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Introduction to Statistical Quality Control, 5th edition

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Title: 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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