Monitoring the Weibull shape parameter by control charts for the sample range

Abstract

In this paper, we propose control charts for monitoring changes in the Weibull shape parameter b. These charts are based on the range of a random sample from the smallest extreme value distribution. The control chart limits depend only on the sample size, the desired stable average run length (ARL), and the stable value of b. We derive control limits for both one-and two-sided control charts. They are unbiased with respect to the ARL. We discuss sample size requirements if the stable value of b is estimated from past data. The proposed method is applied to data on the breaking strengths of carbon fibers. We recommend one-sided charts for detecting specific changes in b because they are expected to signal out-of-control sooner than the two-sided charts.

Monitoring the Weibull Shape Parameter with Type II Censored Data

“…For scheme 2, we refer to Equation (11) and Tables XVI (p The scheme 2 high-side and two-sidedσ control charts are given in Figures 8 and 9, respectively. The OOC patterns here are exactly the same as in scheme 1 except for period 19 in the retrospective high-side chart.…”

“…To derive warning and control lines, we use Equation (14) for high side and (15) for low side with Equations (10) and (11) . Unlike the two-sided chart, in general, the one-sided chart is unbiased for any p WU or p WL chosen inside 0; 1…”

“…That minimax ARL value in our results is equal to L 0 . The warning lines and control limits are given by Equations (10) and (11).…”

“…We consider two control chart schemes, namely, charts with control limits only and those with added warning lines. We will show that our methods perform better, in terms of ARL, than the range-based method in 11 which uses only the maximum and minimum sample values. We also demonstrate that the proposed methods are at least as good and have simpler and more direct interpretations than the method in.…”

Moving range EWMA control charts for monitoring the Weibull shape parameter

“…These charts are based on the sample range of the logarithms of the Weibull data and an unbiased estimator of 1/β. Pascual and Zhang [13] considered control schemes, which are based on the range of a random sample from a smallest extreme value distribution derived from Weibull processes, for monitoring the shape parameter β in uncensored Weibull samples. Finally, Pascual and Li [14] used the sample ranges of smallest extreme value distributions to construct control charts for monitoring the Weibull shape parameter under Type II (failure) censoring.…”