Effect of Measurement Errors on One‐Sided Synthetic‐RZ Control Charts for Monitoring the Ratio of Two Normal Variables

Abstract

Monitoring the ratio of two normally distributed quality characteristics is important in a range of industrial applications. This study quantifies the effect of measurement error on two one‐sided Synthetic‐RZ charts for detecting downward and upward shifts in that ratio. The proposed framework combines a correlated linear covariate error model with the conforming‐run‐length mechanism of the Synthetic chart and evaluates both zero‐state and cyclical steady‐state average run length (). We derive the state‐dependent parameters of the observed ratio without imposing the generally invalid identity , where denotes the shift in the true ratio. A finite‐state Markov chain is then used to compute the run‐length distribution, and a fully specified discrete–continuous optimization procedure selects the conforming‐run‐length limit and the corresponding control limit. The numerical study shows that measurement error generally weakens detection and that increasing the number of repeated readings per inspected unit provides little improvement relative to reducing the underlying measurement‐error components.