On the design of Shewhart control charts monitoring dispersion in processes with random shifts

A key issue for the deployment of control charts in a manufacturing process is the selection of their design parameters: an economic objective can drive the choice of these parameters. Usually, the economic design of a control chart is obtained under the assumption that the shift to the out-of-control condition is deterministic. However, this can result in a too restrictive hypothesis which limits the economic models implementation and needs to be removed in many manufacturing environments. In this paper the optimal costs related to the implementation of two Shewhart control charts monitoring the sample standard deviation S in processes with normal random shifts have been compared. One of the charts considered in this paper monitors a statistic based on a logarithmic transformation of the sample standard deviation. The chart designs have been determined in the respect of a statistical constraint related to the in-control chart Average Time to Signal (ATS0) and an operational constraint related to the available workforce to perform the SPC activities. The obtained results show that on the average the logarithmic transformed chart outperforms the S Shewhart even if some process operating parameters can affect the entity of cost savings; secondly, the accuracy of the expected inspection cost estimation of process dispersion has been computed for several distributions fitting the true normal random shift behaviour.