{"ID":3083742,"CreatedAt":"2026-06-05T06:46:15.197025399Z","UpdatedAt":"2026-06-07T05:49:02.101151534Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.06137","arxiv_id":"2606.06137","title":"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%.","short_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...","url_abs":"https://arxiv.org/abs/2606.06137","url_pdf":"https://arxiv.org/pdf/2606.06137v1","authors":"[\"Kim Duc Tran\"]","published":"2026-06-04T13:19:32Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}