International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 1, Pages 1-15

Optimal Cusum schemes for monitoring variability

S. Poetrodjojo,1 M. A. Abdollahian,1 and Narayan C. Debnath2

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.


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.