An Adaptive Upper One-Sided Cumulative Sum Control Chart with Joint Parameter Optimization for Monitoring the Ratio of Two Normal Variables in Short Production Runs

Abstract

Monitoring the ratio of two correlated normal variables is increasingly important in statistical process control, since many quality characteristics are expressed in relative rather than absolute form. Memory-type ratio charts have mostly been developed for long production runs, while their finite-horizon counterparts rely on a fixed reference value $k$ derived from a specified shift. Such fixed-$k$ designs are not optimal at a given out-of-control magnitude and, in low-variability regimes, yield boundary solutions for which the in-control truncated average run length (TARL${}_0$) is unattainable. This paper proposes an upper one-sided cumulative sum (CUSUM) control chart for the ratio $Z=X/Y$ in short production runs, denoted CUSUM-RZ${}^{+}$ (RZ standing for the ratio $Z$), with fully adaptive joint optimization of $k$ and the decision interval $h$. Given a target TARL${}_0=I$ and a target shift $\ddot{I}„$, a bilevel problem calibrates $h(k)$ by inner root-finding to satisfy the TARL${}_0$ constraint and selects $k^{\ast }$ by outer line search to minimize the out-of-control TARL${}_1$. Both use a finite-state Markov-chain framework with an accurate ratio approximation; the inner step recovers boundary cases that fixed-$k$ designs cannot. The chart is assessed through matched-horizon benchmarks against Shewhart-RZ, exponentially weighted moving average (EWMA-RZ), and fixed-$k$ CUSUM-RZ${}^{+}$ charts, Monte Carlo robustness studies, and a Phase I estimation analysis. All memory-type charts outperform the Shewhart-RZ baseline; the adaptive design matches them under stable correlation and improves appreciably when correlation rises from Phase I to Phase II. It is insensitive to symmetric heavy tails yet mildly anti-conservative under contamination, and $m\ge 100$ subgroups keep the TARL${}_0$ relative bias near 1%.