Synthetic Phase II Shewhart-type Attributes Control Charts When Process Parameters are Estimated

The performance of attributes control charts (such as c and np charts) is usually evaluated under the assumption of known process parameters (i.e., the nominal proportion of nonconforming units or the nominal number of nonconformities). However, in practice, these process parameters are rarely known and have to be estimated from an in-control phase I data set. In this paper, we derive the run length properties of the phase II synthetic c and np charts with estimated parameters, and we investigate the number m of phase I samples that would be necessary for these charts in order to obtain similar in-control average run lengths as in the known parameters case. We also propose new specific chart parameters that allow these charts to have approximately the same in-control average run lengths as the ones obtained in the known parameter case.