國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
Chapter 1. Introduction
1.1 Research Motivation
Sample median is a statistic better than the sample average to measure the population center when the underlying distribution is skewed, since its robustness to the outliers, see Graham, et al. (2010). When we are dealing with process control, in reality the distribution of the quality characteristic may be skewed, therefore it is reasonable to use median type control charts for process monitoring.
The Taguchi loss function, which is an important indicator in many industries, is a useful tool to measure the loss caused by deviation of the quality characteristic from its target value. The expectation of Taguchi loss consists of the population variance and the squared difference between population mean and the target value. Therefore it can be used for process control for change of variance and the difference between process mean and the target value.
So we consider using median loss (ML) statistic to construct several ML related control charts to monitor the change of process mean and variance simultaneously, assuming that the underlying distribution of the quality characteristic is skewed. We also construct sample average loss (AL) related control charts in order to compare the performance with those of the ML type control charts, and we will discuss the advantages and disadvantages of those proposed control charts, respectively.
1.2 Literature Review
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
studied. When the distribution of the quality characteristic is skewed, the use of those control charts may be misleading. Bai and Choi (1995) proposed a weighted variance (WV)
X
and R charts using the in-control probability that X smaller than its population mean to adjust the control limits of Shewhart typeX
and R charts when the distribution is skewed. Chan and Cui (2007) proposed a skewness correction (SC) method for constructingX
and R chart taking into consideration the degree of skewness of the process distribution, with no assumptions on the distribution. They found that when the process distribution is Weibull, lognormal, Burr or binomial distribution, simulation shows that the SC control chart has real false alarm rate closer to its theoretical value than the WV and Shewhart type charts.Graham, Human & Chakraborti (2010) proposed a nonparametric median type control chart. The monitoring statistic for each sample is the count of Xi less than the pooled median. The pooled median is an estimator of population median calculated from in-control samples during Phase I. They found that the median chart has in-control robustness to the outliers while
X
chart does not when the underlying distribution is skewed. They also found that for right-skewed distributions the median chart may perform better for positive shifts of process location. However, the median chart has poorer detection ability for shift of process location than theX
chart when the distribution is symmetric. Yang, Pai, & Wang (2010) compare some median type control charts, such as median ( X~) chart, EWMA- X~
chart and the CUSUM- X~ chart, to those of the
X
type control charts under normality assumption. They also found that the median control charts are robust to outliers while mean control charts are very sensitive to outliers.To measure the performance of a control chart, average run length (ARL) is widely used. ARL is unbiased for symmetrically distributed‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
monitoring statistic. However, the ARL1 of the control chart may exceed ARL0 when the underlying distribution is skewed. Pignatiello, Acosta, and Rao (1995) defined the concept of an ARL-unbiased control chart. That is, the control chart is said to be ARL-unbiased if its ARL curve achieves its maximum when the process parameter is equal to its in-control value. Sven and Manuel (2013) proposed an ARL-unbiased
S
2 chart and an EWMA-S
2 chart to monitor process dispersion. Pascual (2013) proposed combined individual and moving range charts with ARL-unbiasedness for monitoring changes in either the Weibull scale or shape parameter.The Taguchi loss function is developed by Taguchi (1986) to measure the loss caused by deviation of the quality characteristic from its target value. Yang (2013a) proposed an average loss chart to simultaneously monitor the process mean and the variability. Yang (2013b) proposed an EWMA average loss chart with variable sampling intervals (VSI) to monitor the changes in the difference of process mean and target and process variance. The out-of-control process diagnosis approach is also discussed. However, those charts were based on the normality assumption of the quality characteristic, and a loss control chart under skewed distribution has yet been discussed.
A control chart with VSI is a control technique that can reduce the out-of-control detection time when the process going out-of-control. The properties of the VSI
X
chart were first studied by Reynolds et al. (1988). Their work has been inspired for several other papers, for example Costa (1995) proposed aX
chart with Variable‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
process going out-of-control. Some other adaptive control schemes related literatures may refer to Yang (2013b).
1.3 Research Method
In this study we propose some median loss (ML) type control charts, assuming that the distribution of the quality characteristic (X) follows skew-normal distribution.
And we consider the distribution to be right-skewed, symmetric or left-skewed.
For fixed sampling interval (FSI) of the chart, we propose ML and EWMA-ML charts respectively to detect the process change of mean and variance simultaneously under skew-normal assumption. However, if the sampling intervals of the chart can be variable, an optimal VSI-ML chart is proposed to save the detection time when process going out-of-control. We also propose some average loss (AL) type control charts, including AL, EWMA-AL,optimal VSI-AL charts to compare the performance with ML type charts.