High quality process monitoring using a class of inter-arrival time distributions of the renewal process

For high-quality processes where the defect rate is very low, e.g., parts per million (ppm), time-between-events (TBE) control charts have several advantages over the ordinary control charts. Most existing TBE control charts are based on the homogeneous Poisson process assumption, so that the distribution of TBE can only be exponential. However, the exponential distribution is not suitable in many applications, especially when the failure rate is not constant. In this article, we introduce a new TBE control chart, based on the renewal process, where the distribution of the TBE belongs to a parametric class of absolutely continuous distributions, which includes some well-known and commonly used lifetime distributions, i.e., exponential, Rayleigh, Weibull, Burr type XII, Pareto and Gompertz. The control structure of the proposed chart is derived analytically in general and numerical examples are presented for the Weibull distribution, due to its relevance in reliability. The performance of the proposed control charts is evaluated in terms of some standard useful measures, including average run length (ARL), the standard deviation of run length, the coefficient of variation of run length, expected quality loss (EQL) and relative ARL (RARL). The effect of parameter estimation, using both maximum likelihood and Bayesian methods, is also discussed. This study also presents an illustrative example and four case studies to highlight the practical aspects of the new TBE chart.