Computation of the probability of a process being in control at the third stage for the TS X chart

Abstract

He et al., in their paper ‘Design of double- and triple-sampling X-bar control charts using genetic algorithm’ [2002, International Journal of Production Research, 40 (6), 1387–1404], used a genetic algorithm to optimize the obtained model of double- and triple-sampling control charts. In order to obtain the type I and II error probabilities, they calculated the probabilities that the TS chart indicates that the process is in control at the first, second and third stages of sampling. In this article, we use a different approach to calculate the probability that the TS chart indicates that the process is in control at the third stage (P a3). The advantage of this new computational approach is that it requires only single integration rather than the double integration required by He et al. The new approach reduces the computational complexity and increases the accuracy of the probability estimation, since such integration normally involves an approximation.