Process Monitoring with Multivariate Data:Varying Sample Sizes and Linear Profiles
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Multivariate control charts are used to monitor a process when more than one quality variable associated with the process is being observed. The multivariate exponentially weighted moving average (MEWMA) control chart is one of the most commonly recommended tools for multivariate process monitoring. The standard practice, when using the MEWMA control chart, is to take samples of fixed size at regular sampling intervals for each variable. In the first part of this dissertation, MEWMA control charts based on sequential sampling schemes with two possible stages are investigated. When sequential sampling with two possible stages is used, observations at a sampling point are taken in two groups, and the number of groups actually taken is a random variable that depends on the data. The basic idea is that sampling starts with a small initial group of observations, and no additional sampling is done at this point if there is no indication of a problem with the process. But if there is some indication of a problem with the process then an additional group of observations is taken at this sampling point. The performance of the sequential sampling (SS) MEWMA control chart is compared to the performance of standard control charts. It is shown that that the SS MEWMA chart is substantially more efficient in detecting changes in the process mean vector than standard control charts that do not use sequential sampling. Also the situation is considered where different variables may have different measurement costs. MEWMA control charts with unequal sample sizes based on differing measurement costs are investigated in order to improve the performance of process monitoring. Sequential sampling plans are applied to MEWMA control charts with unequal sample sizes and compared to the standard MEWMA control charts with a fixed sample size. The steady-state average time to signal (SSATS) is computed using simulation and compared for some selected sets of sample sizes. When different variables have significantly different measurement costs, using unequal sample sizes can be more cost effective than using the same fixed sample size for each variable. In the second part of this dissertation, control chart methods are proposed for process monitoring when the quality of a process or product is characterized by a linear function. In the historical analysis of Phase I data, methods including the use of a bivariate $T^2$ chart to check for stability of the regression coefficients in conjunction with a univariate Shewhart chart to check for stability of the variation about the regression line are recommended. The use of three univariate control charts in Phase II is recommended. These three charts are used to monitor the $Y$-intercept, the slope, and the variance of the deviations about the regression line, respectively. A simulation study shows that this type of Phase II method can detect sustained shifts in the parameters better than competing methods in terms of average run length (ARL) performance. The monitoring of linear profiles is also related to the control charting of regression-adjusted variables and other methods.
- Doctoral Dissertations