The monitoring of mean vectors with VCS charts for multivariate processes

ABSTRACT

We propose to control the mean vector by taking larger samples but inspecting fewer quality characteristics if we work with samples of size n to control bivariate processes, then 2n observations are usually collected, half are observations of X₁ and the other half are observations of X₂; alternatively, we might work with samples of size 2n if only observations of X₁ (or X₂) are collected; in both cases the cardinality c of the sample data set is c = 2n. If only one of the two quality characteristics, X₁ or X₂, is measured at each sampling time, then only one of the two statistics (${\overline{X}}_1\text{ or }{\overline{X}}_2$) is computed. The variable charting statistic (VCS) chart works as follows: if a ${\overline{X}}_1$ point falls in the central region, then the statistic for the next sample changes to ${\overline{X}}_2$ and vice-versa; yet, if a ${\overline{X}}_1$ point reaches the warning region, then the statistic for the next sample remains ${\overline{X}}_1$.

KEYWORDS:

• Variable charting statistic
• alternate charting statistic
• X-bar control chart
• T² Hotelling
• multivariate processes

Acknowledgment

This version of our paper is substantially better than the first one, thanks to the reviewers. We are in debt with them. This research was supported by National Council for Scientific and Technological Development (CNPq), grants 304599/2015–8 and 402559/2016–9.

Disclosure statement

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

ORCID

Antonio Fernando Branco Costa http://orcid.org/0000-0001-6620-4573

Additional information

Funding

Conselho Nacional de Desenvolvimento Científico e Tecnológico grants 304599/2015–8 and 402559/2016–9.