basic assumptions are the individual observations are independent and normally distributed. • The distributions of R, S, and S 2 differ considerably from a normal distribution. • Many process characteristics will not be well approximated by normal distribution. (diameter, roundness, mold dimensions, customer waiting time, leakage from a fuel injector, flatness, runout, and percent contamination) • Non-normality is not a serious problem unless there is a considerable deviation from normality.

4. 17. 2 Independence •

4. 17. 2 Example of Invalid Assumption of Independence •

1. 30 1. 59 0. 17 0. 01 0. 07 -1. 18 -3. 36 -3. 35 0. 50 -0. 26 -2. 07 -2. 02 -1. 77 -0. 12 0. 32 -1. 16 -0. 89 -0. 80 0. 08 1. 74 0. 06 -1. 46 -1. 75 -1. 46 0. 19 -0. 60 -0. 67 1. 10 1. 15 1. 30 1. 61 0. 12 1. 11 0. 76 -0. 21 -0. 73 0. 17 0. 45 0. 27 -0. 78 -1. 63 0. 03 0. 52 1. 21 0. 87 1. 23 0. 78 0. 86 2. 71 1. 68 -0. 52 -1. 40 -0. 59 -1. 15 -0. 48 0. 13 0. 33 0. 07 0. 34 0. 78 1. 28 0. 48 -0. 50 0. 99 1. 00 -0. 05 -1. 55 -0. 88 0. 79 1. 04 2. 52 1. 79 3. 72 2. 65 1. 92 0. 90 -0. 60 -1. 70 -0. 75 0. 76 0. 52 -0. 29 2. 22 1. 21 -0. 20 0. 59 0. 42 0. 63 0. 52 -0. 89 -0. 50 -0. 99 -2. 02 -1. 40 0. 39 1. 36 1. 24 -1. 10 -1. 98 -0. 41

Figure 4. 7

4. 17. 2 Remedy for Correlated Data •

4. 18 Measurement Error • Assume measurement variability is independent of product variability, and that repeatability variability and reproducibility variability are independent, then observations = product + repeatability + reproducibility • Reproducibility variability: Determined by the performance of the measurement process under changing conditions (DOE) • Using control chart to determine if reproducibility is in a state of statistical control • Repeatability variability: Estimated using at least a moderately large number of measurements under identical conditions. Var(S 2) = 2 4/(n-1)

4. 18. 1 Monitoring Measurement Systems • Separate monitoring of variance components for repeatability and reproducibility, or a simultaneous procedure.