Monitoring the ratio of two normal variables using CUSUM type control charts in short production runs

Monitoring the ratio of two normal variables has gained increasing attention in statistical process control (SPC) because many industrial quality characteristics are naturally expressed as ratios rather than absolute values. Although various ratio-type control charts have been developed for long-run production, they are unsuitable for short production runs where the horizon is finite, and the traditional average run length (ARL) becomes inadequate.This paper proposes an upper one-sided cumulative sum control chart, CUSUM-RZ+, for detecting increases in the ratio of two correlated normal variables in short runs. The chart is based on the ratio of sample means and relies on a precise approximation of the ratio statistic’s distribution under joint normality.Performance in finite-horizon settings is evaluated using the truncated average run length (TARL) as the primary measure. The chart is calibrated through a Markov-chain framework that accurately computes TARL and efficiently determines the decision interval for a prescribed in-control target. Monte Carlo experiments validate the distributional approximation and benchmark the proposed chart against a competing short-run ratio chart under matched horizons.Extensive numerical studies examine the effects of coefficients of variation, correlation structure, subgroup size, and shift magnitude. Results show that the CUSUM-RZ+ provides a flexible and effective framework for monitoring ratio-type quality characteristics in short production runs. While small upward shifts are detected gradually, the chart efficiently identifies moderate-to-large shifts across practical conditions, with sensitivity increasing as shift magnitude and sample size grow.