Sample size and the Binomial CUSUM Control Chart: the case of 100% inspection

Abstract

The Binomial CUSUM is used to monitor the fraction defective p of a repetitive process, particularly for detecting small to moderate shifts. The number of defectives from each sample is used to update the monitoring CUSUM. When 100% inspection is in progress, the question arises as to how many sequential observations should be grouped together in forming successive samples. The tabular form of the CUSUM has three parameters: the sample size n, the reference value k, and the decision interval h, and these parameters are usually chosen using statistical or economic-statistical criteria, which are based on Average Run Length (ARL). Unlike earlier studies, this investigation uses steady-state ARL rather than zero-state ARL, and the occurrence of the shift can be anywhere within a sample. The principal ®nding is that there is a signi®cant gain in the performance of the CUSUM when the sample size n is set at one, and this CUSUM might be termed the Bernoulli CUSUM. The advantage of using n 1 is greater for larger shifts and for smaller values of in-control ARL.

Monitoring change in spatial patterns of disease: comparing univariate and multivariate cumulative sum approaches

“…Another alternative is to transform the binomial or Poisson variable to normality [17]. Finally, it is possible to use a binomial or Poisson cusum chart [18,19]. Binomial and Poisson cusum charts are designed for the case where the underlying probability (p), or mean intensity ( ) is constant (and the purpose is to detect a sudden shift to values greater than p or ).…”

Detecting improvement using Shewhart attribute control charts when the lower control limit is zero

“…We consider binomial and poisson models. For the binomial distribution the CUSUM reference values are (Gan, 1993;Bourke, 2001):…”

“…Two reasons that this is reasonable are that: (i) in most of the situations that we have encountered the time over which samples are taken is a small fraction of the production process so there is only a small probability that a level shift will occur during a sample; and (ii) even if a shift occurs within a sample it can only have a small effect on the ARL because the only change in distribution is caused by the change in probability of a nonzero observation in the part of the sample taken before the shift. Situations where a shift occurring within a sample can have a larger effect have been considered by Reynolds and Stoumbos (2000) and by Bourke (2001).…”

“…We also include a Bernoulli CUSUM (with ARL values provided by Reynolds (2001)). Bourke (2001) proved that a Bernoulli CUSUM is optimal for continuous production when shifts can occur at any time. However, a Bernoulli CUSUM will not perform as well as a binomial CUSUM when shifts occur between samples.…”

Modified Shewhart Charts for High Yield Processes

“…Research on CUSUM charts for monitoring fraction non-conforming can be found in Reynolds & Stoumbos (1999, 2000, Gan (1993) and Bourke (1991Bourke ( , 2001. Although these charts were not specifically developed for high yield process monitoring, they can be modified for this purpose.…”