International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 1, Pages 1-15
doi:10.1155/S0161171202202239
Optimal Cusum schemes for monitoring variability
1Department of Statistics and Operation Research, Royal Melbourne Institute of Technology, GPO Box 2476 V, Melbourne Victoria 3001, Australia
2Computer Science Department, Winona State University, Winona 55987, MN, USA
Received 5 February 2002
Copyright © 2002 S. Poetrodjojo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Cumulative Sum (Cusum) Control Schemes are widely used in
industry for process and measurement control. Most Cusum
applications have been in monitoring shifts in the mean level of a
process rather than process variability. In this paper, we study
the use of Markov chain approach in calculating the
average run length (ARL) of a Cusum scheme when controlling
variability. Control statistics S and S2, where S is the
standard deviation of a normal process are used. The optimal
Cusum schemes to detect small and large increases in the
variability of a normal process are designed. The control
statistic S2 is then used to show that the Cusum scheme is
superior to the exponentially weighted moving average (EWMA) in
terms of its ability to quickly detect any large or small
increases in the variability of a normal process. It is also
shown that Cusum with control statistics sample variance (S2)
and sample standard deviation (S) perform uniformly better
than those with control statistic logS2. Fast initial response (FIR) Cusum properties are also presented.