Determining Optimal Number of Samples for Constructing Multivariate Control Chart

Usually, there are two phases in constructing a multivariate control chart. Phase I is to estimate the in-control process parameters and to establish control limits using historical data. Phase II is when the control limits are used to monitor the process. This paper focuses on determining the optimal number of samples in Phase I since number of samples affect the cost of quality control in practice. The number of samples in Phase I directly affect the performance of the 2 T control chart in Phase II when using the sample mean vector and the successive difference matrix. An average run length (ARL) is used as a performance measure for evaluating the 2 T control charts in Phase II. Normally, a longer ARL is preferred for an in-control process and a smaller ARL is preferred for an out-of-control process. But, our simulation results indicate that, with a smaller number of samples, both the in-control and out-of-control ARLs are larger than those 2 T control charts with greater sample sizes. In this paper, a relative error of the ARL is defined as the difference between the ARL for the 2 T control chart using estimated process parameters and the ARL for the 2 T control chart using true parameters divided by the ARL for the 2 T control chart using true parameters. Hence, the optimal number of samples can be determined as long as the decreasing of relative error reaches a steady-state in its scree plot.