Monitoring and diagnosing the mean vector and covariance matrix of multistage processes

In a manufacturing environment, quality improves reliability and increases production. Fewer defects translate to fewer warranty claims and increased customer satisfaction. Process improvement also eliminates waste, improves flow and enhances workplace safety, all contributing to the bottom line. As customer retention and loyalty becomes more and more critical to the conduct of business in today’s competitive marketplace, service industries face a quality challenge as well: Meeting customer needs and maintaining high-quality personal interaction between service employees and customers. In practice, Statistical Process Control (SPC) techniques have been widely utilized in a variety of industries for the purpose of quality improvement by identifying root cause and reducing variability. Since Dr. W. A. Shewhart devised the first control chart in 1920’s, a large variety of control charts have been proposed and discussed. However, most existing statistical process control research focuses on monitoring the mean or variance of a single-stage process or the quality dimensions of the finished products. Limited works have been done on multistage processes monitoring and diagnosis. In this thesis, an engineering model that is capable of incorporating engineering background and knowledge is adopted for describing multistage processes. The problem of monitoring and diagnosing the mean vector and covariance matrix of multistage processes is formulated as a multiple hypotheses testing problem. However, as the number of stages increases, the detection power of conventional multiple hypotheses testing methods that aim at controlling type I error rate reduces dramatically. To maintain the detection power, we utilize a false discovery rate (FDR) control approach that has impacted the microarray research area in the last decade. Two FDR-adjusted multistage monitoring and fault identification schemes, FDR-adjusted Shewhart chart and FDR-adjusted CUSUM chart, are established. To apply the FDR approach, the distribution of the CUSUM statistics, is obtained based on Markov chain theory and Brownian motion with drift models. Three methods for approximating the limiting distribution of CUSUM statistics are provided and compared. Average power is used to evaluate the fault detection and identification capability of the new methods. Monte Carlo simulation results show that the novel FDR-adjusted approaches are more capable at identifying faulty stages than conventional approaches, especially when there are more than one out-of-control stages. In addition to the control charts for monitoring step changes in process mean vectors, detecting the dynamic zero-mean disturbances that affects the process variability is of equal importance. In the presence of changes in the covariance matrix of a multistage process, it is highly desirable to develop a control chart that is specifically tailored for variability change detection by taking the known process model into consideration. However, most of the existing methods either are restricted to the analysis of static data set or use the generalized variance which is calculated from the sample covariance matrix. Online monitoring and diagnosis schemes are still unavailable. In this thesis, a control chart is developed based on multivariate statistical theory. The sample covariance matrix that contains the information of possible faults is transformed into a vector by stacking its columns. Based on the unique cascading property of multistage processes, the possible change direction of the vectorized covariance matrix can be obtained as long as the model of the multistage process is known. With the vectorized sample covariance matrix and the possible change directions, we formulate the covariance matrix monitoring problem into a hypothesis testing problem too. Hotelling T2 control statistics are built on the vector and an Exponentially Weighted Moving Average (EWMA) type control chart is established for covariance matrix monitoring and diagnosis. The simulation results show that the new covariance matrix monitoring scheme has a clear advantage over traditional methods. The new control chart responds to the change in covariance matrix in a much quicker fashion and has a higher diagnosis capability. In addition to the EWMA type control chart, CUSUM type control schemes could also be readily established