CUSUM Charts With Variable Sampling Intervals

Abstract

A standard cumulative sum (CUSUM) chart for controlling the process mean takes samples from the process at fixed-length sampling intervals and uses a control statistic based on a cumulative sum of differences between the sample means and the target value. This article proposes a modification of the standard CUSUM scheme that varies the time intervals between samples depending on the value of the CUSUM control statistic. The variable sampling interval (VSI) CUSUM chart uses short sampling intervals if there is an indication that the process mean may have shifted and long sampling intervals if there is no indication of a change in the mean. If the CUSUM statistic actually enters the signal region, then the VSI CUSUM chart signals in the same manner as the standard CUSUM chart. A Markov-chain approach is used to evaluate properties such as the average time to signal and the average number of samples to signal. Results show that the proposed VSI CUSUM chart is considerably more efficient than the standard CUSUM chart. Tables are provided to aid in the design of VSI CUSUM charts.

Synthetic-np chart for attributes

“…If the process shift is detected at the end of the shifting interval (or by the first nonconforming sample after the p shift), ATS is equal to [h · (1/g 1 -0.5)]. Here, the random time of process shift is assumed to have a uniform distribution between two samples (Reynolds et al [18]). The probability G s of this event is equal to On the other hand, if the process shift has not been detected by the first nonconforming sample but by a following one, ATS equals [h · (1/g 1 -0.5) + ATS zero (p 1 )].…”

The optimal sample sizes of the Xbar and CUSUM charts

“…The out-of-control ATS will be calculated under the steadystate mode. It is assumed that the process has reached its steady state or stationary distribution at the random time when a mean shift occurs and the random time of mean shift has a uniform distribution within the sampling interval [6]. Then,…”

“…The sampling interval h is determined based on the sample size n and the fixed sampling cost B using Equation (6). Table 1 shows the performance comparison of the three charts.…”

New design algorithm of the Shewhart&CUSUM chart for monitoring process mean shifts

“…It assumes that the process has reached its stationary distribution at the time when the process shift occurs and the random time of process shift has a uniform distribution within the sampling interval [11]. Since production processes often operate in incontrol condition for most or relatively long periods of time [12], the steady-state mode is therefore more realistic than the zero-state mode.…”