A first look at the performances of a Bayesian chart to monitor the ratio of two Weibull percentiles

TL;DR

This paper evaluates a Bayesian control chart designed to monitor the ratio of Weibull percentiles between two processes, assessing its responsiveness, robustness, and diagnostic ability through simulation studies.

Contribution

It introduces a Bayesian chart for Weibull percentile ratios and analyzes its performance under various data and prior information conditions.

Findings

The chart's responsiveness improves with more training data.

It remains robust despite poor prior information.

The chart effectively detects shifts in process stability.

Abstract

The aim of the present work is to investigate the performances of a specific Bayesian control chart used to compare two processes. The chart monitors the ratio of the percentiles of a key characteristic associated with the processes. The variability of such a characteristic is modeled via the Weibull distribution and a practical Bayesian approach to deal with Weibull data is adopted. The percentiles of the two monitored processes are assumed to be independent random variables. The Weibull distributions of the key characteristic of both processes are assumed to have the same and stable shape parameter. This is usually experienced in practice because the Weibull shape parameter is related to the main involved factor of variability. However, if a change of the shape parameters of the processes is suspected, the involved distributions can be used to monitor their stability. We first tested the effects of the number of the training data on the responsiveness of the chart. Then we tested the robustness of the chart in spite of very poor prior information. To this end, the prior values were changed to reflect a 50% shift in both directions from the original values of the shape parameter and the percentiles of the two monitored processes. Finally, various combinations of shifts were considered for the sampling distributions after the Phase I, with the purpose of estimating the diagnostic ability of the charts to signal an out-of-control state. The traditional approach based on the Average Run Length, empirically computed via a Monte Carlo simulation, was adopted.