Improving the performance of the attribute charts: X¯tn$\bar{X}_{tn}$ and Stn2$S^2_{tn}

• Published in: Quality and reliability engineering international 2022-03, Vol.38 (2), p.703-732
• Main Authors: Yamauchi, Tsubasa, Lee Ho, Linda, da Costa Quinino, Roberto
• Format: Article
• Language: English
• Subjects: average run length; Control charts; Control limits; Normal distribution; Performance enhancement; Sample variance; simplified variable sample size; simulation; Switches; truncated normal distribution; Variance
• ISSN: 0748-8017; 1099-1638
• DOI: 10.1002/qre.3009

Summary This paper proposes a procedure to improve the performance for the attribute control charts: X¯tn$\bar{X}_{tn}$ and Stn2$S^2_{tn}$. In these control charts, a simple size n$n$ is collected periodically and each item is allocated into one of the five categories defined by a go‐no‐go gauge. After that, a random value is generated for each item according to a truncated normal distribution with its upper and lower limits defined by the cut points of the go‐no‐go gauge. With the n$n$ generated values, the sample mean X¯tn$\bar{X}_{tn}$ and the sample variance Stn2$S^2_{tn}$ are calculated and compared with the their respective control limits. In this paper, a procedure was implemented in order to improve the performance of these control charts, which consists of the sequential switches between two sample sizes na$n_a$ and nb$n_b$ (with na$n_a$ >$>$ nb$n_b$) and checking whether the sample mean (or sample variance depending) exceeds the control limits or not. The proposal is relatively easy for practitioners to implement it. Extensive computational experiments showed that the procedure improved both control charts to detect small shifts in term of ARL1$ARL_1$. With this procedure, the “Improved X¯tn$\bar{X}_{tn}$” (shortly X¯tn(I)$\bar{X}_{tn(I)}$) control chart performs similarly to X¯$\bar{X}$ chart for small shifts in the process mean provided that the sample sizes are raised by approximately two additional sample units and the “Improved Stn2$S^2_{tn}$” (shortly Stn(I)2$S^2_{tn(I)}$) control chart also has a similar performance to S2$S^2$ chart with more or less double sample size in the range of small shifts of the variance. Two numerical examples are presented to illustrate the use of the procedure.