Multivariate EWMA Control Chart and Application to a Semiconductor Manufacturing Process

Abstract

<p>The multivariate cumulative sum (MCUSUM) and the multivariate exponentially weighted moving average (MEWMA) control charts are the two leading methods to monitor a multivariate process. This thesis focuses on the MEWMA control chart. Specifically, using the Markov chain method, we study in detail several aspects of the run length distribution both for the on- and off- target cases. Regarding the on-target run length analysis, we express the probability mass function of the run length distribution, the average run length (ARL), the variance of run length (V RL) and higher moments of the run length distribution in mathematically closed forms. In previous studies, with respect to the off-target performance for the MEWMA control chart, the process mean shift was usually assumed to take place at the beginning of the process. We extend the classical off-target case and introduce a generalization of the probability mass function of the run length distribution, the ARL and the V RL. What Prabhu and Runger (1996) proposed can be derived from our new model. By evaluating the off-target ARL values for the MEWMA control chart, we determine the optimal smoothing parameters by using the partition method that provides an easy algorithm to find the optimal smoothing parameters and study how they respond as the process mean shift time changes. We compare the ARL performance of the MEWMA control chart with that of the multivariate Shewhart control chart to see whether the MEWMA chart is still effective in detecting a small mean shift as the process mean shift time changes. In order to apply the model to semiconductor manufacturing processes, we use a bivariate normal distribution to generate sample data and compare the MEWMA control chart with the multivariate Shewhart control chart to evaluate how the MEWMA control chart behaves when a delayed mean shift happens. We also apply the variation transmission model introduced by Lawless et al. (1999) to the semiconductor manufacturing process and show an extension of the model to make our application to semiconductor manufacturing processes more realistic. All the programming and calculations were done in R</p>