T2 Control Charts with Variable Dimension
Francisco Aparisi1, Eugenio Epprecht2
1Estadistica e I. O. Aplicadas y Calidad, Universidad Politecnica de Valencia, Valencia, Spain; 2Departamento de Engenharia Industrial, Pontificia Universidade Catolica de Rio de Janeiro, Rio de Janeiro, Brazil

When the quality of a process is measured by a p-dimensional vector of correlated variables, one of the most employed options for statistically controlling this vector is the T2 chart. We consider the situation where this set of variables can be partitioned in two subsets, of dimensions p1 and p2, p1+p2=p, such that the variables in the first subset are easy and cheap to measure, but measuring the remaining variables (in the second subset) presents a high cost; for instance, this may require a destructive or otherwise expensive test. An idea that naturally comes to the mind in these cases is to regularly monitor the “cheap” subset of variables and to measure the additional, expensive ones only when there is a suspicion that the process may be out-of-control.

According to this idea, we propose T2 control chart schemes where the dimension (number of variables that are measured) is variable. The first p1 variables are measured and the respective T2 statistic is computed. If this statistic exceeds a pre-established threshold (warning limit), then the additional variables are measured and the T2 statistic is computed for the full-dimension vector. A fundamental underlying concept is that the sensitivity of the control scheme to assignable causes is increased with the full-dimension T2 statistic (if this were not the case, it would make no sense to measure the additional p2 variables).

Two new charts are proposed: a Variable-Dimension T2 control chart (VDT2) and a Double-Dimension T2 control chart (DDT2). With the VDT2 control chart, when the T2 statistic of the first p1 variables exceeds the warning limit, the next sample is taken with full dimension (p1+p2 variables); with the DDT2 control chart, when the T2 statistic of the first p1 variables exceeds the warning limit, the additional p2 variables are immediately measured; these schemes can be regarded, respectively, as analogous to variable-sample-size and to double-sampling schemes, except for the fact that what is allowed to vary is the dimension of the vector and not the sample size.

We have developed the expressions for computing the in-control and out-of-control ARL of both charts, VDT2 and DDT2. In addition, genetic algorithms were employed to optimize the design of the charts.

References:

Aparisi, F. (1996). Hotelling's T2 control chart with adaptive sample sizes. International Journal of Production Research, 34, 2853-2862.

Champ, C. W. and Aparisi, F. (2008). Double Sampling Hotelling's T2 Charts. Quality and Reliability Engineering International, 24, 153-166.

Keywords: SPC; variable dimension; genetic algorithms; double sampling

Biography: Dr Aparisi received his PhD on Industrial Statistics by Universidad Politecnica de Valencia (Spain) in 1995. His main rearch area is SPC (Statistical Process Control).