Economical operation of (x̄, s) control chart indexed by Taguchi's loss function

Assume that lot quality characteristics obey a normal distribution. Kanagawa et al. have proposed the (x̄, s) control chart which enable the user to monitor both changes of the mean and variance simultaneously. Further, the results of Watakabe and Arizono enable the user to evaluate the performance for out-of-control state in the case of using the (x̄, s) control chart. On the other hand, Taguchi has presented an approach to quality improvement in which reduction of deviation from the target value is the guiding principle. In this approach, the loss is expressed as a quadratic form with respect to the difference between the measured value x of a product characteristic X and the target value T of a product quality characteristic. Then, we can evaluate the process quality based on the Taguchi's loss criterion. We consider here the economical operation of the (x̄, s) control chart in conformity with the expected total operation cost function based on the sampling cost and the loss due to derivation in the process quality. First, we consider the economical operation of the (x̄, s) control chart in the situation that the loss to be considered is known. Further, the economical operation of the (x̄, s) control chart is also discussed under any loss instead of a known loss. Then, the relationship between the two economical operations proposed here corresponds to the relationship between the lot tolerance per cent defective plans under the fixed fraction defective and the average outgoing quality limit plans under any fraction defective in rectifying inspection plans.