A new distribution-free scheme for simultaneous Phase-I analysis of four process aspects and its application in monitoring customers' waiting times.

In industrial process monitoring and control, Phase-I or retrospective analysis plays an essential role in assessing historical information and setting up a benchmark for future quality practices. It is vital to analyse four critical aspects of a process, including location, scale, skewness and kurtosis. Assuming or fitting a parametric distribution and estimating process parameters are not recommended before conducting a Phase-I analysis. Otherwise, extreme values may influence the estimation, and the fit could be misleading. However, no distribution-free method can monitor the four aspects in tandem. In recent years, the RS/P schemes, and some rank-based schemes, including Lepage-type combined statistics for location and scale shifts, have been introduced for retrospective study in industrial operations. However, they are not designed to handle complex process shifts involving skewness and kurtosis. To this end, we propose a new scheme using subgroup samples, the Legendre polynomial and an omnibus rank globe statistic, referred to as the Phase-I globe chart and abbreviated as the GLC scheme. The GLC scheme can recognise out-of-control subsamples in respect of four aspects of a process, and hence it may be looked upon as a quad-aspect scheme which is a further improvement over the existing bi and tri-aspect schemes. Computer-intensive Monte Carlo simulations are used to evaluate charting constants for a given false alarm probability. The correct alarm probability of our scheme is compared with that of various existing distribution-free Phase-I schemes. Various shift patterns in one or more subsamples from different various out-of-control situations in a class of in-control distributions are considered in our investigation. The proposed Phase-I scheme is found to have an excellent performance in a wide range of scenarios. Some applications in monitoring service processes are also discussed.