A two-stage decision procedure for monitoring processes with low fraction nonconforming

Abstract

Decision procedures for monitoring industrial processes can be based on application of control charts. The commonly used p-chart and np-chart are unsatisfactory for monitoring high-quality processes with a low fraction nonconforming. To overcome this difficulty, one may develop models based on the number of items inspected until r (⩾1) nonconforming items are observed. The cumulative count control chart (CCC-chart) is such an example. Like many other control charts, the CCC-charts suggested in the literature are one-stage control charts in which a decision is made when a signal for out of control appears. A CCC-chart with a small value of r requires less items inspected in order to obtain a signal for out of control, but is less reliable in detecting shifts of p than a CCC-chart with a large value of r (because the standard deviation of the number of items inspected in order to observe the rth nonconforming item, when divided by the mean, is proportional to 1/ r ). In the present paper, inspired by the idea of double sampling procedures in acceptance sampling, a two-stage CCC-chart is proposed in order to improve the performance of the one-stage CCC-chart. Analytic expressions for the average number inspected (ANI) of this two-stage CCC-chart is obtained, which is important for further studies of the chart. As an application of this result, an economic model is used to calculate the optimal values of probabilities of false alarm set at the first and second stages of the two-stage CCC-chart so that an expected total cost can be minimized.