Phase II control charts for monitoring the depth-ratio of ball-bearings involving three normal variables.

This paper investigates the problem of monitoring the ratio involving three variables, jointly distributed as trivariate normal. The Shewhart-type and two exponentially weighted moving average (EWMA) type schemes for monitoring depth ratio are proposed. The ratio of a normal variable to the average of two other normal variables has wide applications in natural science, production, and engineering. It is defined with slightly different terminology in various contexts, such as depth or aspect ratios. In modern bearing manufacturing, the aspect ratio of width to the average of inner and outer diameters can be an essential indicator of product quality and process stability. While there are many helpful existing charts for monitoring the three components separately or jointly when these characteristics follow a normal distribution, the ratio aspect is often ignored. The Shewhart-type schemes' exact and approximated control limits are considered and analyzed. Numerical results based on Monte-Carlo are conducted using the average run length as a metric with different values of in-control ratio and correlation between the three variables. An application based on the parts manufacturing data illustrates the implementation design of the two control charts. The real-life data analysis shows the efficacy of the proposed monitoring schemes in practice.