Comparison of Individual and Moving Range Chart Combinations to Individual Charts in Terms of ARL after Designing for a Common “All OK” ARL
Journal of Modern Applied Statistical Methods
November
Comparison of Individual and Moving Range Chart Combinations to Individual Charts in Terms of ARL after Designing for a Common “All OK” ARL
Dewi Rahardja 0
0 U.S. Department of Defense , Indianapolis, IN , USA
Follow this and additional works at: http://digitalcommons.wayne.edu/jmasm Part of the Applied Statistics Commons, and the Industrial Engineering Commons Recommended Citation
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Article 19
Comparison of Individual and Moving Range
Chart Combinations to Ind ividual Charts in
Terms of ARL after Designing for a Common
“All OK” ARL
Dewi Rahardja
U.S. Department of Defense
Indianapolis, Indiana
In some process monitoring situations, consecutive measurements are spaced widely apart
in time, making monitoring process aim and spread difficult. This study uses three cases to
compare the effectiveness of two such monitoring schemes, i.e., the X chart alone (X-only
chart) and the Individuals and Moving Range Chart Combination (X/MR chars), in terms
of Average Run Length (ARL) after designing for a common “all OK” (in-control) ARL.
The study finds that X chart alone is sufficient (and hence, recommended) in detecting
changes in all the 3 cases: changes in the process mean, changes in the process standard
deviation, and changes in both process mean and standard deviation.
Introduction
In some process monitoring situations, consecutive measurements are spaced
widely apart in time. For example, an engineering process may allow only one
measurement per day. In some cases, a series of individual items are produced in
such a way that no natural subgrouping is possible
(Crowder, 1987a)
. When this
happens, exactly how to monitor process aim and spread is not completely obvious.
One sensible possibility is to simply plot individual observations on their own chart
(X-only chart). Another possibility is to plot a combination of a chart for individual
measurements and a moving range chart based on two consecutive observations.
Duncan (1974) outlines such a procedure.
The purpose of this study is to compare the effectiveness of these two
monitoring schemes, i.e., the X chart alone (X-only chart) and the Individuals and
Moving Range Chart Combination (X/MR chars), in terms of Average Run Length
(ARL) after designing for a common “all OK” (in-control) ARL.
The run length of any process monitoring procedure is the number of
sampling periods before an out-of-control signal is given. An out-of-control signal
suggests that some change in the process has occurred and that action should be
taken to find and correct any assignable causes. The average run length (ARL) is
often used to describe the likely performance of a control procedure. A large ARL
is desired when the process is stable or in control, and a small ARL otherwise
(Crowder, 1987a)
.
Comparison of monitoring schemes will be made under three sets of
circumstances. The first case is where the process mean changes from its standard
value, the second case is where the process variability changes, and third case is
where both process mean and process variability change from standard values. In
each of these three cases, a small ARL is desired, since it will indicate quick
detection of the out-of-control situation.
Literature Review
Vardeman and Jobe (1999)
discussed the charting of individuals and moving ranges
and some other process monitoring techniques that improve on Shewhart charts in
situations where it is important to quickly detect small process changes. That is,
they also considered EWMA and CUSUM process monitoring schemes. Four types
of process monitoring schemes were originally considered in the present study: The
X chart alone, Individuals and Moving Range Chart Combinations, EWMA and
CUSUM process monitoring schemes. However, because EWMA and CUSUM
schemes are known to be better than an X chart alone for detecting small process
changes, no further analysis is needed (for EWMA and CUSUM) if the X chart
alone is better than Individual and Moving Range Chart Combinations.
Crowder (1987a, 1987b) discussed the Computation of ARLs for Combined
Individual Measurement and Moving Range Charts. Numerical procedures and a
control chart design strategy are presented. ARLs are given for various choices of
the control limits and shifts in the level of the process mean and standard deviation.
Also, a Fortran computer program was presented that allows inputting control limits
for combined individual measurement and moving range charts and then returns the
approximate average run length (ARL) for the normal case with standard deviation
1 and various shifts in the process mean.
Roes, et al. (1993)
discussed several options in designing a Shewhart-type
control chart for Individual Observations. A number of possible estimators of the
standard deviation were considered and a two-stage procedure is suggested for
retrospective testing. It was argued that adding a Moving Range Chart has no real
added value and, t (...truncated)