Confirmation Sample Control Charts

Transcription

1 Confirmation Sample Control Charts Stefan H. Steiner Dept. of Statistics and Actuarial Sciences University of Waterloo Waterloo, NL 3G1 Canada Control charts such as X and R charts are widely used in industry to monitor quality. These charts are effective in detecting large departures from the in-control condition. Other process monitoring methods, such as Cumulative Sum (CUSUM) charts are sequential in nature as they accumulate information from previous observations. CUSUM charts are good at detecting more moderate presistent process shifts, but are more complicated to implement. As a compromise, a number of researchers have investigated adapting the simple Shewhart type control charts ( X and R ) to take into account some but not all the previous observations. Variable Sampling Interval (VSI) control charts adjust the sampling interval based on the current observation. If there is some evidence (but no out-of-control signal) that the process may have shifted the sampling interval is shortened. This has the effect of reducing the number of non-conforming parts produced if there moderate process shift. VSI X charts are discussed by Reynolds, Amin, Arnold, Nachlas The VSI approach is most effect for sequential charts such as CUSUM charts since sample statistics plotting close to the control limit have accumulated significant evidence against the in-control hypothesis (Reynolds, Amin, and Arnold 1990). However, VSI charts have also been created for Shewhart charts (Cui, Reynolds, 1988, Reynolds, Amin, Arnold, Nachlas, 1988), np charts (Vaughan, 1993), and combined SAhewhart and CUSUM charts (Amin, Ncube, 1991). Other previous work considered the adjustment of the sample size in response to the observed statistics. Variable sample size (VSS) control charts are discussed by Sawalapurkar, Reynolds, and Arnold (1990) and Prabhu, Montgomery and Runger (1994). Croasdale (1974) and Daudin (1994) suggest X charts that occasionally require a second sample. These two approaches suggest adding warning limits to the standard control

2 charts. Any points falling between the warning limits and the control limits lead to taking an additional sample. In the Croasdale (1974) procedure the accept/reject decision is based solely on the second sample, whereas Daudin bases the decision on the combined sample. In both cases the first and second sample sizes need not be the same. VSI, VSS and double sampling control charts are compared in Costa (1994). More on VSS In this article a specific special case of the Daudin procedure called a confirmation control chart that has intuitive appeal to production personnel is proposed and explored in detail. Unlike the Daudin approach the proposed procedure has the advantage of being a very simple variation of the standard X chart. The proposed procedure adapts to the natural inclination of people to wish to verify bad news. Whenever a sample of poor or questionable quality is observed a confirmation sample is immediately taken to verify that the original observation was not a fluke. In addition, the article discusses the creation of a confirmation control chart to detect process standard deviation shifts. Typically the process mean is monitored assuming a stable process standard deviation. Thus, a complete quality monitoring procedure requires the simultaneous monitoring of the process mean and standard deviation. The advantage of the confirmation sample control chart over standard X and R charts is the greater power of the tests. This increased power arises since the confirmation control chart is partially sequential since the amount of sampling is related to the outcome of the sampling. Confirmation Sample Control Chart A confirmation sample control chart can be setup in one of two ways. First, the chart can be run solely using confirmation control limits, or it can be run combining both confirmation control limits and traditional three sigma limits. The combined confirmation sample control chart is operated in the following manner:

3 3 Stand Alone Confirmation Sample Procedure take a rational sample of size n plotting the sample mean X and the range R or the standard deviation s. If the point lies outside the confirmation sample limit take an additional sample of size n immediately, plotting the new sample mean and range (or standard deviation) together with the first points. If the second point also lies outside the same confirmation sample limit then stop process and look for an assignable cause. Combined Confirmation Sample Procedure take a rational sample of size n plotting the sample mean X and the range R or the standard deviation s. If the plotted point lies outside the control limits stop the process and look for an assignable cause. If the point lies between a control limit and the confirmation sample limit take an additional sample of size n immediately, plotting the new sample mean and range (or standard deviation) together with the first points. If the second point also lies outside the same confirmation sample limit then stop process and look for an assignable cause. Figure 3 in the Example section shows how the confirmation sample control chart may look. The figure shows the confirmation sample control chart that utilizes solely the confirmation sample control limit. The derivation of an appropriate setting for the confirmation sample limit depends on the type of chart ( X or R or s) and the desired operating characteristics. For standard X charts the usual control limits are placed at X ±3 ˆσ, where X and ˆσ are the estimated process mean and standard deviation respectively. This setting leads to a false alarm rate (assuming normality) of

4 4 P( X > 3σ n)+ P X < 3σ n ( ) = 0.007, in other words when the process is in control 0.7% of the time the chart will signal. This false alarm rate has been found to provide good results in industry application. As in the design of standard control charts for confirmation sample control charts there is a choice regarding the appropriate position of the confirmation control limit. One reasonable possibility is to set the decision limits so that the false alarm rate is still at or near In other words find k so that P( X > k σ n) + P X < kσ n ( ) = (1) Assuming symmetry, and ignoring the standard control limits solving equation (1) for k yields k = This suggests that k = 1.8 is a good choice for the stand alone confirmation control chart. For the combined confirmation control chart k = is a reasonable choice. Clearly with the combined approach the chance of signaling is increased. This is similar to addition of run rules to standard Shewhart control charts. Using combined confirmation sample limits at ±σ, in control, assuming normality and an in-control process, only about 4.5% of samples would require a confirmation sample and only about 0.% of samples lead to a false alarm. This results in a very small increase in the sampling effort required. If this proves to be too onerous the sample size required for each sample could be increased. It should be noted that these theoretical calculation, as with standard control charts, requires independence between individual observations and thus between two subgroups. Typically samples are gathered in such a way to ensure this. With the confirmation sample control chart there is an additional concern due to the potential requiring an immediate confirmation sample. A confirmation sample control chart to monitor the process variation (R or s chart) can be design in the same way. However, unlike the distribution of the sample mean, the distribution of the sample R and s are quite skewed. As a result the use of probability limits (as discussed by

5 5 Ryan, 1989) are recommended. The values in Table 1 are based on the distribution of the sample range and were derived from Harter (1960) using 0.05 tail probabilities. This size of tail probability yields a false alarm rate of close to that obtained theoretically with traditional control charts. Table 1: Confirmation Control Limits for Confirmation R chart n R lcl R ucl For the s chart the confirmation sample control limits can be determined using the chisquare distribution since if X ~ N µ, σ ( ) then (n 1)s σ ~ χ n 1 standard deviation. Set control limits at LCL = ˆσ χ.0316, where s is the sample ( n 1) and UCL = ˆσ χ.9684 ( n 1), where ˆσ is an estimate for the process standard deviation derived from previous data. The limits shown in Table were derived using the function chiinv in the statistics toolbox of MATLAB. We use tail probabilities of 0.08 to provide a combined false alarm rate of approximately

6 6 Table : Confirmation Control Limits for Confirmation Sample s chart n χ.08,n 1 ( n 1) χ.0367,n 1 ( n 1) χ.977,n 1 ( n 1) χ.9633,n ( n 1) Assume that the confirmation sample is independent. As proposed the confirmation sample control chart would require a slight larger average sample size than a traditional control chart. If the process is out-of-control this additional sampling requirement is significant since a confirmation sample would often be required, however, typically rapid detection of an out-of-control situation is the highest priority. In-control the confirmation sample control chart would occasionally require a confirmation sample. This additional sampling effort is undesirable since the process is operating normally. Fortunately however, the increasing in sampling effort is quite modest. When the sample mean is centered at the nominal value around 4.5% of observations will fall outside the confirmation sample limits and thus require an additional sample. Comparison with Standard Control Charts In the previous section the confirmation control charts were design to closely match the false alarm rates of standard control charts. This section shows that power of the confirmation control charts are significantly higher than traditional control charts. This design philosophy could be altered to provide charts with smaller false alarms and similar power, but the size of the false alarm that yields good results in practice is well established.

7 7 By matching the false alarm rates and making an adaptive scheme that takes an additional sample if there is some reasonable doubt that the process is still in control the power of the confirmation control chart is improved. Figure 1 shows the operating characteristics of a standard X control chart and a confirmation sample control chart both with sample sizes of five. Clearly the confirmation sample control chart has a better ability to detect deviations from nominal. In fact since the confirmation sample X chart was setup with ±σ limits it also has a slightly smaller false alarm rate. The comparison given in Figure 1 shows the increased power of the confirmation sample X chart, but due to the nature of the confirmation strategy it requires on average a slightly larger sample size since occasionally it requires a second sample of size five. Similar results are obtained for the R and s charts. 1 Probability of Acceptance Confirmation Chart and Çombined Chart X chart Mean Shift Figure 1: Comparison of the Standard X and Confirmation Sample X Control Charts, n = 5

8 8 mean Table 3: Probability of observing a signal P signal X chart P signal confirmation chart with k = 1.8 P signal combination chart with k = Example Point falling above the warning limits lead to an addition sample. The confirmation sample statistic is plotted together with the first sample point and the points are joined with a line. sample mean sample number sample standard deviation sample number Figure 4:

9 9 Conclusions A confirmation sample control chart is a good alternative to a traditional X and R control chart procedure if another independent sample can be immediately obtained. Implementation is straight-forward. If we wish to design the confirmation sample control chart to have the same (or close to the same) false alarm rate than confirmation sample limits should be placed at plus or minus two sigma. appeals to the desire to confirm out-ofcontrol signals with an additional observation(s). Power of test is increased while the sampling work is only marginally increased. Easier to implement than previous work since the sample size for the first and second samples is the same. There is some reduction in efficiency for the confirmation sample control chart when compared with Daudin s Two Sample control chart since decisions are made solely on the second sample. However, this slight loss of efficiency is compensated by the simplicity of implementation. Simplification that is more likely to be used in practice. second sample taken and plotted for the same time period References Amin, R., Ncube, M.M. (1991), Variable sampling interval combined Shewhart-cumulative score control procedure, Journal of Applied Statistics, 40, 1-1. Croasdale, R. (1974), Control charts for a double-sampling scheme based on average production run lengths, International Journal of Production Research, 1, Costa A.F.B. (1994), X Charts with Variable Sample Size, Journal of Quality Technology, 6, Cui, R, Reynolds, M.R., X charts with runs rules and variable sampling intervals, Communications in Statistics Series B 17,

10 10 Daudin, J.J. (1994), Double Sampling X Charts, Journal of Quality Technology, 4, Harter, H. L. (1960), Tables of the range and studentized range, The Annals of Mathematical Statistics, 31, Pitt, H. (1987), The Resampling Syndrome, Quality Progress, April 7-9. Prabhu, S.S, Montgomery, D.C., and Runger, C.C. (1994), A Combined Adaptive Sample Size and Sampling Interval Control Scheme, Journal of Quality Technology, 6, Reynolds Jr., M. R., Amin, R. W., Arnold, J., Nachlas, J.A., (1988), X charts with variable sampling intervals, Technometrics, 30, Reynolds Jr., M. R., Amin, R. W., Arnold, J. (1990), CUSUM charts with variable sampling intervals, Technometrics, 3, Ryan, T.P. (1989), Statistical Methods for Quality Improvement, John Wiley & Sons, New York. Sawalapurkar, U., Reynolds, M.R. Jr., and Arnold, J.C. (1990), Variable Sample Size X Control Charts, Presented at the Winter Conference of the American Statistical Association, Orlando, FL. Vaughan T.S., (1993), Variable sampling interval np process control charts, Communications in Statistics Series A,,