Modified Simple Robust Control Chart Based on Median Absolute Deviation

The control limits derived for the Median Absolute Deviation (MAD) based Standard deviation (S) control chart proposed by Abu-Shawiesh was for monitoring quality characteristics when a standard value of sigma (σ) is known or given by the management/ engineers. When sigma (σ) is unknown and we are interested in monitoring past/nonnormal data, then there is the need to modify the simple robust control limits. In this paper, the control limits for the Shewhart X̄ and S control chart based on median absolute deviation were modified using the concept of three sigma (3σ) limits. An evaluation performance tool was also developed to evaluate the efficiency of the modified control chart. An algorithm implemented on S-Plus programming language was developed to compute the two evaluation parameters used in this study. The results show that the control limits interval and the average run length for the modified control charts is smaller than that of the existing control charts. Therefore, the modified control limits is more efficient than the existing control limits. It is recommended that the modified control limits be used when monitoring past/non-normal data or when there is no standard value of sigma specify by the process engineer/ management.


Introduction
Statistical process control is commonly used as a tool in improving product quality through the achievement of process stability and capability.Monitoring and reducing variability in the process is a goal of statistical process control.With control charts, one of an on-line process control technique, changes of quality characteristics caused by common causes or assignable causes will be investigated.Then further corrective actions can be taken in order to remove those causes and return the process to the target with stable operation.Determination of the common or assignable causes of variation in control chart is possible with the use of control limits.Therefore, incorrect estimation of the control limits will lead to incorrect inference and thus wrong action.
Specifying the control limits is the most important step in designing a control chart.Improper estimation of the process dispersion which results in narrower or wider limits can increase the probability of type I error or the probability of type II error.When the limits are narrow, the risk of a point falling beyond the limits increases, falsely indicating that the process is out of control (Shahriari et al., 2009).When the limits are wider the risk increases the points falling within the limits, this falsely indicates that the process is out of control (Shahriari et al., 2009).The limits on a control chart can either be 0.001 probability limits or 3-sigma limits (Oakland 2008).The 0.001 probability limits were determined so that, if chance causes alone were at work the probability of a point falling above the upper limit would be one out of a thousand and the probability of points falling below the lower limit would be one out of a thousand.Therefore, 3-sigma limits are the practical equivalent of the 0.001 probability limits.In this paper, the 3-sigma limits approach will be adopted.
Several authors have worked on robust control charts, among them are Figueiredo and Gomes (2006;2009), Rocke (1992).Abu-Shawiesh (2008) derived the control limits for the standard deviation control chart using the median absolute deviation.However, the derived control limits was for situation when a standard value of sigma is known or specified by the management.Using such control limits to monitor a process where the standard value of sigma is unknown will lead to misplacement of control limits and hence false alarm of out of control.In this paper, we consider situation when the standard value of sigma is unknown and we are interested in monitoring past data.Thus, we modify the control limits earlier presented by Abu-Shawiesh (2008) for the S-control chart and Adekeye et al. (2012) for the X control chart.

Method
Let X i j represent a random sample of size n taken over m subgroup, i = 1, 2, ..., n and j = 1, 2, ..., m.The sample are assumed to be independent and taken from a continuous identical distribution functions.
If σ 2 is unknown, then an unbiased estimate of σ 2 is the sample variance (S 2 ) evaluated by If the underline distribution is normal, then an unbiased estimator of S is C 4 σ, where ).Therefore, the 3-sigma control limits for S-chart will be For simplification of Equation ( 1), let Then the parameter of Equation ( 1) becomes Suppose we use σ = S C 4 which is an unbiased estimator of σ, then the control limits for the S -chart in Equation (1) will be reduced to where S = 1 m m j=1 S j .For simplification of Equation (3), let , then the control limits for S -Chart for analyzing past data in Equation (3) becomes To monitor the process average, the 3-sigma control limits for monitoring past data (retrospective analysis) are where X = 1 m m j=1 X j .When S /C 4 is used to estimate σ, then the control limits on the corresponding X-charts will be Control Limits Interval (CLI) and the Average Run Length (ARL) for the X and S control charts using the modified control limits and the control limits of Abu-Shawiesh ( 2008) and Adekeye et al. (2012) are presented in Table 2.

Discussion
The evaluation parameters used in this study (the control limits interval and the average run length) show insiginificant difference for the S control charts when compared with the earlier results by Abu-shawiesh ( 2008) and Adekeye et al. (2012).However, for the X control charts, there is considerable improvement as seen in the control limits interval and the average run length for all the data set considered in this study.Furthermore, it can be seen that the average run length decreases as the data set deviates from normality.This is an indication that the median absolute deviation is more efficient and hence more appropriate than the standard deviation for highly skewed data.Thus, researchers or control chart users are advised to first carry out an exploratory data analysis on process data (real life data) before the use of control chart.This will help to guide against misplacement of control limits.

Conclusion
We have modified the simple robust control limits for monitoring process data when the standard value of sigma is not given and Median Absolute Deviation (MAD) is used to give an estimate.From the results in Table 2, it is clear that the average run length for the X chart for the modified control limits is more efficient than the earlier control limits for all the distributional data sets under consideration.Therefore, the modified control limits derived in this work for the X chart can be used when monitoring past/non-normal data or when there is no standard value of sigma specify by the process engineer/ management.

Table 2 .
Average Run Length(ARL) and Control Limits Interval (CLI)