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Mathematical Expression Editor
We study basic ideas behind the use of control charts in industry.
An Application of X-bar Charts to Manufacturing
In thermoform production, plastic sheets are fed into machines which can turn them
into molded plastic objects which are used in everyday life. As an example, consider
the molded plastic container holding the “surprise” inside of the chocolate candy in
the picture below.
In the following Desmos graph we can view 16 samples of 5 diameters measured
during thermoform production in Taiwan. From prior production runs it was known
that the mean diameter was mm, and the standard deviation was mm. Let’s assume
that the samples are taken every hour. For the “surprise” to fit inside the plastic
shell, and for the plastic shell to fit inside the chocolate candy, specification limits are
set to mm. Any thermoform with a diameter greater than 35.3 mm or less than 34.2
mm is unusable.
We can see that after the first 12 hours of production, most of the diameters that
were sampled were too big, and so many thermoforms had to be scrapped. The point
of Statistical Process Control is to detect signs of a process being out of control early,
so that corrections can be made and waste can be reduced.
Observe that in the first 12 hours most diameters were within the spec limits. This
may seem like a reassuring sign, but is it possible that even the early samples pointed
to this thermoform production process being out of control? In this section we will
see how control charts could have been used to detect an out of control process early,
and make adjustments before scrap is produced.
When is it time to investigate?
There will always be some variation in any process. One of the fundamental
principles of statistical process control is that if a process is acting in a non-random
way, the process can be improved. Control charts are used to detect non-random
behavior. In the previous section we discussed several general trends seen in control
charts that indicate an out-of-control process. We now formalize our previous
observations with a set of rules. The Nelson rules for control charts is a common
set of rules used in manufacturing to determine when a process may be
out-of-control.
Recall, from the previous section, that control limits are typically set at three
standard deviations away from the mean, where standard deviation, is given by the
Central Limit Theorem to be
where is population standard deviation and is the sample size.
Nelson Rules
Stop and examine the process if one of the following occurs.
Rule 1: One point falls above UCL or below LCL.
Rule 2: Two out of three consecutive points are on the same side of the center line
and above or below .
Rule 3: Four out of five consecutive points are on the same side of the center line
and above or below .
Rule 4: Eight consecutive points are on the same side of the center line.
Rule 5: Six consecutive points are ascending/decending.
Rule 6: Fifteen consecutive points are between and . This is known as “hugging the
center line”.
Rule 7: Fourteen consecutive points form an alternating up/down pattern.
Rule 8: Eight consecutive points are above or below .
See if you can apply the Nelson rules in the following problem.
Suppose that a manufacturing process is known to have a normal distribution with a
mean , and standard deviation . A random sample of size is collected every hour to
monitor the manufacturing process. The distribution of sample means (sampling
distribution) will be normal and
Determine upper and lower control limits (), for the -control chart for this
manufacturing process.
The GeoGebra interactive below shows the means of consecutive samples through
taken over the course of two days. Use the scroll bar to navigate all samples and
answer the questions below.
What can you say about Samples 1-3:
Everything looks good, keep the
process going.Stop the process due to Rule 1.Stop the process due
to Rule 2.Stop the process due to Rule 3.Stop the process due to
Rule 4.Stop the process due to Rule 5.Stop the process due to Rule
6.Stop the process due to Rule 7.Stop the process due to Rule 8.
What can you say about Samples 4-7:
Everything looks good, keep the
process going.Stop the process due to Rule 1.Stop the process due
to Rule 2.Stop the process due to Rule 3.Stop the process due to
Rule 4.Stop the process due to Rule 5.Stop the process due to Rule
6.Stop the process due to Rule 7.Stop the process due to Rule 8.
What can you say about Samples 8-10:
Everything looks good, keep the
process going.Stop the process due to Rule 1.Stop the process due
to Rule 2.Stop the process due to Rule 3.Stop the process due to
Rule 4.Stop the process due to Rule 5.Stop the process due to Rule
6.Stop the process due to Rule 7.Stop the process due to Rule 8.
What can you say about Samples 11-20:
Everything looks good, keep the
process going.Stop the process due to Rule 1.Stop the process due
to Rule 2.Stop the process due to Rule 3.Stop the process due to
Rule 4.Stop the process due to Rule 5.Stop the process due to Rule
6.Stop the process due to Rule 7.Stop the process due to Rule 8.
What can you say about Samples 21-32:
Everything looks good, keep the
process going.Stop the process due to Rule 1.Stop the process due
to Rule 2.Stop the process due to Rule 3.Stop the process due to
Rule 4.Stop the process due to Rule 5.Stop the process due to Rule
6.Stop the process due to Rule 7.Stop the process due to Rule 8.
CASE STUDY: Thermoform production
We return now to the application that we began this section with, and apply control
charts to the thermoform production example. Let’s begin by determining the control
limits for this example. Recall that population mean and standard deviation were
determined to be mm, and mm.
For samples of size 5, we have
Therefore,
Below we see an X-bar control chart constructed by computing the mean diameter
of each sample of size we obtained earlier. Individual measurements from
each sample are shown in different color, sample means are shown in black.
Let’s see how the Nelson rules may have helped us to reduce waste in this
example.
HOLD EVERYTHING!!! The very first sample mean of 34.51 mm is more than
three standard deviations () from the population mean, so it is below the LCL.
Following Nelson Rule 1, we halt production and inspect, correcting anything that
may seem to be wrong. Then we resume production.
WAIT A MINUTE!!! Our ninth sample mean is more than three standard
deviations above the center line mm. We should stop and inspect the process.
THIS PROCESS SEEMS OUT OF CONTROL!!! Our tenth sample mean is
also more than three standard deviations above the center line. We should stop and
inspect the process, and take corrective action.
The point here is that by following the Nelson rules, we would investigate during the
first hour of production and again after 9 or 10 hours. Hopefully this would provide
us ample opportunity to inspect and make modifications to the process before
continuing, so that the diameters don’t continue to grow to the point where they are
unusable.
References
CASE STUDY on Thermoform Production is modified from:
