Chapter 4

Control Charts for Measurements with Subgrouping

(for One Variable)

4.9 Determining the Point of a Parameter Change

• A change in the mean: The max. likelihood estimator is the max over t of , with T denoting the time of a signal occurs.(“Identifying the time of a step change with X-bar control chart” by Samuel, Pignatiello, and Calvin, QE, 1998)

• A change in the variance:(“Identifying the time of a step change in a normal process variance” by Samuel, Pignatiello, and Calvin, QE, 1998)

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• p1: acceptable % of units falling outside the spec.• p2: 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 with Control Limits

• The acceptance chart can be constructed with the -chart control Limits either shown or not shown on the chart.

• Not in accordance with contemporary views on quality improvement

4.10.1.1 Acceptance Chart Example

• Use data from Table 4.2• Assume the specification limits are at (USL=150.6495, LSL=-31.7745)• p1 =.00001, =.05

=98.3165 =20.5585

• Recall the -chart UCL=82.2552, LCL=36.6198

Table 4.2 Data in Subgroups Obtained at Regular Intervals

Subgroup X1 X2 X3 X41 72 84 79 492 56 87 33 423 55 73 22 604 44 80 54 745 97 26 48 586 83 89 91 627 47 66 53 588 88 50 84 699 57 47 41 46

10 13 10 30 3211 26 39 52 4812 46 27 63 3413 49 62 78 8714 71 63 82 5515 71 58 69 7016 67 69 70 9417 55 63 72 4918 49 51 55 7619 72 80 61 5920 61 74 62 57

X-bar R S71.00 35 15.4754.50 54 23.6452.50 51 21.7063.00 36 16.8557.25 71 29.6881.25 29 13.2856.00 19 8.0472.75 38 17.2347.75 16 6.7021.25 22 11.3541.25 26 11.5342.50 36 15.7669.00 38 16.8767.75 27 11.5367.00 13 6.0675.00 27 12.7359.75 23 9.9857.75 27 12.4268.00 21 9.8363.50 17 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

• The 3-sigma control limits are

Where and are avg. range for the reference lot and the current lot

• Difference Control Chart replaces -chart, with ( plotted• It is comparable to a pooled-t test for the equality of

population means with a significance level of .0027 provided that ( is approximately normal distribution.

4.12 R-chart of Difference

• The 3-sigma control limits on an R-chart are

• ) are plotted

4.12 Paired Difference Control Charts

• The 3-sigma control limits are

WhereR = largest difference between pairs – smallest difference

• Difference Control Chart replaces -chart, with ( plotted

4.13 Other Charts

• Median chart is a substitute for an -chart.• It is not as efficient as the average• Midrange (average of the largest and the smallest)

chart • Coefficient of variation (/) chart

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)

• With 3-sigma limits, the in-control ARL is 1/.0027=370.37

• Assume 1 increase in the mean, the parameter-change ARL is 43.89

• When the parameters are estimated, both the in-control ARL and the parameter-change ARL are inflated.

4.14.1 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

• White noise model (when process is in control): • Normality• Independence

4.17.1 Normality

• For R-, S-, and S2-charts, the basic assumptions are the individual observations are independent and normally distributed.

• The distributions of R, S, and S2 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

• The estimation of by is appropriate only when the data are independent.

• The appropriate expression for can be determined from the time series model

4.17.2 Example of Invalid Assumption of Independence

• First-order autoregressive (AR) process

Where and

Table 4.6 100 Consecutive Values from AR(1) and

1.30 0.06 -1.63 1.28 0.521.59 -1.46 0.03 0.48 -0.290.17 -1.75 0.52 -0.50 2.220.01 -1.46 1.21 0.99 1.210.07 0.19 0.87 1.00 -0.20-1.18 -0.60 1.23 -0.05 0.59-3.36 -0.67 0.78 -1.55 0.42-3.35 1.10 0.86 -0.88 0.630.50 1.15 2.71 0.79 0.52-0.26 1.30 1.68 1.04 -0.89-2.07 1.61 -0.52 2.52 -0.50-2.02 0.12 -1.40 1.79 -0.99-1.77 1.11 -0.59 3.72 -2.02-0.12 0.76 -1.15 2.65 -1.400.32 -0.21 -0.48 1.92 0.39-1.16 -0.73 0.13 0.90 1.36-0.89 0.17 0.33 -0.60 1.24-0.80 0.45 0.07 -1.70 -1.100.08 0.27 0.34 -0.75 -1.981.74 -0.78 0.78 0.76 -0.41

Figure 4.7

4.17.2 Remedy for Correlated Data

• Fitting a time-series model to the data and using the residuals from the model in monitoring the process

• Drawbacks of residual chart: – poor ARL properties– Harder to relate to a residual– Solution: -chart plotted with residual chart

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(S2) = 24/(n-1)

4.18.1 Monitoring Measurement Systems

• Separate monitoring of variance components for repeatability and reproducibility, or a simultaneous procedure.