Cumulative Sum charts, or Cusum charts, are
an alternative to Shewhart control charts. While
Shewhart control charts are widely used and
the control violations well documented, there
may be conditions to which they are insensitive.
This technique determines an average value for
the subgroups and determines the expected variability
about the mean. These charts are good at detecting
distinct, steady and intermittent shifts in
the process.
Cusum charts are better suited to detecting
small, sustained shifts in a process. These
charts measure a cumulative deviation from the
mean or a target value. Depending on the type
of test used, the chart either displays the
standardized deviation from the target or the
mean value of the subgroup size.
The types of tests used to evaluate an out-of-control
condition are the run-sum and the V-mask. The
V-mask standardizes the deviations from the
mean, or target value, and plots the deviations
from this value. If the process remains in control,
then the deviations will scatter around the
target. On the Cusum chart, this will produce
a straight line or a random shift around the
target with a mean of zero.
The run-sum method of evaluating a Cusum chart
plots the subgroup average, similar to a Shewhart
chart. However, instead of using control chart
rules to detect a violation, the user sets a
limit, or score, of cumulative deviation from
target as the signal of a violation. For example,
if the first subgroup average is 1 unit above
the target, the score is set to 1. If the next
subgroup average is 1 unit below the target,
the subgroup score is –1. The cumulative
sum becomes 1+(-1)=0. This is what would be
expected with random variation about the target.
If the process mean is shifting up or down,
then the score continues to increase or decrease.
Violations are identified when the cumulative
score exceeds a threshold value set by the user.
Once a violation has occurred, a decision must
be made regarding whether the cumulative score
should be reset. With this method, the user
has the option of resetting the score once a
problem has been detected and resolved. If the
score is not reset, violations in the opposite
direction may not become apparent or may trigger
warnings in the same direction as the first
violation that are not actual violations.
In both the run-sum and V-mask methods, the
user has the option to specify the criteria
for determining deviation. By default, the process
mean is used as the target value. The alternative
is to enter the desired target. For the run-sum
technique, the user also has the option to specify
the magnitude of the cumulative deviation from
the target before a violation is identified.
In the following example, a Cusum chart depicts
mean oven temperature for subgroups of 20 runs.
This chart uses the run-sum test. The run-sum
score that defines a violation is set to 8 and
is reset upon reaching that threshold. This
technique suggests that a significant upward
shift in the mean occurs by run 20. Since we
are using the run-sum test, the scores must
be steadily increasing in order for the score
to reach the threshold value. If the mean were
fluctuating above and below the overall mean,
the score would alternately increment and decrement
and remain below the threshold value.
This chart is set to reset the score after
a violation occurs. If the subgroup means do
not continue to shift either above or below
the overall mean, we would not expect to see
further violations unless another significant
drift occurs. In this chart, there is a second
violation identified by run 30. This violation
indicates a downward shift in the mean. This
example clearly illustrates the results of a
2-sided test. The maximum score allowed before
identifying a violation is used for both positive
and negative drifts in the subgroup means. While
there is no need to specify that the absolute
value of the score is being tested, the chart
successfully evaluates the drift of subgroup
means in both directions.
The next example uses the same data but with
a V-mask test. The values plotted on this chart
are standardized values. This means that the
difference between the subgroup mean and the
overall mean is plotted in lieu of the actual
subgroup mean. This method provides visibility
into the magnitude of the temperature drift.
This test is also a 2-sided test. Upward shifts
in the mean are detected by points plotted above
the upper V-mask line. Downward shifts are detected
by points plotted below the lower V-mask line.
Using the V-mask test also shows that there
is significant upward drift in the subgroup
means that occurs by run 20. It is possible
to modify features of this test. Options exist
to change the slope of the V, alter the probability
of a Type 1 error and other options that are
beyond the scope of this discussion.
Note that this test does not show a significant
decrease in temperature from the overall average,
at least not yet. If the trend continues, it
appears that the points will soon fall below
the lower line.
The final chart uses the same data in subgroups
of 20 to produce a Shewhart X-bar chart. The
X-bar chart detects a violation at run 19. The
rule that is violated is 4 of 5 successive points
are in lower Zone B or beyond. The subgroup
average does not fall outside of the 3-sigma
limit until run 30. These results indicate that
the suggestion of increased temperature in not
significant, but the decrease in subgroup temperature
is significant.
For the sake of completeness, the S-chart for
this data follows. The S-chart shows no violations
that could cause concern over the temperature
variability within subgroups.
The decision for choosing Shewhart charts over
Cusum charts depends on the process. If small
changes negatively impact the process and must
be resolved quickly, then Cusum charts may be
critical tools. If the process works well within
the parameters defined in Shewhart charts, then
they are the appropriate choice. It is important
to know what options are available. There may
be situations when a combination of these tools
is warranted to detect necessary process details.
