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 $ τ$, a bilevel problem calibrates $ h(k) $ by inner root-finding to satisfy the TARL$ _0 $ constraint and selects $ k^* $ 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 \geq 100 $ subgroups keep the TARL$ _0 $ relative bias near 1%.