Determining Out-of-control Variable(S) in a Multivariate Quality Control Chart for Individual Observations
CHAPTER ONE
AIM AND OBJECTIVES OF THE STUDY
Much research has been done on multivariate process control for variable data in various situations. Moreover, interpretation of out-of-control signals and how to identify the quality characteristics contributing to out-of-control signals have also been discussed. However, identifying influential variable(s) that contribute to out-of-control signal is still a difficult task especially when the quality characteristics are beyond three.
The aim of this research work is to:
Determine variable(s) that significantly contributes to an out-of-control signal in a multivariate quality control chart. This is achieved through the following objectives:
- Application of Hotelling’s T2 components using MYT decomposition
- Identifying out-of-control condition and out-of-control variable(s)
- Illustration of invariance property of the Hotelling’s T2 statistic from the derived
CHAPTER TWO
LITERATURE REVIEW
INTRODUCTION
This chapter seeks to review related literatures on the works of different scholars regarding multivariate statistical quality control chart. Many authors have proposed quality control procedures for several related variables that led to the development of multivariate quality control.
MULTIVARIATE CHART
Many control charts have been proposed for multivariate data, with the most popular being Hotelling’s T2 or χ 2 chart, the Multivariate Exponentially-Weighted Moving Average
(MEWMA) chart and the Multivariate Cumulative Sum (MCUSUM) chart. Mason et al., (1997b) presented an assessment of many multivariate techniques. In their paper, they recommended when multivariate charts should be used and also discussed some problems associated with the selection of multivariate charts. The problems involve the violation of the assumption of multivariate normality that is required for many charts, the estimation of the covariance matrix, missing data, and the effect of autocorrelation. Bersimis et al.,(2007) provided a review on some multivariate charts.
Hotelling’s T2 Control Chart
Hotelling’s T2 control chart is the most common multivariate control chart which was proposed by Hotelling (1947). Hotelling’s T2 control chart has attracted much attention in the area of multivariate quality control by researchers. Hotelling’s T2 control chart is purposely used for monitoring the mean vector of quality characteristics of a process. There are two types of T2 chart, one for subgroup data and the other for individual observations.
CHAPTER THREE METHODOLOGY
INTRODUCTION
Hotelling’s control chart is the most commonly used multivariate control charts, when all the quality characteristics are normally distributed.
CHAPTER FOUR
RESULTS AND DISCUSSIONS
INTRODUCTION
This chapter seeks to apply the derived Hotelling’s T2 decomposition for using four variables and to identify the variable(s) that contribute to an out-of-control condition. The Q-Q plot is first constructed to assess the normality assumption on the four variables.
CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATION
INTRODUCTION
In this chapter, we present the summary, conclusion and recommendation based on the results obtained from this research work.
SUMMARY
This research work was aimed at deriving the T2 decomposition model which can be used to identify variable(s) that contribute significantly to out-of-control signal. The model was derived and the invariance property of the Hotelling’s T2 statistic was demonstrated to show that all the 24 decompositions have the same overall T2 value. This technique is extremely useful in production supervision by providing information as to when and where to control %FFA (percentage Free Fatty Acid) for quality improvement.
Phase I Hotelling’s T2 control chart indicated that the process is in control using the historical dataset while on drawing further samples after the 180th sample, Phase II Hotelling’s T2 control chart shows that three points falls above the control limit and these points are located in sample 9, 30 and 75.
The Hotelling’s T2 control chart is able to account simultaneously for variation in several variables since it requires both an overall measure of deviation from the targets as well as an assessment of the data correlation structure. This was noticed in the relationship between variable 1 and 2 conditioned on other variables as seen in sample 9 and 30. Also variable 4 (X4) conditioned on the other quality characteristics indicates deviation in the variance-covariance structure with reference to the historical dataset.
It is an established fact that the most difficult part of applying multivariate control chart is the interpretation of out-of-control variable. In this research work, the T2 decomposition approach is employed. This is because it provides a comprehensive interpretation of the situation with respect to the original variable, though it is known to be computationally intensive.
In this study, the direct comparison between the values of each unconditional component is used. The quality characteristic with the highest value of unconditional component is the characteristic that significantly contributes to an out-of-control signal. Also, the quality characteristic with the highest value of conditional component is considered to have significantly contributed to an out-of-control signal.
CONCLUSION
This research is an extension of the T2 decomposition of the Hotelling’s T2 statistic into orthogonal components from three variables to four variables with the aid of the MYT decomposition approach. The T2 decomposition is aimed at identifying the variable(s) that significantly contribute to an out-of-control condition. The T2 decomposition is chosen in this research work due its ability to comprehensively display the potential cause of out-of-control condition.
The use of multivariate control chart is very effective than the univariate control chart, although the interpretation of out-of-control signal in a multivariate control chart is more complicated than the univariate control chart. Thus, the T2 decomposition is being derived and applied in this research. This is to encourage quality control engineers on the significance of the T2 decomposition in detecting variables that significantly contribute to out-of-control signal.
RECOMMENDATION
Whenever a multivariate statistical process is to be monitored, it is important to construct the Phase I and Phase II control chart so as not to rush into wrong conclusion. The Phase II
control chart is very sensitive in detecting shifts unlike the Phase I. In this research, we noticed that our Phase I control chart using the historical dataset was unable to detect an out-of-control condition, but the Phase II chart did.
CONTRIBUTION TO KNOWLEDGE
In this work, the following contributions are made which are vital for academic and industrial consumption.
- This research work has succeeded in deriving the T2decomposition when four variables are considered for monitoring any process control.
- Thedecomposition model will help quality control engineers as well as researchers in the field of quality control for monitoring process control.
- This work has also provided a guide on how to derive models for the T2decomposition for monitoring process control beyond three variables.
FURTHER RESEARCH
- Anextension on this work can be made regarding the decomposition of the Hotelling’s T2
- Aprogram/algorithm can be design for easy computing of the Hotelling’s T2
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