Dr. Dipayan Das Assistant Professor Dept. of Textile Technology Indian Institute of Technology Delhi Phone: Statistical Quality Control in Textiles Module 6: Non-Shewhart Control Charts

Introduction

Non-Shewhart Control Charts  Cumulative sum control chart (Cusum control chart)  Moving average control chart (MA control chart)  Exponentially weighted moving average control chart (EWMA control chart)

Cusum Control Chart

Overview [1, 2] The Shewhart control charts are relatively insensitive to the small shifts in the process, say in the order of about 1.5  or less. Then, a very effective alternative is an advanced control chart: Cumulative sum (CUSUM) control chart.

Illustration: Yarn Strength Data Consider the yarn strength (cN.tex -1 ) data as shown here. The first twenty of these observations were taken from a normal distribution with mean  =10 cN.tex -1 and standard deviation  =1cN.tex -1. The last ten observations were taken from a normal distribution with mean  =11 cN.tex-1 and standard deviation  =1cN.tex -1. The observations are plotted on a Basic Control Chart as shown in the next slide Sample No. i Strength x i (cN.tex -1 ) Sample No. i Strength x i (cN.tex -1 )

Illustration: Basic Control Chart CL UCL LCL UCL= LCL= CL= Process average is in control.

CUSUM: What is it? The cumulative sum (CUSUM) of observations is defined as When the process remains in control with mean at , the cumulative sum is a random walk with mean zero. When the mean shifts upward with a value  0 such that  >  0 then an upward or positive drift will be developed in the cumulative sum. When the mean shifts downward with a value  0 such that  <  0 then a downward or negative drift will be developed in the CUSUM.

Illustration: CUSUM i x i (cN.tex -1 ) C i (cN.tex -1 ) i x i (cN.tex -1 ) C i (cN.tex -1 )

Tabular CUSUM Upper CUSUM: Lower CUSUM: where K is called as reference value or the allowance. The tabular CUSUM works by accumulating deviations from  (the target value) that are above the target with one statistic C + and accumulating deviations from  (the target value) that are below the target with another statistic C -. These statistics are called as upper CUSUM and lower CUSUM, respectively.

Tabular CUSUM… If the shift  in the process mean value is expressed as where  1 denotes the new process mean value and  and  indicate the old process mean value and the old process standard deviation, respectively. Then, K is the one-half of the magnitude of shift. If either and exceed the decision interval H, the process is considered to be out of control. A reasonable value for H is five times the process standard deviation,

Illustration: Tabular CUSUM The counter records the number of successive points since rose above the value of zero.

Illustration: Tabular CUSUM The counter records the number of successive points since rose above the value of zero.

Illustration: CUSUM Status Chart Process average is not under control.

Concluding Remarks Although we have discussed the development of CUSUM chart for individual observations ( n =1), it is easily extended to the case of averages of samples where ( n >1). Simply replace by and by We have concentrated on cusums for sample averages, however, it is possible to develop cusums for other sample statistics such as ranges, standard deviations, number of defectives, proportion of defectives, and number of defects.

MA Control Chart

Moving Average: What is it? [2] Suppose the individual observations are x 1, x 2, x 3,… The moving average of span w at time t is defined as That is, at time period i, the oldest observation in the moving average set is dropped and the newest one added to the set. The variance of the moving average M i is

Control Limits If  denotes the target value of the mean used as the center line of the control chart, then the three-sigma control limits for M i are The control procedure would consists of calculating the new moving average Mi as each observation xi becomes available, plotting Mi on a control chart with upper and lower limits given earlier and concluding that the process is out of control if Mi exceeds the control limits. In general, the magnitude of the shift of interest and w are inversely related; smaller shifts would be guarded against more effectively by longer-span moving averages, at the expense of quick response to large shifts.

Illustration The observations x i of strength of a cotton carded rotor yarn for the periods 1  i  30 are shown in the table. Let us set-up a moving average control chart of span 5 at time i. The targeted mean yarn strength is 4.5 cN and the standard deviation of yarn strength is 0.5 cN.

Data Observations i Original Strength value (cN) x i Moving average of strength (cN) M i UCL (cN)LCL(cN)

Data (Contd.) Observations i Original Strength value (cN) x i Moving average of strength (cN) M i UCL (cN)LCL(cN)

Calculations The statistic M i plotted on the moving average control chart will be for periods i  5. For time periods i <5 the average of the observations for periods 1,2,…, i is plotted. The values of these moving averages are also shown in the table. The control limits for M i apply for periods i  5. They are For periods i <5 the control limits are given by The control limits are CL= LCL= UCL=

Control Chart i MiMi UCL LCL

Conclusion Note that there is no point that exceeds the control limits. Also note that for the initial periods i < w the control limits are wider than their final steady-state value. Moving averages that are less than w periods apart are highly correlated, which often complicates interpreting patterns on the control chart. This is clearly seen in the control chart.

Comparison Between Cusum Chart and MA Chart The MA control chart is more effective than the Shewhart control chart in detecting small process shifts. However, it is not as effective against small shifts as cusum chart. Nevertheless, MA control chart is considered to be simpler to implement than cusum chart in practice.

EWMA Control Chart

Exponentially Weighted Moving Average (EWMA): What is it? [2] Suppose the individual observations are x 1, x 2, x 3,… The exponentially weighted moving average is defined as where 0<  1 is a constant and z 0 = , where  is process mean.

What is it called “Exponentially Weighted MA”? It is known that Then, If we continue to substitute recursively for, we obtain The weights decrease geometrically with the age of sample mean. Furthermore, the weights sum to unity, since If =0.2 the weight assigned to current mean is 0.2 and the weights given to the preceding means are 0.16, 0.128, , and so forth.

What is it called “Exponentially Weighted MA”? The control limits are where L is known to be the width of the control chart. The choice of the parameters L and will be discussed shortly.

Choice of and L The choice of and L are related to average run length (ARL). ARL is the average number of points that must be plotted before a point indicates an out-of-control condition. So, ARL=1/ p, where p stands for the probability that any point exceeds the control limits. As we know, for three-sigma limit, p =0.0027, so ARL=1/0.0027=370. This means, even if the process is in control, an out-of-control signal will be generated every 370 samples, on the average. In order to detect a small shift in process mean, which is what is the goal behind the set-up of an EWMA control chart, the parameters and L are required to be selected to get a desired ARL performance. The following table illustrates this.

Choice of and L (Continued) Shift in mean (multiple of σ) Average Run Length (ARL) L =3.054 =0.40 L =2.998 =0.25 L =2.962 =0.20 L =2.814 =0.10 L =2.615 =

Choice of and L (Continued) As a rule of thumb, should be small to detect smaller shifts in process mean. It is generally found that work well in practice. It is also found that L=3 (3-sigma control limits) works reasonably well, particularly with higher values of. But, when is small, that is,  0.1, the choice of L between 2.6 and 2.8 is advantageous.

Illustration Let us take our earlier example of yarn strength in connection with MA control chart. Here, the process mean is taken as and process standard deviation is taken as We choose =0.1 and L =2.7. We would expect this choice would result in an in-control average run length equal to 500 and an ARL for detecting a shift of one standard deviation in the mean of ARL=10.3. The observations of yarn strength, EWMA values, and the control limit values are shown in the following table.

Table Observations i Original Strength value (cN) x i EWMAof strength (cN) M i UCL (cN)LCL(cN)

Table (Continued) Observations i Original Strength value (cN) x i EWMAof strength (cN) M i UCL (cN)LCL(cN)

Graph i MiMi UCL LCL

Conclusion Note that there is no point that exceeds the control limits. We therefore conclude that the process is in control.

Frequently Asked Questions & Answers

Frequency Asked Questions & Answers Q1: Why the non-Schwhart control charts are required to be set up? A1: The Shewhart control charts are relative insensitive to small shifts in the process mean, but the non-Shewhart control charts are expected to detect the small shifts in the process mean. Q2: Which one is the least effective in detecting small shift in the process mean: Cusum control chart, MA control chart, and EWMA control chart? A2: The MA control chart is generally found to be the least effective among the three in detecting small shift in the process mean. Q3: Which one is the most effective in detecting small shift in the process mean: Cusum control chart, MA control chart, and EWMA control chart? A3: The EWMA control chart and Cusum control charts can be made comparable depending upon the choice of parameters of the control charts.

Frequency Asked Questions & Answers Q4: What is the equivalence of MA control chart and EWMA control charts? A4: If =2/(w+1) for the EWMA control chart then it is equivalent to a w-period moving average control chart in the sense that the control limits are identical in steady state.

References 1.Leaf, G. A. V., Practical Statistics for the Textile Industry: Part II, The Textile Institute, UK, Montgomery, D. C., Introduction to Statistical Quality Control, John Wiley & Sons, Inc., Singapore, 2001.

Sources of Further Reading 1.Leaf, G. A. V., Practical Statistics for the Textile Industry: Part I, The Textile Institute, UK, Leaf, G. A. V., Practical Statistics for the Textile Industry: Part II, The Textile Institute, UK, Gupta, S. C. and Kapoor, V. K., Fundamentals of Applied Statistics, Sultan Chand & Sons, New Delhi, Gupta, S. C. and Kapoor, V. K., Fundamentals of Mathematical Statistics, Sultan Chand & Sons, New Delhi, Montgomery, D. C., Introduction to Statistical Quality Control, John Wiley & Sons, Inc., Singapore, Grant, E. L. and Leavenworth, R. S., Statistical Quality Control, Tata McGraw Hill Education Private Limited, New Delhi, Montgomery, D. C. and Runger, G. C., Applied Statistics and Probability for Engineers, John Wiley & Sons, Inc., New Delhi, 2003.