The Variable Sampling Interval EWMA X¯ Chart with Estimated Process Parameters

The exponentially weighted moving average (EWMA) X, ¯, , , , , chart with the variable-sampling-interval (VSI) feature is usually scrutinized under the assumption of known process parameters. However, in practice, process parameters are usually unknown, and they need to be estimated from the in-control Phase-I data set. With this in mind, this article proposes the VSI EWMA X, ¯, , , , chart in which the process parameters are estimated. A Markov Chain approach is adopted to derive the run-length properties of the VSI EWMA X, ¯, , , , chart with estimated process parameters. The standard deviation of the average time to signal (SDATS) is employed to measure the practitioner-to-practitioner variation in the control chart’s performance. This variation occurs because different Phase-I datasets are used among practitioners to estimate the process parameters. Based on the SDATS criterion, this article provides recommendations regarding the minimum number of required Phase-I samples. For an optimum implementation, this article develops two optimization algorithms for the VSI EWMA X, ¯, , , , chart with estimated process parameters, i.e., by minimizing the (i) out-of-control expected value of the average time to signal (AATS) and (ii) out-of-control expected value of the AATS (EAATS) for the cases of deterministic and unknown shift sizes, respectively. With the implementation of these new design procedures, the VSI EWMA X, ¯, , , , chart with estimated process parameters is not only able to achieve a desirable in-control performance, but it is also able to quickly detect changes in the process.