Due to the rapid development of technology, quality control charts have attracted more attention from manufacturing industries in order to monitor quality characteristics of interest more effectively. Among many control charts, my research work has focused on the multivariate exponentially weighted moving average (MEWMA) and the univariate exponentially weighted moving average (EWMA) control charts by using the Markov chain method.
The performance of the chart is measured by the optimal average run length (ARL). My Ph.D. thesis is composed of the following three contributions.
My first research work is about differential smoothing. The MEWMA control chart proposed
by Lowry et al. (1992) has become one of the most widely used charts to monitor
multivariate processes. Its simplicity, combined with its high sensitivity to small and
moderate process mean jumps, is at the core of its appeal. Lowry et al. (1992) advocated
equal smoothing of each quality variable unless there is an a priori reason to weigh quality
characteristics differently. However, one may have situations where differential smoothing
may be justified. For instance: (a) departures in process mean may be different across
quality variables, (b) some variables may evolve over time at a much different pace than
other variables, and (c) the level of correlation between variables could vary substantially.
For these reasons, I focus on and assess the performance of the differentially smoothed
MEWMA chart. The case of two quality variables (BEWMA) is discussed in detail. A bivariate
Markov chain method that uses conditional distributions is developed for average run length
(ARL) calculations. The proposed chart is shown to perform at least as well as Lowry et al.
(1992)'s chart, and noticeably better in most other mean jump directions. Comparisons with the
recently introduced double-smoothed BEWMA chart and the univariate charts for the
independent case show that the proposed differentially smoothed BEWMA chart has
superior performance.
My second research work is about monitoring skewed multivariate processes.
Recently, Xie et al. (2011) studied monitoring bivariate exponential quality
measurements using the standard MEWMA chart originally developed to
monitor multivariate normal quality data. The focus of my work is on situations
where, marginally, the quality measurements may follow not only exponential
distributions but also other skewed distributions such as Gamma or Weibull,
in any combination. The joint distribution is specified using the Gumbel copula
function thus allowing for varying degrees of correlation among the quality measurements. In
addition to the standard MEWMA chart, charts based on the largest or smallest
of the measurements and on the joint cumulative distribution function or the joint
survivor function, are studied in detail. The focus is on the case of two quality
measurements, i.e., on skewed bivariate processes. The proposed charts avoid
an undesirable feature encountered by Xie et al. (2011) for the standard MEWMA
chart where in some cases the off-target average run length turns out to be larger
than the on-target one. Using the optimal average run length, our extensive
numerical results show that the proposed methods provide an overall good
detection performance in most directions. Simulations were performed to
obtain the optimal ARL results but the Markov chain method using the empirical
CDF of the statistics involved verified the accuracy of the ARL results. In addition,
an examination of the effect of correlation on chart performance was undertaken
numerically. The methods are easily extendable to more than two variables.
Final study is about a new ARL criterion for univariate
processes studied in detail in this thesis. The traditional ARL is calculated
assuming a given fixed process mean jump and a given time point where
the jump occurs, usually taken to be from the very beginning in most chart
performance studies. However, Ryu et al. (2010) demonstrated that the
assumption of a fixed mean shift might lead to poor performance of control
charts when the actual size of the mean shift is significantly different and
therefore suggested a new ARL-based performance measure, called
expected weighted run length (EWRL), by assuming that the size of the
mean shift is not specified but rather it follows a probability distribution.
The EWRL becomes the expected value of the weighted ordinary ARL
with respect to this distribution. My methods generalize this criterion by
allowing the time at which the mean shift occurs to also vary according to
a probability distribution. This leads to a joint distribution for the size of the
mean shift and the time the shift takes place, then the EWRL is calculated
as the weighted expected value with respect to this joint distribution.
The benefit of the generalized EWRL is that one can assess the performance
of control charts more realistically when the process starts on-target
and then the mean shift occurs at some later random time. Moreover, I also
propose the effective EWRL, which measures the number of additional
process runs that on average are needed to detect a jump in the mean
after it happens. I evaluate several well-known univariate control charts
based on their EWRL and effective EWRL performance. The numerical
results show that the choice of control chart depends on the additional
information on the transition point of the mean shift. The methods can
readily be extended to other control charts, including multivariate charts. / Thesis / Doctor of Philosophy (PhD) / Since the introduction of the standard multivariate exponentially weighted moving average (MEWMA) procedure (Lowry et al. 1992), equal smoothing on all quality variables has been conveniently adopted. In this thesis, a bivariate exponentially weighted moving average (BEWMA) control statistic with unequal smooth- ing parameters is introduced with the aim of improving performance over the standard BEWMA chart. Extensive numerical comparisons reveal that the proposed chart enhances the efficiency and flexibility of the control chart in many mean-shift directions. Recently, Xie et al. (2011) proposed a chart for bivariate Exponential data when the quality measures follow Gumbel’s bivariate Exponential distribution (Gumbel 1960). However, when the process means experience a downward shift (D-D shift), the control charts are shown to break down. In other words, we encounter the strange situation where the out-of-control ARL becomes larger than the in-control ARL. To address this issue, we have proposed two methods, the MAX-MIN and CDF methods and applied them to the univariate EWMA chart. Our numerical results show that not only do our proposed methods prevent the undesirable behaviour from happening, but they also offer substantial improvement in the ARL over the approach proposed by Xie et al. (2011) in many mean shifts. Finally, in general, when it comes to designing a control chart, it is assumed that the size of the mean shift is fixed and known. However, Ryu et al. (2010) proposed a new general performance measure, EWRL, by modelling the size of the mean shift with a probability distribution function. We further generalize the measure by introducing a new random variable, T, which is the transition point of the mean shift. Based on that, we propose several ARL-based criteria to measure the chart performance and try them on several univariate control charts.
| Identifer | oai:union.ndltd.org:mcmaster.ca/oai:macsphere.mcmaster.ca:11375/15959 |
| Date | 11 1900 |
| Creators | Huh, Ick |
| Contributors | Viveros, Roman, Balakrishnan, Narayanaswamy, Mathematics and Statistics |
| Source Sets | McMaster University |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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