Marion R. Reynolds

The State of Statistical Process Control as We Proceed into the 21st Century

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Optimal one-sided shewhart control charts with variable sampling intervals

When a control chart is used to detect changes in a process the usual practice is to take samples from the process using a fixed sampling interval between samples. This paper considers the properties of Shewhart control charts when the sampling interval used after each sample is not tixed but instead depends on what is observed in the sample. The basic rationale is that the sampling interval should be short if there is some indication of a change in the process and long if there is no indication of a change. If the indication of a change is strong enough then the chart signals in the same way as the fixed sampling interval Shewhart chart. The result is that samples will be taken more frequently when there is a change in the process, and this process change can be detected much more quickly than when fixed sampling intervals are used. Expressions for properties such as the average time to signal and the average number of samples to signal are developed. It is shown that if the sampling interval must be cho...

Should observations be grouped for effective process monitoring

When control charts are used to monitor processes to detect special causes, it is usually assumed that a special cause will produce a sustained shift in a process parameter that lasts until the shift is detected and the cause is removed. However, some special causes may produce a transient shift in a process parameter that lasts only for a short period of time. Control charts are usually based on samples of n ≥ 1 observations using a sampling interval of fixed length, say d. When n > 1, the usual practice, based on the so-called rational subgroups concept, is to take a concentrated sample at one time point at the end of the sampling interval d, but another option is to disperse the sample over the interval d. In this paper, we investigate the question of whether it is better to use n = 1, or to use n > 1 and either concentrated or dispersed samples. The objective of monitoring is assumed to be the detection of special causes that may produce either a sustained or transient shift in the process mean %mU and/or process standard deviation σ. It is assumed that the sampling rate in terms of the number of observations per unit time is fixed, so that the ratio n/d is fixed. The best sampling strategy depends on the type of control chart being used, so Shewhart and cumulative sum (CUSUM) charts are considered. For both types of control charts, a combination of two charts is investigated; one chart is designed to monitor μ, and the other is designed to monitor σ. The overall conclusion is that it is best to take samples of n = 1 and use a CUSUM chart combination. The Shewhart chart combination with the best overall performance is based on n > 1, but this combination has inferior statistical performance compared with the CUSUM chart combination.

Nonparametric quality control charts based on the sign statistic

Nonparametric control chart are presented for the problem of detecting changes in the process median (or mean), or changes in the process variability when samples are taken at regular time intervals. The proposed procedures are based on sign-test statistics computed for each sample, and are used in Shewhart and cumulative sum control charts. When the process is in control the run length distributions for the proposed nonparametric control charts do not depend on the distribution of the observations. An additional advantage of the non-parametric control charts is that the variance of the process does not need to be established in order to set up a control chart for the mean. Comparisons with the corresponding parametric control charts are presented. It is also shown that curtailed sampling plans can considerably reduce the expected number of observations used in the Shewhart control schemes based on the sign statistic.

Evaluating properties of variable sampling interval control charts

Standard fixed sampling interval (FSI) control charts take samples from a process at fixed length sampling intervals for purposes of detecting changes in the peocess that may affect the quality of the output. Variable sampling interval (VSI) control charts vary the sampling interval as a function of what is observed from the process and can detect process changes faster than FSI control charts. Evaluation of properties of VSI control charts requires extensions of methods that have been developed for FSI control Control charts. A unified treatment of two widely used methods, the Markov chain method and the integral equation method. is given for VSI control charts.This unified treatment includes some results which are new in the FSI case. The new methods are used for the numerical evaluation of propertics of exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) control charts.Some general optimality results for the choice fo the sampling intervals are also given.

Robustness to non-normality and autocorrelation of individuals control charts

This paper studies the effects of non-normality and autocorrelation on the performances of various individuals control charts for monitoring the process mean and/or variance. The traditional Shewhart X chart and moving range (MR) chart are investigated as well as several types of exponentially weighted moving average (EWMA) charts and combinations of control charts involving these EWMA charts. It is shown that the combination of the X and MR charts will not detect small and moderate parameter shifts as fast as combinations involving the EWMA charts, and that the performana of the X and MR charts is very sensitive to the normality assumption. It is also shown that certain combinations of EWMA charts can be designed to be robust to non-normality and very effective at detecting small and moderate shifts in the process mean and/or variance. Although autocorrelation can have a significant effect on the in-control performances of these combinations of EWMA charts, their relative out-of-control performances under independence are generally maintained for low to moderate levels of autocorrelation.

A general approach to modeling CUSUM charts for a proportion

This paper considers two CUmulative SUM (CUSUM) charts for monitoring a process when items from the process are inspected and classified into one of two categories, namely defective or non-defective. The purpose of this type of process monitoring is to detect changes in the proportion p of items in the first category. The first CUSUM chart considered is based on the binomial variables resulting from counting the total number of defective items in samples of n items. A point is plotted on this binomial CUSUM chart after n items have been inspected. The second CUSUM chart considered is based on the Bernoulli observations corresponding to the inspection of the individual items in the samples. A point is plotted on this Bernoulli CUSUM chart after each individual inspection, without waiting until the end of a sample. The main objective of the paper is to evaluate the statistical properties of these two CUSUM charts under a general model for process sampling and for the occurrence of special causes that change the value of p. This model applies to situations in which there are inspection periods when n items are inspected and non-inspection periods when no inspection is done. This model assumes that there is a positive time between individual inspection results, and that a change in p can occur anywhere within an inspection period or a non-inspection period. This includes the possibility that a shift can occur during the time that a sample of n items is being taken. This model is more general and often more realistic than the simpler model usually used to evaluate properties of control charts. Under our model, it is shown that there is little difference between the binomial CUSUM chart and the Bernoulli CUSUM chart, in terms of the expected time required to detect small and moderate shifts in p, but the Bernoulli CUSUM chart is better for detecting large shifts in p. It is shown that it is best to choose a relatively small sample size when applying the CUSUM charts. As expected, the CUSUM charts are substantially faster than the traditional Shewhart p-chart for detecting small shifts in p. But, surprisingly, the CUSUM charts are also better than the p-chart for detecting large shifts in p.

Variable Sampling Interval X Charts in the Presence of Correlation

Traditional applications of control charts use fixed sampling interval (FSI) charts in which the time interval between samples is fixed. Recently, more efficient variable sampling interval (VSI) charts have been developed in which the next observation i..

Economic Design of a Variable Sampling Rate X̄ Chart

An economic model is developed for a variable sampling rate (VSR) Shewhart X-bar chart in which sample size and sampling interval for the next sample depend on the current sample mean. The model expresses long-run cost per hour of the VSR chart as a fu..

Variable sampling intervals for multiparameter shewhart charts

This paper considers the problem of using control charts to simultaneously monitor more than one parameter with emphasis on simultaneously monitoring the mean and variance. Fixed sampling interval control charts are modified to use variable sampling intervals depending on what is being observed from the data. Two basic strategies are investigated. One strategy uses separate control charts for each parameter, A second strategy uses a proposed single combined statistic which is sensitive to shifts in both the mean and variance. Each procedure is compared to corresponding fixed interval procedures. It is seen that for both strategies the variable sampling interval approach is substantially more efficient than fixed interval procedures.