Traditional control charts for process monitoring are based on taking samples of fixed size from the process using a fixed sampling interval. Variable sample size (VSS) and variable sampling interval (VSI) control charts vary the sampling rate from the process as a function of the data from the process. By sampling at a higher rate when there is an indication of a change in the process, VSS and VSI control charts can detect process changes faster than traditional control charts. Previous research has considered the properties of CUSUM charts which use the VSI feature. This paper considers CUSUM charts with the VSS feature and with both the VSS and VSI features. Two ways of developing the control statistic of these charts are considered. It is shown that using either the VSS or VSI feature in a CUSUM control chart will improve the ability to detect all but very large process shifts. The VSI feature usually gives more improvement in detection ability than the VSS feature, but using both features together will give more improvement than either one separately. Guidelines are given for choosing the possible sample sizes and the possible sampling intervals for these charts. Methods for setting up these charts for practical applications are also given.

Keywords: Cumulative Sum Control Charts, Variable Sample Sizes, Variable Sampling.

*by* **JESSE C. ARNOLD and MARION R. REYNOLDS, JR.,
Virginia Polytechnic Institute and State University, Blacksburg,
VA 24061 **

*INTRODUCTION*

Control charts are widely used for monitoring quality characteristics of production processes. The traditional practice in using these control charts to monitor a process is to use a fixed sampling rate (FSR) which takes samples of fixed sample size with a fixed sampling interval. Control chart statistics are then plotted, and the quality characteristic is regarded as in control as long as the statistic does not fall into the signal region of the chart.

A recent development in control charts is the variable sampling interval (VSI) control chart which allows the time intervals between samples to vary depending on the value of the control statistic. Examples of studies of VSI Shewhart charts include Reynolds, Amin, Arnold and Nachlas (1988), Reynolds and Arnold (1989), Chengalur-Smith, Arnold, and Reynolds (1989), Runger and Pignatiello (1991), Reynolds, Arnold, and Baik (1996), and Reynolds (1996a). These studies mostly consider processes with independent normal observations and demonstrate that the best procedure generally uses only two possible sampling intervals. Similar investigations of VSI CUSUM charts are presented in Reynolds, Amin, and Arnold (1990) and Reynolds (1989, 1995, 1996b). VSI EWMA charts are considered by Saccucci, Amin, and Lucas (1992), Arnold, Reynolds, and Sawalapurkar-Powers (1993), and Reynolds (1995, 1996a, 1996b).

The investigations of VSI control charts demonstrate that
using VSI charts instead of FSR charts can provide a considerable
reduction in detection time and yet not increase the false
alarm rate or the average in-control sampling rate. Alternately,
VSI control charts can be used to reduce sampling costs while
maintaining the same detection time as FSR charts. Baxley
(1995) discusses an application of VSI charts at Monsanto
in which the objective was to reduce sampling costs while
maintaining an acceptable detection time.Another recent development
in control charts is the variable sample size (VSS) control
chart which allows the sample size used at each sampling point
to vary depending on the previous value of the control statistic.
A preliminary report on VSS control charts is given by Sawalapurkar-Powers,
Arnold and Reynolds (1990). VSS Shewhart _{}
charts are studied by Prabhu, Runger, and Keats (1993), Park
and Choi (1993), Costa (1994), Park and Reynolds (1994), and
Zimmer, Montgomery, and Runger (1998). Annadi, Keats, Runger,
and Montgomery (1998) consider VSS CUSUM charts. Reynolds
(1996b) compares the performance of VSS CUSUM and EWMA charts
to the performance of VSI CUSUM and EWMA charts.

The VSS and VSI features can be combined to give a VSSVSI
control chart which allows both the sample size and the sampling
interval to vary. VSSVSI Shewhart _{}
charts are considered by Prabhu, Montgomery, and Runger (1994),
Costa (1997), and Park and Reynolds (1999). Rendtel (1990)
considered VSSVSI CUSUM charts for monitoring the process
proportion defective. For the case of the mean of normal observations,
Arnold and Reynolds (1994) consider VSSVSI CUSUM charts, and
Arnold, Reynolds, and Sawalapurkar (1993) consider VSSVSI
EWMA charts.

The VSS control charts that have been developed determine
the sample size at the current sample using the data from
past samples. This means that the sample size at the current
sample is known before sampling starts for the sample. For
applications in which the results for individual observations
can be determined quickly, it may be feasible to allow the
sample size at the current sample to depend on the data at
the current sample. For this situation, Stoumbos and Reynolds
(1996, 1997) and Reynolds and Stoumbos (1998) have developed
control charts based on applying a sequential probability
ratio test (SPRT) at each sampling point. In the context of
evaluating this SPRT chart, they give numerical results for
VSS, VSI, and VSSVSI _{}
and CUSUM charts, and show that the SPRT chart gives faster
detection of process shifts than these other charts. Tagaras
(1998) is a general review paper covering VSI, VSS, and VSSVSI
control charts.

The objective of this paper is to give a detailed investigation of VSSVSI CUSUM control charts for the mean of normal observations for the usual situation, in which it is desirable to know the sample size for the current sample before sampling is started. Two ways to define the CUSUM control statistic are considered and compared. The effect of the choice of the sample sizes and sampling intervals is investigated, and recommendations are given for the practical application of VSSVSI charts.

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