The Shewhart attribute chart with alternated charting statistics to monitor bivariate and trivariate mean vectors

Abstract

In this article, we combined the Alternated Charting Statistic (ACS) scheme with the traditional attribute np chart to control mean vectors of bivariate and trivariate normal processes. With the bivariate ACS scheme in use (the trivariate scheme is similar), the two quality characteristics (X, Y) are controlled in an alternating fashion. If the current sample point is the number of disapproved items with respect to the X discriminating limits, then the next sample point will be the number of disapproved items with respect to the Y discriminating limits. The strategy of using the X discriminating limits to classify the items of one sample and the Y discriminating limits to classify the items of the next sample instead of using jointly the X and Y discriminating limits to classify the items of all samples might be compensated with the adoption of larger samples. In other words, the proposed bivariate (trivariate) ACS chart might work with samples as large as 2n (3n); n is the sample size of the competing Hotelling and Max D charts. The proposed chart resembles an np chart with alternated charting statistic; because of that, it is called the ACS mp chart. The ACS mp chart always outperforms the Max D chart and, in comparison with the standard T2 chart and with the combined Max D − T2 chart, it has a better overall performance. With the ACS scheme, the items are classified as approved or disapproved regarding only one of the two quality characteristic, X or Y; with the Max D chart the complexity increases, once the items are classified into four different categories: approved (disapproved) regarding both, the X and Y discriminating limits, or approved (disapproved) regarding the X discriminate limits and disapproved (approved) regarding the Y discriminate limits. The T2 chart always requires the measurement of the two quality characteristics. The additional advantage of inspecting only one quality characteristic of the sample items lies in the fact that the XY-correlation doesn't need to be estimated.