On Improvement of Control Chart using Repetitive Sampling for Monitoring Process Mean

TL;DR

This work tackles improving control charts for monitoring a process mean by exploiting repetitive sampling and auxiliary information. It introduces two enhanced estimators, a ratio–product exponential type $M_{rep}$ and a difference–cum–exponential type $M_{rwp}$, and embeds them into chart statistics $T_2$ and $T_3$ to supersede the traditional $M_r$ chart. Monte Carlo simulations under a bivariate normal framework show that $T_2$ and $T_3$ detect mean shifts more rapidly and robustly, evidenced by lower ARL and better $EQL$, $RARL$, and $PCI$ metrics. The findings highlight the practical value of using auxiliary data and advanced estimators to boost the responsiveness of SPC/QC charts in industrial monitoring. This work provides actionable design guidance for implementing more efficient mean-shift detection in quality control settings.

Abstract

In the practical industry, the most commonly used application of statistical analysis for monitoring the process mean is the control chart. Control charts are generated based on the presumption that we have a sample from a stable process. The control chart then provides a graphical display to test this presumption. In the existing estimator \textcolor{red}{$Mr$}, researchers use a technique involving repetitive sampling along with an auxiliary variable for detecting and monitoring the statistical process mean. The existing control chart, namely \textcolor{red}{$Mr$}, is based on the regression estimator of the mean using a single auxiliary variable $X$. We propose the \textcolor{red}{$Mrep$} chart using a ratio-product exponential type estimator, and the \textcolor{red}{$Mrwp$} chart with a more efficient difference-cum-exponential type estimator used in quality control for improving the process mean in terms of $ARL$. Then we compare the proposed charts \textcolor{red}{$Mrep$} and \textcolor{red}{$Mrwp$} with the existing \textcolor{red}{$Mr$} chart in terms of $ARL$. Using $ARL$ as a performance measure, better results of the proposed charts are observed for detecting shifts in the mean level of the characteristic of interest. Moreover, Monte Carlo simulation in terms of repetitive sampling is used for quality control charting and statistical process control for the betterment of the process mean.

Figures (6)

• Figure 1: $T_0-T_3$ based charts: (i) n=5, ARL=500, $\rho_{xy}=0.90$; (ii) n=5, ARL=371, $\rho_{xy}=0.90$; (iii) n=5, ARL=200, $\rho_{xy}=0.90$; (iv) n=10, ARL=371, $\rho_{xy}=0.60$; (v) n=15, ARL=500, $\rho_{xy}=0.90$.
• Figure 2: ARL curve of $T_0-T_3$ at: (a) ARL=200, n=5, $l_{xy}=0.30$; (b) ARL=200, n=5, $l_{xy}=0.60$; (c) ARL=200, n=10, $l_{xy}=0.90$; (d) ARL=200, n=10, $l_{xy}=0.30$; (e) ARL=200, n=10, $l_{xy}=0.60$; (f) ARL=200, n=10, $l_{xy}=0.90$.
• Figure 3: ARL curve of $T_0-T_3$ at: (g) ARL=200, n=15, $l_{xy}=0.30$; (h) ARL=200, n=15, $l_{xy}=0.60$; (i) ARL=200, n=15, $l_{xy}=0.90$; (j) ARL=371, n=5, $l_{xy}=0.30$; (k) ARL=371, n=5, $l_{xy}=0.60$; (l) ARL=371, n=5, $l_{xy}=0.90$.
• Figure 4: ARL curve of $T_0-T_3$ at: (m) ARL=371, n=10, $l_{xy}=0.30$; (n) ARL=371, n=10, $l_{xy}=0.60$; (o) ARL=371, n=10, $l_{xy}=0.90$; (p) ARL=371, n=15, $l_{xy}=0.30$; (q) ARL=371, n=15, $l_{xy}=0.60$; (r) ARL=371, n=15, $l_{xy}=0.90$.
• Figure 5: ARL curve of $T_0-T_3$ at: (s) ARL=500, n=5, $l_{xy}=0.30$; (t) ARL=500, n=5, $l_{xy}=0.60$; (u) ARL=500, n=5, $l_{xy}=0.90$; (v) ARL=500, n=10, $l_{xy}=0.30$; (w) ARL=500, n=10, $l_{xy}=0.60$; (x) ARL=500, n=10, $l_{xy}=0.90$.