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

In this paper, we present a method to monitor count data so as to be able to detect improvement when the counts are low enough to cause the lower limit to be zero. The method, which is proposed as an add-on to the conventional Shewhart control chart, consists in counting the number of samples in which zero defectives or zero defects per unit occur and signaling an increase in quality if k-in-a-row or 2-in-t samples have zero counts of defectives or zero defects per unit. This method enjoys some similarities to the very popular Shewhart control chart in that it is easy to design, understand and use. It is flexible, robust, and, like the Shewhart chart, yields detection frequencies that are optimal for very large shifts and good for other shifts. Some comparisons with traditional CUSUM charts are provided. Figures enabling Shewhart control chart users to easily design low-side add-on control charts are given for c and np charts.

Acknowledgement

The authors wish to thank the referees for their many suggestions on how to improve an earlier draft of this paper.

Notes

**Indicates ARL values > 100 000.

*For the np chart, P(0) = (1 − p) ⁿ , and for the c chart, P(0) = e ^{− c} .

**The ARL for 2-in-a-row (equivalent to 2-in-2) reduces to ARL = [1 + P(0)]/P(0)².

***β = P(X ≥ LCL).

Notes: **Indicates ARL values > 100 000.

***ARL values reported for the Bernoulli CUSUM are calculated as (average number of observations to signal)/50 in order to provide a consistent comparison with the other methods given in the table.

^sValues estimated using simulation.

Indicates ARL values > 100 000.

**Indicates ARL values > 100 000.

**Indicates ARL values > 100 000.

**Indicates ARL values > 100 000.