Moving range charts for monitoring the Weibull shape parameter with single observation samples

This paper proposes an approach to monitor shifts in the Weibull shape parameter bfβ via control charts based on the moving range of single-point samples from a smallest extreme value distribution. The average run length (ARL) of the proposed charts are computed using Fredholm integral equations of the second kind. The derived control limits for one-sided and two-sided control charts are unbiased in the sense that the ARL when β has shifted is shorter than the desired stable-process ARL. These control limits depend only on the desired stable-process ARL and the stable value of β. The paper also discusses the sample size requirements for Phase I so that the run length distributions are similar under standards-given scenario (β is given) and retrospective scenario (β is estimated from past data). The proposed methods are then applied to data on the breaking strengths of carbon fibers. The results suggest that one-sided control charts can detect small shifts in β sooner than two-sided charts.