Chapter III1

Chapter III : Other statistical process-monitoring and control techniques

3. Other statistical process-monitoring and control techniques3.1. Cumulative sum and Exponentially weighted Moving average control charts3.2. Other univariate statistical process monitoring and control techniques3.3. Multivariate process monitoring and control

Shewhart conventional control chart, have been in use for well over fifty years. It is a basic method of statistical process control. In quality control, there are many situations which have complicate and sometimes the conventional charts will not in use and we need to have some special charts for some occasions. However the increasing emphasis on variability reduction leads enhancement, and process improving has led to the development of many innovative techniques for statistical process monitoring and for control.

Now we can list some of the control charts that are used in special situations:

1. Cumulative sum chart (CUSUM- chart)2. Exponentially Weighted Moving Average Control charts (EWMA- control chart)3. Statistical process control for short production runs4. Modified and Acceptance control charts5. Control charts for Multiple stream processes6. Statistical process control with Auto correlated Process Data7. Adaptive sampling procedures8. Economic design of control charts9. Over of other procedures10. Multivariate quality control problem11. Hotelling T2 control chart12. Multivariate EWMA control chart13. Regression Adjustment 14. Control charts for Monitoring variability15. Latent structure Methods

CUSUM control chart and EWMA control charts

A major disadvantage of any shewhart control chart is, it uses the information about the process contained in the last plotted point, further, it ignores any information given by the sequence of points. This feature makes the shewhart control chart with small shifts in process, the criteria for runs and the warning limits provides supplemented sensitizing rules reduces the simplicity . It helps to interpretation of the control charts in easy way.

Two effective and alternatives to dealt small shifts are Cumulative sum control chart and exponentially weighted Moving average control chart.

A.R.Muralidharan. SQC Lecture Notes( Stat3142) Wollega UniversityPage 1


THE CUSUM control chart

Shewhart control charts and it failed to detect the shift. The reason for this failure is the relatively small magnitude of the shift. This shewhart control charts for average is not very effective in small shifts and it is very effective in the magnitude of the shift is 1.5 sigma to 2sigma or over.

There are major differences between CUSUM charts and other control (Shewhart) charts:

•A Shewhart control chart plots points based on information from a single subgroup sample. In cusum charts, each point is based on information from all samples taken up to and including the current subgroup.

•On a Shewhart control chart, horizontal control limits define whether a point signals an out-of-control condition. On a cusum chart, the limits can be either in the form of a V-mask or a horizontal decision interval.

•The control limits on a Shewhart control chart are commonly specified as 3σ limits. On a cusum chart, the limits are determined from average run length, from error probabilities, or from an economic design.

Cumulative Sum (Cusum) charts display cumulative sums of subgroup or individual measurements from a target value. Cusum charts are graphical and analytical tools for deciding whether a process is in a state of statistical control and for detecting a shift in the process mean. The CUSUM control charts is good in small shifts, It directly taken all the information in the sequence of sample values by plotting the cumulative sums of the deviations of the sample values from the target value.

The CUSUM control chart were first proposed by Page in 1954 and developed by some others. It is possible to devise cumulative sum procedures for other variables and we can used for monitoring process variability. We concentrate on the cumulative sum chart for the process mean.

CUSUM works as follows: Let us collect m samples, each of sizen, and compute the mean of each sample. Then the cumulative sum (CUSUM) control chart is formed by plotting one of the following quantities:

against the sample number m, where is the estimate of the in-control mean and is the known (or estimated) standard deviation of the sample means. The choice of which of these two quantities is plotted is usually determined by the statistical software package.

In either case, as long as the process remains in control centered at , the CUSUM plot will show variation in a random pattern centered about zero. If the process mean shifts upward, the charted CUSUM points will eventually drift upwards, and vice versa if the process mean decreases.



Suppose if the given sample size is greater than or equal to one and the mean of the jth sample is given by x j , then the target for the process mean is μ0. The CUSUM control chart is formed by plotting the

quantity C i=∑j=1

i

( x¿¿ j−μ0)¿ against the sample i. Cj is called the cumulative sum up to and including

the ith sample. Because they combine information from several samples, cumulative sum charts are more effective than shewhart charts for detecting small process shifts. There are more effective with samples of size as one. This makes CUSUM charts as a good tool where rational subgrous are frequently of size one and in discrete parts manufacturing also in on-line control . the CUSUM defined in above is a random walk with mean zero if the process remains in control at the target value.

If the mean shifts upward to some valueμ1>μ0, then an upward or positive drift will develop in the CUSUM cj. Conversely, if the mean shifts downward to some μ1<μ0 then a downward or negative drift in Cj will develop. Thus, if a trend develops in the plotted points either upward or downward, we should consider this as evidence that the process mean has shifted, and a search for some assignable cause should be performed. The starting value of CUSUM will be Zero.

There are two ways to represent CUSUMs

1. Tabular or algorithmic CUSUMs2. The V-mask form of CUSUMs

Among the two forms “The tabular CUSUMs” is preferable.

A. Tabular or Algorithmic CUSUM for monitoring the process mean

Most users of CUSUM procedures prefer tabular charts over theV-Mask. The V-Mask is actually a carry-over of the pre-computer era. The tabular method can be quickly implemented by standard spreadsheet software.

CUSUMs may be constructed both for individual observations and for the averages of rational subgroups. It is very often in practice, the case of individual observations.

To generate the tabular form we use the h and k parameters expressed in the original data units. It is also possible to use sigma units.

The following quantities are calculated:

Shi(i) = max(0, Shi(i-1) + xi - - k) ; Slo(i) = max(0, Slo(i-1) + - k - xi) )

where Shi(0) and Slo(0) are 0. When either Shi(i) or Slo(i) exceeds h, the process is out of control.

As individual observation cusums are described as below:



Let xi be the ith observation on the process. When the process is in control, xi has a normal distribution with mean μ0with standard deviationσ . We assume that the standard deviation is known or estimated and it is available. The target value μ 0 is the value for the quality characteristics x. if the process drifts or shifts off this target value, the CUSUMs will signal, and an adjustment made to some manipulatable variable to bring the process on target. Further in some cases a signal from a cusums indicates the presence of an assignable cause that must be investigated just as in the Sherhart charts .

The tabular CUSUM works by accumulating derivations from the target value that are above target with

one statistics C +

and accumulating derivations form target value that are below target with one statistics

C-

these statistics are called one-sided upper and lower CUSUMs respectively.

They are computed as follows:

In the above table K is usually called the reference value or allowance value or slack value and it is often chosen about halfway between the target value and out-of-control value of the mean μ 1 that we are interested in detecting quickly. It is noted that C+ and C- accumulate deviations from the target value that are greater thank, with both quantities reset to Zero on becoming negative. If either C+ or C- exceed the decision interval H, the process is considered to be out of control. It is very important that the two parameters have to be selected properly, as it has substantial impact on the performance of the CUSUMs. A reasonable value for H is five times the process standard deviation.

The tabular cusum also indicates when the shift probably occurred. The counter N+ records the number of consecutive periods since the upper-side cusum C+ rose above the value of zero. It is useful to present a graphical display for the tabular cusum. These charts are sometimes called Cusum status charts. They are constructed by plotting C+ and C- versus the sample number. Each vertical bar represents the value of C+ and C- in period i. with the decision interval plotted on the chart, the cusum status chart resembles a shewhart control chart. We have also plotted the observations xi for each period on the CUSUM status charts as the solid dots. This frequently helps the user of the control chart visualize the actual process performance that has led to a particular value of the cusums. It is identical to that with any chart, that the action taken a decision of an out-of-control signal on a CUSUM chart schemes; one should search non-random causes and to take any corrective action required, then reinitialized the cusum at Zero. This chart is in determining when the assignable cause has occurred. Just count backward from the out of control signal to the time period when the cusum lifted above zero to find the first period following the process sifts. The counters N+ and N- are used in this capacity

An estimate of the new process mean may be helpful when an adjustment to some manipulatable variable is required in order to bring the process back to the target value. This can be computed from

μ̂=¿

Finally, we can discuss the runs tests, in CUSUM and other sensitizing rules:



The Zone rules, cannot be safely applied, because successive values of C+ and C- are dependent. In fact, the cusum can be thought of as a weighted average, where the weights ae random.

Recommendation for CUSUM Designs

The important parameters to design the tabular cusum chart s are the reference K and the decision interval H. It is usually recommended that these parameters be selected to provide good average run length performance. There have been many innovative study of cusum ARL performance. Now some discussion based on selecting H and K.

Define H and K as shown in above table as to provide a cusum that has good ARL properties against a shift of about 1sigma in the process mean.

We will construct a CUSUM tabular chart for the example described above. For this example, the parameter are h = 4.1959 and k = 0.3175. Using these design values, the tabular form of the example is

h k

325 4.1959 0.3175

Increase in mean Decrease in mean Group x x-325 x-325-k Shi 325-k-x Slo CUSUM

1 324.93 -0.07 -0.39 0.00 -0.24 0.00 -0.0072 324.68 -0.32 -0.64 0.00 0.01 0.01 -0.403 324.73 -0.27 -0.59 0.00 -0.04 0.00 -0.674 324.35 -0.65 -0.97 0.00 0.33 0.33 -1.325 325.35 0.35 0.03 0.03 -0.67 0.00 -0.976 325.23 0.23 -0.09 0.00 -0.54 0.00 -0.757 324.13 -0.88 -1.19 0.00 0.56 0.56 -1.628 324.53 -0.48 -0.79 0.00 0.16 0.72 -2.109 325.23 0.23 -0.09 0.00 0.54 0.17 -1.8710 324.60 -0.40 -0.72 0.00 0.08 0.25 -2.2711 324.63 -0.38 -0.69 0.00 0.06 0.31 -2.6512 325.15 0.15 -0.17 0.00 0.47 0.00 -2.5013 328.33 3.32 3.01 3.01 -3.64 0.00 0.8314 327.25 2.25 1.93 4.94* -0.57 0.00 3.0815 327.83 2.82 2.51 7.45* -3.14 0.00 5.9016 328.50 3.50 3.18 10.63* -3.82 0.00 9.40



17 326.68 1.68 1.36 11.99* -1.99 0.00 11.0818 327.78 2.77 2.46 14.44* -3.09 0.00 13.8519 326.88 1.88 1.56 16.00* -2.19 0.00 15.7320 328.35 3.35 3.03 19.04* -3.67 0.00 19.08

* = out of control signalThe Average Run Length of Cumulative Sum Control ChartsThe operation of obtaining samples to use with a cumulative sum (CUSUM) control chart consists of taking samples of size n and plotting the cumulative sums

versus the sample number r, where is the sample mean and k is a reference value.

In practice, k might be set equal to ( + 1)/2, where is the estimated in-control mean, which is sometimes known as the acceptable quality level, and 1 is referred to as the rejectable quality level.If the distance between a plotted point and the lowest previous point is equal to or greater than h, one concludes that the process mean has shifted (increased).Hence, h is referred to as the decision limit. Thus the sample size n, reference value k, and decision limit h are the parameters required for operating a one-sided CUSUM chart. If one has to control both positive and negative deviations, as is usually the case, two one-sided charts are used, with respective values k1, k2, (k1 > k2) and respective decision limitsh and -h.The shift in the mean can be expressed as - k. If we are dealing with normally distributed measurements, we can standardize this shift by

Similarly, the decision limit can be standardized by

The average run length (ARL) at a given quality level is the average number of samples (subgroups) taken before an action signal is given. The standardized parameters ks and hs together with the sample size n are usually selected to yield approximate ARL's L0 and L1 at acceptable and rejectable quality levels 0 and 1 respectively. We would like to see a high ARL, L0, when the process is on target, (i.e. in control), and a low ARL, L1, when the process mean shifts to an unsatisfactory level.In order to determine the parameters of a CUSUM chart, the acceptable and rejectable quality levels along with the desired respective ARL ' s are usually specified. The design



parameters can then be obtained by a number of ways. Unfortunately, the calculations of the ARL for CUSUM charts are quite involved.There are several nomographs available from different sources that can be utilized to find the ARL's when the standardized h and k are given. Some of the nomographs solve the unpleasant integral equations that form the basis of the exact solutions, using an approximation of Systems of Linear Algebraic Equations (SLAE). This Handbook used a computer program that furnished the required ARL's given the standardized h and k. An example is given below:

mean shift Shewart(k = .5) 4 5

0 336 930 371.00.25 74.2 140 281.14.5 26.6 30.0 155.22.75 13.3 17.0 81.221.00 8.38 10.4 44.01.50 4.75 5.75 14.972.00 3.34 4.01 6.302.50 2.62 3.11 3.243.00 2.19 2.57 2.004.00 1.71 2.01 1.19

If k = .5, then the shift of the mean (in multiples of the standard deviation of the mean) is obtained by adding .5 to the first column. For example to detect a mean shift of 1 sigma at h = 4, the ARL = 8.38. (at first column entry of .5).The last column of the table contains the ARL's for a Shewhart control chart at selected mean shifts. The ARL for Shewhart = 1/p, where p is the probability for a point to fall outside established control limits. Thus, for 3-sigma control limits and assuming normality, the probability to exceed the upper control limit = .00135 and to fall below the lower control limit is also .00135 and their sum = .0027. (These numbers come from standard normal distribution tables or computer programs, setting z = 3). Then the ARL = 1/.0027 = 370.37. This says that when a process is in control one expects an out-of-control signal (false alarm) each 371 runs.When the means shifts up by 1 sigma, then the distance between the upper control limit and the shifted mean is 2 sigma (instead of 3 ). Entering normal distribution tables with z = 2 yields a probability of p = .02275 to exceed this value. The distance between the shifted mean and the lower limit is now 4 sigma and the probability of < -4 is only .000032 and



can be ignored. The ARL is 1 / .02275 = 43.96 .The conclusion can be drawn that the Shewhart chart is superior for detecting large shifts and the CUSUM scheme is faster for small shifts. The break-even point is a function of h, as the table shows.

V-Mask CUSUM Procedure

A visual procedure proposed by Barnard in 1959, known as the V-Mask, is sometimes used to determine whether a process is out of control. An alternative procedure to use of a tabular CUSUM is the V-mask control scheme. The V-mask is applied to successive values of the

cusum statistic C i=∑j=1

i

y j= y i+Ci−1 where yi is the standardized observation y i=x i−μ0

σ. A typical value

v-mask is shown below. The decision procedure consists of placing the V-mask on the CUSUM chart with the point as shown in the diagram. More often, the tabular form of the V-Mask is preferred A V-Mask is an overlay shape in the form of a V on its side that is superimposed on the graph of the cumulative sums. The origin point of the V-Mask diagram below is placed on top of the latest cumulative sum point and past points are examined to see if any fall above or below the sides of the V. As long as all the previous points lie between the sides of the V, the process is in control. Otherwise (even if one point lies outside) the process is suspected of being out of control.




In the diagram above, the V-Mask shows an out of control situation because of the point that lies above the upper arm. By sliding the V-Mask backwards so that the origin point covers other cumulative sum data points, we can determine the first point that signaled an out-of-control situation. This is useful for diagnosing what might have caused the process to go out of control. From the diagram it is clear that the behavior of the V-Mask is determined by the distance k (which is the slope of the lower arm) and the rise distance h. These are the design parametersof the V-Mask. Note that we could also specify d and the vertex angle (or, as is more common in the literature, θ = 1/2 of the vertex angle) as the design parameters, and we would end up with the same V-Mask.

In practice, designing and manually constructing a V-Mask is a complicated


Cusum Charts Compared with Shewhart Charts

Although cusum charts and Shewhart charts are both used to detect shifts in the proces mean, there are important differences in the two methods.

Each point on a Shewhart chart is based on information for a single subgroup sample or measurement. Each point on a cusum chart is based on information from all samples (measurements) up to and including the current sample (measurement).

On a Shewhart chart, upper and lower control limits are used to decide whether a point signals an out-of-control condition. On a cusum chart, the limits take the form of a decision interval or a V-mask.

On a Shewhart chart, the control limits are commonly computed as 3 limits. On a cusum chart, the limits are determined from average run length specifications, specified error probabilities, or an economic design.

A cusum chart offers several advantages over a Shewhart chart. A cusum chart is more efficient for detecting small shifts in the process mean, in particular, shifts of 0.5

to 2 standard deviations from the target mean. Lucas (1976) noted that "a V-mask designed to detect a shift will detect it about four times as fast as a competing Shewhart chart."

Shifts in the process mean are visually easy to detect on a cusum chart since they produce a change in the slope of the plotted points. The point at which the slope changes is the point at which the shift has occurred.These advantages are not as pronounced if the Shewhart chart is augmented by the tests for special causes described by Nelson (1984,1985). Also see Tests for Special Causes. Moreover,

cusum schemes are more complicated to design. a cusum chart can be slower to detect large shifts in the process mean. it can be difficult to interpret point patterns on a cusum chart since the cusums are correlated.




http://support.sas.com/documentation/cdl/en/qcug/63922/HTML/default/qcug_shewhart_a0000003912.htm

http://support.sas.com/documentation/cdl/en/qcug/63922/HTML/default/qcug_cusum_sect001.htm#nels_l_85

http://support.sas.com/documentation/cdl/en/qcug/63922/HTML/default/qcug_cusum_sect001.htm#nels_l_84

http://support.sas.com/documentation/cdl/en/qcug/63922/HTML/default/qcug_cusum_sect001.htm#luca_j_76


The Exponentially Weighted Moving Average Control charts( EWMA)

The exponentially weighted moving average (EWMA) is a statistic for monitoring the process that averages the data in a way that gives less and less weight to data as they are further removed in time from the current measurement. The data Y1, Y2, ... , Yt are the check standard measurements ordered in time. The EWMA statistic at time t is computed recursively from individual data points, with the first EWMA statistic, EWMA1, being the arithmetic average of historical data.

The EWMA control chart can be made sensitive to small changes or a gradual drift in the process by the choice of the weighting factor, . A weighting factor of 0.2 - 0.3 is usually suggested for this purpose (Hunter), and 0.15 is also a popular choice.

Because it takes time for the patterns in the data to emerge, a permanent shift in the process may not immediately cause individual violations of the control limits on a Shewhart control chart. The Shewhart control chart is not powerful for detecting small changes, say of the order of 1 - 1/2 standard deviations. The EWMA (exponentially weighted moving average) control chart is better suited to this purpose

This chart is a good alternative to the Shewhart control chart when we are in detecting small shifts. The process of this chart is mostly same of the CUSUM chart and in some ways it is easy to construct and operate. As the same of the CUSUM , the EWMA s typically used with individual observations and It is a type of control chart used to monitor either variables or attributes-type data using the monitored business or industrial process's entire history of output.[1] While other control charts treat rational subgroups of samples individually, the EWMA chart tracks the exponentially-weighted moving average of all prior sample means. EWMA weights samples in geometrically decreasing order so that the most recent samples are weighted most highly while the most distant samples contribute very little.

Although the normal distribution is the basis of the EWMA chart, the chart is also relatively robust in the

face of non-normally distributed quality characteristics. There is, however, an adaptation of the chart that

accounts for quality characteristics that are better modeled by the Poisson distribution. The chart monitors

only the process mean; monitoring the process variability requires the use of some other technique.

The EWMA control chart requires a knowledgeable person to select two parameters before setup:





1. The first parameter is λ, the weight given to the most recent rational subgroup mean. λ must

satisfy 0 < λ ≤ 1, but selecting the "right" value is a matter of personal preference and experience.

One source recommends 0.05 ≤ λ ≤ 0.25,[2]:411while another recommends 0.2 ≤ λ ≤ 0.3.

2. The second parameter is L, the multiple of the rational subgroup standard deviation that

establishes the control limits. L is typically set at 3 to match other control charts, but it may be

necessary to reduce L slightly for small values of λ.

Instead of plotting rational subgroup averages directly, the EWMA chart computes successive

observations zi by computing the rational subgroup average, , and then combining that new subgroup

average with the running average of all preceding observations, zi - 1, using the specially–chosen weight, λ,

as follows:

.

The control limits for this chart type are where T and S

are the estimates of the long-term process mean and standard deviation established during control-

chart setup and n is the number of samples in the rational subgroup. Note that the limits widen for

each successive rational subgroup, approaching .[2]:407

The EWMA chart is sensitive to small shifts in the process mean, but does not match the ability of

Shewhart-style charts (namely the and R and and s charts) to detect larger shifts.[2]:412 One author

recommends superimposing the EWMA chart on top of a suitable Shewhart-style chart with widened

control limits in order to detect both small and large shifts in the process mean

The target or center line for the control chart is the average of historical data. The upper (UCL) and lower (LCL) limits are

where s times the radical expression is a good approximation to the standard deviation of the EWMA statistic and the factor k is chosen in the same way as for the Shewhart control chart -- generally to be


http://en.wikipedia.org/wiki/Xbar_and_R_chart

http://en.wikipedia.org/wiki/Xbar_and_s_chart


2 or 3.The implementation of the EWMA control chart is the same as for any other type of control procedure. The procedure is built on the assumption that the "good" historical data are representative of the in-control process, with future data from the same process tested for agreement with the historical data. To start the procedure, a target (average) and process standard deviation are computed from historical check standard data. Then the procedure enters the monitoring stage with the EWMA statistics computed and tested against the control limits. The EWMA statistics are weighted averages, and thus their standard deviations are smaller than the standard deviations of the raw data and the corresponding control limits are narrower than the control limits for the Shewhart individual observations chart.Data collectionDepiction of check standard measurements with J = 4 repetitions per day on the surface of a silicon wafer over K days where the repetitions are randomized over position on the wafer

K days - 4 repetitions

2-level design for measurements on a check standard

For J measurements on each of K days, the measurements are denoted by

The check standard value for the kth day is

The accepted value, or baseline for the control chart, is



The process standard deviation is

Check standard measurements should be structured in the same way as values reported on the test items. For example, if the reported values are averages of two measurements made within 5 minutes of each other, the check standard values should be averages of the two measurements made in the same manner.Averages and short-term standard deviations computed from J repetitions should be recorded in a file along with identifications for all significant factors. The best way to record this information is to use one file with one line (row in a spreadsheet) of information in fixed fields for each group. A list of typical entries follows:

1. Month2. Day3. Year4. Check standard identification5. Identification for the measurement design (if applicable)6. Instrument identification7. Check standard value8. Repeatability (short-term) standard deviation from J repetitions9. Degrees of freedom10. Operator identification11. Environmental readings (if pertinent)

Monitoring bias and long-term variability

Once the baseline and control limits for the control chart have been determined from historical data, and any bad observations removed and the control limits recomputed, the measurement process enters the monitoring stage. A Shewhart control chart and EWMA control chart for monitoring a mass calibration process are shown below. For the purpose of comparing the two techniques, the two control charts are based on the same data where the baseline and control limits are computed from the data taken prior to 1985. The monitoring stage begins at the start of 1985. Similarly, the control limits for both charts are 3-standard deviation limits. The check standard data and analysis are explained more fully in another section.


http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3521.htm




In the EWMA control chart below, the control data after 1985 are shown in green, and the EWMA statistics are shown as black dots superimposed on the raw data. The EWMA statistics, and not the raw data, are of interest in looking for out-of-control signals. Because the EWMA statistic is a weighted average, it has a smaller standard deviation than a single control measurement, and, therefore, the EWMA control limits are narrower than the limits for the Shewhart control chart shown above.

The control strategy is based on the predictability of future measurements from historical data. Each new check standard measurement is plotted on the control chart in real time. These values are expected to fall within the control limits if the process has not changed. Measurements that exceed the control limits are probably out-of-control and require remedial action. Possible causes of out-of-control signals need to be understood when developing strategies for dealing with outliers.The control chart should be viewed in its entirety on a regular basis] to identify drift or shift in the process. In the Shewhart control chart shown above, only a few points exceed the control limits. The small, but significant, shift in the process that occurred after 1985 can only be identified by examining the plot of control measurements over time. A re-analysis of the kilogram check standard data shows that the control limits for the Shewhart control chart should be updated based on the the data after 1985. In the EWMA control chart, multiple violations of the control limits occur after 1986. In the




calibration environment, the incidence of several violations should alert the control engineer that a shift in the process has occurred, possibly because of damage or change in the value of a reference standard, and the process requires review.

Remedial actions

There are many possible causes of out-of-control signals.

A. Causes that do not warrant corrective action for the process (but which do require that the current measurement be discarded) are:

1. Chance failure where the process is actually in-control2. Glitch in setting up or operating the measurement process3. Error in recording of data

B. Changes in bias can be due to:

1. Damage to artifacts2. Degradation in artifacts (wear or build-up of dirt and mineral deposits)

C. Changes in long-term variability can be due to:

1. Degradation in the instrumentation2. Changes in environmental conditions3. Effect of a new or inexperienced operator

An immediate strategy for dealing with out-of-control signals associated with high precision measurement processes should be pursued as follows:1. Repeat the measurement sequence to establish whether or not the out-of-control signal was

simply a chance occurrence, glitch, or whether it flagged a permanent change or trend in the process.

2. With high precision processes, for which a check standard is measured along with the test items, new values should be assigned to the test items based on new measurement data.

3. Examine the patterns of recent data. If the process is gradually drifting out of control because of degradation in instrumentation or artifacts, then:

o Instruments may need to be repairedo Reference artifacts may need to be recalibrated.

4. Reestablish the process value and control limits from more recent data if the measurement



process cannot be brought back into control.

When to Use an EWMA Chart

EWMA (or Exponentially Weighted Moving Average) Charts are generally used for detecting small shifts in the process mean. They will detect shifts of .5 sigma to 2 sigma much faster than Shewhart charts with the same sample size. They are, however, slower in detecting large shifts in the process mean. In addition, typical run tests cannot be used because of the inherent dependence of data points.

EWMA Charts may also be preferred when the subgroups are of size n=1. In this case, an alternative chart might be the Individual X Chart, in which case you would need to estimate the distribution of the process in order to define its expected boundaries with control limits. The advantage of Cusum, EWMA and Moving Average charts is that each plotted point includes several observations, so you can use the Central Limit Theorem to say that the average of the points (or the moving average in this case) is normally distributed and the control limits are clearly defined.

When choosing the value of lambda used for weighting, it is recommended to use small values (such as 0.2) to detect small shifts, and larger values (between 0.2 and 0.4) for larger shifts. An EWMA Chart with lambda = 1.0 is an X-bar Chart.

EWMA charts are also used to smooth the affect of known, uncontrollable noise in the data. Many accounting processes and chemical processes fit into this categorization. For example, while day to day fluctuations in accounting processes may be large, they are not purely indicative of process instability. The choice of lambda can be determined to make the chart more or less sensitive to these daily fluctuations.

A modified EWMA control charts may be used for autocorrelated processes with a slowly drifting mean. The wandering mean case has been presented by Montgomery and Mastrangelo (Journal of Quality Technology, July 1991, vol. 23, No. 3, pp. 179-193) for processes that are positively autocorrelated and the mean does not drift too fast. Subgroup size for the wandering mean case is limited to n=1, since the subgroup range would not provide a meaningful indicator of process variation when observations are autocorrelated.

As with other control charts, EWMA charts are used to monitor processes over time. The x-axes are time based, so that the charts show a history of the process. For this reason, you must have data that is time-ordered; that is, entered in the sequence from which it was generated. If this is not the case, then trends or shifts in the process may not be detected, but instead attributed to random (common cause) variation.

Other Univariate Statistical Process Monitoring and Control Techniques

As we seen the basics of SPC methods and some other special methods and we see an overview of some of the more useful recent developments. We can start with a discussion of SPC methods for short production runs and modified for situations. Although there are other techniques that can be applied to the short-run scenario, this approach seems to be most widely used in practice. These techniques find some application in situations where process capability is high, such as the “Six-sigma” manufacturing environment. Multiple stream processes are encountered in many industries.


http://www.qualityamerica.com/Knowledgecenter/qualityimprovementtools/autocorrelation_charts.asp


SPC methods have found wide applications in almost every type of Industrial applications. Some of the most interesting applications occur in job-shop manufacturing systems, or generally in any type of system characterized by short production runs. Some SPC methods for these situations are straight forward adoptions of the standard concepts and require no new methodology. In these situations one of the basic techniques of control charting used in the short-un environment using deviation from the nominal dimension as the variable on the control chart. Now we present a summary of several techniques of univariate SPC monitoring and control techniques that have proven successful in short production run situations.

A. x∧R charts for short productionruns

Statistical process-control methods have found wide application in almost every type of business. Some of the most interesting applications occur in job-shop manufacturing systems, or generally in any type of system characterized by short production runs. Some of the SPC methods for these situations are straightforward adaptations of the standard concepts and require no new methodology.The simplest technique for using and R charts in the short production run situation was introduced as deviation from nominal instead of the measured variable control chart. This is sometimes called the deviation from normal (DNOM) control chart.

If Mi represents the ith actual sample measurement in millimeters, then would be the deviation from nominal. The control limits have been calculated using the data from all 10 samples. In practice, we would recommend waiting until approximately 20 samples are available before calculating control limits. However, for purposes of illustration we have calculated the limits based on 10 samples to show that, when using deviation from nominal as the variable on the chart, it is not necessary to have a long production run for each part number. It is also customary to use a dashed vertical line to separate different products or part numbers and to identify clearly which section of the chart pertains to each part number.Three important points should be made relative to the DNOM approach:1. An assumption is that the process standard deviation is approximately the same forall parts. If this assumption is invalid, use a standardized and R chart 2. This procedure works best when the sample size is constant for all part numbers.3. Deviation from nominal control charts have intuitive appeal when the nominal specificationis the desired target value for the process

Standardized and R Charts. If the process standard deviations are different for different part numbers, the deviation from nominal (or the deviation from process target) control charts described above will not work effectively. However, standardized and R charts will handle this situation easily. Consider the jth part number. Let − and Tj be the average range and nominal value of x for this part number. Then for all the samples from this art number, plot



on a standardized R chart with control limits at LCL = D3 and UCL = D4, and plot

on a standardized chart with control limits at LCL = − A2 and UCL = + A2. Note that thecenter line of the standardized chart is zero because is the average of the original measurements for subgroups of the jth part number. We point out that for this to be meaningful, there must be some logical justification for “pooling” parts on the same chart.

Attributes Control Charts for Short Production RunsDealing with attributes data in the short production run environment is extremely simple; theproper method is to use a standardized control chart for the attribute of interest. This methodwill allow different part numbers to be plotted on the same chart and will automatically compensate for variable sample size. All standardized attributes control charts havethe center line at zero, and the upper and lower control limits are at +3 and −3, respectively.Other MethodsA variety of other approaches can be applied to the short-run production environment. Forexample, the cusum and EWMA control charts discussed already have potential application to short production runs, because they have shorter average run-length performancethan Shewhart-type charts, particularly in detecting small shifts. Since most production runs in the short-run environment will not, by definition, consist of many units, the rapid shift detection capability of those charts would be useful. Furthermore, cusum and EWMA controlcharts are very effective with subgroups of size one, another potential advantage in the shortrun situation.



The “self-starting” version of the cusum is also a useful procedure for the short-run environment. The self-starting approach uses regular process measurements for both establishing or calibrating the cusum and for process monitoring.Thus it avoids the phase I parameter estimation phase. It also produces the Shewhart control statistics as a by-product of the process. The number of subgroups used in calculating the trial control limits for Shewhart charts impacts the false alarm rate of the chart; in particular, when a small number of subgroups are used, the false alarm rate is inflated. Hillier (1969) studied this problem and presented a table of factors to use in setting limits for and R charts based in a small number of subgroups for the case of n = 5 [see also Wang and Hillier (1970)]. Quesenberry (1993) has investigated a similar problem for both and individuals control charts. Since control limits in the short-run environment will typically be calculated from a relatively small number of subgroups, these papers present techniques of some interest.Quesenberry (1991a, b, c) has presented procedures for short-run SPC using a transformationthat is different from the standardization approach discussed above. He refers to theseas Q-charts, and notes that they can be used for both short or long production runs. TheQ-chart idea was first suggested by Hawkins (1987). Del Castillo and Montgomery (1994)have investigated the average run-length performance of the Q-chart for variables and showthat in some cases the average run length (ARL) performance is inadequate. They suggestsome modifications to the Q-chart procedure and some alternate methods based on theEWMA and a related technique called the Kalman filter that have better ARL performancethan the Q-chart. Crowder (1992) has also reported a short-run procedure based on theKalman filter. In a subsequent series of papers, Quesenberry (1995a, b, c, d) reports somerefinements to the use of Q-charts that also enhance their performance in detecting process



shifts. He also suggests that the probability that a shift is detected within a specified numberof samples following its occurrence is a more appropriate measure of the performance of ashort-run SPC procedure than its average run length. The interested reader should refer to theJuly and October 1995 issues of the Journal of Quality Technology that contain these papersand a discussion of Q-charts by several authorities. These papers and the discussion include a number of useful additional references.

Some guidelines for univariate control chart selection



Figure : The situations where various types of control charts are useful.

Multivariate process monitoring and control

We have addressed process monitoring and control primarily from the univariate perspective; that is, we have assumed that there is only one process output variable or quality characteristic of interest. In practice, however, many if not most process monitoring and control scenarios involve several related variables. Although applying univariate control charts to each individual variable is a possible solution, we will see that this is inefficient and can lead to erroneous conclusions. Multivariate methods that consider the variables jointly are required.In this chapter, control charts that can be regarded as the multivariate extensions of some of the univariate charts. The Hotelling T2 chart is the analog of the Shewhart x chart. We will also discuss a multivariate version of the EWMA control chart and some methods for monitoring variability in the multivariate case. These multivariate control charts work well when the number of process variables is not too large—say, 10 or fewer. As the number of variables grows, however, traditional multivariate control charts lose efficiencywith regard to shift detection. A popular approach in these situations is to reduce thedimensionality of the problem. We show how this can be done with principal components



The Multivariate Quality-Control Problem

Multivariate analysis is a branch of statistics concerned with the analysis of multiple measurements, made on one or several samples of individuals. For example, we maywish to measure length, width and weight of a product. Multiple measurement, or observation, as row or column vectorA multiple measurement or observation may be expressed as

x = [4 2 0.6]

referring to the physical properties of length, width and weight, respectively. It iscustomary to denote multivariate quantities with bold letters. The collection ofmeasurements on x is called a vector. In this case it is a row vector. We could havewritten x as a column vector.

Matrix to represent more than one multiple measurements If we take several such measurements, we record them in a rectangular array of numbers.For example, the X matrix below represents 5 observations, on each of three variables. By convention, rows typically represent observations and columns represent variablesIn this case the number of rows, (n = 5), is the number of observations, and the number ofcolumns, (p = 3), is the number of variables that are measured. The rectangular array isan assembly of n row vectors of length p. This array is called a matrix, or, morespecifically, a n by p matrix. Its name is X. The names of matrices are usually written inbold, uppercase letters, as in previous section.We could just as well have written X as a p

(variables) by n (measurements) matrix as follows:

Definition of Transpose: A matrix with rows and columns exchanged in this manner is called the

transpose of the original matrix.



Hotelling Control Charts

Definition of Hotelling's T 2 "distance" statistic The Hotelling T 2 distances is a measure that accounts for the covariance structure of amultivariate normal distribution. It was proposed by Harold Hotelling in 1947 and iscalled Hotelling T 2. It may be thought of as the multivariate counterpart of the Student's-tstatistic. The T 2 distance is a constant multiplied by a quadratic form. This quadratic form isobtained by multiplying the following three quantities: 1. The vector of deviations between the observations and the mean m, which isexpressed by (X-m)',

2. The inverse of the covariance matrix, S,

3. The vector of deviations, (X-m)-1.

It should be mentioned that for independent variables, the covariance matrix is a diagonalmatrix and T 2 becomes proportional to the sum of squared standardized variables. In general, the higher the T 2 value, the more distant is the observation from the mean.The formula for computing the T 2 is:

The constant c is the sample size from which the covariance matrix was estimated. T 2 readily graph able: The T 2 distances lend themselves readily to graphical displaysand as a result the T 2-chart is the most popular among the multivariate control charts.

Estimation of the Mean and Covariance Matrix

Mean and Covariance matrices

Let X1,...X be n p-dimensional vectors of observations that are sampled independentlyfrom Npn(m, ) with p < n-1, with the covariance matrix of X. The observed meanvector and the sample dispersion matrix

Are the unbiased estimators of m and , respectively. Principal component control charts

Problems with T 2 charts: Although the T 2 chart is the most popular,easiest to use and interpret method for handling multivariate process data, and isbeginning to be widely accepted by quality engineers and operators, it is not a panacea.First, unlike the univariate case, the scale of the values displayed on the chart is notrelated to the scales of any of the monitored variables. Secondly, when the T



statistic exceeds the upper control limit (UCL), the user does not know which particularvariable(s) caused the out-of-control signal. Run univariate charts along with the multivariate ones With respect to scaling, we strongly advise to run individual univariate charts in tandemwith the multivariate chart. This will also help in honing in on the culprit(s) that mighthave caused the signal. However, individual univariate charts cannot explain situationsthat are a result of some problems in the covariance or correlation between the variables.This is why a dispersion chart must also be used.

Another way to monitor multivariate data: Principal Components control chartsAnother way to analyze the data is to use principal components. For each multivariatemeasurement (or observation), the principal components are linear combinations of thestandardized p variables (to standardize subtract their respective targets and divide bytheir standard deviations). The principal components have two important advantages: 1. The new variables are uncorrelated (or almost) 2. Very often, a few (sometimes 1 or 2) principal components may capture most of the variability in the data so that we do not have to use all of the pprincipal components for control. Eigen values: Unfortunately, there is one big disadvantage: The identity of the originalvariables is lost! However, in some cases the specific linear combinations correspondingto the principal components with the largest Eigen values may yield meaningfulmeasurement units. What is being used in control charts are the principal factors. A principal factor is the principal component divided by the square root of its eigenvalue. Multivariate EWMA Control Chart Univariate EWMA model: - The model for a univariate EWMA chart is given by:

where Z i is the ith EWMA, Xi is the ith observation, Z is the average from the historicaldata, and 0 < 1.

Multivariate EWMA model:- In the multivariate case, one can extend this formula to

where Zi is the ith EWMA vector, Xi0 is the the ith observation vector i = 1, 2, ..., n, Z is the vector of variable values from the historical data, is the diag ( 1

, ) which is a diagonal matrix with 1, 2, ... , on the main diagonal, and p is the number ofvariables; that is the number of elements in each vector.

Illustration of multivariate EWMA:- The following illustration may clarify this. There arep variables and each variable contains n observations. The input data matrix looks like:



The quantity to be plotted on the control chart is

Simplification:- It has been shown (Lowry et al., 1992) that the (k, l)th element of the covariance matrix of the ith EWMA, , is

where is the (k,l)th element of , the covariance matrix of the X's. If 1 = 2 = ... = p = , t

then the above expression simplifies to

where is the covariance matrix of the input data.

Further simplification:- There is a further simplification. When i becomes large, the

covariance matrix may be expressed as:

The question is "What is large?". When we examine the formula with the 2i in it,

we observe that when 2i becomes sufficiently large such that (1 - ) 2i becomes almost zero,then

we can use the simplified formula.



Table for selected values of and iThe following table gives the values of (1- ) 2i

for selected values of and i. 2i 1 - 4 6 8 10 12 20 30 40 50 .9 .656 .531 .430 .349 .282 .122 .042 .015 .005 .8 .410 .262 .168 .107 .069 .012 .001 .000 .000 .7 .240 .118 .058 .028 .014 .001 .000 .000 .000 .6 .130 .047 .017 .006 .002 .000 .000 .000 .000 .5 .063 .016 .004 .001 .000 .000 .000 .000 .000 .4 .026 .004 .001 .000 .000 .000 .000 .000 .000 .3 .008 .001 .000 .000 .000 .000 .000 .000 .000 .2 .002 .000 .000 .000 .000 .000 .000 .000 .000 .1 .000 .000 .000 .000 .000 .000 .000 .000 .000

Specified formula not required:-

It should be pointed out that a well-meaning computer program does not have to adhere to the simplified formula, and potential inaccuracies for low values for and i can thusbe avoided. MEWMA computer out put for Lowry data:- Here is an example of the application of an MEWMA control chart. To facilitate comparison with existing literature, we used data from Lowry et al. The data were simulated from a bivariate normal distribution with unit variances and a correlation coefficient of 0.5. The value for = .10 and the values for were obtained by the equation given above. The covariance of the MEWMA vectors was obtained by using the non-simplified equation. That means that for each MEWMA control statistic, the computer computed a covariance matrix, where i = 1, 2, ...10. The results of the computer routine are:

***************************************************** * Multi-Variate EWMA Control Chart ******************************************************

DATA SERIES MEWMA Vector MEWMA 1 2 1 2 STATISTIC



-1.190 0.590 -0.119 0.059 2.1886 0.120 0.900 -0.095 0.143 2.0697-1.690 0.400 -0.255 0.169 4.8365 0.300 0.460 -0.199 0.198 3.4158 0.890 -0.750 -0.090 0.103 0.7089 0.820 0.980 0.001 0.191 0.9268-0.300 2.280 -0.029 0.400 4.0018 0.630 1.750 0.037 0.535 6.1657 1.560 1.580 0.189 0.639 7.8554 1.460 3.050 0.316 0.880 14.4158

VEC XBAR MSE Lamda 1 .260 1.200 0.100 2 1.124 1.774 0.100

The UCL = 5.938 for = .05. Smaller choices of are also used. Sample MEWMA plots

The following is the plot of the above MEWMA


Documents

Chapter III1