Monitoring the Ratio of two Normal Variables using EWMA Type Control Charts in Short Production Runs

Abstract

In many industrial and engineering applications, process performance is characterized by the ratio of two normally distributed quality characteristics. Monitoring such ratios is particularly challenging in short production runs, where conventional control charts often suffer from limited sensitivity due to the small number of available inspections. This paper proposes an exponentially weighted moving average (EWMA) control chart for monitoring the ratio of two normally distributed random variables under short production run (SPR) conditions. The statistical distribution of the ratio is first reviewed, adopting the corrected closed-form density of Nadarajah (2020) rather than the approximation used in earlier studies. The control limit of the proposed chart is calibrated to a prescribed in-control truncated average run length (TARL$ _0 $) over a finite horizon $ I $ of inspections, using a Markov-chain representation of the EWMA recursion. The detection performance of the chart is then assessed through a large factorial study covering the smoothing constant $ λ$, the in-control correlation $ ρ_0 $, the coefficients of variation $ (γ_X, γ_Y) $, the sample size $ n $, and the magnitude of the shift $ τ$. Numerical results show that the proposed EWMA-RZ chart provides substantially better detection of small and moderate shifts than the recently developed Shewhart-type short-run ratio chart (ShRZ) of Tran et al. (2021), especially for $ |τ- 1| \le 0.05 $. An illustrative example based on a beverage filling process is included to demonstrate the practical implementation of the method.