In this paper we investigate control charts for monitoring a process to detect changes in the mean and/or variance of a normal quality variable when an individual observation is taken at each sampling point. The traditional X chart and moving range (MR) chart are evaluated. Also evaluated are the exponentially weighted moving average (EWMA) chart of the observations and the EWMA chart of the squared deviations of the observations from the target. 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. The ability of charts to diagnose the type of parameter shift produced by a special cause is also investigated. It is shown that combinations involving the EWMA charts are just as effective at diagnosing the type of parameter shift as the traditional combination of the X and MR charts. The effect of adding the variable sampling interval (VSI) feature is also evaluated for some of the combinations of charts. The VSI feature allows the sampling interval to be varied as a function of the values of the statistics being plotted. It is shown that adding the VSI feature to the combinations of charts results in very substantial reductions in the expected time required to detect shifts in process parameters.

Keywords: Average Number of Observations to Signal, Average Time to Signal, Exponentially Weighted Moving Average Control Charts, Moving Range Control Charts, Shewhart Control Charts, Statistical Process Control, Steady State, Variable Sampling Interval Control Charts, X Control Charts.

*by* **Marion R. Reynolds, JR. Virginia Polytechnic
Institute and State University, Blacksburg, VA 24061-0439**
and **Zachary G. Stoumbos, Rutgers, The State University
of New Jersey, Newark, NJ 07102-1895 **

*INTRODUCTION*

A control chart is used to monitor a process for the purpose
of detecting special causes of process variation that may
result in lower-quality process output. The statistic plotted
on a control chart is usually based on samples (or subgroups)
of *n* observations that are taken at regular sampling
intervals. For example, a sample of *n* = 4 observations
might be taken every hour. There are, however, many applications
in which the control chart is based on individual observations
(*n* = 1) rather than samples of *n* > 1. In
these applications, sampling may be expensive, destructive,
and/or time consuming, and the sampling interval may be relatively
long. It is not unusual to find industrial examples in which
there is only one sample per shift. In some situations, for
example when an instrument is being monitored for accuracy
and precision, the observations obtained for monitoring may
be days or even weeks apart.

In most situations in which a continuous quality variable
is being measured, it is assumed that this variable, say *X*,
has a normal distribution. If *X* is normal and a special
cause affects the distribution of *X*, then the special
cause will change the mean , the standard
deviation , or both
and . When samples of *n* >
1 are taken, the usual practice is to use two control charts,
one for detecting changes in and the
other for detecting changes in .
The control chart for monitoring is
usually based on the sample means. For example, the Shewhart
chart is based on plotting the sample
means. The control chart for is
usually based on a measure of within-sample dispersion, such
as the sample variance or sample range. For example, the Shewhart
R chart is based on plotting the sample ranges.

When individual observations are taken, a chart for monitoring
would use these individual observations
in place of the sample means. For example, the Shewhart *X*
chart would be used in place of the Shewhart
chart. Monitoring is now more difficult,
however, because the usual within-sample measures of dispersion
no longer apply. Thus, there is the question of how to monitor
when *n* = 1. A traditional
choice is to base a chart on the moving range (MR), which
is the range of successive individual observations, but there
is now considerable evidence that there is little or no benefit
to using the Shewhart MR chart when a Shewhart *X* chart
for is also being used (see, e.g.,
Nelson (1982), Roes, Does, and Schurink (1993), Rigdon, Cruthis,
and Champ (1994), Albin, Kang, and Shea (1997), and Stoumbos
and Reynolds (2000)).

Shewhart charts are widely used for process monitoring, and they are effective for detecting large changes in process parameters; however, a Shewhart chart may take a very long time to detect a small persistent shift in a process parameter. The ability to detect smaller parameter shifts can be improved by using a chart based on a statistic that incorporates information from past samples in addition to current samples. For example, the exponentially weighted moving average (EWMA) chart is based on a weighted average of current and past sample statistics, and the cumulative sum (CUSUM) chart is based on a sum of current and past sample statistics. When individual observations are taken at widely spaced intervals, there is relatively little data from the process, so using a chart such as the EWMA chart or the CUSUM chart will provide efficient use of the scarce data.

A useful approach to improving the ability to detect process
changes is to use a variable sampling interval (VSI) control
chart instead of the traditional fixed sampling interval (FSI)
control chart. In a VSI control chart, the sampling interval
is varied as a function of the control statistic. A short
sampling interval is used whenever there is some indication
that a process parameter may have changed, and a long sampling
interval is used if there is no indication of a parameter
change. Most work on developing VSI control charts has been
done for the problem of monitoring
(see, e.g., Reynolds Amin, Arnold, and Nachlas (1988), Reynolds,
Amin, and Arnold (1990), and Reynolds (1996a, b), and Stoumbos
and Reynolds (1996, 1997, 2001)). Very little work has been
done on VSI charts for monitoring and
. Chengular, Arnold, and Reynolds
(1989) considered VSI Shewhart charts for monitoring and when
samples of *n* > 1 are taken. Shamma and Amin (1993)
considered the performance of a single VSI EWMA chart that
can be used to monitor and .

We have two main objectives in this paper. The first objective
is to consider various combinations of Shewhart and EWMA charts
in order to determine which combinations are most effective
for detecting shifts in and/or
when single observations are taken. We show that there are
much better alternatives than the traditional approach of
using the *X* and MR charts. The second objective is
to show how the use of the VSI feature can offer substantial
additional improvements in the ability to detect parameter
shifts.

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