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

Abstract

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.

Keywords:

• Charting schemes
• parts manufacturing
• ratio involving three variables
• phase-II process monitoring
• trivariate normal

Acknowledgments

The authors are grateful to the Editor-in-chief, Associate Editor and three reviewers for various constructive comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 https://products.emersonbearing.com/viewitems/deep-groove-radial-ball-bearings/6300-series-deep-groove-radial-ball-bearings

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [Grant Nos. 12171328,12201429]; the Beijing Natural Science Foundation [Grant No. Z210003]; the Liaoning BaiQianWan Talents Program; the Natural Science Foundation of Liaoning Province [Grant Nos. 2020-MS-139, 2023-MS-142]; the Scientific Research Fund of Liaoning Provincial Education Department of China [Grant No. LJC202006]; the Project of Science and Research of Hebei Educational Department of China [Grant No. ZD2022020]; the Doctoral Research Start-up Fund of Liaoning Province [Grant No. 2021-BS-142]; the Research on Humanities and Social Sciences of the Ministry of Education [Grant No. 22YJC910009]; and the Research of economic and social development in Liaoning Province [Grant No. 20231s1ybkt-103].