A closer look at the equivalence of Bernoulli and geometric CUSUM control charts

• Published in: Quality and reliability engineering international 2022-11, Vol.38 (7), p.3470-3484
• Main Authors: Saccucci, Michael S., Lucas, James M., Bourke, Patrick D., Davis, Darwin J., Saniga, Erwin M.
• Format: Article
• Language: English
• Subjects: average number inspected (ANI); average run length (ARL); Bernoulli CUSUM; Chains; Control charts; CUSUM charts; Equivalence; Evaluation; fixed‐shift and random‐shift models; geometric CUSUM; Markov chains; Monitoring
• ISSN: 0748-8017; 1099-1638
• DOI: 10.1002/qre.3145

Some researchers have incorrectly concluded that the geometric CUSUM is superior to the Bernoulli CUSUM as a procedure for monitoring a repetitive process, even though the two procedures have been proved to be equivalent for detecting an upward shift in the proportion of nonconforming items. We use an exact Markov‐chain‐based methodology to re‐examine the relationship between geometric CUSUM and Bernoulli CUSUM control charts. Exact methods allow us to differentiate between similar but different‐valued quantities that have contributed to some misunderstandings in the literature. We show that for a random‐shift model, evaluations of steady‐state average number of inspected items until a signal (ANIs) are identical for both geometric CUSUMs and Bernoulli CUSUMs, provided the correct choices of return levels are made. We also show that a steady‐state geometric CUSUM based on a fixed‐shift model only uses the geometric CUSUM states, while a steady‐state geometric CUSUM based on a random‐shift model will reach the states of a Bernoulli CUSUM after a long series of zeros. We note that our conclusions are contrary to the published results of other researchers, and we examine these differences in detail. Layman's : Since, their introduction by Walter Shewhart in 1931, control charts such as the Shewhart p‐chart have had widespread application for monitoring the output quality from manufacturing processes, such as the proportion (p) in the stream of manufactured items that are nonconforming. But the Shewhart p‐chart is not very effective for monitoring processes when the proportion p is less than about 4 percent. Other charts, such as the geometric CUSUM (proposed in 1991) and the Bernoulli CUSUM (proposed in 1999) have been shown to be superior at identifying changes in p when p is small. Some researchers have relied on simulation‐based investigations to compare these two CUSUMs, and have incorrectly concluded that the geometric CUSUM is superior to the Bernoulli CUSUM, even though the two procedures have been proved to be equivalent for detecting an upward shift in the proportion p. Using exact, Markov‐chain‐based methodology, we re‐examine the relationship between the geometric CUSUM and the Bernoulli CUSUM control charts and demonstrate that these two charts are equivalent when evaluated correctly. Exact methods allow us to precisely compare these two monitoring schemes and correct some erroneous conclusions that have appeared in the literature.