Two memory-based monitoring schemes for the ratio of two normal variables in short production runs

In many production processes, monitoring the ratio of two normal random variables (RZ) plays an important role in ensuring product quality. In recent years, flexible manufacturing has become increasingly important to meet the ever-changing market demands, making small batch production very common in real industrial processes. However, it is worth noting that few studies have been conducted on monitoring the RZ in short production runs (SPR) processes. To address this issue, two popular memory-based schemes, i.e. the SPR exponentially weighted moving average (EWMA) and the SPR cumulative sum (CUSUM), are proposed to monitor the RZ. The truncated run length performance measures, i.e. truncated average run length (TARL) and truncated standard deviation of the run length (TSDRL), of the proposed monitoring schemes are obtained by using the Markov chain method. Furthermore, by comparing their statistical performance with that of the existing SPR Shewhart-RZ scheme, the superiority of the proposed schemes is demonstrated. The findings show that the advantages of the EWMA and CUSUM schemes over the Shewhart scheme increase with an increase in the number of batches and subgroups of samples. Finally, the implementation of SPR EWMA-RZ and SPR CUSUM-RZ schemes in the small batch production process is illustrated with a real data example.