1Multivariate Process Variability Monitoring through Projection Based on a Process Model SHUGUANG HAO and SHIYU ZHOU University of Wisconsin, Madison, WI 53706 YU DING Texas A&M University, College Station, TX 77843 Inspired by the recently developed projection chart such as the U2 chart for monitoring multivariate mean shift, this article proposes a multivariate projection chart for monitoring process variability. In the engineering practice, people can often build a linear process model to connect the multivariate quality measurements with a set of model fixed assignable causes. The column space of the process model naturally provides a subspace for projection and subsequent monitoring, and was indeed used as the projection subspace in the recently developed projection control charts for monitoring mean shift. For the purpose of monitoring variability, however, we will show that such a projection may not be advantageous. We propose an alternative projecting statistic, labeled as VS, to be used for constructing a multivariate variability monitoring chart. We show through extensive numerical studies that the VS chart entertains several advantages over other competing methods, such as its less restrictive requirements on the process model and generally improved detection performance. Key Words: Control Chart; Generalized Variance; Multivariate Process Variability; Statistical Process Control. PROCESS variability monitoring has been an important field in the statistical quality control (Woodall and Montgomery (1999)). Several control charts have been developed specifically to detect the process variability change. For example, R chart and S chart are used to detect the variance change Mr. Shuguang Hao is a graduate student in the Department of Industrial and Systems Engineering. His email address is [email protected]. Dr. Shiyu Zhou is an Assistant Professor in the Department of Industrial and Systems Engineering. His email address is [email protected]. Dr. Yu Ding is an Associate Professor in the Department of Industrial and Systems Engineering. His email address is [email protected]. in univariate measurements. For multivariate quality characteristic measurements, the existing charts for monitoring variability are mainly based on the statistic of the generalized variance, calculated from the sample covariance matrix S , such as ||S (Alt (1985)), log | |S (Montgomery and Wadsworth (1972)), and ||S (Alt and Smith (1988)). Recently, Reynolds and Cho (2006) proposed a multivariate exponential weighted moving average (MEWMA) chart for monitoring process variability. Besides these variability monitoring charts, some other multivariate charts such as Hotelling’s 2T chart, although designed for detecting mean shifts, could also signal variability changes because 2T chart has the sample2 covariance matrix S in its statistic. For these control charts, measurements of product quality characteristics are taken from the finished or intermediate product, and they are treated as random variables and their distributions are compared with the corresponding distributions under normal conditions. If the measurements show that there are some quality characteristics “out of control” (e.g., deviation from the target or variability is too large), an alarm is generated to show that some faults happen in the process. Clearly, these charts are easy to use, but do not take extra process information such as the relationship between process faults and product quality characteristics into consideration. There is some recent advancement in SPC to improve the monitoring performance through a subspace projection method. Particularly, Runger (1996) proposed a projection chart, called 2U chart, which appears to perform much better than a regular 2T chart. This chart projects the quality measurement y into a predefined lower dimensional space and the projected vector is then monitored. The project directions can be identified by selecting a subset of quality variables which only certain assignable causes affect or by adopting a process model which link the model-fixed assignable causes with the quality variables. Quite a few such process models have been developed recently, for instance, the process-oriented basis representation model (Barton and Gonzalez-Barreto (1996)), the physical models for assembly processes (Mantripragada and Whitney (1999), Jin and Shi (1999), Ding et al. (2000)), and the state space models for machining processes (Zhou et al. (2003b), Djurdjanovic and Ni (2001), and Huang et al. (2000)). These models are in a common linear mixed model form as =+yAfε (1) where y is a vector consisting of product quality measurements, A is a coefficient matrix determined by process/product design, f is a vector representing the model-fixed process variation sources, and ε includes the measurement noise. In the implementation of the 2U chart, the measurement y is projected onto the column space of matrix A and the projected results are monitored. It has been shown that 2U chart is quite sensitive to detect the mean changes in f. In a more recent paper, Runger et al. (2007) showed that the 2U chart is actually equivalent to a 2T chart on the weighted least squares estimation of f based on the model (1). Zhou et al. (2005) also proposed a directionally variant chart to take the advantages of the known shift directions when assignable causes occur in the system. It is not surprising that these projection charts outperform the generic charts in detecting the changes in the root causes because extra information, the process model, is considered in the chart design. The above mentioned 2U chart and the directionally variant chart are designed for mean shift detection. Although the 2U chart will also signal variance changes, further investigation is needed to differentiate whether the root cause is a mean shift or a variance change. However, in many practical situations, people are very interested in the variability change for variation reduction purposes. Thus, it is highly desirable to develop a technique which is specifically tailored for variability change detection. In other words, we want a control chart which is sensitive only to the variance change of f , and it should outperform generic variability chart such as the ||S chart by taking the known process model into consideration. To achieve this goal, we can follow exactly