The X control chart for monitoring process shifts in mean and variance

Abstract

Control charts are widely used in statistical process control (SPC) to monitor the quality of products or production processes. When dealing with a variable (e.g., the diameter of a shaft, the hardness of a component surface), it is necessary to monitor both its mean and variability (Montgomery 2009[Montgomery, D.C., 2009. Introduction to statistical quality control. New York: John Wiley & Sons.]). This article studies and compares the overall performances of the X chart and the 3-CUSUM chart for this purpose. The latter is a combined scheme incorporating three individual CUSUM charts and is considered as the most effective scheme for detecting mean shift and/or standard deviation shift in current SPC literature. The results of the performance studies reveal two interesting findings: (1) the best sample size n for an " X chart is always n ¼ 1, in other words, the simplest X chart (i.e., the " X chart with n ¼ 1) is the most effective "X chart for detecting and/or ; (2) the simplest X chart often outperforms the 3-CUSUM chart from an overall viewpoint unless the latter is redesigned by a difficult optimisation procedure. However, even the optimal 3-CUSUM chart is only slightly more effective than the X chart unless the process shift domain is quite small. Since the X chart is very simple to understand, implement and design, it may be more suitable in many SPC applications, in which both the mean and variance of a variable need to be monitored.

Empirical Comparisons of X-bar Charts when Control Limits are Estimated

“…Figures 3 and 4 analyze the performance of the various X charts in terms of the empirical measures ARL 1 and ARL 2 defined in Section 2. Note that the values of ARL 1 and ARL 2 should be close to 370.4, which is the theoretical value of the measure ARL defined by (3). We observe that the measure ARL 1 performs generally better than the measure ARL 2 .…”

“…Recent research indicates that the charts are very simple to understand, implement and design, and may be more suitable in many SQC applications (see Montgomery and Yang et al …”

“…Recent research indicates that the X charts are very simple to understand, implement and design, and may be more suitable in many SQC applications (see Montgomery 2 and Yang et al 3 ). When the parameters of the process are unknown, and they need to be estimated in phase I, the most common statistics used to estimate the variability of the process are based on the sample standard deviations, the sample ranges or the sample variances.…”

An Improved Maximum Exponentially Weighted Moving Average Control Chart for Monitoring Process Mean and Variability

“…Since the estimators of mean and variance in both MaxEWMA and MaxGWMA charts are based on simple random sampling (SRS), therefore, we name these control charts as MaxEWMA-SRS and MaxGWMA-SRS control charts. Some important literature on the joint monitoring of process mean and variability may be seen in [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] and references cited therein.…”

A bibliometric analysis of 50 years of worldwide research on statistical process control

“…Significant advances can also be seen in the literature regarding the development of new methods and strategies for SPC, such as models for auto-correlated data and multivariate processes (Costa & Castagliola, 2011;Costa & Machado, 2011;Dokouhaki & Noorossana, 2013;Franco et al, 2014;Leoni et al, 2015), control charts allowing for varying the sampling interval and the sample size (Reynolds et al, 1988;Reynolds, 1996 Yang et al, 2012;Mahadik, 2013), improvement in the performance of control charts by double sampling (Costa & Castagliola, 2011;Teoh et al, 2015;Inghilleri et al, 2015), the design of control charts that minimize operational costs (Michel & Fogliatto, 2002;Celano et al, 2011;Lupo, 2014;Franco et al, 2014), the integration of statistical process control and automatic/engineering process control, avoiding over adjustement of the process (Holmes & Mergen, 2011;Siddiqui et al, 2015); application of the Bernoulli control charts in the field of medicine (Szarka & Woodall, 2011), application of SPC to image data (Megahed et al, 2011;Wells et al, 2013), and strategies for monitoring the variability of small batches (Celano et al, 2012;Castagliola et al, 2013).…”