The Phase I Dispersion Charts for Bivariate Process Monitoring

Self Cite

A phase‐I study is generally used when population parameters are unknown. The performance of any phase‐II chart depends on the preciseness of the control limits obtained from the phase‐I analysis. The performance of phase‐I bivariate dispersion charts has mainly been investigated for bivariate normal distribution. However, this assumption is seldom fulfilled in reality. The current work develops and studies the performance of phase‐I |S| and |G| charts for monitoring the process dispersion of bivariate non‐normal distributions. The necessary control charting constants are determined for the bivariate non‐normal distributions at nominal false alarm probability (FAP0). The performance of these charts is evaluated and compared in a situation when samples are generated by bivariate logistic, bivariate Laplace, bivariate exponential, or bivariate t5 distribution. The analysis shows that the proper consideration to underlying bivariate distribution in the construction of phase‐I bivariate dispersion charts is very important to give a real picture of in or out of control process status. Copyright © 2016 John Wiley & Sons, Ltd.

“…The population parameters are generally unknown, and their estimates are calculated from the historical dataset. Two unbiased point estimators for

{|normalΣ|}^{\frac{1}{2}}

(refer to Saghir et al

{|\overset{Σ}{true}|}_{S}^{1 true/ 2} = \frac{2 ((), n - 1) \overset{{(||, S)}^{1 / 2}}{true}}{a_{1}} = \frac{2 ((), n - 1)}{a_{1}} \cdot ((), \frac{1}{m} {\sum_{i = 1}^{m} |S|}_{i}^{1 true/ 2})

{|\overset{Σ}{true}|}_{G}^{1 true/ 2} = \frac{2 ((), n - 1) \overset{{(||, S)}^{1 / 2}}{true}}{b_{0}} = \frac{2 ((), n - 1)}{b_{0}} \cdot ((), \frac{1}{m} {\sum_{i = 1}^{m} |G|}_{i}^{1 true/ 2})

respectively, where

{|S|}_{i}^{1 true/ 2}

and

{|G|}_{i}^{1 true/ 2}

are the square roots of the sample generalized variance and Gini's generalized variance (refer to Saghir for more details).…”

Section: Phase‐i Bivariate Dispersion Chartsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Performance of Phase‐I Bivariate Dispersion Charts to Non‐Normality

Saghir

Chakraborti

Ahmad

2016

Self Cite

“…Multivariate charts are usually designed based on the assumption that process observations follow the multivariate normal distribution. Other notable works such as those by Khoo and Quah 8 Surthadi et al 9 Vargas and Lagos 10 Riaz and Does 11 Costa and Machado 12 Saghir et al 13 Costa and Neto 14 etc. The performance of the control charts (based on normality assumption) can be greatly affected in the absence of normality.…”

Section: Introductionmentioning

confidence: 99%

Efficient bivariate EWMA charts for monitoring process dispersion

Osei-Aning

2019

To ensure high quality standards of a process, the application of control charts to monitor process performance has become a regular routine. Multivariate charts are a preferred choice in the presence of more than one process variable.In this article, we proposed a set of bivariate exponentially weighted moving average (EWMA) charts for monitoring the process dispersion. These charts are formulated based on a variety of dispersion statistics considering normal and non-normal bivariate parent distributions. The performance of the different bivariate EWMA dispersion charts is evaluated and compared using the average run length and extra quadratic loss criteria. For the bivariate normal process, the comparisons revealed that the EWMA chart based on the maximum standard deviation (SMAX E ) was the most efficient chart when the shift occurred in one quality variable. It also performed well when the sample size is small and the shift occurred in both quality variables. The EWMA chart based on the maximum average absolute deviation from median (MDMAX E ) performed better than the other charts in most situations when the shift occurred in the covariance matrix for the bivariate non-normal processes. An illustrative example is also presented to show the working of the charts.

Monitoring bivariate and trivariate mean vectors with a Shewhart chart

Leoni¹,

Costa

2017

In this article, we propose the use of the mean chart to control multivariate processes. The basic idea is to control the mean vector of bivariate (X, Y) and trivariate (X, Y, Z) processes by alternating the charting statistic of the Shewhart chart. If the mean of X observations was the charting statistic to obtain the current sample point, then the mean of Y observations will be the charting statistic to obtain the next sample point (for the trivariate case, the mean of Z observations will be the charting statistic to obtain the sample point subsequent to the next one). As a Shewhart chart, the signal is given anytime a sample point is plotted beyond the control limits, independent of the charting statistic in use. A fair comparison between the proposed chart and the Hotelling chart is based on an equal number of measurements per sample. The Shewhart chart with alternated charting statistic (ACS) always outperforms the Hotelling chart, except for specific types of disturbances in quality characteristics highly correlated (ρ = 0.7). The ACS chart is substantially easier to operate and faster than the Hotelling chart in signaling changes in the mean vector of bivariate and trivariate processes. Even with fewer measurements per sample, the trivariate ACS chart outperforms the Hotelling chart.