Joint EWMA charts for multivariate process control: Markov chain and optimal design

Abstract

This paper deals with the optimal design of a set of univariate exponentially weighted moving average (EWMA) quality control charts to monitor several correlated variables. The aim is to find the values for the parameters of the set of charts that minimise the average run length (ARL) for a given process shift, according to three different problem formulations, in terms of the symmetry of the directions of shift that may be expected in the particular process. The first step to achieve this objective has been the development of a multidimensional Markov chain model to compute the ARLs of the set of EWMA control charts. A genetic algorithm has been employed for the optimisation. Finally, a (non-exhaustive) performance comparison is presented between the joint EWMA charts and the equivalent multivariate EWMA (MEWMA) control chart. No scheme uniformly outperforms the other one. In some cases, the univariate charts largely outperform the MEWMA chart for the shift they are optimised but perform much worse for other shifts. Therefore, the tools described in this paper help the user to make an informed decision considering the shifts that may be expected in each particular case.

Keywords:

• statistical process control
• MEWMA
• genetic algorithms
• ARL
• shift directions
• multiple control charts

Acknowledgements

This work has been supported by CNPq – Conselho Nacional de Desenvolvimento Científico e Tecnológico, Ministry of Science and Technology, Brazil, research project number 201811/2007-3 and later 302326/2008-1; by the Research Chairs in Design of Supply Chains in Emergent Markets: Program of Research at Tecnológico de Monterrey (ITESM – Campus Guadalajara), and also by the Ministry of Education and Science of Spain, research project number DPI2009-09925 including European FEDER funding.