Performance of Moving Average (MA) Chart Under Three Delta Control Limits and Six Delta Initiatives

Published online: March 6, 2019 An SQC chart is a graphical tool for representation of the data for knowing the extent of variations from the expected standard. This technique was first suggested by W.A. Shewhart of Bell Telephone Company based on 3σ limits. M. Harry, the engineer of Motorola has introduced the concept of six sigma in 1980. In 6σ limits, it is presumed to attain 3.4 or less number of defects per million of opportunities. Naik V.D and Desai J.M proposed an alternative of normal distribution, which is named as moderate distribution. The parameters of this distribution are mean and mean deviation. Naik V.D and Tailor K.S. have suggested the concept of 3-delta control limits and developed various control charts based on this distribution. Using these concepts, control limits based on 6-delta is suggested in this paper. Also the moving average chart is studied by using 6-delta methodology. A ready available table for mean deviation is prepared for the quality control experts for taking fast actions.


Introduction
The conventional quality control charts developed by (Shewhart 1931) were based on normality assumptions and control limits are calculated using 3 times standard deviation distance from the expected level of quality. The six sigma approach was first suggested Mikel Harry in 1980. At that time he was working as an engineer in Motorola. (Radhakrishnan and Balamurugan 2010, 2011 have developed various types of control charts based on six sigma approach. (Naik and Desai 2015) have proposed an alternative of normal distribution called moderate distribution, which has location parameter as mean ( µ 0 ) and scale parameter as mean deviation( δ ). (Naik and Tailor 2015) have suggested 3 δ (3 mean deviation) control limits based on moderate distribution. On the basis of 3 δ control limits, they have developed X -chart, R-chart, s-chart and d-chart (Tailor 2016) has also developed moving average and moving range chart and exponentially moving average chart under moderateness assumption.
Similar to six sigma concept, six delta concepts is suggested by Tailor K.S. The six sigma control limits are based normality assumption and the control limits are determined by using standard deviation ( σ -sigma) of the statistic, whereas the six delta control limits are based on moderateness assumption and the control limits are determined by using mean deviation ( δ -delta) of the statistic. In six sigma approach, it is presumed to attain 3.4 or less number of defects per million of opportunities whereas in six delta approach, it is presumed to attain 1.7 or less number of defects per million of opportunities.  has proposed sample standard deviation(s) chart, sample mean deviation (d) chart and exponentially weighted moving .average (EWMA) chart based on six delta initiatives. In this paper, control limits based on 6-delta is suggested. Also the moving average chart under moderateness assumption is studied by using 6-delta methodology. A ready available table for mean deviation is also prepared for the quality control experts for taking fast actions.

A. upper specification limit (U.S.L)
It is the acceptable maximum value of an item suggested by the quality control expert.

B. lower specification limit (L.S.L)
It is the acceptable minimum value of an item suggested by the quality control expert.

F. Quality Control Constant (S md )
The constant S md is introduced to computzze six delta based control limits for the said chart.

G. Span (w)
This is a value, computed from the two subsequent moving averages.

Three Delta Control Limits for Moving Average Chart
Suppose that the main variable of the process x follows moderate distribution. The mean of x is E(x) = µ 0 and mean deviation of x is δ δ , ,…… are the sample observations taken from the production process. The moving average of span w at time i is defined as The variance of the moving average M i is derived as, Hence, its standard deviation = σ w So, its mean deviation = π δ 2 w On the basis of 3 δ criteria suggested by Naik and Tailor the control limits for proposed chart can be represented as follows.
Where µ 0 is specified value of the mean and δ is the process mean error.

Six Delta Based Control Limits for Moving Average Chart
First of all, we have to fix the level of tolerance (T.L) and process capability (C p ) to determine the process mean deviation δ (termed as δ δ 6 ) , which is calculated from Cp =

An Empirical Study for Moving Average Chart and Comparison of Three Delta Limits Against Six Delta Initiatives
To illustrate the use of moving average chart with three delta and six delta limits, a data set is taken from (Montgomery 2007). The data, together with the corresponding moving averages of span five (w = 5) are shown in Table 1. The target mean is taken to be 0 and process mean deviation is taken to be 1. Three delta and six delta control limits are computed from this data set, and control charts are plotted under these two limits.

(a) Three delta control limits for M. A chart:
Here the target mean ( µ 0 ) is taken as 10, process mean deviation ( δ ) is taken to be 1 and w = 5. The three delta control limits are computed using equations (7) , . ,

Summary and Conclusion
In this paper, moving average chart is discussed under three delta and six delta control limits with an illustration. From figure 1, it can be seen that the production process is in statistical control when we are applying 3-delta control limits but the process is out of control in six-delta control limits. If we compare the UCL and LCL of both the types of charts, six delta control limits are always smaller than the three delta control limits. So it can be concluded that the chart based on six delta control limits are more effective towards detecting the shift in the value of moving averages than the charts under three delta control limits. This is a next generation control chart technique and it will replace existing six sigma technique. So it is recommended that the control charts under six delta control limits should be used for the best results.