Run sum control chart for monitoring the ratio of population means of a bivariate normal distribution

This article proposes a two-sided run sum ratio chart for monitoring the ratio of two normal variables. A Markov chain procedure is applied to evaluate the statistical performance of the chart by using both average run length (ARL) and expected average run length (EARL) criteria. A numerical comparison with the Shewhart ratio and synthetic ratio charts for the zero state analysis reveals that the run sum ratio chart has a better sensitivity in most cases. In particular, for the values of the coefficients of variation (γX,γY) ∈ {(0.2, 0.2), (0.2, 0.01)}, the run sum ratio chart outperforms the two charts in contest for almost all shift sizes in the ratio of the two variables. In terms of the steady state analysis, the results indicate that the run sum ratio chart outperforms the synthetic ratio chart almost uniformly. The run sum ratio chart also surpasses the exponentially weighted moving average (EWMA) ratio chart in detecting all decreasing shifts when (γX,γY) = (0.2, 0.2), while the former outperforms the latter for (γX,γY) = (0.01, 0.2), when the sample size is small. An illustrative example of a real quality issue in a food industry is presented to demonstrate the implementation of the proposed chart.