Introduction to Statistical Quality Control, 5th edition

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Introduction

• Chapters 4 through 6 focused on Shewhart control
  charts.
• Major disadvantage of Shewhart control charts is
  that it only uses the information about the
  process contained in the last plotted point.
• Two effective alternatives to the Shewhart
  control charts are the cumulative sum (CUSUM)
  control chart and the exponentially weighted
  moving average (EWMA) control chart. Especially
  useful when small shifts are desired to be
  detected.

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Consider the data in table 8-1, Column(a). The
first 20 of these observations were drawn at
random from normal distribution with µ10 and
s1 UCL µ3s 13 CL µ 10 LCL µ-3s 7
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8-1.2 The Tabular or Algorithmic Cusum
for Monitoring the Process Mean

• Let xi be the ith observation on the process
• If the process is in control then
• Assume ? is known or can be estimated.
• Accumulate derivations from the target ?0 above
  the target with one statistic, C
• Accumulate derivations from the target ?0 below
  the target with another statistic, C
• C and C-- are one-sided upper and lower cusums,
  respectively.

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8-1.2 The Tabular or Algorithmic Cusum
for Monitoring the Process Mean

• Selecting the reference value, K
• K is often chosen halfway between the target ?0
  and the out-of-control value of the mean ?1 that
  we are interested in detecting quickly.
• Shift is expressed in standard deviation units as
  ?1 ?0??, then K is

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8-1.2 The Tabular or Algorithmic Cusum
for Monitoring the Process Mean

• The statistics are computed as follows
• The Tabular Cusum
• starting values are
• K is the reference value (or allowance or slack
  value)
• If either statistic exceed a decision interval
  H, the process is considered to be out of
  control. Often taken as a H 5?

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8-1.2 The Tabular or Algorithmic Cusum for
monitoring the process Mean

• Example 8-1
• The cusum control chart indicates the process is
  out of control. (Table 8/2 shows that upper side
  cusum at period 29 is . Since this
  is the first period at which ).
  Tabular cusum also indicates when the shift
  probably occured. The counter N records the
  number of consecutive periods since the
  rose above the calue of zero. Since N 7 at
  period 29, we would conclude that the process was
  last in control at period 29-722. So the shift
  is likely to occur between periods 22 and 23.
• The next step is to search for an assignable
  cause, take corrective action required, and
  reinitialize the cusum at zero.
• If an adjustment has to be made to the process,
  may be helpful to estimate the process mean
  following the shift.

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Cusum Status Chart (Figure 8-3a)
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MINITAB Version of Cusum Status Chart
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Recommendations for Cusum Design
The tabular cusum is designed by shoosing values
for the reference value K and the decision
interval H. It is usually recommended that these
paramters be selected to provide good ARL
performance. Consider the two sided ARLs shown
below where Hhs Kks
Note that a 1 s shift would be detected in either
8.38 samples or 10.4 samples with k0.5 and h5.
By comparison a shewart CC for individual
measurements would require 43.96
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Generally, we want to choose k relative tro the
size of the shift we want to detect that is,
k1/2d, where d is the size of the shift in s
units. This approach comes very close to
minimizing the ARL1 value for detecting a shift
of size d for fixed ARL0
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• Several techniques can be used to calculate the
  ARL of a cusum. Woodal and adams recommended the
  ARL approximation given by Siegmond because of
  its simplicity.
• For a one sided cusum with parameters h and k,
  siegmunds approxiamation is
• for ? ? 0, where ? d- k fot the upper one
  sided cusum , ? d- k for the lower one
  sided cusum , bh1.166, and d (µ1-µ0)/s.
  If ? 0, one can use ARLb2. The quantity d
  represents the shift in the mean, in the units of
  s, for which the ARL is to be calculated.
  Therefore, if d 0, we would calculate ARL0
  from equation 8.6, whereas if d ? 0, wwe sould
  calculate the value of ARL1 corresponding to a
  shift of size d.

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• To obtain ARL of the two sided cusum from the
  ARLs of the two one sided statistics - Say ARL
  and ARL- - use
• To illustrate, consider the two sided cusum with
  k1/2 and h5. To find ARL0 we sould first
  calculate the ARL0 values for the two-sided
  schemes. Set d 0, then ? d- k 0 ½ -
  ½, b h 1.166 51.166 6.166 and from
  equation 8.6
• By symmetry, we have , and
  so from equation 8.7 the in control ARL for the
  two-sided cusum is
• This is very close to the true ARL0 value of 465
  shown in table 8.3

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The Standardized Cusum
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Improving CusumResponsiveness for large shifts

• Cusum control chart is not as effective in
  detecting large shifts in the process mean as the
  Shewhart chart.
• An alternative is to use a combined
  cusum-Shewhart procedure for on-line control.
• The combined cusum-Shewhart procedure can improve
  cusum responsiveness to large shifts.

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The Fast Initial Response or headstart feature

• These procedures were introduced to increase
  sensitivity of the cusum control chart upon
  start-up.
• The fast initial response (FIR) or headstart sets
  the starting values equal to some
  nonzero value, typically H/2.
• Setting the starting values to H/2 is called a 50
  percent headstart.

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The Fast Initial Response (FIR) Cusum
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More on Cusums

• Cusums are often used to determine if a process
  has shifted off a specified target because it is
  easy to calculate the required adjustment
• One-sided cusums are often useful
• Cusums can also be used to monitor variability
• Cusums are available for other sample statistics
  (ranges, standard deviations, counts,
  proportions)
• Rational subgroups and cusums

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One-Sided Cusums

• There are practical situations where a single
  one-sided cusum is useful.
• If a shift in only one direction is of interest
  then a one-sided cusum would be applicable.

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Rational Subgroups

• Shewhart chart performance is improved with
  rational subgrouping
• Cusum is not necessarily improved with rational
  subgrouping
• Only if there is significant economy of scale or
  some other reason for taking larger samples
  should rational subgrouping be considered with
  the cusum

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A Cusum for Monitoring process Variability

• Let
• The standardized value of xi is
• A new standardized quantity (Hawkins (1981)
  (1993)) is given by
• Hawkins suggest that the ?i are sensitive to
  variance changes rather than mean changes.

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A Cusum for Monitoring process Variability

• ?I N(0, 1), two one-sided standardized scale
  cusums are
• The Scale Cusum
• where
• if either statistic exceeds h, the process is
  considered out of control.

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The V-Mask Procedure

• The V-mask procedure is an alternative to the
  tabular cusum.
• It is often strongly advised not to use the
  V-mask procedure for several reasons.
• The V-mask is a two-sided scheme it is not very
  useful for one-sided process monitoring problems.
• The headstart feature, which is very useful in
  practice, cannot be implemented with the V-mask.
• It is sometimes difficult to determine how far
  backwards the arms of the V-mask should extend,
  thereby making interpretation difficult for the
  practitioner.
• Ambiguity associated with with ? and ?

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The Cusum V-Mask
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EWMA control chart is also a good alternative to
the shewart control chart when we are interested
in detecting small shifts. The performance of the
EWMA control chart is aproximately equivalent to
that of the cumulative sum control chart. EWMA is
defined as
Starting value is the process target, so that
Sometimes the average of prelimnary data is used
as the starting value of the EWMA so that z0
. If the observations xi are independent random
variables with variance s2, then the variance of
zi is
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Design of an EWMA Control Chart

• The design parameters of the chart are L and ?.
• The parameters can be chosen to give desired ARL
  performance.
• In general, 0.05 ? ? ? 0.25 works well in
  practice.
• L 3 works reasonably well (especially with the
  larger value of ?.
• L between 2.6 and 2.8 is useful when ? ? 0.1
• Similar to the cusum, the EWMA performs well
  against small shifts but does not react to large
  shifts as quickly as the Shewhart chart.
• EWMA is often superior to the cusum for larger
  shifts particularly if ? 0.1

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Design of the EWMA
Hunter has also stucked the EWMA and suggested
choosing ? so that the weight given to current
and previous observations watches as closely as
possible the weight given to these observations
by a shewart chart with the western Electric
rules. Then results in a recommended value of
?0.4. If ?3.054, then table 8.10 indicates that
this chart would have ARL0500 and for detecting
a shift of 1s in the process mean the ARL114.3
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Robustness of the EWMA to Non-normality

• As discussed in Chapter 5, the individuals
  control chart is sensitive to non-normality.
• A properly designed EWMA is less sensitive to the
  normality assumption.

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Robustness of EWMA toNon-normal Process Data
The ARL0 if the shewart individuals chart and
several EWMA control charts for these non-normal
distributions are given in table 8.11 and 8.12.
Two aspects of the information in these tables
are very striking. First, even moderately
non-normal distributions have the effect of
greately reducing the in control ARL of the
shewart individual chart. This will dramatically
increase the rate of false alarms. Second, on
EWMA with ?0.05 or ?0.05 and an approximately
chosen control limit will perform very well
against both normal and non-normal distributions.
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Extensions of the EWMA

• Fast initial response feature
• Monitoring variability
• Monitoring count data
• The EWMA as a predictor of process level

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Moving average control chart is more effective
than the shewart chart in detecting small process
shifts. However, it is generally not as effective
against small shifts as either the cusum or the
EWMA. The moving average cc is considered by some
to be simpler to implement than the cusum.
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