AbstractsMathematics

Due to the rapid development of technology, quality control charts have attracted more attention from manufacturing industries in order to monitor quality characteristics of interest more effectively. Among many control charts, my research work has focused on the multivariate exponentially weighted moving average (MEWMA) and the univariate exponentially weighted moving average (EWMA) control charts by using the Markov chain method. The performance of the chart is measured by the optimal average run length (ARL). My Ph.D. thesis is composed of the following three contributions. My first research work is about differential smoothing. The MEWMA control chart proposed by Lowry et al. (1992) has become one of the most widely used charts to monitor multivariate processes. Its simplicity, combined with its high sensitivity to small and moderate process mean jumps, is at the core of its appeal. Lowry et al. (1992) advocated equal smoothing of each quality variable unless there is an a priori reason to weigh quality characteristics differently. However, one may have situations where differential smoothing may be justified. For instance: (a) departures in process mean may be different across quality variables, (b) some variables may evolve over time at a much different pace than other variables, and (c) the level of correlation between variables could vary substantially. For these reasons, I focus on and assess the performance of the differentially smoothed MEWMA chart. The case of two quality variables (BEWMA) is discussed in detail. A bivariate Markov chain method that uses conditional distributions is developed for average run length (ARL) calculations. The proposed chart is shown to perform at least as well as Lowry et al. (1992)'s chart, and noticeably better in most other mean jump directions. Comparisons with the recently introduced double-smoothed BEWMA chart and the univariate charts for the independent case show that the proposed differentially smoothed BEWMA chart has superior performance. My second research work is about monitoring skewed multivariate processes. Recently, Xie et al. (2011) studied monitoring bivariate exponential quality measurements using the standard MEWMA chart originally developed to monitor multivariate normal quality data. The focus of my work is on situations where, marginally, the quality measurements may follow not only exponential distributions but also other skewed distributions such as Gamma or Weibull, in any combination. The joint distribution is specified using the Gumbel copula function thus allowing for varying degrees of correlation among the quality measurements. In addition to the standard MEWMA chart, charts based on the largest or smallest of the measurements and on the joint cumulative distribution function or the joint survivor function, are studied in detail. The focus is on the case of two quality measurements, i.e., on skewed bivariate processes. The proposed charts avoid an undesirable feature encountered by…