A modified individual measurements q-chart for monitoring the average of multiple stream processes

Abstract

An increasing number of manufacturing processes involve multiple streams where the same type of item is produced in a parallel fashion. Traditionally, streams need to be monitored using separate control charts. The number of charts becomes unrealistic as the number of streams increases. Group control charts (GCC) were developed by Boyd (1951) to address this issue. However, the streams were assumed to be independent, and the in-control average run length $(ARL_0)$ is reduced with the number of streams. This effect translates to a large number of false alarms causing unacceptable interruptions to the production process. This research proposed modified control limits for the Q-charts, originally developed by Quesenberry (1991b). These are charts for individual measurements suited for monitoring the average of a single stream. The modified charts, however, can be used to monitor the average of multiple streams and account for the level of correlation between the streams. The results of simulation studies were used to develop a mathematical model representing the relationship between the in-control average run length $(ARL_0)$, the number of streams (m), the level of correlation between them $(\rho)$, and the half-width of the control limits (L). The model was confirmed and used to generate tables of recommended values of the half-width (L) to be used in constructing the modified Q-charts to achieve a widely accepted level of $ARL_0$ equal to 370.4. Another set of simulated studies was performed to evaluate the shift detection capability of the modified Q-charts. A second model representing the relationship between the out-of-control average run length $(ARL_1)$, the shift magnitude $(\delta)$, the number of streams (m), and the level of correlation $(\rho)$ was developed and confirmed. Statistical analysis of the $ARL_1$ of the modified Q-charts indicated an improved shift detection capability compared to the alternative charting schemes proposed by Epprecht et al. (2011) and Mortell & Runger (1995).