Upper and Lower Control Limits for Monitoring Markoff’s Model and Non-Normal Variation with Known CV

AbstractControl charts are essential tools for quality monitoring and construction of this charts were based on the assumption that observations were independent and normally distributed. The assumption of independence of quality characteristic is questionable, as autocorrelation among the observations becomes an inherent characteristic in quality control data. This phenomenon of inherent autocorrelation can cause significant deterioration of control charting performance. Autocorrelation had a substantial and detrimental effect on the statistical properties of control charts. The setting of control limits to utilize on a control chart assumes the assumption of normality. However, in many situation the condition does not hold. The objective of this paper is to investigate the upper and lower control limit properties of the X control charts when the Markoff's model and non-normality exists in the process.


INTRODUCTION
Control charts were first proposed by Shewhart [32], and have become effective tools for improving process quality and productivity. A basic assumption in the applications of control charts is that the observations from the process at different times are independent random variables. However, the assumption of independence is often violated for processes in the chemical and pharmaceutical industries. Observations from these processes are always autocorrelated. When the control charts are developed under the independence assumption, the autocorrelated process results in decreasing the in-control average run length (ARL). For effective monitoring autocorrelated processes, one popular approach is to constructing control charts using the residuals from the time series model of the process data [2][3][4][5][6][7][8][9][10][11][12][13]. The properties of the proposed residual charts and their performance were investigated by Harris and Ross [12], Longnecker and Ryan [16], Yashchin [39], Kramer and Schmid [14], Schmid [27], Lin and Adams [15], Schmid [28], Padgett, Thombs and Padgett [23], Runger, Willemain and Prabhu [26], Vander Weil [34], Timmer, Pignatiello and Longnecker [33], Schmid [29], Zhang [40], Schmid and Schone [30], Alwan LC [2],Alwan and Roberts [3] and Lu and Reynolds [17]. Most of the papers on performance of control charts based on residuals has focused on the Shewhart control chart of residuals. The above papers on autocorrelated processes assume that imprecise measurement devices for process measurements are impossible. However, the performance of control charts and other statistical process control tools could be seriously affected when the process measurement includes the errors due to the measurement instrument. The effect of measurement error on the operating characteristics of and X chart, in cases where only the process mean shifts, is discussed by Bennett [5], Mizuno [20], Abraham and Kartha [1]. Kanazuka [13] and Mittag [18] investigate the effect of measurement error on the power characteristics of the X -R control chart where both the process mean and process spreadchange. Mittag and Stemann [19] extend the results of Mittag [18], referring to the X -S control chart. Rahim [24] analyzed the effects of imprecise measurement devices on the design parameters of the economic X control chart, Bhayare and Singh [8] investigated the effect of measurement error on the power of X charts with known coefficient of variation.Yang [38] is worked the effect of measurement error on the design parameters of the economic asymmetric X and S control charts. The above papers on control charts consider a single process control. Today, many industrial products are produced in several dependent processes. Consequently, it is not appropriate to monitor these processes by utilizing a control chart for each individual process. Zhang [41] proposes a simple cause selecting chart to monitor the second process of the two dependent processes. Wade and Woodall [35] reviewed the basic principles of the cause-selecting chart for two dependent processes and modify Zhang's approach, and give an example to illustrate the use of the individual X chart and the simple cause selecting chart. It is shown that their approach is better than that of Zhang for dependent processes control

II. THE MARKOFF'S PROCESS FOR A CONTROL CHART
Suppose that observations are taken from Markoff's process. The t th observation x t can be written as where  is the random process mean at sampling time t and  t is the random error. It is further assumed that  t can be defined as the Markoff's model process and can be expressed by where  1 is the autoregressive parameter which is the correlation measure between  t and  t-1 . The process is stationary if | 1 | <1. If  1 = 0, the random process means are independent. The variable  t is a random shock which is normally distributed with mean 0 and variance 2   and is independent of the other random errors and random shocks.
In which  1 and  2 are given by

IV. NUMERICAL ILLUSTRATION AND CONCLUSION
For various non-normal populations with various non-normality parameters ( 3 ,  4 ) and for different values of autocorrelation with varying known cv the values of upper and lower control limits are given in Table 1 to 4. For many continuous processes, there may correlation between observations that are closely spaced in time. It is observed that a control chart using various sample sizes always leads to different level of autocorrelation between consecutive samples. As the level of autocorrelation increases the difference between the limits are larger for positive autocorrelation, which indicates that it takes more time and more sample to detect the mean shifts. Based on the study, for the case of  1 close to 0, i.e., close to the independent case, the limits of X -chart is generally better than Markoff's model in detecting small mean shift. When autocorrelation is negative, the difference between the limits are reduced rapidly, which is good feature for apply X -chart. By comparing entries of Table 1 for independent case ( 1 = 0) with Table 2 to Table 4 when v=0, 2, 4, 6, one can easily see that the effect of autocorrelation is quite serious on the lower and upper control limits.

V. CONCLUSION
Thus we see that either the values of the upper limits or lower limits will serve the purpose. Just to avoid negative sign it is advisable to tabulate values of upper limits, since lower limits have always negative values. However, when the autocorrelation is high in the process, the control limits feature is worthless in the detection ability of the chart, and the advantage of the mean chart is reduced. Non-normality parameters are less effective than the Markoff's process.